Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topolgy

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Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology

NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme. The Series is published by IOS Press, Amsterdam, and Springer in conjunction with the NATO Public Diplomacy Division

Sub-Series I. II. III. IV.

Life and Behavioural Sciences Mathematics, Physics and Chemistry Computer and Systems Science Earth and Environmental Sciences

IOS Press Springer IOS Press Springer

The NATO Science Series continues the series of books published formerly as the NATO ASI Series. The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council. The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, and the NATO Science Series collects together the results of these meetings. The meetings are co-organized by scientists from NATO countries and scientists from NATO’s Partner countries – countries of the CIS and Central and Eastern Europe. Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field. Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action. As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series was re-organized to the four sub-series noted above. Please consult the following web sites for information on previous volumes published in the Series. http://www.nato.int/science http://www.springer.com http://www.iospress.nl

Series II: Mathematics, Physics and Chemistry – Vol. 216

Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology edited by

Paul Biran Tel-Aviv University, Israel

Octav Cornea University of Montreal, QC, Canada and

Franç ois Lalonde University of Montreal, QC, Canada

Published in cooperation with NATO Public Diplomacy Division

Proceeding of the NATO Advanced Study Institute on Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology Montré al, Canada July 2004

ISBN-10 ISBN-13 ISBN-10 ISBN-13 ISBN-10 ISBN-13

1-4020-4273-6 (PB) 978-1-4020-4273-7 (PB) 1-4020-4272-8 (HB) 978-1-4020-4272-0 (HB) 1-4020-4266-3 (e-book) 978-1-4020-4266-9 (e-book)

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

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All Rights Reserved © 2006 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in the Netherlands.

CONTENTS

Preface

xi

Contributors

xiii

A. Abbondandolo and P. Majer/ Lectures on the Morse Complex for Infinite-Dimensional Manifolds 1 A few facts from hyperbolic dynamics 1.1 Adapted norms 1.2 Linear stable and unstable spaces of an asymptotically hyperbolic path 1.3 Morse vector fields 1.4 Local dynamics near a hyperbolic rest point 1.5 Local stable and unstable manifolds 1.6 The Grobman – Hartman linearization theorem 1.7 Global stable and unstable manifolds 2 The Morse complex in the case of finite Morse indices 2.1 The Palais – Smale condition 2.2 The Morse – Smale condition 2.3 The assumptions 2.4 Forward compactness 2.5 Consequences of compactness and transversality 2.6 Cellular filtrations 2.7 The Morse complex 2.8 Representation of ∂∗ in terms of intersection numbers 2.9 How to remove the assumption (A8) 2.10 Morse functions on Hilbert manifolds 2.11 Basic results in transversality theory 2.12 Genericity of the Morse – Smale condition 2.13 Invariance of the Morse complex 3 The Morse complex in the case of infinite Morse indices 3.1 The program 3.2 Fredholm pairs and compact perturbations of linear subspaces 3.3 Finite-dimensional intersections

v

1 2 2 4 8 9 11 12 19 21 21 22 22 23 25 27 28 33 37 38 40 42 47 52 52 54 54

vi

CONTENTS

3.4 Essential subbundles 3.5 Orientations 3.6 Compactness 3.7 Two-dimensional intersections 3.8 The Morse complex Bibliographical note

57 59 61 65 67 69

A. Abbondandolo and M. Schwarz/ Notes on Floer Homology and Loop Space Homology 75 1 Introduction 75 2 Main result 77 2.1 Loop space homology 77 2.2 Floer homology for the cotangent bundle 80 3 Ring structures and ring-homomorphisms 85 3.1 The pair-of-pants product 85 3.2 The ring homomorphisms between free loop space Floer homology and based loop space Floer homology 89 and classical homology 4 Morse-homology on the loop spaces ΛQ and ΩQ, and the 95 isomorphism 5 Products in Morse-homology 102 5.1 Ring isomorphism between Morse homology and 105 Floer homology J.-F. Barraud and O. Cornea/ Homotopical Dynamics in Symplectic Topology 1 Introduction 2 Elements of Morse theory 2.1 Connecting manifolds 2.2 Operations 3 Applications to symplectic topology 3.1 Bounded orbits 3.2 Detection of pseudoholomorphic strips and Hofer’s norm

109 109 110 113 125 127 127 130

R. L. Cohen/ Morse Theory, Graphs, and String Topology 1 Graphs, Morse theory, and cohomology operations 2 String topology 3 A Morse theoretic view of string topology 4 Cylindrical holomorphic curves in the cotangent bundle

149 152 165 173 178

M. Farber/ Topology of Robot Motion Planning 1 Introduction 2 First examples of configuration spaces 3 Varieties of polygonal linkages

185 185 186 189

CONTENTS

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

3.1 Short and long subsets 3.2 Poincar´e polynomial of M(a) Universality theorems for configuration spaces A remark about configuration spaces in robotics The motion planning problem Tame motion planning algorithms The Schwarz genus The second notion of topological complexity Homotopy invariance Order of instability of a motion planning algorithm Random motion planning algorithms Equality theorem An upper bound for TC(X) A cohomological lower bound for TC(X) Examples Simultaneous control of many systems Another inequality relating TC(X) to the usual category Topological complexity of bouquets A general recipe to construct a motion planning algorithm How diﬃcult is to avoid collisions in Rm ? The case m = 2 TC F(Rm , n) in the case m ≥ 3 odd Shade Illuminating the complement of the braid arrangement A quadratic motion planning algorithm in F(Rm , n) Configuration spaces of graphs Motion planning in projective spaces Nonsingular maps TC(RPn ) and the immersion problem Some open problems

K. Fukaya/ Application of Floer Homology of Langrangian Submanifolds to Symplectic Topology 1 Introduction 2 Lagrangian submanifold of Cn 3 Perturbing Cauchy – Riemann equation 4 Maslov index of Lagrangian submanifold with vanishing second Betti number 5 Floer homology and a spectral sequence 6 Homology of loop space and Chas – Sullivan bracket 7 Iterated integral and Gerstenhaber bracket 8 A∞ deformation of de Rham complex

vii 190 190 194 195 195 198 199 200 201 201 202 203 207 207 208 209 210 211 212 213 214 216 217 219 220 221 222 224 226 228 231 231 232 235 240 243 246 251 254

viii

CONTENTS

9 10 11 12 13 14

S 1 equivariant homology of loop space and cyclic A∞ algebra L∞ structure on H(S 1 × S n ; Q) Lagrangian submanifolds of C3 Aspherical Lagrangian submanifolds Lagrangian submanifolds homotopy equivalent to S 1 × S 2m Lagrangian submanifolds of CPn

M. Izydorek/ The LS-Index: A Survey 1 Introduction 2 The LS-index 2.1 Basic definitions and facts 2.2 Spectra 2.3 The LS-index 3 Cohomology of spectra 4 Attractors, repellers and Morse decompositions 5 Equivariant LS-flows and the G-LS-index 5.1 Symmetries 5.2 Isolating neighbourhoods and the equivariant LS-index 6 Applications 6.1 A general setting 6.2 Applications of the LS-index 6.3 Applications of the cohomological LS-index 6.4 Applications of the equivariant LS-index Y.-G. Oh/ Lectures on Floer Theory and Spectral Invariants of Hamiltonian Flows 1 Introduction 2 The free loop space and the action functional 2.1 The free loop space and the S 1 -action in general 2.2 The free loop space of symplectic manifolds 2.3 The Novikov covering 2.4 Perturbed action functionals and their action spectra 2.5 The L2 -gradient flow and perturbed Cauchy – Riemann equations 2.6 Comparison of two Cauchy – Riemann equations 3 Floer complex and the Novikov ring 3.1 Novikov – Floer chains and the Novikov ring 3.2 Definition of the Floer boundary map 3.3 Definition of the Floer chain map 3.4 Semi-positivity and transversality 3.5 Composition law of Floer’s chain maps 4 Energy estimates and Hofer’s geometry 4.1 Energy estimates and the action level changes

259 260 265 266 271 272 277 277 282 282 285 288 290 293 296 296 296 303 303 305 309 312 321 321 326 326 327 328 330 332 336 338 338 341 346 347 348 352 352

CONTENTS

5

6

7

8

9

10 A

4.2 Energy estimates and Hofer’s norm 4.3 Level changes of Floer chains under the homotopy 4.4 The -regularity type invariants Definition of spectral invariants and their axioms 5.1 Floer complex of a small Morse function 5.2 Definition of spectral invariants 5.3 Axioms of spectral invariants The spectrality axiom 6.1 A consequence of the nondegenerate spectrality axiom 6.2 Spectrality axiom for the rational case 6.3 Spectrality for the irrational case Pants product and the triangle inequality 7.1 Quantum cohomology in the chain level 7.2 Grading convention 7.3 Hamiltonian fibrations and the pants product 7.4 Proof of the triangle inequality Spectral norm of Hamiltonian diﬀeomorphisms 8.1 Construction of the spectral norm 8.2 The -regularity theorem and its consequences 8.3 Proof of nondegeneracy Applications to Hofer geometry of Ham(M, ω) 9.1 Quasi-autonomous Hamiltonians and the minimality conjecture 9.2 Length minimizing criterion via ρ(H; 1) 9.3 Canonical fundamental Floer cycles 9.4 The case of autonomous Hamiltonians Remarks on the transversality for general (M, ω) Proof of the index formula

L. Polterovich/ Floer Homology, Dynamics and Groups 1 Hamiltonian actions of finitely generated groups 1.1 The group of Hamiltonian diﬀeomorphisms 1.2 The no-torsion theorem 1.3 Distortion in normed groups 1.4 The No-Distortion Theorem 1.5 The Zimmer program 2 Floer theory in action 2.1 A brief sketch of Floer theory 2.2 Width and torsion 2.3 A geometry on Ham(M, ω) 2.4 Width and distortion 2.5 More remarks on the Zimmer program

ix 355 358 359 361 361 362 365 367 368 370 374 377 377 380 382 385 386 386 389 394 399 399 401 403 404 406 408 417 417 417 418 420 421 422 423 423 425 425 426 426

x

CONTENTS

3

The Calabi quasi-morphism and related topics 3.1 Extending the Calabi homomorphism 3.2 Introducing quasi-morphisms 3.3 Quasi-morphisms on Ham(M, ω) 3.4 Distortion in Hofer’s norm on Ham(M, ω) 3.5 Existence and uniqueness of Calabi quasi-morphisms 3.6 “Hyperbolic” features of Ham(M, ω)? 3.7 From π1 (M) to Diﬀ 0 (M, Ω)

428 428 429 430 431 433 434 435

C. Viterbo/ Symplectic topology and Hamilton – Jacobi equations 1 Introduction to symplectic geometry and generating functions 1.1 Uniqueness and first symplectic invariants 2 The calculus of critical level sets 2.1 The case of GFQI 2.2 Applications 3 Hamilton – Jacobi equations and generating functions 4 Coupled Hamilton – Jacobi equations

439 439 446 447 450 453 454 456

Index

461

PREFACE

The papers collected in this volume are contributions to the 43rd session of the S´eminaire de math´ematiques sup´erieures (SMS) on “Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology.” This session took place at the Universit´e de Montr´eal in July 2004 and was a NATO Advanced Study Institute (ASI). The aim of the ASI was to bring together young researchers from various parts of the world and to present to them some of the most significant recent advances in these areas. More than 77 mathematicians from 17 countries followed the 12 series of lectures and participated in the lively exchange of ideas. The lectures covered an ample spectrum of subjects which are reflected in the present volume: Morse theory and related techniques in infinite dimensional spaces, Floer theory and its recent extensions and generalizations, Morse and Floer theory in relation to string topology, generating functions, structure of the group of Hamiltonian diﬀeomorphisms and related dynamical problems, applications to robotics and many others. We thank all our main speakers for their stimulating lectures and all participants for creating a friendly atmosphere during the meeting. We also thank Ms. Diane B´elanger, our administrative assistant, for her help with the organization and Mr. Andr´e Montpetit, our technical editor, for his help in the preparation of the volume. The ASI was made possible by the financial support from the Scientific and Environmental Aﬀairs Division of NATO, the “Centre de recherches math´ematiques” from Montr´eal and the Universit´e de Montr´eal. We are most grateful to all three organizations. Paul Biran Octav Cornea Franc¸ois Lalonde

xi

CONTRIBUTORS

Alberto Abbondandolo Dipartimento di Matematica Universit`a di Pisa via Buonarroti 2 56127 Pisa Italy [email protected] Jean-Franc¸ois Barraud UFR de Math´ematiques Universit´e de Lille 1 59655 Villeneuve d’Ascq France [email protected] Octav Cornea D´epartement de math´ematiques et statistique Universit´e de Montr´eal Montr´eal, QC H3C 3J7 Canada [email protected]

Kenji Fukaya Department of Mathematics Kyoto University Japan [email protected] Marek Izydorek Faculty of Applied Physics and Mathematics Gda´nsk University of Technology ul. Gabriela Narutowicza 11/12 80-952 Gda´nsk Poland [email protected] Pietro Majer Dipartimento di Matematica Universit`a di Pisa via Buonarroti 2 56127 Pisa Italy [email protected]

Ralph L. Cohen Department of Mathematics Stanford University 450 Serra Mall, Bldg. 380 Stanford, CA 94305-2125 USA [email protected]

Yong-Geun Oh Department of Mathematics University of Wisconsin Madison, WI 53706 USA [email protected]

Michael Farber Department of Mathematics University of Durham Durham HH1 3LE UK [email protected]

Leonid Polterovich School of Mathematics Tel Aviv University Tel Aviv 69978 Israel [email protected]

xiii

xiv

CONTRIBUTORS

Matthias Schwarz Mathematisches Institut Fakult¨at f¨ur Mathematik und Informatik Universit¨at Leipzig Augustusplatz 10/11 04109 Leipzig Germany [email protected]

Claude Viterbo Centre de Math´ematiques Laurent Schwartz ´ Ecole Polytechnique 91128 Palaiseau Cedex France [email protected]

LECTURES ON THE MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS ALBERTO ABBONDANDOLO AND PIETRO MAJER Universit`a di Pisa

Abstract. After reviewing some classical results about hyperbolic dynamics in a Banach setting, we describe the Morse complex for gradient-like flows on an infinite-dimensional Banach manifold M, under the assumption that rest points have finite Morse index. Then we extend these ideas to rest points with infinite Morse index and co-index, by using a suitable subbundle of the tangent bundle of M as a comparison object.

Introduction These lectures consist of three parts. In the first one we review some results about the dynamics of diﬀerentiable flows with hyperbolic rest points, in a Banach space setting. In particular, we prove the local stable manifold theorem, the Grobman – Hartman linearization theorem, and we describe the global stable and unstable manifolds in the case of a flow admitting a Lyapunov function. In the second part we study the Morse complex of gradient-like flows on Banach manifolds, assuming that all the rest points have finite Morse index. We introduce this chain complex as the cellular chain complex of a suitable cellular filtration of the underlying manifold M. In particular, the homology of the Morse complex is isomorphic to the singular homology of M (or to the singular A), in the relative case, in which we consider a grahomology of the pair ( M, with a positively invariant open set A, and we consider dient like flow on M, \ A¯ in the construction of the Morse complex). Then the rest points in M = M we describe the chain boundary operator in terms of the intersection numbers of the unstable and stable manifolds of pairs of rest points with index diﬀerence equal to 1. Finally, we specialize the analysis to the negative gradient flow of a Morse function on a Riemannian Hilbert manifold. In this case, we prove that the Morse – Smale transversality assumption holds for generic perturbations of the metric, and that the isomorphism class of the Morse complex does not depend on the metric. These results provide an alternative approach to infinite-dimensional Morse theory, as developed by Palais and Smale in the sixties, see Palais (1963) and Smale (1964a; 1964b). 1 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 1–74. © 2006 Springer. Printed in the Netherlands.

2

A. ABBONDANDOLO AND P. MAJER

The third part is an exposition of recent results by the authors (see Abbondandolo and Majer, 2003b) about the Morse complex approach for gradient-like flows whose rest points have infinite Morse index and co-index. The framework is that of a Hilbert manifold M with a fixed infinite-dimensional and -codimensional subbundle V of the tangent bundle. When the gradient-like flow satisfies suitable compatibility conditions with respect to V, each rest point x can be given a relative Morse index m(x, V), and the unstable and stable manifolds of pairs of critical points x, y intersect in submanifolds of finite dimension m(x, V) − m(y, V). The study of the Hilbert Grassmannian, and in particular of the determinant bundle on the space of Fredholm pairs of subspaces of a Hilbert space, allow to prove that these intersections carry coherent orientations. Finally, suitable integrability assumptions on V, together with compactness assumptions on the flow, imply that the above intersections have compact closure in M. These facts allow to define the Morse complex. The first two parts contain fairly detailed proofs of all the statements, most of which — especially in the second part — are folklore results, for which we could not find appropriate reference in the literature. The style of the third part is diﬀerent: proofs are only sketched, or given in a simplified framework. We refer to Abbondandolo and Majer (2003b) for a more complete presentation. 1. A few facts from hyperbolic dynamics 1.1. ADAPTED NORMS

Let E be a real Banach space. A bounded linear operator L on E is said hyperbolic if its spectrum does not meet the imaginary axis1 : σ(L) ∩ iR = ∅. In this case, the decomposition of the spectrum of L into the disjoint closed subsets σ+ (L) = σ(L) ∩ {Re z > 0} and σ− (L) = σ(L) ∩ {Re z < 0} induces the splitting E = E u ⊕ E s into L-invariant closed linear subspaces, such that σ(L|E u ) = σ+ (L) and σ(L|E s ) = σ− (L), with projectors Pu = χ{Re z>0} (L), Ps = χ{Re z<0} (L). The spaces E u = E u (L) and E s = E s (L) are often called the positive (or unstable) and the negative (or stable) eigenspaces of L (although they may not consist of eigenvectors). An L-adapted norm is an equivalent norm · on E such that: ξ = max{Pu ξ, Ps ξ},

∀ξ ∈ E,

(1)

e−tL ξ ≤ e−λt ξ ∀ξ ∈ E u .

(2)

and there is λ > 0 such that for every t ≥ 0 etL ξ ≤ e−λt ξ ∀ξ ∈ E s ,

1 In the framework of discrete dynamical systems, a hyperbolic operator is a bounded operator whose spectrum does not meet the unit circle. In that context, an operator L satisfying σ(L)∩iR = ∅ should be called infinitesimally hyperbolic.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

3

As a consequence, also e−tL ξ ≥ eλt ξ ∀ξ ∈ E s ,

etL ξ ≥ eλt ξ ∀ξ ∈ E u ,

(3)

for every t ≥ 0. Such an adapted norm exists. Actually, for every λ in the interval ]0, min | Re σ(L)|[ there is a norm · satisfying (1), (2), and (3). The construction is based on the following lemma, applied to L|E s and to −L|E u . LEMMA 1.1. Let L be a bounded operator on the Banach space (E, · 0 ), and let λ be a real number such that λ > max Re σ(L). Then there exists a norm · on E equivalent to · 0 such that etL ξ ≤ etλ ξ for every ξ in E and t ≥ 0. Proof. Up to replacing L by L − λI, we may assume that λ = 0, so α := max Re σ(L) is negative, and we must find an equivalent norm · for which etL ξ ≤ ξ, for every ξ ∈ E and t ≥ 0. Still denoting by · 0 the operator norm induced by · 0 , the spectral radius formula and the spectral mapping theorem imply lim etL 01/t = max |σ(eL )| = max |eσ(L) | = eα < 1. t→+∞

Therefore, there exists c0 > 0 such that etL 0 ≤ c0 eαt/2 for every t ≥ 0, so +∞ 2c0 ξ0 ∀ξ ∈ E, etL ξ0 dt ≤ ξ := α 0 defines a norm on E not finer than · 0 . On the other hand, by compactness e−tL 0 ≤ c1 for every t ∈ [0, 1], so 1 1 etL ξ0 dt ≥ ξ0 ∀ξ ∈ E, ξ ≥ c 1 0 and the norm · is equivalent to · 0 . Finally, for every t ≥ 0 and ξ ∈ E, +∞ +∞ tL (s+t)L e ξ0 ds = e sL ξ0 ds ≤ ξ, e ξ = 0

t

concluding the proof.

EXERCISE 1.2. Find an adapted norm for the hyperbolic operator on E = R2 defined by the matrix −1 µ L= , 0 −1 where µ ∈ R, and draw the corresponding unit ball when µ is large. EXERCISE 1.3. Prove that if L is a normal operator on a Hilbert space H, that is L commutes with its adjoint L∗ , then the Hilbert norm is L-adapted.

4

A. ABBONDANDOLO AND P. MAJER

1.2. LINEAR STABLE AND UNSTABLE SPACES OF AN ASYMPTOTICALLY HYPERBOLIC PATH

Let A: [0, +∞] → L(E) be a continuous path of bounded linear operators on the Banach space E, such that A(+∞) is hyperbolic. Let XA : [0, +∞[ → L(E) be the solution of the linear Cauchy problem XA (t) = A(t)XA (t), XA (0) = I. EXERCISE 1.4. Prove that XA (t) is an isomorphism for every t, and find a linear Cauchy problem solved by its inverse. The linear subspace of E

WAs = ξ ∈ E lim XA (t)ξ = 0 t→+∞

is said the linear stable space of the asymptotically hyperbolic path A. Similarly, if A: [−∞, 0] → L(E) is a continuous path of operators such that A(−∞) is hyperbolic, the linear unstable space of A is defined as

WAu = ξ ∈ E lim XA (t)ξ = 0 . t→−∞

EXERCISE 1.5. Prove that if A(t) ≡ L is constant (and hyperbolic), then WAs = E s (L), the negative eigenspace of L, and WAu = E u (L), the positive eigenspace of L. A consequence of the hyperbolicity of A(+∞) is that the linear subspaces WAs and WAu are closed and complemented in E, and they are isomorphic to E s A(+∞) and to E u A(−∞) , respectively. Indeed, one can prove that if A is close enough to A(+∞) in the L∞ norm, then WAs is the graph of a bounded operator from E s A(+∞) to E u A(+∞) . The statement for a general asymptotically hyperbolic path A follows, because s WAs = XA (t)−1 WA(t+·) . See for instance Abbondandolo and Majer (2003c, Proposition 1.2) for a complete proof (the case of a Hilbert space is treated in that reference, but the proof in the Banach setting presents no diﬀerence). Denote by C0k ([0, +∞[, E) the Banach space of all C k curves u: [0, +∞[ → E such that lim u(h) (t) = 0 ∀h ∈ {0, 1, . . . , k}. t→+∞

PROPOSITION 1.6. Let A ∈ C 0 [0, +∞], L(E) be a path of bounded linear operators on the Banach space E such that L = A(+∞) is hyperbolic.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

5

(i) The bounded linear operator F A+ : C01 ([0, +∞[, E) → C00 ([0, +∞[, E),

u → u − Au,

is a left inverse. Moreover, F A+ admits a right inverse R+A such that

WAs + R+A v(0) v ∈ C00 ([0, +∞[, E), v(0) = 0 = E. (ii) The evaluation map

ker F A+ → E,

(4)

u → u(0),

is a right inverse. Proof. We endow E with a Banach norm · adapted to L. (i) Let us start by considering the case of the constant path A(t) ≡ L. By (2), the operator valued piecewise continuous function G: R → L(E),

G(t) = etL (R+ (t)Ps − R− (t)Pu ),

satisfies G(t) ≤ e−λ|t| , in particular it is integrable on R. Let v ∈ C00 ([0, +∞[, E). It is readily seen that the curve +∞ u(t) = (G ∗ v)(t) = G(t − τ)v(τ) dτ 0

is continuously diﬀerentiable and solves the equation u (t) − Lu(t) = v(t).

(5)

u(t) ≤ GL1 (R,L(E)) v∞,[s,+∞[ + GL1 (]t−s,t[,L(E)) v∞ ,

(6)

Moreover, the inequality

shows that u ∈ C00 ([0, +∞[, E), so by (5), u ∈ C01 ([0, +∞[, E). We conclude that the operator R+L : C00 ([0, +∞[, E) → C01 ([0, +∞[, E),

v → G ∗ v,

is a right inverse of F L+ . Indeed, such a linear map is continuous by (5) and (6) with s = 0. Let us check that the operator v → R+L v(0) maps v ∈ Cc∞ (]0, +∞[, E) onto E u ; since E u is a direct complement of E s = WLs this implies that the right inverse R+L satisfies (4). Let ϕ be a smooth real function with supp ϕ ⊂ ]0, +∞[ so small that the operator +∞ U := ϕ(τ)e−τL dτ ∈ L(E) 0

6

A. ABBONDANDOLO AND P. MAJER

is an isomorphism. The operator U preserves the splitting E = E u ⊕ E s . If ξ ∈ E u , setting v = −ϕU −1 ξ, there holds +∞ +∞ R+L v(0) = e−τL Pu ϕ(τ)U −1 ξ dτ = ϕ(τ)e−τL dτ U −1 ξ = ξ, 0

0

proving the claim. Let us now consider the general case. Setting A s (t) = A(s + t), we have that lim F A+s = F L+

s→+∞

in the operator norm of L(C01 , C00 ). Since the set of left inverses is open, by our previous case we deduce that F A+s has a right inverse R+As for s large, such that R+As → R+L in the operator norm for s → +∞. Since the space of surjective operators is open, R+As satisfies (4) for s large. Fix such a large s. We can now define a right inverse R+A of F A+ by setting R+A v = u, where u is the solution of the linear Cauchy problem u − Au = v, u(s) = R+As v s (0). The continuity of R+A is easily seen by the formula t + −1 + XA (τ)−1 v(τ) dτ . (RA v)(t) = XA (t) XA (s) RAs v s (0) + s

Finally, the fact that R+As satisfies (4) implies that also R+A satisfies (4). Indeed, let ξ ∈ E, and let v ∈ C00 ([0, +∞[, E) with v(0) = 0 be such that XA (s)ξ ∈ R+As v(0) + WAs s . Since v(0) = 0, the curve

(7)

w(t) =

v(t − s) if t ≥ s, 0 if 0 ≤ t ≤ s,

belongs to C00 ([0, +∞[, E). Since w vanishes on [0, s], R+A w solves the equation u − Au = 0 on [0, s], so R+As v(0) = R+A w(s) = XA (s)R+A w(0). By (7) and (8), ξ ∈ R+A w(0) + XA (s)−1 WAs s = R+A w(0) + WAs , concluding the proof of (i).

(8)

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

7

(ii) The kernel of F A+ is ker F A+ = {XA (t)ξ | ξ ∈ WAs }, so if Q ∈ L(E) is a projector onto WAs , the linear map E → ker F A+ ,

ξ → XA (·)Qξ,

is a left inverse of the evaluation at 0, ker F A+ → E,

u → u(0).

REMARK 1.7. If P is a projector onto WAs , it can be shown that +∞ R+A v(t) = XA (t) R+ (t − τ)P − R− (t − τ)(I − P) XA (τ)−1 v(τ) dτ 0

defines a right inverse of F A+ . See Abbondandolo and Majer (2003c) for a more extensive discussion of the topics of this section. We conclude this section by establishing some properties of the operator d/dt − A(t) on the whole real line. PROPOSITION 1.8. Assume that A ∈ C 0 R, L(E) has hyperbolic asymptotic operators A(−∞) and A(+∞), both with finite-dimensional positive eigenspace. Then the bounded linear operator F A : C01 (R, E) → C00 (R, E),

u → u − Au,

is Fredholm of index ind F A = dim E u A(−∞) − dim E u A(+∞) . Moreover, WAu + WAs is closed and ker F A WAu ∩ WAs , coker F A E/(WAu + WAs ). (9) Proof. Since WAu E u A(−∞) and WAs E s A(+∞) , the first space is finitedimensional and the second one is finite-codimensional, with dim WAu = dim E u A(−∞) , codim WAs = dim E u A(+∞) . (10) Therefore, WAu + WAs is (closed and) finite-codimensional, and dim WAu ∩ WAs − codim(WAu + WAs ) = dim WAu − codim WAs .

(11)

8

A. ABBONDANDOLO AND P. MAJER

The kernel of F A is the linear subspace ker F A = {XA (t)ξ | ξ ∈ WAu ∩ WAs }, so dim ker F A = dim WAu ∩ WAs .

(12)

By Proposition 1.6(i), the operators F A+ : C01 ([0, +∞[, E) → C00 ([0, +∞[, E),

u → u − Au,

F A− : C01 (]−∞, 0], E) → C00 (]−∞, 0], E),

u → u − Au,

have right inverses R+A and R−A . If v is an element of C00 (R, E), any solution of u − Au = v has the form u(t) = XA (t) u(0) − R+A v(0) + R+A v(t), ∀t ≥ 0, u(t) = XA (t) u(0) − R−A v(0) + R−A v(t), ∀t ≤ 0. Such a curve u belongs to C01 (R, E) if and only if u(0) − R+A v(0) ∈ WAs and u(0) − R−A v(0) ∈ WAu . Therefore, v belongs to the range of F A if and only if the aﬃne subspaces R+A v(0) + WAs and R−A v(0) + WAu have nonempty intersection, that is if and only if R+A v(0) − R−A v(0) belongs to WAs + WAu . So the range of F A is the linear subspace

ran F A = v ∈ C00 (R, E) R+A v(0) − R−A v(0) ∈ WAu + WAs . Such a linear subspace is closed. By the second assertion in Proposition 1.6(i), the operator E C00 (R, E) → u , v → [R+A v(0) − R−A v(0)], WA + WAs is onto, so codim ran F A = codim(WAu + WAs ). All the statements follow from (10) – (13).

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1.3. MORSE VECTOR FIELDS

Let M be a Banach manifold of class C 2 , i.e., a paracompact Hausdorﬀ topological space, locally homeomorphic to a Banach space E, endowed with an atlas whose transition maps are of class C 2 . See Lang (1999) for foundational results on Banach manifolds. A C 1 tangent vector field X on M defines a local flow φ solving ∂t φ(t, p) = X φ(t, p) , φ(0, p) = p, ∀p ∈ M, −∞ ≤ t− (p) < t < t+ (p) ≤ +∞,

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

9

where ]t− (p), t+ (p)[ denotes the maximal interval of existence of the above Cauchy problem. The functions t− and t+ are upper and lower semi-continuous, respectively. Denote by Ω(X) the subset of R × M which lies strictly between the graph of t− and the graph of t+ . Then Ω(X) is an open neighborhood of {0} × M, and the map φ: Ω(X) → M is of class C 1 . The vector field X is said complete (resp. positively complete, resp. negatively complete) if Ω(X) = R × M (resp. Ω(X) ⊃ [0, +∞[ × M, resp. Ω(X) ⊃ ]−∞, 0] × M). Let A be a positively invariant subset of M: this means that if p ∈ A then φ(t, p) ∈ A for every t ∈ [0, t+ (p)[. The vector field is said positively complete with respect to A if for every p ∈ M such that t+ (p) < +∞ there exists t ∈ [0, t+ (p)[ such that φ(t, p) ∈ A. Similarly, one defines a negatively complete vector field with respect to a negatively invariant subset. A rest point of X is a point x ∈ M such that X(x) = 0. The set of rest points of X will be denoted by rest(X). The Jacobian of X at a rest point x is the bounded linear operator on T x M defined by ∇X(x)ξ = [X, Y](x), where ξ ∈ T x M and Y is a tangent vector field on M such that Y(x) = ξ. Indeed, the fact that X(x) = 0 implies that this definition does not depend on the choice of extension Y of ξ. EXERCISE 1.9. Give an alternative definition of the Jacobian of a vector field at a rest point in terms of a local chart: if ϕ: U → E maps a neighborhood U of x ∈ rest(X) diﬀeomorphically onto an open subset of the Banach space E, define the operator ∇X(x) on T x M by ϕ∗ (∇X(x)ξ) = D(ϕ∗ X) ϕ(x) [ϕ∗ ξ],

∀ξ ∈ T x M,

where, for η ∈ T p M, ϕ∗ η is the vector in E defined by ϕ∗ η = Dϕ(p)[η]. Show that such a definition does not depend on the choice of the chart ϕ. A rest point x of X is said hyperbolic if the Jacobian of X at x is a hyperbolic operator. The corresponding splitting of the tangent space at x will be denoted by T x M = E ux ⊕ E sx . By the inverse mapping theorem, the hyperbolic rest points are isolated in rest(X). The Morse index m(x) ∈ N ∪ {+∞} of the hyperbolic rest point x is the dimension of the subspace E ux . The Morse co-index is the dimension of E sx . If all the rest points of X are hyperbolic, the vector field X is said a Morse vector field. 1.4. LOCAL DYNAMICS NEAR A HYPERBOLIC REST POINT

Let U be an open neighborhood of 0 in the Banach space E, and let X ∈ C 1 (U, E) be a vector field having 0 as a hyperbolic rest point. Denote by φ: Ω(X) → U the local flow of X. Let L := ∇X(0) = DX(0), with splitting E = E u ⊕ E s and projectors Pu , Ps , and let us endow E with an L-adapted norm · . If V ⊂ E is a

10

A. ABBONDANDOLO AND P. MAJER

closed linear subspace, V(r) will denote the closed ball in V of radius r centered in 0, and ∂V(r) will be the relative boundary of V(r) in V. Consider the cones C u = {ξ ∈ E | Ps ξ ≤ Pu ξ} ⊃ E u ,

C s = {ξ ∈ E | Pu ξ ≤ Ps ξ} ⊃ E s .

We recall that if A ⊂ B ⊂ U the set A is said positively (negatively) invariant with respect to B if for every ξ ∈ A and for every t > 0, φ([0, t] × {ξ}) ⊂ B implies φ([0, t] × {ξ}) ⊂ A (resp. for every ξ ∈ A and for every t < 0, φ([t, 0] × {ξ}) ⊂ B implies φ([t, 0] × {ξ}) ⊂ A). LEMMA 1.10. For every r > 0 small enough there holds: (i) the set C u ∩ E(r) is positively invariant with respect to E(r); (ii) the set C s ∩ E(r) is negatively invariant with respect to E(r); (iii) if ξ belongs to the set C u ∩ ∂E(r) = ∂E u (r) × E s (r) then Pu φ(t, ξ) > r for every t ∈ ]0, 1], and Pu φ(t, ξ) < r for every t ∈ [−1, 0[; (iv) if ξ belongs to the set C s ∩ ∂E(r) = E u (r) × ∂E s (r) then Ps φ(t, ξ) < r for every t ∈ ]0, 1], and Ps φ(t, ξ) > r for every t ∈ [−1, 0[. Proof. Since t+ (0) = +∞ and t− (0) = −∞, we have t+ (ξ) > 1 and t− (ξ) < −1 for ξ small enough. A first order expansion of φ(t, ·) at 0 yields to φ(t, ξ) = etL ξ + o(ξ)t

for ξ → 0,

uniformly in t ∈ [−1, 1]. Therefore, if r > 0 is small enough, for every ξ ∈ C s ∩ E(r) and t ∈ [0, 1], (2) implies Ps φ(t, ξ) = Ps etL ξ + o(ξ)t = etL Ps ξ + o(Ps ξ)t ≤ e−λt Ps ξ + o(Ps ξ)t ≤ e−λt/2 Ps ξ, and similarly, for every ξ ∈ C u ∩ E(r) and t ∈ [0, 1], (3) implies Pu φ(t, ξ) ≥ eλt/2 Pu ξ. All the statements follow from the above inequalities and from the analogous inequalities holding for t ∈ [−1, 0]. REMARK 1.11. In the language of Conley theory, E(r) is an isolating neighborhood for the invariant set {0}, and ∂E u (r) × E s (r) is its exit set.

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11

1.5. LOCAL STABLE AND UNSTABLE MANIFOLDS

Given r > 0, the local unstable manifold and the local stable manifold of 0 are the sets

u Wloc,r (0) = ξ ∈ E(r) t− (ξ) = −∞, φ(]−∞, 0] × {ξ}) ⊂ E(r), lim φ(t, ξ) = 0 , t→−∞

s + Wloc,r (0) = ξ ∈ E(r) t (ξ) = +∞, φ([0, +∞[ × {ξ}) ⊂ E(r), lim φ(t, ξ) = 0 . t→+∞ When r is small, these sets are actually graphs of regular maps. THEOREM 1.12 (Local (un)stable manifold theorem). Assume that 0 is a hyperbolic rest point of the C k vector field X: U → E, k ≥ 1. For any r > 0 small s (0) is the graph of a C k map σs : E s (r) → E u (r) such that σs (0) = 0 enough, Wloc,r u (0) is the graph of a C k map σu : E u (r) → E s (r) and Dσs (0) = 0. Similarly, Wloc,r u u such that σ (0) = 0, Dσ (0) = 0. See Shub (1987, Chapter 5) for a proof based on the graph transform method. Here we will present a proof based on the study of the orbit space and on Proposition 1.6. Proof. We shall prove the conclusion for the local stable manifold, the case of the unstable one following by considering the vector field −X. The map Φ: C01 ([0, +∞[, U) → C00 ([0, +∞[, E),

u → u − X ◦ u,

is of class C k , and its diﬀerential at u ∈ C01 ([0, +∞[, U) is DΦ(u): C01 ([0, +∞[, E) → C00 ([0, +∞[, E), v → v − DX(u)v. Since DX u(t) converges to L = DX(0) for t → +∞, statement (i) of Proposition 1.6 implies that DΦ(u) is a left inverse, so Φ is a C k submersion. In particular, its set of zeros Φ−1 ({0}) is a C k submanifold of C01 ([0, +∞[, U). The set of zeros is nonempty, because it contains the curve 0. Actually T 0 Φ−1 ({0}) = ker DΦ(0) = ker(v → v − Lv) = {etL ξ | ξ ∈ E s }.

(14)

By statement (ii) of Proposition 1.6, the evaluation map ev0 : u → u(0) subordinates an immersion ev0 : Φ−1 ({0}) → U, which is injective by the uniqueness of the solution of Cauchy problems. Therefore, W s (0) := ev0 Φ−1 ({0}) = ξ ∈ U t+ (ξ) = +∞, lim φ(t, ξ) = 0 t→+∞

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A. ABBONDANDOLO AND P. MAJER

is the image of an injective C k immersion. The point 0 belongs to W s (0), and by (14), T 0 W s (0) = D ev0 (0)T 0 Φ−1 ({0}) = ev0 T 0 Φ−1 ({0}) = E s . By the implicit function theorem, if r is small enough the path-connected component of W s (0) ∩ E(r) containing 0 — call it Zr — is the graph of a C k map σs : E s (r) → E u (r) such that σs (0) = 0 and Dσs (0) = 0. We claim that if r is so small that the conclusions of Lemma 1.10 hold, and s (0), which concludes that the Lipschitz norm of σs is less than 1, then Zr = Wloc,r s s (0) to the proof. Indeed, by definition Wloc,r (0) ⊂ Zr , a path connecting ξ ∈ Wloc,r s 0 within W (0) ∩ E(r) being provided by the orbit of ξ. On the other hand, notice that by definition Zr is positively invariant with respect to E(r). So if there exists s (0), by Lemma 1.10 there is some t > 0 for which φ(t, ξ) ∈ ∂E u (r) × ξ ∈ Zr \ Wloc,r E s (r) (the latter is the exit set of E(r)) and φ(t, ξ) ∈ Zr (Zr is positively invariant with respect to E(r)). Therefore Zr ∩ ∂E u (r) × E s (r) is nonempty, contradicting the fact that Zr is the graph of a map whose Lipschitz constant is less than 1, taking value 0 at 0.

1.6. THE GROBMAN – HARTMAN LINEARIZATION THEOREM

The Grobman – Hartman theorem says that up to a change of variables, the dynamics near a hyperbolic point is the dynamics given by a linear vector field. We will deduce this fact from the analogous statement for discrete dynamical systems. The proof is adapted from Shub (1987). Let us start with a result about the existence, uniqueness, and H¨older regularity of a semi-conjugacy between two perturbations of a linear operator. PROPOSITION 1.13. Let E = E u ⊕ E s be an invariant splitting for the bounded invertible operator T . Let Pu and Ps be the corresponding projectors, and assume that there exists µ < 1 such that max{Ps T Ps , Pu T −1 Pu } ≤ µ. Let ϕ and ψ be Lipschitz continuous maps from E to E such that: (i) ϕ − ψ∞ < +∞; (ii) lip ϕ < 1 − µ; (iii) lip ψ < 1/T −1 .

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13

Then there exists a unique bounded map g: E → E such that (T + ϕ) ◦ (I + g) = (I + g) ◦ (T + ψ). Moreover, g∞ ≤ and setting

(15)

ϕ − ψ∞ , 1 − (µ + lip ϕ)

θ := max T + lip ψ,

T −1 , 1 − T −1 lip ψ

g is α-H¨older continuous for every α<

− log(µ + lip ϕ) . log θ

Notice that if E u (0), then T ≥ Pu T Pu ≥ 1/µ, while if E s (0), then T −1 ≥ Ps T −1 Ps ≥ 1/µ. Therefore log θ ≥ − log µ, so the quantity − log(µ + lip ϕ)/ log θ appearing in the above proposition does not exceed 1. In general, the map g is not locally Lipschitz, even when ϕ and ψ are smooth. Proof. For an E-valued map f , we denote by fu and fs its components with respect to the splitting E = E u ⊕ E s , that is fu := Pu f , fs := Ps f . By applying the projectors Pu and Ps , (15) is equivalent to (T u + ϕu ) ◦ (I + g) = (Pu + gu ) ◦ (T + ψ), (16) (T s + ϕs ) ◦ (I + g) = (Ps + gs ) ◦ (T + ψ). Since lip T −1 ψ ≤ T −1 lip ψ < 1, the map T + ψ = T (I + T −1 ψ) is a homeomorphism of E onto E. Actually, its inverse is Lipschitz continuous with lip(T + ψ)−1 = lip (I + T −1 ψ)−1 T −1 ≤

T −1 . 1 − T −1 lip ψ

(17)

By a simple algebraic manipulation, (16) is equivalent to the fixed point problem F(g) = g, where F(g)u = T u−1 (gu ◦ (T + ψ) − ϕu ◦ (I + g) + ψu ), F(g)s = (T s gs + ϕs ◦ (I + g) − ψs ) ◦ (T + ψ)−1 . Since F(g)u ∞ ≤ ≤ F(g)s ∞ ≤ ≤

µ (1 + lip ϕ)g∞ + ϕ − ψ∞ (µ + lip ϕ)g∞ + µϕ − ψ∞ , (T s + lip ϕ)g∞ + ϕ − ψ∞ (µ + lip ϕ)g∞ + ϕ − ψ∞ ,

(18) (19)

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F maps B(E, E), the Banach space of bounded maps from E to E, into itself. Actually, (18) and (19) imply that if g∞ ≤ R, with R :=

ϕ − ψ∞ , 1 − (µ + lip ϕ)

then F(g)∞ ≤ R. Moreover, the maps Fu = Pu F: B(E, E) → B(E, E u ) and Fs = Ps F: B(E, E) → B(E, E s ) are Lipschitz with lip Fu ≤ T u −1 (1 + lip ϕ) ≤ µ(1 + lip ϕ) < 1, lip Fs ≤ T s + lip ϕ ≤ µ + lip ϕ < 1, so F: B(E, E) → B(E, E) is a contraction, proving that there exists a unique g ∈ B(E, E) satisfying (15). Since F maps the closed R-ball of C 0 ∩ B(E, E) into itself, the fixed point g is continuous and bounded by R. If h ∈ B(E, E) has modulus of continuity2 ω, then F(h)u has modulus of continuity t → µω (T + lip ψ)t + µ lip ϕω(t) + µ(lip ϕ + lip ψ)t, (20) while by (17), F(h)s has modulus of continuity t → (µ + lip ϕ)ω(σt) + (lip ψ + lip ϕ)σt,

(21)

where σ := T −1 /(1−T −1 lip ψ). Comparing (20) and (21), we find that setting a := (lip ϕ + lip ψ)σ, the function t → (µ + lip ϕ)ω(θt) + at is a modulus of continuity for F(h). If moreover h∞ ≤ R, we have that F(h)∞ ≤ R, so F(h) has modulus of continuity t → min{(µ + lip ϕ)ω(θt) + at, 2R}. Therefore, if a modulus of continuity ω satisfies min{(µ + lip ϕ)ω(θt) + at, 2R} ≤ ω(t) ∀t ∈ [0, +∞[,

(22)

we deduce that the nonempty closed subset of B(E, E) {h ∈ B(E, E) | h∞ ≤ R, h has modulus of continuity ω} is F-invariant, hence the fixed point g has modulus of continuity ω. A function of the form ω(t) = ctα satisfies (22) if (µ + lip ϕ)θα < 1, 2

Here, moduli of continuity are always assumed to be nondecreasing.

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and c is large enough. The conclusion follows.

15

If we symmetrize the assumptions of the above proposition, a standard argument involving uniqueness yields to the following global version of the Grobman – Hartman theorem for discrete dynamical systems. COROLLARY 1.14. Let T be an invertible bounded operator on E satisfying the same assumptions of Proposition 1.13. Let ϕ and ψ be Lipschitz continuous maps from E to E such that (i) ϕ − ψ∞ < +∞; (ii) lip ϕ < min{1 − µ, 1/T −1 }, lip ψ < min{1 − µ, 1/T −1 }. Then there exists a unique bounded map g: E → E such that (T + ϕ) ◦ (I + g) = (I + g) ◦ (T + ψ). Moreover, g is H¨older continuous, and I + g is homeomorphism of E onto E. Proof. Applying Proposition 1.13 to the pair (ϕ, ψ) and to the pair (ψ, ϕ), we find H¨older continuous bounded maps g: E → E and h: E → E such that (T + ϕ) ◦ (I + g) = (I + g) ◦ (T + ψ), (T + ψ) ◦ (I + h) = (I + h) ◦ (T + ϕ). It follows that (I + g) ◦ (I + h), which is of the form I + k with k ∈ B(E, E), satisfies (T + ϕ) ◦ (I + k) = (I + k) ◦ (T + ϕ). By the uniqueness statement of Proposition 1.13 applied to the pair (ϕ, ϕ), k must be the zero map, that is (I + g) ◦ (I + h) = I. Similarly, (I + h) ◦ (I + g) = I, so I + g is a homeomorphism of E onto E with inverse I + h. Now we can derive a global version of the Grobman – Hartman theorem for flows. THEOREM 1.15. Let L be a hyperbolic operator on E. Let · be an L-adapted norm on E, satisfying (1) and (2) for some positive λ. Let B1 : E → E and B2 : E → E be Lipschitz continuous maps such that (i) B1 − B2 ∞ < +∞;

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(ii) lip B1 < λ, lip B2 < λ. Then the flows φ1 , φ2 : R × E → E of the vector fields X1 (ξ) = Lξ + B1 (ξ) and X2 (ξ) = Lξ + B2 (ξ) are conjugated. More precisely, there is a unique bounded map g: E → E such that φ1 t, (I + g)(ξ) = (I + g) φ2 (t, ξ ∀(t, ξ) ∈ R × E, and I + g is a homeomorphism of E onto E. Moreover, g is α-H¨older continuous for every λ − lip B1 . α< L + lip B2 Proof. A c-Lipschitz vector field X produces a globally defined flow φ, with lip φ(t, ·) ≤ ec|t| . If two c-Lipschitz vector fields X1 , X2 have bounded distance, then φ1 (t, ·) − φ2 (t, ·)∞ ≤ X1 − X2 ∞ |t|ec|t| . Let ψi (t, ξ) = φi (t, ξ) − etL ξ, for i = 1, 2. By our initial considerations, for every t the maps ψ1 (t, ·) and ψ2 (t, ·) are Lipschitz and have bounded distance. The maps ψi satisfy t e(t−s)L Bi (e sL ξ + ψi (s, ξ)) ds ∀(t, ξ) ∈ R × E. (23) ψi (t, ξ) = 0

By (23), t e(t−s)L e sL + lip ψi (s, ·) ds lip ψi (t, ·) ≤ lip Bi 0 ≤ |t| lip Bi sup e sL sup e sL + sup lip ψi (s, ·) |s|≤|t|

|s|≤|t|

|s|≤|t|

= |t| lip Bi 1 + sup lip ψi (s, ·) 1 + o(1) ,

|s|≤|t|

for t → 0. Taking the supremum for all |t| ≤ τ, we obtain sup lip ψi (t, ·) ≤ lip Bi |τ| 1 + o(1) for τ → 0. |t|≤τ

(24)

Since lip Bi < λ, the last inequality implies that there exists τ > 0 such that lip ψi (t, ·) < 1 − e−λ|t| , lip ψi (t, ·) < 1/e−tL ,

∀ 0 < |t| ≤ τ, i = 1, 2.

By Corollary 1.14 applied to T = etL , µ = e−λ|t| , ϕ = ψ1 (t, ·), and ψ = ψ2 (t, ·), for every 0 < |t| ≤ τ there exists a unique gt ∈ B(E, E) such that (25) φ1 t, (I + gt )(ξ) = (I + gt ) φ2 (t, ξ) ,

17

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and I +gt is a homeomorphism of E onto E. If n ∈ Z\{0} and |nt| ≤ τ, (25) implies φ1 nt, (I + gt )(·) = (I + gt ) φ2 (nt, ·) , so by uniqueness gt = gnt . If p, q are rational numbers in [−τ, τ] \ {0}, they have a common sub-multiple, so g p = gq . Therefore g p = g for every rational p ∈ [−τ, τ] \ {0}. By the continuity of φ1 and φ2 with respect of t, φ1 t, (I + g)(·) = (I + g) φ2 (t, ·) holds for every |t| ≤ τ, hence by taking iterates, for every t ∈ R. There remains to estimate the H¨older exponent of g. Let 0 < t ≤ τ. By Proposition 1.13 gt is α-H¨older for every − log e−λt + lip ψ1 (t, ·) α< , log θ(t) where

θ(t) = max etL + lip ψ2 (t, ·),

e−tL . 1 − e−tL lip ψ2 (t, ·)

Since gt = g, g is α-H¨older for every α < β := lim sup t→0+

− log(e−λt + lip ψ1 (t, ·)) . log θ(t)

By (24), − log e−λt + lip ψ1 (t, ·) ≥ − log e−λt + lip B1 t + o(t) = (λ − lip B1 )t + o(t) (26) for t → 0+ . Since

e±tL ≤ etL = 1 + Lt + o(t),

for t → 0+ , by (24) there holds etL + lip ψ2 (t, ·) ≤ 1 + (L + lip B2 )t + o(t), 1 + Lt + o(t) e−tL ≤ = 1 + (L + lip B2 )t + o(t), −tL 1 − lip B2 t + o(t) 1 − e lip ψ2 (t, ·) for t → 0+ . Therefore

log θ(t) ≤ log 1 + (L + lip B2 )t + o(t) = (L + lip B2 )t + o(t),

(27)

for t → 0+ . The inequalities (26) and (27) imply that − log e−λt + lip ψ1 (t, ·) λ − lip B1 (λ − lip B1 )t + o(t) ≥ lim+ = β = lim sup , t→0 + log θ(t) (L + lip B )t + o(t) L + lip B2 2 t→0 concluding the proof. It is then straightforward to deduce the following local linearization result.

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COROLLARY 1.16. Assume that 0 is a hyperbolic rest point of the C 1 vector field X: U → E, and let L = DX(0). If r > 0 is small enough, the local flow φ restricted to E(r) is conjugated to the linear flow (t, ξ) → etL ξ by a bi-H¨older continuous homeomorphism. More precisely, there exists a bi-H¨older continuous homeomorphism h: E(r) → h E(r) ⊂ E such that h φ(t, ξ) = etL h(ξ) ∀(t, ξ) ∈ Ω(X|E(r) ). We conclude the discussion about the local dynamics at a rest point with the following proposition. PROPOSITION 1.17. For every r > 0 small enough there holds: for every sequence (ξn ) ⊂ E converging to 0 and for every sequence (tn ) ⊂ [0, +∞[ such that φ([0, tn ] × {ξn }) ⊂ E(r) and φ(tn , ξn ) ∈ ∂E(r), there holds u dist φ(tn , ξn ), Wloc,r (0) ∩ ∂E(r) → 0. Proof. If the vector field is linear, X(ξ) = Lξ, the conclusion is immediate: u (0) = E u (r), and for any (ξ ) ⊂ E converging to 0 and indeed in this case Wloc,r n any (tn ) ⊂ [0, +∞[, by (2) we have lim sup dist(etn L ξn , E u ) = lim sup Ps etn L ξn ≤ lim sup e−tn λ Ps ξn = 0. n→∞

n→∞

n→∞

By the Grobman – Hartman theorem, if r0 > 0 is small enough the local flow φ restricted to E(r0 ) is conjugated to its linearization (t, ξ) → etL ξ, by a biuniformly continuous homeomorphism. By the local (un)stable manifold theorem, u we may also assume that r0 is so small that Wloc,r (0) is the graph of a uniformly 0 u u s continuous map σ : E (r0 ) → E (r0 ). Let r < r0 and set ηn := φ(tn , ξn ) ∈ ∂E(r), with ξn → 0 and tn ≥ 0. By Lemma 1.10, Pu ηn = r. By the linear case and by the uniform continuity of the u conjugacy, there exists (η n ) ⊂ Wloc,r (0) such that ηn −η n is infinitesimal. Setting 0 u u u u u η n = P ηn , σ (P ηn ) ∈ Wloc,r (0) ∩ ∂E(r), by the uniform continuity of σ we have u (0) ∩ ∂E(r) dist ηn , Wloc,r ≤ ηn − η n s s ≤ ηn − η n + Pu η n − Pu η n + P ηn − P ηn = ηn − η n + Pu η n − Pu ηn + σu (Pu η n ) − σu (Pu ηn ) → 0, concluding the proof.

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1.7. GLOBAL STABLE AND UNSTABLE MANIFOLDS

Let us assume that X is a C 1 tangent vector field on the Banach manifold M, and that x is a hyperbolic rest point of X. We shall identify a neighborhood of x with a neighborhood of 0 in the Banach space E x = T x M, identifying x with 0. We shall consider a ∇X(x)-adapted norm on E x , and we will use the notation introduced in the above sections: for instance, E x (r) ⊂ M will denote the closed r ball centered in x. The unstable and the stable manifolds of the rest point x are the subsets of M

W u (x) = p ∈ M t− (p) = −∞ and lim φ(t, p) = x , t→−∞

s + W (x) = p ∈ M t (p) = +∞ and lim φ(t, p) = x . t→+∞

THEOREM 1.18. Let x ∈ M be a hyperbolic rest point of the C k vector field X, k ≥ 1, on the Banach manifold M. Then W u (x) and W s (x) are the images of injective C k immersions of manifolds which are homeomorphic to E ux and E sx , respectively. Proof. By Theorem 1.12, if r is small enough the local unstable manifold u (x) is the graph of a C k map σu : E u (r) → E s (r). Since Wloc,r

u (x), 0 ≤ t < t+ (p) , W u (x) = φ(t, p) p ∈ Wloc,r u (x) by the the set W u (x) inherits the structure of a C k manifold from that of Wloc,r maps {φ(t, ·)}, and the inclusion of W u (x) into M is a C k injective immersion. u (x) is the C k diﬀeomorphism θ(ξ) = ξ + σu (ξ), the map If θ: E u (r) → Wloc,r

A := ξ ∈ E ux log ξ < t+ θ(rξ/ξ) → W u (x),

ξ → φ log ξ, θ(rξ/ξ) ,

is a homeomorphism from a star-shaped open subset of E ux - thus homeomorphic to E ux itself - onto W u (x). The analogous results for W s (x) follow by considering the vector field −X. REMARK 1.19. If M is a Hilbert manifold, then the regularity of the norm implies that W u (x) and W s (x) are actually images of C k immersions of E ux and E sx , respectively. In general W u (x) and W s (x) need not be embedded submanifolds: actually, they need not be locally closed. A Lyapunov function for X is a C 1 function f : M → R such that D f (p)[X(p)] < 0 for every p ∈ M \ rest(X). In this case, of course crit( f ) ⊂ rest(X). If X is

20

A. ABBONDANDOLO AND P. MAJER

a Morse vector field, the two sets actually coincide. Indeed, we can assume that M = E is a Banach space, so if x ∈ rest(X) and t ∈ R we have D f (x+tv)[X(x+tv)] = D f (x)[X(x+tv)]+o(t) = tD f (x)[DX(x)v]+o(t) for t → 0. The principal part of the right-hand side is an odd function of t. Since the above quantity has to be negative for every t 0, such a principal part has to be identically zero. Since DX(x) is an isomorphism, we deduce that D f (x) = 0. THEOREM 1.20. Assume that X admits a Lyapunov function f , and that for every r0 > 0 small enough there holds

u s p ∈ Wloc,r sup f (p) p ∈ Wloc,r (x) ∩ ∂E (r ) < inf f (p) (x) ∩ ∂E x (r0 ) . (28) x 0 0 0 Then if r > 0 is small enough: (i) for every p ∈ M, the closed set I = {t ∈ ]t− (p), t+ (p)[ | φ(t, p) ∈ E x (r)} is an interval, and its interior is {t ∈ ]t− (p), t+ (p)[ | φ(t, p) ∈ E˚ x (r)}; (ii) if I is upper bounded, then φ(max I, p) ∈ ∂E ux (r) ∩ E s (r); conversely, if φ(t, p) ∈ ∂E ux (r) ∩ E s (r), then t = max I; (iii) if I is lower bounded, then φ(min I, p) ∈ E ux (r) ∩ ∂E s (r); conversely, if φ(t, p) ∈ E ux (r) ∩ ∂E s (r), then t = min I; u (x), and W s (x) ∩ E (r) = W s (x); (iv) W u (x) ∩ E x (r) = Wloc,r x loc,r

(v) W u (x) and W s (x) are submanifolds of M. Proof. Let r0 be so small that (28) and the conclusions of Lemma 1.10, Theorem 1.12, and Proposition 1.17 hold. Since f is of class C 1 , up to choosing a smaller r0 we may also assume that f is uniformly continuous on E x (r0 ). We shall prove the first assertion in (i) arguing by contradiction. In fact, assume that there exist an infinitesimal sequence of positive numbers (ρn ), ρn < r0 , a sequence of points (ξn ) ∈ ∂E x (ρn ), and a sequence of positive numbers (tn ) such that φ(tn , ξn ) ∈ ∂E x (ρn ) and φ(]0, tn [ × {ξn }) ∩ E x (ρn ) = ∅. Lemma 1.10(iv) implies that (at least for n large) ξn ∈ ∂E ux (ρn ) × E sx (ρn ) ⊂ C u ∩ E x (r0 ). By Lemma 1.10(i), C u ∩ E x (r0 ) is positively invariant with respect to E x (r0 ), and by (iii) if t > 0 and φ([0, t] × {ξn }) ⊂ C u ∩ E x (r0 ) then φ(t, ξn ) E x (ρn ). Therefore, there exists an ∈ ]0, tn [ such that φ([0, an ] × {ξn }) ⊂ E x (r0 ) and φ(an , ξn ) ∈ ∂E x (r0 ). Similarly, there exists bn ∈ [an , tn [ such that φ(bn , ξn ) ∈ ∂E x (r0 ) and φ([b, tn ] × {ξn }) ⊂ E x (r0 ). Since ξn → 0 and φ(tn , ξn ) → 0, Proposition 1.17 implies that u s dist φ(an , ξn ), Wloc,r (x)∩∂E x (r0 ) → 0, dist φ(bn , ξn ), Wloc,r (x)∩∂E x (r0 ) → 0. 0 0 Therefore, by (28), taking into account the fact that f is uniformly continuous on E x (r0 ), lim sup f (φ(an , ξn )) ≤ n→∞

sup u Wloc,r (x)∩∂E x (r0 ) 0

f <

inf

s Wloc,r (x)∩∂E x (r0 ) 0

f ≤ lim inf f (φ(bn , ξn )), n→∞

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21

a contradiction because an ≤ bn implies f φ(an , ξn ) ≥ f φ(bn , ξn ) . The second statement in (i), and statements (ii), (iii) are immediate consequences of Lemma 1.10. The inclusions u Wloc,r (x) ⊂ W u (x) ∩ E x (r),

s Wloc,r (x) ⊂ W s (x) ∩ E x (r),

are obvious. The opposite inclusions follow from statement (i). Then W u (x) and W s (x) are submanifold of M, because W u (x) = {φ(t, p) | p ∈ W u (x) ∩ E(r), 0 ≤ t < t+ (p)}, W s (x) = {φ(t, p) | p ∈ W s (x) ∩ E(r), t− (p) < t ≤ 0}, and because φ(t, ·) is a diﬀeomorphism.

REMARK 1.21. The weak inequality always holds in (28). The strict inequality holds if either: (i) x has finite Morse index; (ii) M is a Hilbert manifold, f is twice diﬀerentiable at x and the second diﬀerential of f at x satisfies D2 f (x)[ξ, ξ] ≤ −λξ2 for every ξ ∈ E ux , for some positive constant λ. u (x) is a compact set, so Indeed, in the first case Wloc,r 0

sup u Wloc,r (x)∩∂E x (r0 ) 0

f =

max

u Wloc,r (x)∩∂E x (r0 ) 0

f < f (x) ≤

inf

s Wloc,r (x)∩∂E x (r0 )

f.

0

In the second case, a second order expansion of f at x yields to the same conclusion.

2. The Morse complex in the case of finite Morse indices 2.1. THE PALAIS – SMALE CONDITION

Assume that f is a Lyapunov function for the vector field X on the Banach manifold M. A Palais – Smale (PS) sequence at level c is a sequence (pn ) ⊂ M such that ( f (pn )) converges to c and (D f (pn )[X(pn )]) is infinitesimal. We shall say that (X, f ) satisfies the Palais – Smale (PS) condition at level c if every Palais – Smale sequence at level c is compact. If (X, f ) satisfies the (PS) condition at every c ∈ [a, b], then rest(X)∩ f −1 ([a, b]) is compact, so if X is also Morse this set is finite.

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A. ABBONDANDOLO AND P. MAJER

REMARK 2.1. Assume that Jn = φ([0, tn ] × {pn }), tn > 0, is contained in a strip {a ≤ f ≤ b}, and that lim

n→∞

f (pn ) − f (φ(tn , pn )) = 0. tn

(29)

Then there is a (PS) sequence qn ∈ Jn . Indeed, by the mean value theorem there is sn ∈]0, tn [ such that f (pn ) − f φ(tn , pn ) D f φ(sn , pn ) X φ(sn , pn ) = , tn and by (29), qn = φ(sn , pn ) is a (PS) sequence. Actually, the above observation could be used to give a weaker formulation of the (PS) condition, which does not require f to be diﬀerentiable, and could be used to study flows in the continuous category. 2.2. THE MORSE – SMALE CONDITION

We recall that two closed linear subspaces V1 , V2 of a Banach space E are said transverse if V1 +V2 = E and V1 ∩V2 is complemented in E. Two C 1 submanifolds M1 and M2 of the Banach manifold M are said transverse if for every p ∈ M1 ∩ M2 the closed linear subspaces T p M1 and T p M2 are transverse in T p M. Let X be a Morse vector field having only rest points with finite Morse index and admitting a Lyapunov function. We will say that X satisfies the Morse – Smale condition up to order k ∈ N if for every pair of rest points x, y satisfying m(x) − m(y) ≤ k, the submanifolds W u (x) and W s (y) are transverse. In this case, the implicit function theorem implies that W u (x) ∩ W s (y) — if nonempty — is a submanifold of dimension m(x) − m(y). Notice that the presence of a Lyapunov function implies that W u (x) ∩ W s (x) = {x}, and such an intersection is always transverse. Notice also that the fact that φ(t, ·) is a diﬀeomorphism implies that if W u (x) ∩ W s (y) meet transversally at some p ∈ M, they meet transversally at every point of the orbit of p. 2.3. THE ASSUMPTIONS

and let X be a C 1 vector field Let M be an open subset of the Banach manifold M, on M (possibly, M = M). Denote by A the open subset M \ M, and denote by X to M. the restriction of X We shall construct the Morse complex for X on M under the following assumptions: and X is positively (A1) A is positively invariant with respect to the flow of X, complete with respect to A;

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

23

(A2) X is a Morse vector field on M; (A3) every rest point of X has finite Morse index; (A4) X admits a Lyapunov function f ∈ C 1 (M) ∩ C 0 (M); (A5) f is bounded below on M; (A6) (X, f ) satisfies the (PS) condition at every level c ∈ f (M); (A7) X satisfies the Morse – Smale condition up to order 0. will be denoted by φ. By (A1), the local flow of X is just The local flow of X the restriction of φ to p ∈ M, φ(t, p) ∈ M . Ω(X) = (t, p) ∈ Ω(X) and A is a sublevel In most applications f is actually defined on the whole M of f . Notice that (A6) and the fact that f ∈ C 0 (M) imply that there are no rest points on the boundary of M: such a rest point would be the limit of a (PS) sequence in M, which does not converge in M. Notice also that (A7) means asking that W u (x) does not meet W s (y) whenever x y are rest points with m(x) ≤ m(y). 2.4. FORWARD COMPACTNESS

The (PS) condition plays a crucial role in the following compactness result. PROPOSITION 2.2. Assume (A1) – (A7). Then (i) for every p ∈ M, φ(t, p) either converges to a rest point of X for t → +∞ or eventually enters A; (tn ) ⊂ [0, +∞[, and φ(tn , pn ) ⊂ M, then (ii) if (pn ) ⊂ M converges to p ∈ M, the sequence φ(tn , pn ) is compact in M. Proof. (i) Let p ∈ M. Assume that φ(t, p) never enters A: by (A1) this implies that t+ (p) = +∞. By Remark 2.1, with pn = p, tn → +∞, a = inf f , b = f (p), and by (PS) we can find a sequence sn → +∞ such that φ(sn , p) converges to a rest point x ∈ M. The function t → f φ(t, p) converges for t → +∞, being monotone, therefore lim f φ(t, p) = lim f φ(sn , p) = f (x). t→+∞

n→∞

Assume by contradiction that φ(t, p) does not converge to x for t → +∞. Then we can find r > 0 (as small as we like), two sequences an ≤ bn ≤ an+1 , an → +∞, such that φ(an , p) ∈ ∂E x (r), φ(bn , p) ∈ ∂E x (2r), φ([an , bn ] × {p}) ⊂ E x (2r) \ E x (r).

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A. ABBONDANDOLO AND P. MAJER

Choosing r so small that X is bounded on E x (2r) ⊂ M, one has that bn − an is bounded away from 0. Since lim f φ(an , p) = lim f φ(bn , p) = lim f φ(t, p) = f (x), n→∞

n→∞

we have lim

n→∞

t→+∞

f φ(bn , p) − f φ(an , p) = 0, bn − an

so by Remark 2.1 there is a (PS) sequence in E x (2r) \ E x (r), converging by (PS) to a rest point. Since r is arbitrarily small, x is not isolated in rest(X), contradicting (A2). (ii) If (tn ) is bounded, then lim sup tn < t+ (p).

(30)

n→∞

Indeed, if by contradiction t+ (p) ≤ lim supn→∞ tn , t+ (p) is finite, so by (A1) there exists s ∈ [0, t+ (p)[ such that φ(s, p) ∈ A. Then φ(s, pn ) eventually belongs to A, so s > tn for n large, and lim supn→∞ tn ≤ s < t+ (p), a contradiction. When (tn ) is bounded, the continuity of φ and (30) imply that φ(tn , pn ) is so we may assume that tn → +∞. compact in M, By Remark 2.1 and (PS) there exists a sequence an ∈ [0, tn ] such that, up to a subsequence, φ(an , pn ) converges to a rest point x ∈ M, with inf f ≤ f (x) ≤ f (p). Since there are finitely many rest points in this strip, we may assume that f (x) is minimal, that is for no sequence a n ∈ [0, tn ], φ(a n , pn ) has a subsequence converging to a rest point y ∈ M with f (y) < f (x). If φ(tn , pn ) converges to x then there is nothing to prove, otherwise up to a subsequence we can find r > 0 (as small as we like) and bn ∈ [an , tn ] such that φ(bn , pn ) ∈ ∂E x (r) and φ([an , bn ] × {pn }) ⊂ E x (r). By Proposition 1.17, the sequence φ(bn , pn ) is compact, since its distance from the compact set u (x) ∩ ∂E (r) tends to 0 (here we are using the fact that x has finite Morse Wloc,r x index). So a subsequence of φ(bn , pn ) converges to a point q with f (q) ≤

max

u (x)∩∂E (r) Wloc,r x

f < f (x).

The sequence tn − bn is bounded: otherwise by Remark 2.1 and (PS) there would exist cn ∈ [bn , tn ] such that a subsequence of φ(cn , pn ) converges to a rest point y with f (y) ≤ f (q) < f (x), contradicting the minimality of f (x). Therefore φ(tn , pn ) = φ tn − bn , φ(bn , pn ) is compact in M. The above result has the following immediate consequence. COROLLARY 2.3. For every x ∈ rest(X), W u (x) ∩ M has compact closure in M.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

25

Another consequence is the following convergence result for forward orbits: if (pn ) ⊂ M converges to p ∈ M, up to a subsequence the forward orbit of pn converges to a “broken orbit” consisting of h + 1 flow lines, h ≥ 0, matching at h rest points xh , . . . , x1 . The first of these flow lines is the forward orbit of p, the last one either converges to a rest point x0 , which is also the common limit of φ(t, pn ) for t → +∞, or eventually enters A, together with all the orbits of pn . More precisely, the situation is described by the following corollary. COROLLARY 2.4. Assume that (pn ) ⊂ M converges to some p ∈ M. Then there exists a subsequence (pkn ) such that one of the following two alternatives holds: (a) t+ (pkn ) = +∞, and there exists x0 ∈ rest(X) such that φ(t, pkn ) converges to x0 for t → +∞, for every n ∈ N; (b) for every n ∈ N, φ(t, pkn ) eventually enters A. Moreover, there exist h ∈ N, a set {x j }1≤ j≤h ⊂ rest(X), with f (x1 ) < · · · < f (xh ), sequences of real numbers t0n > t1n > · · · > thn = 0, and points q0 , q1 , . . . , qh = p in M such that: (i) q j ∈ W s (x j ) ∩ W u (x j+1 ) for every 1 ≤ j ≤ h − 1; (ii) qh = p ∈ W s (xh ), unless case (b) holds and h = 0, in which case φ(t, qh ) = φ(t, p) eventually enters A; (iii) q0 ∈ W u (x1 ) if h ≥ 1; in case (a) q0 ∈ W s (x0 ), in case (b) φ(t, q0 ) eventually enters A; j

(iv) limn→∞ φ(tn , pkn ) = q j for every 0 ≤ j ≤ h. The proof is an easy application of Proposition 2.2, together with an induction argument. Details are left to the reader. 2.5. CONSEQUENCES OF COMPACTNESS AND TRANSVERSALITY

we will denote by φ([0, +∞[ × B) its forward evolution, Given a subset B ⊂ M, although this set should more properly be indicated by . φ ([0, +∞[ × B) ∩ Ω(X) The Morse – Smale condition up to order zero, assumption (A7), has the following consequence. LEMMA 2.5. Assume (A1) – (A7). Let x, y be distinct rest points of X, with m(x) ≤ m(y). Then there exists r > 0 such that φ [0, +∞[ × E x (r) ∩ Ey (r) = ∅.

26

A. ABBONDANDOLO AND P. MAJER

Proof. Assume the contrary: there exist a sequence (pn ) ⊂ M converging to x and a sequence (tn ) ⊂ [0, +∞[ such that φ(tn , pn ) converges to y. By Corollary 2.4, a subsequence of the sequence of forward orbits of pn converges to a “broken orbit” passing through x and y. In particular, there are pairwise distinct rest points z1 = x, z2 , . . . , zk = y, k ≥ 2, such that W u (zi ) ∩ W s (zi+1 ) ∅ for every 1 ≤ i ≤ k − 1. The Morse – Smale condition up to order zero implies that m(x) = m(z1 ) > · · · > m(zk ) = m(y), a contradiction. In particular, the closure of the unstable manifold of a rest point x of index k does not contain rest points of index greater than or equal to k, other than x itself. Let us state a stronger assumption, which will be later removed: (A8) every rest point y does not belong to the closure of the union of the unstable manifolds of rest points x y with m(x) ≤ m(y): y W u (x). x ∈ rest(X) \ {y} m(x) ≤ m(y)

Since the closure of a finite union is the union of the closures, by Lemma 2.5 condition (A8) is implied by the Morse – Smale condition up to order zero (A7) when X has finitely many rest points of index k, for every k ∈ N. In general it is strictly more restrictive. Assumption (A8) implies the following result. PROPOSITION 2.6. Assume (A1) – (A8). Then there exists a positive function ρ: rest(X) → ]0, +∞[ such that φ [0, +∞[ × E x ρ(x) ∩ Ey ρ(y) = ∅ for all pairs of rest points x y with m(x) ≤ m(y). Proof. By (A8) there exists a function σ: rest(X) → ]0, +∞[ such that W u (x) = ∅ ∀y ∈ rest(X). Ey σ(y) ∩

(31)

x ∈ rest(X) \ {y} m(x) ≤ m(y)

Let us prove that for every x ∈ rest(X) there is a positive number θ(x) such that φ [0, +∞[ × E x θ(x) ∩ Ey σ(y) = ∅ ∀y ∈ rest(X) \ {x}, m(y) ≥ m(x). (32) Then the function ρ(x): rest(X) → ]0, +∞[, x → min{σ(x), θ(x)}, will satisfy the requirements. We argue by contradiction, assuming that there exists x ∈ rest(X) for which (32) does not hold, no matter how small θ(x) is. Since there are finitely many rest

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

27

points in {p ∈ M | f (p) ≤ f (x)}, we can find a sequence (pn ) ⊂ M converging to x and a sequence (tn ) ⊂ [0, +∞[ such that φ(tn , pn ) ∈ Ey (σ(y)), for some y ∈ rest(X) \ {x} with m(y) ≥ m(x). By Corollary 2.4, a subsequence of the sequence of the forward orbits of pn converges to a “broken orbit” starting from x and passing through Ey σ(y) . In particular, there are rest points z1 = x, . . . , zk y, k ≥ 1, such that W u (zi ) ∩ W s (zi+1 ) ∅ for 1 ≤ i ≤ k − 1, and W u (zk ) ∩ Ey σ(y) ∅. (33) By the Morse – Smale condition up to order 0, m(zk ) ≤ m(x) ≤ m(y), and since zk y, (33) contradicts (31).

2.6. CELLULAR FILTRATIONS

Cellular filtrations are a useful tool to compute the singular homology of a topological space. See Dold (1980, Section V.1) for a more extensive discussion and for the proof of the results stated in this section. Let T be a topological space. A sequence F = {Fn }n∈Z of subsets of T is said a cellular filtration of T if: (i) Fn ⊂ Fn+1 for every n ∈ Z; (ii) every singular simplex in T is a simplex in Fn for some n; (iii) the k-th singular homology group Hk (Fn , Fn−1 ) vanishes for every k n. Notice that (ii) is fulfilled when T is the union of the family {Fn } and each Fn is open. The space F−1 may be empty. The spaces Fn for n ≤ −2 will be actually irrelevant in the construction. Singular homology is always meant to have integer coeﬃcients. If F = {Fn }n∈Z is a cellular filtration of T , we denote by Wk F the Abelian group Wk F := Hk (Fk , Fk−1 ). The homomorphism ∂k : Wk F → Wk−1 F is given by the composition Hk (Fk , Fk−1 ) → Hk−1 (Fk−1 ) → Hk−1 (Fk−1 , Fk−2 ), where the first map is the boundary homomorphism of the pair (Fk , Fk−1 ), and the second map is induced by the inclusion. It is readily seen that ∂k ∂k+1 = 0, so W∗ F is a chain complex of Abelian groups, said the cellular complex of the filtration F. A cellular map g: (T, F ) → (T , F ) is a continuous map from T to T mapping each Fn into Fn . Such a map induces homomorphisms Wk g: Wk F → Wk F ,

Wk g = g∗ : Hk (Fk , Fk−1 ) → Hk (Fk , Fk−1 ),

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A. ABBONDANDOLO AND P. MAJER

which are readily seen to form a chain map W∗ g: W∗ F → W∗ F . This makes W a functor from the category of cellular filtrations and cellular maps to the category of chain complexes of Abelian groups and chain maps. THEOREM 2.7. If F = {Fn }n∈Z is a cellular filtration of the topological space T , then there is an isomorphism Hk ({W∗ F , ∂∗ }) Hk (T, F−1 ). Such isomorphisms form a natural transformation between the functor HW and the singular homology functor H, in the sense that if g: (T, F ) → (T , F ) is a cellular map, then the diagram Hk ({W∗ F , ∂∗ }) Hk Wk g

Hk ({W∗ F , ∂∗ })

/ Hk (T, F−1 )

g∗

/ Hk (T , F ) −1

commutes. A cellular homotopy h between two cellular maps g0 , g1 : (T, F ) → (T , F ) is cellular map h: ([0, 1] × T, F) → (T , F ), F being the cellular filtration {[0, 1] × Fn }n∈Z , such that h(0, ·) = g0 and h(1, ·) = g1 . If there is a cellular homotopy between g and g , the homotopy invariance of singular homology implies that W∗ g = W∗ g . A cellular map g: (T, F ) → (T , F ) is said a cellular homotopy equivalence if there are a cellular map g : (T , F ) → (T, F ), said a cellular homotopy inverse of g, and cellular homotopies h between g ◦ g and id(T,F ) and h between g ◦ g and id(T ,F ) . By functoriality and homotopy invariance, if g is a cellular homotopy equivalence then W∗ g is an isomorphism. 2.7. THE MORSE COMPLEX

Denote by restk (X) the set of rest points of X of Morse index k, and let Ck (X) be the free Abelian group generated by the elements of restk (X). Let ρ: rest(X) → ]0, +∞[ be a function satisfying φ [0, +∞[ × E x ρ(x) ∩ Ey ρ(y) = ∅, ∀x y ∈ rest(X), m(x) ≤ m(y), (34) whose existence is established by Proposition 2.6. Consider the subsets of M Mk = Mk (ρ) := A ∪ φ([0, +∞[ × E˚ x (ρ(x))) ∀k ≥ 0, Mk = A ∀k < 0, x ∈ rest(X) m(x) ≤ k

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

29

and M∞ = M∞ (ρ) := k∈Z Mk . Each Mk is open and positively invariant. We shall denote by Dk the closed unit ball of Rk , and by ωk the generator of Hk (Dk , ∂Dk ) corresponding to the standard orientation of Rk . Here is the main result of this second part. THEOREM 2.8. Assume (A1) – (A8). Let ρ: rest(X) →]0, +∞[ be a function satisfying (34), and let Mk be the sets defined above. Then: A) is a homotopy equivalence. (i) The inclusion (M∞ , A) → ( M, (ii) M = M(ρ) := {Mk }k∈Z is a cellular filtration of M∞ , with Wk M = Hk (Mk , Mk−1 ) Ck (X),

∀k ∈ N.

More precisely, the choice of an orientation of each unstable manifold W u (x) determines an isomorphism Θk (ρ): Ck (X) Wk M(ρ), x → θ∗x (ωk ),

∀x ∈ restk (X),

where θ x : (Dk , ∂Dk ) → (Mk , Mk−1 ) is a map of the form θ x (ξ) = φ t(ξ), w(ξ) , with w an orientation preserving embedding of Dk onto an open neighbor hood of x in W u (x), and 0 ≤ t < t+ so large that φ t(ξ), w(ξ) ∈ Mk−1 for every ξ ∈ ∂Dk . (iii) If ρ ≤ ρ, then the inclusion j = jρ ρ : M∞ (ρ ) → M∞ (ρ) is a cellular homotopy equivalence with respect to the cellular filtrations {Mk (ρ )}k∈Z and {Mk (ρ)}k∈Z . Moreover, the diagram W9 k M(ρ ) s

Θk (ρ ) sss

sss sss

Ck (X)

Θk (ρ)

Wk j

(35)

/ Wk M(ρ)

commutes. By (iii), the isomorphism class of the cellular chain complex Wk M(ρ) does not depend on the choice of the function ρ satisfying (34). In order to fix a standard representative, we can define W∗ (X) := lim W∗ M(ρ), ρ↓0

the limit of the direct system of chain complexes {W∗ M(ρ), W∗ jρ ρ }. The chain complex W∗ (X) is said the Morse complex of X. By Theorem 2.7, the homology of such a chain complex is isomorphic to the singular homology of (M∞ , A), which

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by statement (i) of the theorem above is isomorphic to the singular homology of A): ( M, A) ∀k ∈ N. Hk W∗ (X) Hk ( M, In particular when A is the empty set (so that X is a positively complete Morse vector field on M admitting a Lyapunov function which is bounded below), the homology of the Morse complex is isomorphic to the singular homology of M. By (ii) and by the commutativity of diagram (35), a choice of an orientation of each unstable manifold allows to identify the groups Ck (X) and Wk (X), by the isomorphism Θk = lim Θk (ρ): Ck (X) Wk (X). ρ↓0

EXERCISE 2.9. Deduce the so called strong Morse relations: there exists a formal series Q with coeﬃcients in N ∪ {+∞} such that ∞ k=0

| restk (X)| tk =

∞

A)tk + (1 + t)Q(t), βk ( M,

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k=0

A) = rank Hk ( M, A) ∈ N ∪ {+∞} is the kth Betti number of ( M, A). where βk ( M, Before proving Theorem 2.8, we recall the semi-continuity properties of the entrance time function into a subset C ⊂ M:

tC (p) := inf t ∈ [0, t+ (p)[ φ(t, p) ∈ C ∈ [0, +∞]. LEMMA 2.10. If C is open, tC is upper semi-continuous. If C is closed, tC is lower semi-continuous. Proof. Assume that C is open. It tC (p) < t, there exists s ∈ [tC (p), t[ such that φ(s, p) ∈ C. By continuity, φ(s, q) ∈ C for every q in a neighborhood of p, so tC (q) ≤ s < t in such a neighborhood. Assume that C is closed. If tC (p) > t, choosing t ∈]t, tC (p)[ we have that \ C for every s ∈ [0, t ]. By continuity and φ(s, p) belongs to the open set M \ C for every s ∈ [0, t ] and every q in a neighborhood compactness, φ(s, q) ∈ M of p. Therefore, tC (q) ≥ t > t in such a neighborhood. either Proof of Theorem 2.8. (i) By Proposition 2.2(i), the orbit of every p ∈ M converges to some rest point x ∈ M for t → +∞, or eventually enters A. Since M∞ the entrance time of is a neighborhood of rest(X) and contains A, for every p ∈ M φ(·, p) in M∞ ,

t M∞ (p) := inf t ∈ [0, t+ (p)[ φ(t, p) ∈ M∞ is finite, and less than t+ (p). Since M∞ is open, by Lemma 2.10 the function t M∞ is upper semi-continuous.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

31

On the other hand, the function t+ is lower semi-continuous. A simple argument with partitions of unity (also known as Dowker theorem; see Dugundji, 1978, VIII.4.3) shows that on a paracompact topological space we can always find a continuous function between an upper semi-continuous function and a lower → R such that semi-continuous one. So we can find a continuous function s : M + t M∞ < s < t . Then the continuous map A) → (M∞ , A), r: ( M,

r(p) = φ(s(p), p),

A), the homotopies is a homotopical inverse of the inclusion i: (M∞ , A) → ( M, id( M,A) ∼ i ◦ r and id(M∞ ,A) ∼ r ◦ i being the map [0, 1] × A) → ( M, A), ([0, 1] × M,

(λ, p) → φ(λs(p), p),

and its restriction to ([0, 1] × M∞ , [0, 1] × A) into (M∞ , A). (ii) Let us prove that M is a cellular filtration. Since M is an open covering of M∞ , we just need to compute the singular homology of (Mk , Mk−1 ). Since Mk is the union of the open sets Mk−1 and φ [0, +∞[ × E˚ x ρ(x) , Uk := x∈restk (X)

by excision the singular homology of (Mk , Mk−1 ) is isomorphic to the singular homology of (Uk , Uk ∩ M k−1 ). Condition (34) implies that the open sets U(x) := ˚ φ [0, +∞[×E x ρ(x)bigr) , x ∈ restk (X), are pairwise disjoint, so H∗ (U(x), U(x) ∩ Mk−1 ). H∗ (Mk , Mk−1 ) H∗ (Uk , Uk ∩ Mk−1 ) x∈restk (X)

We shall prove that (U(x), U(x) ∩ Mk−1 ) is homotopically equivalent to a kdimensional disc modulo its boundary, so that 0 if j k, H j (U(x), U(x) ∩ Mk−1 ) = Z if j = k, proving that M is a cellular filtration. Set for simplicity ρ = ρ(x). By Lemma 1.10(iii), E ux (ρ) × E˚ sx (ρ) ⊂ U(x). Let p ∈ U(x) \ E ux (ρ) × E˚ sx (ρ). By Proposition 2.2(ii), the orbit of p either eventually enters A, or converges to a rest point y for t → +∞. In the latter case, y x because of Theorem 1.20(i), so by (34) m(y) ≤ k − 1. In both cases, the orbit of p eventually enters Mk−1 . The upper semi-continuous function 0 if p ∈ E˚ x (ρ), a˜ : U(x) → R, p → t Mk−1 (p) if p ∈ U(x) \ E˚ x (ρ),

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A. ABBONDANDOLO AND P. MAJER

is strictly less than the lower semi-continuous function t+ , so we can find a continuous function a: U(x) → [0, +∞[ such that a˜ < a < t+ , so that φ(a(p), p) ∈ Mk−1

∀p ∈ U(x) \ E˚ x (ρ).

Then we can define the continuous map α: E ux (ρ) × E˚ sx (ρ), ∂E ux (ρ) × E˚ sx (ρ) → (U(x), U(x) ∩ Mk−1 ),

(37)

p → φ(a(p), p).

By Theorem 1.20(i), for every p ∈ U(x) there holds

b(p) := sup t ∈ ]t− (p), 0] φ(t, p) ∈ E˚ x (ρ) = max t ∈ ]t− (p), 0] φ(t, p) ∈ E x (ρ) , so by Lemma 2.10 the function b: U(x) → ]−∞, 0] is both lower and upper semicontinuous, hence continuous. The map p → φ(b(p), p) is the identity on E ux (ρ) × E˚ sx (ρ), and maps all the other points of U(x) into ∂E ux (ρ) × E˚ sx (ρ). Since by (34) E x (ρ) ∩ Mk−1 = ∅, the continuous map β: (U(x), U(x) ∩ Mk−1 ) → E ux (ρ) × E˚ sx (ρ), ∂E ux (ρ) × E˚ sx (ρ) , p → φ(b(p), p), is well-defined. It is easy to check that α and β are homotopy inverses. Indeed, (λ, p) → β φ(λa(p), p) is a homotopy between β ◦ α and the identity map on u E x (ρ) × E˚ sx (ρ), ∂E ux (ρ) × E˚ sx (ρ) . On the other hand, by (37), (λ, p) → φ a φ(λb(p), p) , φ(λb(p), p) is a homotopy between α ◦ β and the map (U(x), U(x) ∩ Mk−1 ) → (U(x), U(x) ∩ Mk−1 ),

p → φ(a(p), p),

which is clearly homotopy equivalent to the identity on (U(x), U(x) ∩ Mk−1 ). We conclude that (U(x), U(x) ∩ Mk−1 ) is homotopy equivalent to u E x (ρ) × E˚ sx (ρ), ∂E ux (ρ) × E˚ sx (ρ) , which is homotopy equivalent to E ux (ρ), ∂E ux (ρ) , a k-dimensional disc modulo u u (x), and the its boundary. The latter pair is homeomorphic to Wloc,ρ (x), ∂Wloc,ρ statement about the form of the isomorphism Θk easily follows. (iii) Since Mk (ρ ) ⊂ Mk (ρ) for every k, j is a cellular map; we will construct a cellular homotopy inverse of j of the form γ(p) = φ(c(p), p),

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33

with c a suitable positive continuous function. Given p ∈ M∞ (ρ), set

κ(p) := min k ∈ N p ∈ Mk (ρ) , and c˜ (p) := t Mκ(p) (ρ ) (p), the entrance time of p into the open set Mκ(p) (ρ ). By (34), every point in Mk (ρ) either eventually enters A or converges to a rest point x with m(x) ≤ k; in both cases, p eventually enters Mk (ρ ). Therefore, c˜ < t+ . Since {Mh (ρ )}h∈Z is a filtration, t Mh (ρ ) is nonincreasing in h, so

c˜ (p) = min t Mh (ρ ) (p) 0 ≤ h ≤ κ(p) = min t Mh (ρ ) (p)χh (p) h ∈ N , where χh (p) = 1 if h ≤ κ(p), i.e., p Mh−1 (ρ), and χh (p) = +∞ otherwise; hence the positive function χh is upper semi-continuous. Since also t Mh (ρ ) is upper semi-continuous and nonnegative, so is the function c˜ . Let c: M∞ (ρ) → R be a continuous function such that c˜ < c < t+ , and let γ: M∞ (ρ) → M∞ (ρ ) be the map defined in (38). By construction, γ maps Mk (ρ) into Mk (ρ ), so it is a cellular map. The cellular homotopies id M∞ (ρ) ∼ j ◦ γ and id M∞ (ρ ) ∼ γ◦ j are given by the cellular map (λ, p) → φ(λc(p), p) on the respective domains. If θ x (ρ): (Dk , ∂Dk ) → Mk (ρ), Mk−1 (ρ) and

θ x (ρ ): (Dk , ∂Dk ) → Mk (ρ ), Mk−1 (ρ )

are the continuous maps appearing in (ii), then j ◦ θ x (ρ ) is homotopic to θ x (ρ), so the diagram (35) commutes.

2.8. REPRESENTATION OF ∂∗ IN TERMS OF INTERSECTION NUMBERS

Let us strengthen the Morse – Smale assumption (A7) by requiring: (A7 ) X satisfies the Morse – Smale condition up to order 1. In this case, the boundary operator ∂k of the Morse complex of X can be expressed in terms of intersection numbers of unstable and stable manifolds of rest points of index diﬀerence 1. First of all notice that if m(x) − m(y) = 1, the assumption (A7 ) implies that W u (x) ∩ W s (y) is a flow-invariant 1-dimensional manifold, that is a discrete set of flow lines. We claim that W u (x)∩W s (y) is compact: otherwise Corollary 2.4 would imply the existence of a “broken orbit” from z0 = x to zh = y, with intermediate

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A. ABBONDANDOLO AND P. MAJER

rest points z1 , . . . , zh−1 , for some h ≥ 2. By the Morse – Smale condition (up to order 0) m(z0 ) > m(z1 ) > · · · > m(zh ), a contradiction because m(z0 ) − m(zh ) = 1. Therefore W u (x) ∩ W s (y) consists of finitely many flow lines. Let us fix an orientation of each unstable manifold W u (x). As we have seen in Section 2.7, this choice determines a preferred isomorphism Θk : Ck (X) Wk (X). Moreover, it determines an orientation of each transverse intersection W u (x) ∩ W s (y). Indeed, the orientation of each unstable manifold determines a coorientation of each stable manifold (that is an orientation of its normal bundle), and the transverse intersection of a finite-dimensional oriented submanifold with a finite-codimensional co-oriented submanifold carries a canonical orientation: if p ∈ W u (x)∩W s (y) and V ⊂ T p W u (x) is a linear complement of T p (W u (x)∩W s (y)) in T p W u (x), by transversality V is also a complement of T p W s (x) in T p M, so it is oriented, and the orientation of W u (x) ∩ W s (y) is the one for which T p W u (x) = T p W u (x) ∩ W s (y) ⊕ V is an oriented sum. In particular, if m(x)−m(y) = 1 each connected component W of W u (x)∩W s (y) is an oriented line, and we can define (W) to be +1 if φ is orientation preserving on W, −1 otherwise. Then we can define the integer n(x, y) = (W), ∀x, y ∈ rest(X), m(x) − m(y) = 1. W connected component of W u (x) ∩ W s (y)

Assume that conditions (A1) – (A6), (A7 ), and (A8) hold. Then we have the following fact. THEOREM 2.11. In terms of the preferred isomorphism Θk : Ck (X) Wk (X), the boundary operator of the Morse complex of X has the form ∂k x = n(x, y)y, ∀x ∈ restk (X) ⊂ Ck (X). (39) y∈restk−1 (X)

Before proving this result, we recall that if σn denotes the generator of Hn (S n ) corresponding to the standard orientation of ∂Dn+1 = S n , that is the one for which Rn+1 = Rζ ⊕T ζ S n is an oriented sum, for every ζ in S n , we have that the boundary homomorphism Hn+1 (Dn+1 , ∂Dn+1 ) → Hn (∂Dn+1 ) maps ωn+1 into σn . EXERCISE 2.12. Let A1 , . . . , Ah be pairwise disjoint closed n-discs in S n , with maps h ai : (Dn , ∂Dn ) → S n , S n A˚ i , i=1

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

35

mapping Dn homeomorphically onto Ai , preserving the standard orientations. Let j: S n → (S n , S n \ hi=1 A˚ i ) be the inclusion. Then j∗ (σn ) =

h

ai∗ (ωn ).

i=1

Proof of Theorem 2.11. Notice first of all that by (A2) and (A6), for every x ∈ rest(X) there are finitely many rest points y ∈ rest(X) with f (y) < f (x), so the sum appearing in (39) is finite. Let ρ: rest(X) → ]0, +∞[ be a function satisfying (34), and let Mk = Mk (ρ), for k ∈ N ∪ {∞}. Let us fix a rest point x of Morse index k. By the naturality of the boundary homomorphism of pairs in singular homology, we have the commutative diagram Hk (Dk , ∂Dk )

Hk−1 (∂Dk )

θ∗x

α∗

/ Hk (Mk , Mk−1 ) / Hk−1 (Mk−1 )

where α: ∂Dk → Mk−1 is the restriction of θ x . The cellular boundary homomorphism ∂k of the cellular filtration {Mk }k∈Z is the composition of the right vertical arrow with the homomorphism induced by the inclusion i: Mk−1 → (Mk−1 , Mk−2 ). On the other hand, the left vertical arrow is an isomorphism mapping ωk into σk−1 . Therefore, ∂k maps the generator θ∗x (ωk ) of Hk (Mk , Mk−1 ) into i∗ α∗ (σk−1 ) ∈ Hk−1 (Mk−1 , Mk−2 ), and we must express the latter element in terms y of the generators θ∗ (ωk−1 ) of Hk−1 (Mk−1 , Mk−2 ), for y ∈ restk−1 (X). By the Morse – Smale condition up to order 1, −1 s α W (y) = {ζ1 , . . . , ζh } y∈restk−1 (X)

is a finite subset of ∂Dk , and α maps all the other points into points which either belong to stable manifolds of rest points of index less than k − 1, or eventually enter A; so the orbit of any point in α(∂Dk \ {ζ1 , . . . , ζh }) eventually enters Mk−2 . Choose r > 0 so small that the closed r-balls Br (ζi ) ⊂ ∂Dk centered in ζi are pairwise disjoint (k − 1)-discs. Let b: ∂Dk → R be a continuous function such that χt Mk−2 ◦ α < b < t+ ◦ α, where χ is the characteristic function of the open set ∂Dk \ hi=1 Br (ζi ), and t Mk−2 is the entrance time function into Mk−2 . Then α is homotopic to the map β: ∂Dk → Mk−1 , ζ → φ b(ζ), α(ζ) ,

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A. ABBONDANDOLO AND P. MAJER

so θ∗x (ωk ) = i∗ α∗ (σk−1 ) = i∗ β∗ (σk−1 ). Denote by γi : (Dk−1 , ∂Dk−1 ) → (Mk−1 , Mk−2 ) the composition of i ◦ β with an orientation preserving homeomorphism (Dk−1 , ∂Dk−1 ) → Bρ (ζi ), ∂Bρ (ζi ) . Then the result of Exercise 2.12 shows that: θ∗x (ωk )

= i∗ β∗ (σk−1 ) =

h

γi ∗ (ωk−1 ).

(40)

i=1

Fix some i ∈ {1, . . . , h}, let y be the rest point of index k − 1 toward which the orbit of α(ζi ), i.e., of β(ζi ), converges for t → +∞, and let Wi be the connected component of W u (x) ∩ W s (y) consisting of such an orbit. We claim that γi is homotopic to either θy , in the case (Wi ) = 1, or to θy ◦ µ, where µ is an orientation reversing automorphism of (Dk−1 , ∂Dk−1 ), in the case (Wi ) = −1. Therefore y

γi ∗ (ωk−1 ) = (Wi )θ∗ (ωk−1 ), and (40) allows to conclude. Let us proof the claim. Up to a small perturbation, we may assume that γi is a C 1 embedding of a closed (k − 1)-disc, meeting W s (y) transversally at the single point p = γi (0). The diﬀeomorphism γi induces an orientation of T p γi (Dk−1 ), the one for which T p W u (x) = RX(p) ⊕ T p γi (Dk−1 ) is an oriented sum. The diﬀerential of the flow D2 φ(t, ·) at p maps the tangent space of γi (Dk−1 ) at p onto a subspace of T φ(t,p) M which converges to T y W u (y) for t → +∞ (see for instance Abbondandolo and Majer, 2003c, Theorem 2.1(iii)). A first consequence is that the orientation of T p γi (Dk−1 ) defined above is (Wi ) times the orientation obtained by seeing T p γi (Dk−1 ) as a complement of T p W s (y) in T p M. A second consequence is that, by the evolution of graphs of Lipschitz maps from Eyu (r) to Eys (r) near the hyperbolic rest point y (see Shub, 1987, or Abbondandolo and Majer, 2001, Proposition A.3 and Addendum A.5), if r > 0 is small and t ≥ 0 is large then φ {t} × γi (Dk−1 ) ∩ Ey (r) is the graph of a map3 τ : Eyu (r) → Eys (r). Let K ⊂ Dk−1 be the closed neighborhood of 0 such that φ({t} × γi (K)) = graph τ. 3 This statement is part of the content of the so called λ-lemma, in the particular case of a gradient-like flow. See Palis (1968) and Palis and de Melo (1982).

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37

Since K is a closed (k − 1)-disc, it is a deformation retract of Dk−1 . Since the local u (y) is also the graph of a map σu : E u (r) → E s (r), it is unstable manifold Wloc,r y y now easy to combine the above maps to construct a homotopy between γi and an embedding of (Dk−1 , ∂Dk−1 ) into (W u (y), W u (y) ∩ Mk−2 ), which is orientation preserving, hence homotopic to θy , if (Wi ) = 1, orientation reversing, hence homotopic to θy ◦ µ, if (Wi ) = −1.

2.9. HOW TO REMOVE THE ASSUMPTION (A8)

If we drop assumption (A8), there need not exist a function ρ satisfying (34), and it becomes more diﬃcult to associate a cellular filtration to X. Nevertheless, we can make the graded group C∗ (X) into a chain complex by taking a direct limit of the Morse complexes on sublevels { f < a}, for a ↑ sup f . On these domains indeed, there are finitely many rest points and condition (A7) guarantees condition (A8). Not being forced to assume (A8) is a positive fact, in that assumption (A7) can be more easily achieved by generic perturbations, as we shall see in Section 2.12. If the supremum of f on M is attained, by (A2) and (A6) X has finitely many rest points, so (A8) is implied by (A7). Thus, we can assume that sup f is not attained. a := A ∪ { f < For a < sup f , let W∗ (X)a be the Morse complex associated to M a}, and if a < b < sup f , let wab : W∗ (X)a → W∗ (X)b a → M b . The Morse complex of X be the chain map induced by the inclusion M is defined to be the chain complex W∗ (X) := lim W∗ (X)a , a↑sup f

the limit of the direct system {W∗ (X), wab }. Notice that if (A8) holds, so that W∗ (X) is the chain complex defined in Section 2.7, the family of chain complexes {W∗ (X)a }a<sup f is identified with an increasing and exhausting family of sub-complexes of W∗ (X), so this definition of the Morse complex agrees with the previous one. Since the homology of a direct limit of chain complexes is the direct limit of the homologies (see Dold, 1980, VIII.5.20), Hk W∗ (X) = lim Hk W∗ (X)a . a↑sup f

Similarly, the singular homology of an increasing union of open subsets is the limit of the singular homologies (see Dold, 1980, VIII.5.22), so A) = lim Hk ( M a , A). Hk ( M, a↑sup f

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A. ABBONDANDOLO AND P. MAJER

We conclude that the homology of the Morse complex of X is isomorphic to the A), singular homology of ( M, A) ∀k ∈ N. Hk W∗ (X) Hk ( M, Finally, having fixed an orientation for each unstable manifold, we have the isomorphisms Θak : Ck (X)a Wk (X)a , Ck (X)a being the subgroup of Ck (X) generated by the rest points x with f (x) < a, and the limit of this direct system defines an isomorphism Θk : Ck (X) Wk (X). REMARK 2.13. Since the boundary of x ∈ rest(X) in C∗ (X) and in C∗ (X)a coincide when f (x) < a < sup f , the formula for the boundary homomorphism under the Morse – Smale condition up to order 1 (Theorem 2.11) holds also without assuming (A8).

2.10. MORSE FUNCTIONS ON HILBERT MANIFOLDS

A particular but important case is the following situation: f is a C 2 Morse function on a smooth Hilbert manifold N, endowed with a C 1 Riemannian metric g, −∞ < = {p ∈ N | f (p) < b}, M = {p ∈ N | a < f (p) < b}, and X = a < b ≤ +∞, M −∇ f , the negative gradient of f with respect to the metric g. Let us see what the assumptions (A1) – (A8) look like in this situation. In this case, of course, rest(X) = crit( f ) ∩ {a < f < b}, the set of critical points of f with values between a and b. Condition (A2) is equivalent to saying that f is a Morse function on M, and in condition (A3) the Morse index is the standard Morse index of a critical point of f | M . The set of critical points of f with index k will be denoted by critk ( f ). Condition (A4) is automatically fulfilled, f itself being a Lyapunov function for −∇ f , and so is condition (A5). In the case of a gradient flow the (PS) condition can be restated in the more familiar way: the pair ( f, g) satisfies the (PS) condition at level c ∈ R if every such that f (pn ) → c and D f (pn ) → 0 is compact (here the sequence (pn ) ⊂ M ∗ norm · on T M is induced by the Riemannian structure g). The assumption (A6) is equivalent to: ( f, g) satisfies the (PS) condition at level c for every c ∈ [a, b[, and a is a regular value for f . As we shall see in Section 2.12, the Morse – Smale condition required in (A7) can be always achieved by perturbing the metric g. g) is complete. Finally, assumption (A1) is automatically fulfilled when ( M, Indeed, the following fact holds.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

39

R) and a ∈ R be such that the strip PROPOSITION 2.14. Let f ∈ C 2 ( M, induced {a ≤ f ≤ c} is complete (with respect to the geodesic distance d on M by the Riemannian metric g), for every c < sup f . Then the vector field −∇ f is positively complete with respect to { f < a}. and consider the curve u: [0, t+ (p)[ → M, u(t) = φ(t, p). If Proof. Let p ∈ M + f (p) = sup f , then p is a critical point of f , so t (p) = +∞. If inf f ◦ u < a then u(t) eventually enters { f < a}. Therefore we can assume that f (p) < sup f and u([0, t+ (p)[) ⊂ {a ≤ f ≤ f (p)}, and we must prove that t+ (p) = +∞. Let 0 ≤ s < t. Then t t D f u(τ) −∇ f u(τ) dτ = − g ∇ f u(τ) , ∇ f u(τ) dτ, f u(t) − f u(s) = s

s

and the Cauchy – Schwarz inequality implies that t g u (τ), u (τ) dτ d u(s), u(t) ≤ s t = g ∇ f u(τ) , ∇ f u(τ) dτ s t √ 1/2 ≤ t−s g ∇ f u(τ) , ∇ f u(τ) dτ s √ √ = t − s f u(s) − f u(t) ≤ t − s f (p) − inf f ◦ u. The above estimate shows that u is uniformly continuous. If by contradiction t+ (p) < +∞, by the completeness of the strip {a ≤ f ≤ f (p)} we deduce that u(t) converges for t → t+ (p). But then the solution u of u = −∇ f (u), u(0) = p, can be extended to a right neighborhood of t+ (p), contradicting the maximality of t+ (p). We summarize the above discussion into the following proposition. PROPOSITION 2.15. Let f be a C 2 function on the smooth Hilbert manifold N, endowed with a C 1 Riemannian metric g, and let −∞ < a < b ≤ +∞. Assume that (B1) a is a regular value of f ; (B2) f is a Morse function on {a < f < b}, and it has only critical points of finite Morse index in such a strip; (B3) for every c < b, the strip {a ≤ f ≤ c} is complete; (B4) f satisfies the (PS) condition at every level c ∈ [a, b[. = { f < b}, X = −∇ f | and M = {a < f < b}, the conditions Then, setting M M (A1) – (A6) are fulfilled.

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A. ABBONDANDOLO AND P. MAJER

Notice that only (B3) and (B4) involve the metric. Moreover, if (B3) and (B4) hold for some metric, they hold also for every uniformly equivalent metric. Under the assumptions (B1) – (B4), the free Abelian group generated by the critical points of f of index k in {a < f < b} will be denoted by Ck ( f )ba . The lower index will be omitted when a < inf f , the upper index will be omitted when b = +∞. If −∇ f satisfies also the Morse – Smale condition up to order 0 on {a < f < b}, the boundary operator of the Morse complex of −∇ f on {a < f < b} will be denoted by ∂k ( f, g)ba : Ck ( f )ba → Ck−1 ( f )ba . Its homology is isomorphic to the singular homology of ({ f < b}, { f < a}): Hk ({C∗ ( f )ba , ∂∗ ( f, g)ba }) Hk ({ f < b}, { f < a}).

2.11. BASIC RESULTS IN TRANSVERSALITY THEORY

In the following lemma we single out a useful family of linear mappings whose kernel is complemented. LEMMA 2.16. Let E, F, G be Banach spaces, and assume that A ∈ L(E, G) has complemented kernel and finite-codimensional range. Then for every B ∈ L(F, G) the kernel of the operator C ∈ L(E × F, G), C(e, f ) = Ae − B f , is complemented in E × F. Proof. Let E0 := ker A, E1 be a closed complement of E0 in E, and P0 , P1 be the associated projectors. Let G1 := ran A, G0 be a (finite-dimensional) complement of G1 in G, and Q0 , Q1 be the associated projectors. Then A induces an isomorphism from E1 onto G1 , whose inverse will be denoted by T ∈ L(G1 , E1 ). The equation C(e, f ) = 0 is equivalent to AP1 e = B f , which is equivalent to the system AP1 e = Q1 B f, Q0 B f = 0, again equivalent to

P1 e = T Q1 B f, Q0 B f = 0.

(41)

Since Q0 B has finite rank, its kernel — say F0 — has a (finite-dimensional) complement F1 . By (41), the kernel of C is

ker C = (e0 + T Q1 B f0 , f0 ) ∈ E × F e0 ∈ E0 , f0 ∈ F0 , and the closed linear subspace E1 × F1 is a complement of ker C.

MORSE COMPLEX FOR INFINITE-DIMENSIONAL MANIFOLDS

41

Let us recall some definitions and basic facts about transversality in a Banach setting. A classical reference for these topics is Abraham and Robbin (1967). If ϕ: M → N is a C k map between Banach manifolds, k ≥ 1, a point q ∈ N is said a regular value for ϕ if for every p ∈ ϕ−1 ({q}) the diﬀerential Dϕ(p): T p M → T q N is a left inverse, i.e., if it is onto and its kernel is complemented. In this case, ϕ−1 ({q}) is a submanifold of class C k . A C 1 map ϕ: M → N between Banach manifolds is said a Fredholm map if its diﬀerential at every point is a Fredholm operator. When the index of the diﬀerential is constant (for instance when M is connected), this integer is said the Fredholm index of ϕ. PROPOSITION 2.17. Let M, N, O be Banach manifolds, and let ϕ ∈ C 1 (M, N), ψ ∈ C 1 (M, O) be maps with regular values p ∈ N and q ∈ O. Then: (i) p is a regular value for ϕ|ψ−1 ({q}) if and only if q is a regular value for ψ|ϕ−1 ({p}) ; (ii) ϕ|ψ−1 ({q}) is a Fredholm map if and only if ψ|ϕ−1 ({p}) is a Fredholm map, in which case the indices coincide. This proposition is a consequence of the following linear statements. PROPOSITION 2.18. Let E, F, G be Banach spaces, and let A ∈ L(E, F), B ∈ L(E, G) be left inverses. Then: (i) A|ker B is a left inverse if and only if B|ker A is a left inverse; (ii) A|ker B is Fredholm if and only if B|ker A is Fredholm, in which case the indices coincide. Proof. Let R ∈ L(F, E) and S ∈ L(G, E) be right inverses of A and B, respectively. (i) If R0 ∈ L(F, ker B) is a right inverse of A|ker B , i.e., a right inverse of A with range in ker B, the map S 0 := (IE − R0 A)S is a right inverse of B, being a perturbation of S by an operator with range in ker B, and it takes value in ker A because AS 0 = AS − AR0 AS = AS − IF AS = 0. Therefore, S 0 is a right inverse of B|ker A . (ii) The kernels of A|ker B and B|ker A coincide: ker A|ker B = ker B|ker A = ker A ∩ ker B. Moreover, since R: F → RF is an isomorphism and since I − RA is a projector onto ker A, coker A|ker B =

R ker A + RF E F RF = . A ker B RA ker B ker A + RA ker B ker A + ker B

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We conclude that the assertions in (ii) are equivalent, each of them being equivalent to the fact that the pair of subspaces (ker A, ker B) is Fredholm, i.e., ker A ∩ ker B is finite-dimensional, and ker A + ker B is finite-codimensional.4 The index of A|ker B and of B|ker A equals the index of (ker A, ker B), ind(ker A, ker B) = dim ker A ∩ ker B − codim(ker A + ker B).

We recall that a subspace T of a topological space T is said residual if it contains a countable intersection of open and dense subspaces of T . Baire theorem guarantees that a residual subspace of a complete metric space is dense. The following Sard – Smale Theorem, combined with Proposition 2.17, is the basic tool to deal with transversality questions. THEOREM 2.19. Let M, N be C h Banach manifolds, h ≥ 1, with M Lindel¨of. Let ϕ: M → N be a C h Fredholm map of index m. If h > max{0, m} then the set of regular values of ϕ is residual in N. The proof can be found in Smale (1965). 2.12. GENERICITY OF THE MORSE – SMALE CONDITION

Let f be a C h+1 real function, h ≥ 1, on the smooth Hilbert manifold N, endowed with a Riemannian metric g of class C h . Let −∞ < a < b ≤ +∞, and assume (B1) – (B4). The aim of this section is to show that it is possible to perturb the metric g obtaining a uniformly equivalent metric such that the associated negative gradient of f has the Morse – Smale property up to order h. We shall assume N to be infinite-dimensional and second countable (in particular, it is modeled on a separable Hilbert space). A well-known theorem by Eells and Elworthy (1970) implies that every infinite-dimensional Hilbert manifold can be smoothly embedded as an open subset of a Hilbert space. So we may assume that N is an open subset of the separable Hilbert space (H, ·, ·).5 Denote by Sym(H) the Banach space of self-adjoint bounded linear operators on H. The metric g can be represented by a C h map G: N → Sym(H) taking values in the cone of positive operators, such that g(p)[ξ, η] = G(p)ξ, η ∀p ∈ N, ∀ξ, η ∈ T p N = H. 4

See also Section 3.2. Viewing N as an open subset of a Hilbert space is useful to simplify the notation (some spaces of maps are Banach spaces and not Banach manifolds, some sections of Banach bundles are just maps between Banach spaces, and so on) but it is by no means necessary. Therefore the results of this section hold also for a finite-dimensional N which is not diﬀeomorphic to an open subset of Rn . 5

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43

We shall always denote by a lower case letter a symmetric bilinear form, and by the corresponding upper case letter the associated self-adjoint operator. The gradient of f with respect to the metric g is ∇g f (p) = G(p)−1 ∇ f (p), where ∇ f denotes the gradient of f with respect to the Hilbert inner product ·, ·. The Morse – Smale property will be achieved by rank 2 perturbations of G. In order to describe the space of such perturbations, let θ: N → [0, +∞[ be a continuous function such that θ(p) ≤

1 G(p)−1

∀p ∈ N.

(42)

The vector space K := K ∈ Cbh N, Sym(H) rank K(p) ≤ 2 ∀p ∈ N,

∃c ≥ 0 such that K(p) ≤ c θ(p) ∀p ∈ N is a Banach space with the norm K(p) . θ(p)0 θ(p)

KK := KC h + sup

As usual, the symbol Cbh denotes the space of maps whose diﬀerentials up to the hth order are continuous and bounded. Notice that the maps K ∈ K vanish on the set of zeroes of θ. By (42), for every p ∈ N G(p)−1 K(p) ≤ G(p)−1 KK θ(p) ≤ KK , so if KK < 1, G + K = G(I + G−1 K) is positive, and defines a metric g + k which is uniformly equivalent to g. Denote by K1 the open unit ball of K. The main result of this section is the following theorem. THEOREM 2.20. Let f be a C h+1 function, h ≥ 1, on the smooth second countable Hilbert manifold N ⊂ H, endowed with a Riemannian metric g of class C h . Let −∞ < a < b ≤ +∞, and assume (B2). Assume that the continuous function θ: N → [0, +∞[ satisfies (42), that its set of zeroes is the closure of an open set, and that it has the following property: if x, y are critical points in {a < f < b} with m(x) − m(y) ≤ h, such that W u (x) and W s (y) (with respect to −∇g f ) have a nontransverse intersection at p, then θ > 0 somewhere on the orbit of p. Then for every K in a residual subspace of K1 , the metric g + k associated to G + K is such that the vector field −∇g+k f satisfies the Morse – Smale property up to order h. Notice that high regularity of f and g is needed if we want to achieve the Morse – Smale property up to a high order. This phenomenon is determined by

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the regularity versus Fredholm index assumption required by the Sard – Smale Theorem 2.19. In a finite-dimensional setting this problem does not occur because there C h functions can always be C h -approximated by smooth ones, while such an approximation may not be possible on an infinite-dimensional Hilbert space (see for instance Nemirovski˘ı and Semenov, 1973 and Lasry and Lions, 1986). Notice that C 2 regularity of f is enough to get the Morse – Smale property up to order 1, which is just what we need in order to have the Morse complex and to represent it by intersection numbers. The possibility of having a function θ which vanishes on some regions where the intersections are already transversal and which can be very small elsewhere will be useful in Section 2.13. Let us set up the proof of Theorem 2.20. Fix two critical points x y in {a < f < b} with m(x) − m(y) ≤ h, and consider the space of curves

C = C(x, y) := u ∈ C 1 (R, N) lim u(t) = x, lim u(t) = y, lim u (t) = 0 . t→−∞ t→+∞ t→±∞ The space C is a smooth Banach manifold, being an open subset of an aﬃne Banach space modeled on C01 (R, H) (the spaces C0h are defined in Section 1.2). Therefore, T u C = C01 (R, H). The map Ψ: C × K1 → C00 (R, H),

(u, K) → u + ∇g+k f (u) = u + (G + K)−1 (u)∇ f (u),

is of class C h , and its zeroes are the pairs (u, K) such that u is a negative gradient flow line of f with respect to the metric g+k, going from x to y. Set Z := Ψ−1 ({0}). The following two lemmas describe some properties of the diﬀerential of Ψ with respect to the first, respectively the second variable. LEMMA 2.21. Let (u, K) ∈ Z. Then: (i) the operator D1 Ψ(u, K): T u C → C00 (R, H) is Fredholm of index m(x) − m(y); (ii) the operator D1 Ψ(u, K) is onto if and only if the unstable manifold of x and the stable manifold of y with respect to the vector field −∇g+k f intersect transversally at u(t) for some (hence all) t ∈ R; (iii) if w ∈ C00 (R, H) and a < b are real numbers, then there exists v ∈ T u C such that D1 Ψ(u, K)[v](t) = w(t) ∀t ∈ ]−∞, a] ∪ [b, +∞[. Proof. The diﬀerential of Ψ with respect to the first variable is of the form D1 Ψ(u, K): C01 (R, H) → C00 (R, H),

v → v − Av,

where A: R → L(H) is defined by A(t) := −(G + K)−1 u(t) D2 f u(t) − D(G + K)−1 ) u(t) ∇ f u(t) .

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45

Since u(t) converges to x, resp. to y, for t → −∞, resp. t → +∞, A(t) converges in norm to the operators A(−∞) = −(G + K)−1 (x)D2 f (x) = −∇2g+k f (x), A(+∞) = −(G + K)−1 (y)D2 f (y) = −∇2g+k f (y), which are hyperbolic, and have positive eigenspaces of dimension m(x) and m(y), respectively. Then (i) follows from Proposition 1.8. Claim (ii) follows from the second identity in (9) , and from the identities T u(0) W u (x) = WAu ,

T u(0) W s (y) = WAs .

As for claim (iii), up to a translation we may assume that a < 0 < b. Then the conclusion follows from Proposition 1.6(i), applied to A|[0,+∞] and to A|[−∞,0] (−·). LEMMA 2.22. Let (u, K) ∈ Z, and let a < b be real numbers such that θ u(t) 0 for every t ∈ [a, b]. Let w ∈ C h (R, H) be a curve with support in [a, b]. Then there exists J ∈ K such that D2 Ψ(u, K)[J] = w. Proof. The diﬀerential of Ψ with respect to the second variable is D2 Ψ(u, K)[J] = −(G + K)−1 (u)J(u)(G + K)−1 (u)∇ f (u). Since (u, K) ∈ Z, the curve u is a flow line of the vector field −∇g+k f going from x to y. In particular, u is a C h+1 embedding of R into N, and ∇ fg+k ◦u never vanishes. It is easy to find a C h curve J0 : R → Sym(H) with support in [a, b] such that for every t ∈ R the symmetric operator J(t) has rank not exceeding 2, and maps the nonzero vector (G + K)−1 u(t) ∇ f u(t) = ∇g+k f u(t) into the vector −(G + K) u(t) w(t). Indeed, one may write an explicit formula for J0 by noticing that if ξ 0 and η are two elements of H, the bounded linear operator on H ζ →

ξ, ζ η, ζ ξ, η ξ, ζ η+ ξ− ξ, 2 2 |ξ| |ξ| |ξ|4

is self-adjoint, has rank not exceeding 2, vanishes when η = 0, maps ξ into η, and depends smoothly on (ξ, η) ∈ (H \ {0}) × H. Since u is a C h+1 embedding, given δ > 0 we can find an open neighborhood U of u(]a − δ, b + δ[) and a C h+1 submersion τ: U → ]a − δ, b + δ[ such that τ u(t) = t for every t ∈ ]a − δ, b + δ[. Since θ is positive on u([a, b]), up to choosing a smaller δ and a smaller U we may assume that inf U θ > 0, and also that τ has bounded derivatives up to order h+1. If ψ ∈ Cb∞ (H, R) is a cut-oﬀ function with support in U and taking value 1 on u([a, b]), the C h map J: N → Sym(H), J(p) = ψ(p)J0 τ(p) , belongs to K and has the required property. The following lemma is the key point in the proof of Theorem 2.20.

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LEMMA 2.23. Let (u, K) ∈ Z. Then the diﬀerential DΨ(u, K): T u C × K → C00 (R, H) is a left inverse. Proof. We must prove that the operator DΨ(u, K)[(v, J)] = D1 Ψ(u, K)[v] + D2 Ψ(u, K)[J] is onto and that its kernel is complemented in T u C × K. By Lemma 2.21(i), the operator D1 Ψ(u, K) is Fredholm, so Lemma 2.16 implies that ker DΨ(u, K) is complemented in T u C × K. Moreover, the range of DΨ(u, K) contains the range of D1 Ψ(u, K), in particular it has finite codimension. If θ ◦ u(t) = 0 for every t ∈ R, also K ◦ u vanishes identically, so ∇g+k f ◦ u = ∇g f ◦ u, and (recalling that the set of zeroes of θ is the closure of an open set) the tangent spaces of the unstable and stable manifolds of x and y along u are the same for −∇g+k f and for −∇g f . Therefore, the assumption of Theorem 2.20 guarantees that these manifolds meet transversally along u. By Lemma 2.21(ii), D1 Ψ(u, K) is onto, and so is DΨ(u, K). If θ ◦ u is not identically zero, we can find real numbers a < b such that θ u(t) 0 for every t ∈ [a, b]. Let w ∈ C00 (R, H) and let > 0. By Lemma 2.21(iii), there exists v ∈ T u C such that D1 Ψ(u, K)[v](t) = w(t)

∀t ∈ ]−∞, a] ∪ [b, +∞[.

The curve w − D1 Ψ(u, K)[v] is continuous and has support in [a, b], and we can find a C h curve z : R → H with support in [a, b] such that z − (w − D1 Ψ(u, K)[v])∞ < . Since z has support in [a, b], where θ ◦ u does not vanish, by Lemma 2.22 there exists J ∈ K such that D2 Ψ(u, K)[J] = z. Hence DΨ(u, K)[(v, J)] − w∞ = D1 Ψ(u, K)[v] + z − w∞ < . Therefore, DΨ(u, K) has dense and finite-codimensional range, so it is onto.

In particular, Z is a C h submanifold of C × K1 . Let π be the restriction to Z of the projection onto the second factor in the product C × K1 . LEMMA 2.24. The map π: Z → K1 is Fredholm of index m(x) − m(y). Proof. Everything follows from Proposition 2.17(ii), applied to M = C × K1 , N = K1 , O = C00 (R, H), ϕ: C×K1 → K1 projection onto the second factor, ψ = Ψ, together with Lemma 2.21(i) and Proposition 2.23. Denote by H(x, y) the set of regular values of π. LEMMA 2.25. The set H(x, y) is residual in K1 .

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47

Proof. Notice that R acts freely on the submanifold Z by t, (u, K) → (u(t + ·), K), and the map π is invariant with respect to this action. Therefore, the → K1 = Z/R is still a C h manifold, and the induced map π: Z quotient space Z h is of class C and Fredholm index m(x) − m(y) − 1 ≤ h − 1, by Lemma 2.24. Moreover, K is a regular value for π if and only if it is a regular value for π. Since N, and thus H, is assumed to be second countable, C × K1 is second Since the level of diﬀerentiability of countable, and so are Z and Z. π is strictly greater than its Fredholm index, the Sard – Smale Theorem 2.19 implies that the set of regular values of π — and thus of π — is residual in K1 . Proof of Theorem 2.20. By Proposition 2.17(i), H(x, y) is also the set of K ∈ K1 for which the map Ψ(·, K) : C(x, y) → C00 (R, H) has 0 as a regular value. By Lemma 2.21 (ii), H(x, y) is also the set of K ∈ K1 such that the unstable manifold of x and the stable manifold of y with respect to −∇g+k f meet transversally. By Lemma 2.25, the countable intersection H(x, y) x, y ∈ crit( f ) ∩ {a < f < b} x y, m(x) − m(y) ≤ h

is the required residual subset of K1 .

2.13. INVARIANCE OF THE MORSE COMPLEX

Let f ∈ C 2 (M) be a Morse function on the smooth second countable Hilbert manifold M, with critical points of finite index. Assume that f is bounded below and that M admits a complete Riemannian metric g such that ( f, g) satisfies the Palais – Smale condition. We know from the previous section that by perturbing g we can achieve also the Morse – Smale property up to order 1. In general, diﬀerent Morse – Smale metrics will produce diﬀerent Morse complexes: the groups Ck ( f ) are the same, but the boundary operators ∂k may vary. Of course the homology of the Morse complex does not vary, being isomorphic to the singular homology of M, but we can say more: varying the metric we obtain isomorphic chain complexes. This fact was observed by Cornea and Ranicki (2003) (together with other interesting rigidity results) for finite-dimensional manifolds, and for some cases of Floer theory. The proof we give here in our infinite-dimensional situation uses an idea from Abbondandolo and Majer (2001) (see also Po´zniak, 1991). THEOREM 2.26. Let f ∈ C 2 (M) be a Morse function, bounded below, having only critical points of finite Morse index. Let g0 and g1 be complete Riemannian

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metrics on M, such that both ( f, g0 ) and ( f, g1 ) satisfy (PS) and the Morse – Smale property up to order 1. Then there is a chain complex isomorphism Φ: {C∗ ( f ), ∂∗ ( f, g0 )} {C∗ ( f ), ∂∗ ( f, g1 )} of the form

Φx = x +

n(x, y)y,

∀x ∈ critk ( f ), k ∈ N,

(43)

y ∈ critk ( f ) f (y) < f (x)

for suitable integers n(x, y). The following lemma will be needed in the proof: LEMMA 2.27. Let a be a nondegenerate continuous symmetric bilinear form on the real Hilbert space (H, ·, ·), with either finite Morse index or finite Morse coindex. Let t0 ≥ 0, and let t → ·, ·t , t ∈ R, be a continuous path of inner products on H — equivalent to ·, · — constant for t ≥ t0 and for t ≤ −t0 . Let A(t) be the ·, ·t -self-adjoint bounded operator on H representing a with respect to the inner product ·, ·t : a(ξ, η) = A(t)ξ, ηt for every ξ, η ∈ H. Then the linear stable and unstable spaces of the path A (see Section 1.2) satisfy H = WAs ⊕ WAu . Proof. The path A is continuous and it is constant for t ≥ t0 and for t ≤ −t0 . Let us assume that a has finite Morse index, the other case being easily reducible to this one. The linear stable space WAs has dimension m(a), the Morse index of a, while the linear unstable space WAu is closed and has codimension m(a). Therefore, it is enough to prove that WAs ∩ WAu = (0). Let u0 ∈ WAs ∩ WAu , and let u: R → H be the solution of the linear Cauchy problem u (t) = A(t)u(t), u(0) = u0 . Since A is constant for t ≥ t0 and for t ≤ −t0 , u(t) = e(t−t0 )A(t0 ) u(t0 ) ∀t ≥ t0 ,

u(t) = e(t+t0 )A(−t0 ) u(−t0 ) ∀t ≤ −t0 .

Since u(t) → 0 for |t| → 0, we deduce that u(t0 ) belongs to the negative eigenspace of A(t0 ), and u(−t0 ) belongs to the positive eigenspace of A(−t0 ). Since both A(t0 ) and A(−t0 ) represent the symmetric form a, we have a u(t0 ), u(t0 ) ≤ 0, a u(−t0 ), u(−t0 ) ≥ 0. (44) On the other hand, since A(t) represent a for every t, the inequality 1d a u(t), u(t) = a u(t), u (t) = a u(t), A(t)u(t) = A(t)u(t), A(t)u(t)t ≥ 0 2 dt

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49

is compatible with (44) if and only if u(t) = 0 for every t ∈ [−t0 , t0 ] (hence for every t ∈ R), proving that u0 = 0. Therefore WAs ∩ WAu = (0). Proof of Theorem 2.26. We introduce the smooth Morse function ϕ: R → R,

ϕ(s) = 2s3 − 3s2 + 1,

which has two critical points, namely a local maximum at 0, with ϕ(0) = 1, and a local minimum at 1, with ϕ(1) = 0. Moreover ϕ (s) diverges for |s| → +∞. = R × M consider the C 2 function On the manifold M → R, f˜: M

f˜(s, p) = ϕ(s) + f (p).

It is a Morse function, with critical points of finite Morse index, and critk ( f˜) = {0} × critk−1 ( f ) ∪ {1} × critk ( f ) , for every k ∈ N. Therefore Ck ( f˜) Ck−1 ( f ) ⊕ Ck ( f ),

∀k ∈ N,

(45)

the first group in the sum corresponding to the critical points in {0}× M, the second one to critical points in {1} × M. If χ: R → [0, 1] is a smooth cut-oﬀ function such that χ(s) = 1 for s ≤ 1/3 and χ(s) = 0 for s ≥ 23 , we can consider the complete Riemannian metric on M g˜ (s, p)[(σ, ξ), (σ , ξ )] = σσ + χ(s)g0 (p)[ξ, ξ ] + 1 − χ(s) g1 (p)[ξ, ξ ], = R ⊕ T p M. for every (σ, ξ), (σ , ξ ) ∈ T (s,p) M Let (sn , pn ) be a (PS) sequence for ( f˜, g˜ ). Since ∇g˜ f˜(s, p)g˜ ≥ |ϕ (s)|, we can find a subsequence of (sn ) which converges either to 0 or to 1. Since ( f, g0 ) and ( f, g1 ) satisfy (PS) and g˜ (s, p)|(0)⊕T M is just g0 for s close to 0 and g1 for s close to 1, we conclude that ( f˜, g˜ ) satisfies (PS). Let us examine the negative gradient flow of f˜ with respect to the metric g˜ . (i) The hypersurfaces {0} × M and {1} × M are flow-invariant, and the restriction of the flow to {i} × M is nothing else but the negative gradient flow of f with respect to the metric gi , for i = 0, 1. Moreover the invariant set {0} × M is a repeller, while {1} × M is an attractor. Therefore: (ii) The only flow lines going from a critical point in {i} × M to a critical point in the same hypersurface are those which are fully contained in {i} × M, for i = 0, 1. (iii) There are no flow lines going from a critical point in {1} × M to a critical point in {0} × M.

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we have If we view f as a function on M, D f (s, p)[−∇g˜ f˜(s, p)] = D f (s, p)[(−ϕ (s), −∇g˜ (s,·) f (p)] = −∇g˜ (s,·) f (p)2g˜ (s,·) .

(46)

This implies that f is almost a Lyapunov function for the vector field −∇g˜ f˜: (iv) f decreases strictly on all the nonconstant orbits, apart from those of the form t → (s(t), x), with x ∈ crit( f ), s (t) = −ϕ s(t) . In particular, up to time shifts there is exactly one flow line going from (0, x) to (1, x), for x ∈ crit( f ), namely the orbit s¯ (t) = −ϕ s¯(t) , t → ( s¯(t), x), with (47) s¯(0) = 12 . We claim that the intersection W u (0, x) ∩ W s (1, x) = ]0, 1[ × {x} is transverse. Indeed, by linearizing along the flow line ( s¯(t), x), we easily see that T (1/2,x) W u ((0, x)) = R ⊕ WAu ,

T (1/2,x) W s ((1, x)) = R ⊕ WAs ,

where the bounded linear operator A(t) : T x M → T x M is minus the Hessian of f at the critical point x with respect to the inner product g˜ ( s¯(t), x)|(0)⊕T x M = χ( s¯(t))g0 + (1 − χ( s¯(t)))g1 . Then A(t) represents the second diﬀerential of f at x with respect to the above inner product, so by Lemma 2.27, T x M = WAs ⊕ WAu . Therefore T (1/2,x) W u (0, x) ⊕ T (1/2,x) W s (1, x) = R ⊕ T x M = T (1/2,x) M, proving transversality. The vector field −∇g˜ f˜ need not satisfy the Morse – Smale condition up to order 1, but the only points where transversality can fail are the intersections of the unstable manifold of a critical point (0, x) with the stable manifold of a critical point (1, y), with x y critical points of f . We can perturb the metric g˜ in order to achieve the Morse – Smale property up to order 1 without loosing the nice features (i) – (iv) of the vector field −∇g˜ f˜. More precisely, by Theorem 2.20, taking into such that account (46), we can find a complete metric g on M (a) ( f˜, g) satisfies (PS); (b) g coincides with g˜ on the sets ]−∞, 13 ] × M, [ 23 , +∞[ × M, and R × U, where U ⊂ M is a neighborhood of crit( f ); (c) D f (s, p)[−∇g f˜(s, p)] < 0 if p crit( f );

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(d) ( f˜, g) satisfies the Morse – Smale property up to order 1. Indeed, the function θ appearing in the statement of Theorem 2.20 can be chosen to vanish on the regions indicated in (b), where the intersections are already transverse, and to be so small that the metrics belonging to the space of perturbations satisfy (c). By property (b), the flow of −∇g f˜ still satisfies (i), (ii), (iii). By (c), it satisfies also (iv). We can now consider the Morse complex of ( f˜, g) relative to the sublevel ˜ { f < inf f − 1}. Notice that this sublevel contains no critical points. The boundary operator ∂k ( f˜, g) can be described by using Theorem 2.11 and Remark 2.13. To this purpose, it is convenient to choose the orientations of the unstable manifolds in the following way: since for every x ∈ crit( f ) there is a privileged isomorphism T x W u (x; −∇g0 f ) T x W u (x; −∇g1 f ), namely the restriction to the first space of the projection onto the first factor in the splitting T x M = T x W u (x; −∇g1 f ) ⊕ T x W s (x; −∇g1 f ), we can endow these two spaces with orientations which are compatible with this isomorphism. Then T (0,x) W u (0, x); −∇g f˜ = R ⊕ T x W u (x; −∇g0 f ) and

T (1,x) W u (1, x); −∇g f˜ = (0) ⊕ T x W u (x; −∇g1 f )

can be given the product orientations by the the standard orientations of R and (0). In this way, we have chosen an orientation for the unstable manifold of each critical point of f˜. With this choice the transverse intersection W u (0, x) ∩ W s (1, x) = ]0, 1[ × {x} (48) is given the orientation corresponding to the vector ∂/∂s, which agrees with the direction of the flow. By (i), (ii), (iii), and (45) the boundary operator ∂k ( f˜, g): Ck−1 ( f ) ⊕ Ck ( f ) → Ck−2 ( f ) ⊕ Ck−1 ( f )

can be written as ∂k ( f˜, g) =

∂k−1 ( f, g0 ) 0 , Φk−1 ∂k ( f, g1 )

for some homomorphism Φk : Ck ( f ) → Ck ( f ). ˜ The fact that ∂∗ ( f , g) is a boundary, i.e., ∂k ( f˜, g) ∂k+1 ( f˜, g) = 0, implies that Φk−1 ∂k ( f, g0 ) = ∂k ( f, g1 )Φk ,

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that is (Φk )k∈N is a chain homomorphism from the Morse complex of ( f, g0 ) to the Morse complex of ( f, g1 ). By (iv), the intersection W u (0, x); −∇g f˜ ∩ W s (1, y); −∇g f˜ can be nonempty only if f (y) < f (x) or x = y, and in the latter case it consists of the single orbit (48). Together with the previous discussion on orientations, this fact implies that Φ has the form (43). So if we order the critical points of f with Morse index k by increasing value of f , we see that the homomorphism Φk is represented by an upper-triangular matrix, with 1 on the diagonal entries. A homomorphism of this form must be an isomorphism: this is well known when Ck ( f ) has finite rank, because in this case Φk is represented by a finite matrix with determinant 1, an invertible element of Z, but it remains true if the rank of Ck ( f ) is infinite. Indeed if x1 , x2 , . . ., are the critical points of index k ordered by increasing value of f , the inverse of Φk is defined inductively by Φ−1 k x1 = x1 ,

Φ−1 k xh = xh −

h−1

n(xh , xi )Φ−1 k xi , ∀h ≥ 2.

i=1

EXERCISE 2.28. Generalize this result to the case of a strip {a < f < b}. EXERCISE 2.29. When f satisfies the condition (A8), it is possible to obtain the same conclusion of Theorem 2.26 by looking directly at the two cellular filtrations induced by the two negative gradient flows. Prove this fact. Then use the limit arguments of Section 2.9 to prove Theorem 2.26 under the hypothesis that ( f, g0 ) and ( f, g1 ) satisfy the Morse – Smale condition only up to order 0.

3. The Morse complex in the case of infinite Morse indices 3.1. THE PROGRAM

In this part we will consider a gradient-like C 1 vector field X on a Hilbert manifold M, whose rest points have infinite Morse index and co-index. In this case, the stable and the unstable manifolds of rest points are infinite-dimensional, and the flow of X does not produce a meaningful cellular filtration of M. Indeed, the infinite-dimensional Hilbert ball is retractable onto its boundary, so the rest points of X are homotopically invisible. However, we may hope that in some cases the unstable and the stable manifolds of pairs of rest points have finite-dimensional intersections. If this holds,

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we could use the formula for the boundary operator of Theorem 2.11 not as a description, but rather as the definition of the Morse complex. Our program is to follow this idea. Of course this program cannot be pursued in full generality. A first reason is that in general the unstable and stable manifolds may not have finite-dimensional intersections. A deeper reason is that the setting of gradient-like flows for a Morse function with critical points of infinite Morse index and co-index has too little rigidity. For instance, the following result was proved in Abbondandolo and Majer (2004). A sketch of the proof will be presented at the end of Section 3.3. THEOREM 3.1. Let f : M → R be a smooth Morse function on a separable Hilbert manifold, whose critical points have infinite Morse index and co-index. Let a: crit( f ) → Z be an arbitrary function. Then there exists a Riemannian metric g on M such that the corresponding negative gradient flow of f has the following property: for every pair of critical points x, y, the intersection W u (x) ∩ W s (y) is transverse and — if nonempty — has dimension a(x) − a(y). Moreover, the metric g can be chosen to be uniformly equivalent to any given metric g0 on M. Finally, if (xi , yi ), i = 1, . . . , k, are pairs of critical points such that xi and yi can be connected by a path ui : [0, 1] → M such that D f ui (t) [u i (t)] is negative for every t ∈ ]0, 1[, the metric g can be chosen in such a way that W u (xi ) ∩ W s (yi ) is not empty. Therefore the situation is drastically less rigid than the case of finite Morse indices, where the Morse index of a critical point does not involve the metric, and where we have seen that the isomorphism class of the Morse complex does not depend on the metric, and that its homology does not even depend on f . Let us examine another example of the lack of rigidity determined by infinite Morse indices and co-indices. We have seen that when the Morse indices are finite, the transverse intersection W u (x) ∩ W s (y) is always orientable, and each of its components has the same dimension m(x) − m(y). On the other hand, if Z is any separable Hilbert manifold (finite-dimensional or not, possibly with components of diﬀerent dimension), there exists a smooth gradient-like flow on the Hilbert space H with exactly two rest points x and y, such that the intersection W u (x) ∩ W s (y) is transverse and diﬀeomorphic to Z × R (see Abbondandolo and Majer, 2003b, Section 4). These phenomena suggest that a Morse theory for functions f : M → R with critical points of infinite Morse index and co-index requires more structure than just the pair (M, f ). Our choice will be to consider a subbundle V of T M, suitably compatible with the gradient-like flow.

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3.2. FREDHOLM PAIRS AND COMPACT PERTURBATIONS OF LINEAR SUBSPACES

Before proceeding, we need to review some facts about the Hilbert Grassmannian Gr(H), the set of all closed linear subspaces of the separable Hilbert space H. See Abbondandolo and Majer (2003a) for a more complete presentation. If V ∈ Gr(H), we shall denote by PV the orthogonal projection onto V. The set Gr(H) is a complete metric space with the distance dist(V, W) := PV − PW . The connected components of Gr(H) are the subspaces Grn,m (H) = {V ∈ Gr(H) | dim V = n, codim V = m}, where n, m ∈ N ∪ {∞}, n + m = ∞. A pair (V, W) ∈ Gr(H)×Gr(H) is a Fredholm pair if V∩W is finite-dimensional and V + W is finite-codimensional. In this case, the number ind(V, W) := dim V ∩ W − codim(V + W) is said the Fredholm index of (V, W). The space of Fredholm pairs, denoted by Fp(H), is an open subset of Gr(H) × Gr(H), and the Fredholm index is a continuous (i.e., locally constant) function on it. See for instance Kato (1980, IV §4). Let W ∈ Gr(H). A closed linear subspace V is a compact perturbation of W if the operator PV − PW is compact. In this case, the pair (V, W ⊥ ) is Fredholm, and its index is said the relative dimension of V with respect to W, denoted by dim(V, W) := ind(V, W ⊥ ) = dim V ∩ W ⊥ − dim V ⊥ ∩ W. If (V, W) is a Fredholm pair and Z is a compact perturbation of V, then (Z, W) is still a Fredholm pair, and its index is ind(Z, W) = ind(V, W) + dim(Z, V).

(49)

3.3. FINITE-DIMENSIONAL INTERSECTIONS

Let M be a smooth Hilbert manifold, and let X be a C 1 Morse vector field on M, with local flow φ: Ω(X) → M. We shall always assume that X has a Lyapunov function f . In view of Remark 1.21(ii), we shall assume that f ∈ C 2 (M) and that it is a nondegenerate Lyapunov function, meaning that for every x ∈ rest(X) the quadratic form ξ → D2 f (x)[ξ, ξ] is coercive on E s ∇X(x) , while ξ → −D2 f (x)[ξ, ξ] is coercive on E u ∇X(x) . Let V be a smooth subbundle of T M, and let P be a projector onto V: P is a smooth section of the bundle of endomorphisms of T M such that for every p ∈ M, P(p) ∈ L(T p M) is a projector onto V(p). We shall assume the following compatibility conditions between X and V:

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(C1) for every x ∈ rest(X), the positive eigenspace E u ∇X(x) of the Jacobian of X at x is a compact perturbation of V(x); (C2) for every p ∈ M, the operator (LX P)(p)P(p) is compact. Here LX P denotes the Lie derivative of P along X. By (C1), we can define the relative Morse index of the rest point x with respect to V to be the integer m(x, V) := dim E u ∇X(x) , V(x) . Condition (C2) depends only on the subbundle V, and not on the choice of the projector P onto it. Notice that the subbundle V is φ-invariant (in the sense that D2 φ(t, p)V(p) = V φ(t, p) for every (t, p) ∈ Ω(X)) if and only if (LX P)P = 0. Condition (C2) is equivalent to the fact that V is φ-essentially invariant: D2 φ(t, p)V(p) is a compact perturbation of V φ(t, p) , for every (t, p) ∈ Ω(X). When M is an open subset of the Hilbert space H, and V is a constant subbundle V ≡ V ∈ Gr(H), so that we can choose P ≡ PV , there holds (LX P)P = [DX, PV ]PV = (I − PV )DXPV .

(50)

These assumptions have the following consequence. PROPOSITION 3.2. Assume that the Morse vector field X satisfies (C1) and (C2) with respect to the subbundle V. Then for every x ∈ rest(X): (i) for every p ∈ W u (x), T p W u (x) is a compact perturbation of V(p), with dim(T p W u (x), V(p)) = m(x, V); (ii) for every p ∈ W s (x), the pair (T p W s (x), V(p)) is Fredholm, with ind(T p W s (x), V(p)) = −m(x, V). So loosely speaking, W u (x) is essentially parallel to V, while W s (x) is essentially normal to V. Let us sketch the proof of the first claim in a simpler case: we assume that M is an open set of the Hilbert space H, and that V ≡ V ∈ Gr(H) is a constant subbundle. Let p ∈ W u (x), and let u(t) := φ(t, p) be the orbit of p. By linearization along u, using the notation of Section 1.2, we have that T p W u (x) = WAu ,

(51)

where A(t) := DX u(t) . By (C1), W := T x W u (x) = E u A(−∞) is a compact perturbation of V. By (C2), the operator [A(t), PV ]PV is compact for every t, and so is the operator [A(t), PW ]PW . Set B(t) := A(t) − [A(t), PW ]PW ,

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so that B(−∞) = A(−∞) = DX(x), E u B(−∞) = W, and B(t)W ⊂ W for every t. These facts easily imply that WBu = W. On the other hand, since A(t) − B(t) is compact for every t, WAu is a compact perturbation of WBu = W, hence of V. By (51), T p W u (x) is a compact perturbation of V. The formula for its relative dimension with respect to V follows by continuity. The proof of claim (ii) is simpler. Since the set of Fredholm pairs is open and the index is locally constant, by (C1) the pair (T p W s (x), V(p)) is Fredholm of index −m(x, V) for every p ∈ W s (x) in a neighborhood of x. The tangent bundle T W s (x) is φ-invariant, and by (C2) the subbundle V is φ-essentially invariant, so these facts remain true for every p ∈ W s (x). By (49), Proposition 3.2 has the following easy corollary. COROLLARY 3.3. Assume that the Morse vector field X satisfies (C1) and (C2) with respect to the subbundle V. Let x, y ∈ rest(X), and assume that W u (x) and W s (y) meet transversally. Then W u (x) ∩ W s (y) is a submanifold of dimension m(x, V) − m(y, V). We conclude this section by sketching the proof of Theorem 3.1. By the already mentioned embedding theorem of Eells and Elworthy (1970), we can embed M as an open subset of the separable Hilbert space H. By modifying this embedding near the critical points of f , and by using the Morse Lemma (see for instance Palais, 1963), we may assume that f is quadratic near every critical point x: f (x + ξ) = f (x) + 12 A(x)ξ, ξ,

for |ξ| small,

for some self-adjoint invertible operator A(x). Fix a closed linear subspace V of H, with infinite dimension and codimension. By a further modification of the embedding, we may also rotate small neighborhoods of the critical points in such a way that the negative eigenspace E s (A(x)) of the operator A(x) is a compact perturbation of V, of relative dimension a(x). Here we actually need to use Kuiper’s theorem (Kuiper, 1965), stating the orthogonal group of H is contractible. It is now easy to build a vector field X having f as a nondegenerate Lyapunov function, and which satisfies (C1) and (C2) with respect to the constant subbundle V. Indeed, near a critical point x one may choose X to be the linear vector field X(x + ξ) = −∇ f (x + ξ) = −A(x)ξ,

for |ξ| small.

(52)

Since the negative eigenspace of A(x) is a compact perturbation of V of relative dimension a(x), X satisfies (C1) and x has relative Morse index m(x, V) = a(x). The linear vector field X satisfies also (C2). Indeed by (50), (LX PV )PV = −(I − PV )A(x)PV = −PV ⊥ PE s (A(x)) A(x)PV − PV ⊥ A(x)PE u (A(x)) PV , and the operators PV ⊥ PE s (A(x)) and PE u (A(x)) PV are compact because E s (A(x)) is a compact perturbation of V. If p ∈ M is not a critical point, we may choose

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X to be the constant vector field X(p + ξ) = −∇ f (p), for every ξ so small that D f (p + ξ)[−∇ f (p)] < 0. Every constant vector field trivially satisfies (C2) with respect to the constant subbundle V. These local definitions of X can be patched together by a smooth partition of unity. In this way one can build a vector field X satisfying (52) near critical points, so that (C1) holds. The set of vector fields satisfying condition (C2) is a module over the ring of real functions, so X satisfies (C2). Having f as a Lyapunov function is a convex condition, so f is a Lyapunov function for X. Up to a small perturbation, we may assume that X also satisfies the Morse – Smale condition. Then Corollary 3.3 implies that W u (x) ∩ W s (y) is a submanifold of dimension m(x, V) − m(y, V) = a(x) − a(y). The fact that X is actually the negative gradient of f near the critical points makes it possible to find a metric g on M such that X = −∇g f . We refer to Abbondandolo and Majer (2004) for details on how to keep g uniformly equivalent to a given metric, and on how to obtain that W u (xi ) ∩ W s (yi ) is nonempty for every i = 1, . . . , k. 3.4. ESSENTIAL SUBBUNDLES

It is readily seen that if X satisfies (C1) and (C2) with respect to a subbundle V, then it satisfies (C1) and (C2) also with respect to a subbundle W which at every point is a compact perturbation of V. This fact suggests the possibility of weakening the structure, fixing only an essential subbundle of T M. In order to make this precise, we need to introduce the essential Grassmannians of a Hilbert space. See again Abbondandolo and Majer (2003b) for a complete discussion. The essential Grassmannian of H is the quotient of Gr(H) by the equivalence relation {(V, W) ∈ Gr(H) × Gr(H) | V is a compact perturbation of W}, and it is denoted by Gre (H). This space can also be seen as the space of symmetric projectors in the Calkin algebra L(H)/Lc (H) (Lc (H) denotes the closed ideal of compact operators). Notice that the finite-dimensional and the finite-codimensional spaces represent two points in Gre (H). We shall actually be interested in the complementary Gr∗e (H) of these two points, that is in the quotient of Gr∞,∞ (H). The (0)-essential Grassmannian Gr(0) (H) is the quotient of Gr(H) by the stronger equivalence relation {(V, W) ∈ Gr(H) × Gr(H) | V is a compact perturbation of W and dim(V, W) = 0}. Again, Gr∗(0) (H) denotes the quotient of Gr∞,∞ (H). The Bott periodicity theorem (see Bott, 1959), and the fact that the group of automorphisms of H which are

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compact perturbations of the identity is homotopy equivalent to the infinite general linear group GL(∞) = lim GL(n) (see Palais, 1965), allow to determine the −→ homotopy type of the essential Grassmannian, proving the following result. THEOREM 3.4. The quotient projection Gr∞,∞ (H) → Gr∗(0) (H) is a fiber bundle with contractible total space. The quotient projection Gr∗(0) (H) → Gr∗e (H) is a universal covering. The space Gr∗e (H) is path connected, its fundamental group is infinite cyclic, and if i ≥ 2,   Z if i ≡ 1, 5 mod 8,    Z2 if i ≡ 2, 3 mod 8, πi (Gr∗e (H)) πi−2 GL(∞) =    0 if i ≡ 0, 4, 6, 7 mod 8. Since the tangent bundle of an infinite-dimensional Hilbert manifold is always trivial (by the already mentioned Kuiper’s theorem; Kuiper, 1965), a subbundle V of T M can be identified with a map V: M → Gr(H). Similarly, an essential subbundle (respectively a (0)-essential subbundle) of T M can be identified with a map E: M → Gre (H) (resp. E: M → Gr(0) (H)). By Theorem 3.4, an essential subbundle E of T M can be lifted to a (0)essential subbundle if and only if the homomorphism E∗ : π1 (M) → π1 Gr∗e (H) = Z vanishes. A (0)-essential subbundle E of T M can be lifted to a true subbundle of T M if and only if all the homomorphisms E∗ : πi (M) → πi Gr∗(0) (H) vanish (a condition which has to be checked only for i ≡ 1, 2, 3, 5 mod 8). If the vector field X satisfies (C1) and (C2) with respect to a (0)-essential subbundle E of T M, then the relative Morse index m(x, E) can still be defined, and the conclusions of Proposition 3.2 and of Corollary 3.3 still hold (with the obvious changes). If the vector field X satisfies (C1) and (C2) with respect to an essential subbundle, there is no relative Morse index. In this case the transverse intersection W u (x) ∩ W s (y) is finite-dimensional, but diﬀerent components may have diﬀerent dimension. More precisely, the dimension of the connected component containing p depends on the homotopy class of the orbit of p, seen as a curve from (R, −∞, +∞) into (M, x, y). It is actually possible to construct an example of a gradient-like vector field on S 1 × H, which satisfies (C1) and (C2) with respect to a nonliftable essential subbundle, and has two critical points x and y such that the intersection W u (x)∩W s (y) is transverse and consists of two connected components of diﬀerent dimension.

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3.5. ORIENTATIONS

We recall that in the case of finite Morse indices, an arbitrary choice of the orientation of all the unstable manifolds — or equivalently of the finite-dimensional spaces T x W u (x) — determines an orientation of each transverse intersection of unstable and stable manifolds. Now T x W u (x) is infinite-dimensional, so it does not carry orientations. The right object to orient turns out to be the Fredholm pair T x W s (x), V(x) . In order to deal with this question, we need to introduce the determinant bundle Det Fp(H) → Fp(H) on the space of Fredholm pairs (see Abbondandolo and Majer, 2003b, for more details). It is a real line bundle, whose fiber at (V, W) ∈ Fp(H) is ∗ Det(V, W) := Det(V ∩ W) ⊗ Det H/(V + W) , where Det(Z) := Λdim Z (Z) denotes the space of top degree in the exterior algebra of the finite-dimensional vector space Z. Defining a bundle structure for this object is not immediate, because the maps (V, W) → V ∩ W and (V, W) → V + W are not continuous. We just mention the key ingredients in the constructions. The intersection map (V, W) → V ∩ W is continuous on the space of transverse pairs, while the sum (V, W) → V+W is continuous on the space of pairs with intersection (0). Then the bundle structure near a Fredholm pair (V0 , W0 ) can be constructed by fixing a finite-dimensional space Z such that Z + V0 + W0 = H and Z ∩ V0 = (0), and by replacing each pair (V, W) in a neighborhood of (V0 , W0 ) by (Z + V, W). Such a replacement turns out to be possible because of the existence of an exact sequence H 0 → V ∩ W → (Z + V) ∩ W → Z → → 0. V +W We recall that an exact sequence of finite-dimensional vector spaces 0 → Z1 → · · · → Zk → 0 induces a natural isomorphism " i odd

Det(Zi )

"

Det(Zi ).

i even

The space of Fredholm operators from H1 to H2 , denoted by F (H1 , H2 ) is “contained” in the space of Fredholm pairs of H1 × H2 . Indeed, the operator A ∈ L(H1 , H2 ) is Fredholm if and only if the pair (graph A, H1 × (0)) ∈ Gr(H1 × H2 ) × Gr(H1 × H2 ) is Fredholm, and the index is the same. The pullback of the determinant bundle on Fp(H) by the map F (H1 , H2 ) → Fp(H1 × H2 ), A → graph A, H1 × (0) ,

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is the determinant bundle on the space of Fredholm operators, as defined by Quillen (1985). Let n ∈ N, and let Det Grn,∞ (H) → Grn,∞ (H) be the real line bundle whose fiber at Z ∈ Grn,∞ (H) is Det(Z). Let S be the set of all Z, (V, W) in Grn,∞ (H) × Fp(H) n∈N

such that Z ∩ V = (0), and let Det(S) → S be the restriction to S of the tensor product of the bundles n∈N Det Grn,∞ (H) and Det Fp(H) . The map S → Fp(H), Z, (V, W) → (Z + V, W), is continuous, and can be lifted to a continuous morphism between the corresponding determinant bundles: S : Det(S) → Det Fp(H) . The construction of such a morphism is based on the exact sequence 0 → V ∩ W → (Z + V) ∩ W →

H H Z+V Z→ → → 0. V V +W Z+V +W

The morphism S is associative, meaning that if Z and Y are finite-dimensional linear subspaces of H such that Z ∩ Y = (Z + Y) ∩ V = (0), the diagram Det(Y) ⊗ Det(Z) ⊗ Det(V, W) S ⊗id

Det(Y + Z) ⊗ Det(V, W)

id⊗S

S

/ Det(Y) ⊗ Det(Z + V, W)

S

/ Det(Y + Z + V, W)

commutes. An orientation of a finite-dimensional space Z can be defined as an orientation of the line Det(Z); similarly, an orientation of the Fredholm pair (V, W) is an orientation of the line Det(V, W). The morphism S allows to sum orientations: if Z, (V, W) ∈ S, the orientations of two objects among Z,

(V, W),

(Z + V, W),

determines an orientation of the other object. Let us go back to the question of orienting the intersections between unstable and stable manifolds. The assumption is that the vector field X satisfies (C1) and (C2) with respect to a subbundle V of T M. By assumption (C1), the pair

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(T x W s (x), V(x)) is Fredholm, for every x ∈ rest(X). Let us choose an orientation o(x) of such a Fredholm pair, for every rest point x, in an arbitrary way. Now let x, y be rest points such that W u (x) and W s (y) have transversal intersection. Let p ∈ W u (x) ∩ W s (y). By Proposition 3.2(ii), the pair (T p W s (y), V(p)) is Fredholm. Choose a closed complement V of T p (W u (x) ∩ W s (y)) in T p W s (y). By transversality, V is also a complement of T p W u (x) in T p M. It is a general fact in this case that the backward evolution of V with respect to the diﬀerential of the flow converges to T x W s (x): lim D2 φ(t, p)V = T x W s (x).

t→−∞

Therefore, the Fredholm pair V, V(p) inherits by continuity an orientation from the orientation o(x) of T x W s (x), V(x) . On the other hand, the Fredholm pair T p W s (y), V(p) inherits an orientation from the orientation o(y) of T y W s (y), V(y) . The last two objects among T p W u (x) ∩ W s (y) , V, V(p) , u T p W (x) ∩ W s (y) + V, V(p) = T p W s (y), V(p) are then oriented, so they induce an orientation of the first space. The construction continuously depends on p, hence it determines an orientation of W u (x) ∩ W s (y). We shall see in Section 3.7 that the orientations defined here satisfy a suitable coherence property. 3.6. COMPACTNESS

In the case of finite Morse indices, we have seen that the (PS) condition together with the positive completeness of X implies that W u (x) ∩ { f ≥ a} is precompact. Now the unstable manifold is infinite-dimensional, so this cannot be true, but we can hope W u (x) ∩ W s (y) to be precompact. However, assumptions (C1) and (C2) are not suﬃcient to get this result: W u (x) ∩ W s (y) may consist, for instance, of infinitely many flow lines going from x to y, with no cluster points besides x and y. We need to strengthen condition (C2), a local assumption, into a more global condition. We recall that that the Hausdorﬀ distance of two subsets A, B of a complete metric space (W, d) is the number distH (A, B) := max sup inf d(a, b), sup inf d(a, b) ∈ [0, +∞], a∈A b∈B

b∈B a∈A

and that the Hausdorﬀ measure of noncompactness of A is the number βW (A) := inf{r > 0 | A can be covered by finitely many balls of radius r} ∈ [0, +∞],

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so that A is precompact if and only if βW (A) = 0. The function β is continuous with respect to the Hausdorﬀ distance. Moreover, βW (A) coincides with the Hausdorﬀ distance of A from the space of compact subsets of W: βW (A) = inf{distH (A, K) | K ⊂ W compact}.

(53)

In the case of a normed vector space W, β has also the following properties: βW (λA) = |λ|βW (A), βW (A + B) ≤ βW (A) + βW (B), βW conv(A) = βW (A).

(54)

Let E be an essential subbundle of T M, diﬀerent from the trivial essential subbundles [(0)] and [T M]. We shall assume that E admits a global presentation: there exists a smooth map Q : M → N into a Hilbert manifold such that for every p ∈ M, DQ(p) has finite-codimensional range, and ker DQ(p) belongs to the equivalence class E(p). For instance, E could be the equivalence class of a subbundle which is the vertical space of a submersion Q. We shall assume that N is endowed with a complete Riemannian metric, and we shall consider the induced metric on T N. The new assumption on the vector field X is: (C3)

(i) DQ ◦ X∞ < +∞; (ii) for every q ∈ N there exists δ = δ(q) > 0 and c = c(q) ≥ 0 such that βT N DQ X(A) ≤ c βN Q(A) ∀A ⊂ Q−1 Bδ (q) .

Let us restate this condition in a simple situation: assume that M is an open set of the Hilbert space H, and that E is the equivalence class of a constant subbundle V ∈ Gr(H). Then we can choose the global presentation to be the orthogonal projector onto W := V ⊥ , Q := PW . Denote by (XV , XW ) be the two components of X with respect to the orthogonal splitting H = V ⊕ W. Condition (C3)(i) says that XW is bounded, while (C3)(ii) is equivalent to: for every ξ ∈ W there exist δ > 0 and c ≥ 0 such that βW XW (A) ≤ c βW (PW A) ∀A ⊂ M ∩ V × (Bδ (ξ) ∩ W) . (55) In particular, if A ⊂ M is such that PW A is precompact, then also XW (A) = PW X(A) is required to be precompact. Thus, for every ξ ∈ M the map η → (I − PV )X(ξ + PV η) is a compact map in a neighborhood of 0. Therefore, the diﬀerential of this map at 0, namely (I − PV )DX(ξ)PV = (LX PV )(ξ)PV is compact. Hence (C3) implies (C2): the simple situation — M ⊂ H, E constant, Q projector — in which we have checked this fact is indeed the general local situation, and (C2) is a local assumption.

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Notice that in general (55) is strictly stronger than the fact that for every ξ ∈ W, XW should map (ξ + V) ∩ M into a precompact set, because (55) involves a Lipschitz control on the measure of noncompactness. However, these conditions are equivalent under a mild Lipschitz assumption on X. See Abbondandolo and Majer (2003b, Proposition 7.9) for a precise statement (in the case of a general map Q). The main result of this section is the following compactness theorem. THEOREM 3.5. Let E be an essential subbundle of T M with a global presentation Q: M → N into a complete Riemannian Hilbert manifold. Assume that the Morse vector field X is complete, has a nondegenerate Lyapunov function f , (X, f ) satisfies (PS). Assume also that X satisfies (C1) – (C3). Then for every pair of critical points x, y, the intersection W u (x) ∩ W s (y) is precompact. Let us sketch the proof. It is useful to introduce the following notion: a subset A ⊂ M is said essentially vertical if Q(A) is precompact. The proof is then based on the following steps: (i) if A is essentially vertical and t ≥ 0, then φ([0, t] × A) is essentially vertical; u (x) is essentially vertical; (ii) each local unstable manifold Wloc,r s (x) has precompact intersection with every (iii) each local stable manifold Wloc,r essentially vertical subset.

Let us prove (i) under the simplifying assumption that the target of the map Q is a Hilbert space E, and that the constants appearing in condition (C3)(ii) are uniform: c does not depend on q, and we can take δ = +∞. So (C3)(ii) becomes βE DQ X(B) ≤ cβE Q(B) ∀B ⊂ M. (56) Let A ⊂ M be an essentially vertical set, that is βE Q(A) = 0. Since Q takes value in a Hilbert space, there holds 1 t DQ φ(s, p) X φ(s, p) ds, Q φ(t, p) = Q(p) + t · t 0 from which we deduce that Q φ([0, t] × A) ⊂ Q(A) + [0, t] conv DQ X(φ([0, t] × A) . Then, by the properties (54) of the measure of noncompactness β and by (56), βE Q φ([0, t] × A) ≤ βE Q(A) + t βE conv DQ X(φ([0, t] × A) = t βE DQ X φ([0, t] × A) ≤ tc βE Q φ([0, t] × A) .

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By the above inequality, βE Q φ([0, t] × A) vanishes for every t < 1/c, and by iteration, for every t ≥ 0. This proves (i). Since (ii) and (iii) are local statements, we may assume that the rest point x is the origin of the Hilbert space H, and that Q is the orthogonal projector with kernel V, a constant local representative of the essential subbundle E. By (C1), E u := E u ∇X(0) is a compact perturbation of V. This fact easily implies that a bounded set A ⊂ H is essentially vertical if and only if its projection Ps A on E s := E s ∇X(0) is precompact. In particular, the graph of a map σ: E u (r) → E s (r) is essentially vertical if and only if the map σ is compact. So (ii) can be restated by saying that the map σu : E u (r) → E s (r) whose graph is the local unstable manifold (see Theorem 1.12) is compact. By the graph transform method (see Shub, 1987, Chapter 5), σu is the fixed point of the contraction F, mapping every 1-Lipschitz map σ ∈ Lip1 E u (r), E s (r) into the map F(σ) ∈ Lip1 E u (r), E s (r) , whose graph is the φ-evolution at time 1 of the graph of σ, intersected with E u (r) × E s (r). So claim (i) implies that the contraction F maps the closed nonempty subspace of compact maps into itself, hence the fixed point σu is a compact map, proving (ii). s (x) is the graph of a Claim (iii) is an immediate consequence of the fact that Wloc,r continuous map σs : E s (r) → E u (r). Let us see how claims (i), (ii), and (iii) allow to conclude, in the case in which there are no rest points in the strip where f (y) < f (p) < f (x). Let (pn ) ⊂ W u (x) ∩ W s (y). We must prove that (pn ) has a converging subsequence. We can assume that x and y are not limit points of (pn ). Then we can find sn < 0 < tn such that u (x) ∩ { f = f (x) − }, φ(sn , pn ) ∈ Wloc,r

s φ(tn , pn ) ∈ Wloc,r (y) ∩ { f = f (y) + },

for some small > 0. The fact that the are no rest points in the strip { f (y) < f < f (x)} implies that (tn − sn ) is bounded: otherwise by Remark 2.1, we could find a sequence rn ∈ [sn , tn ] such that D f φ(rn , pn ) X φ(rn , pn ) tends to zero, and by (PS) the sequence φ(rn , pn ) would have a subsequence converging to a rest point in the strip { f (y) + ≤ f ≤ f (x) − }, a contradiction. By claim (ii), the set {φ(sn , pn ) | n ∈ N} is essentially vertical. By claim (i) and by the fact that (tn − sn ) is bounded, also the set {φ(tn , pn ) | n ∈ N} is essentially vertical. But the latter set is contained in the local stable manifold of y, so by claim (iii) it is precompact. Since (tn ) is bounded, also the sequence (pn ) is compact. In the general case, one needs the following stronger versions of (ii) and (iii): there exist arbitrarily small neighborhoods U of the rest point x such that if (pn ) converges to x then: (ii ) if tn ≥ 0 and φ(tn , pn )∂U then the set {φ(tn , pn ) | n ∈ N} is essentially vertical; (iii ) if sn ≤ 0 and φ(sn , pn )∂U then the set {φ(sn , pn ) | n ∈ N} has compact intersection with any essentially vertical subset.

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The proof of (ii ) and (iii ) makes use of Proposition 1.17. Then a combination of the argument shown above and the argument in the proof of Theorem 2.2(ii) allows to conclude the proof of Theorem 3.5. REMARK 3.6. The requirement that the essential subbundle E should have a global presentation can be weakened, by replacing the map Q by a suitable family of maps Qi : Mi → Ni , i ∈ I, where {Mi }i∈I is an open covering of M. Besides allowing more general essential subbundles, this fact has also the advantage of localizing even more the constants appearing in assumption (C2)(ii).

3.7. TWO-DIMENSIONAL INTERSECTIONS

Assume that the Morse vector field X is complete, has a nondegenerate Lyapunov function f , and that (X, f ) satisfies (PS). Assume also that X satisfies (C1) – (C3) with respect to a subbundle V of T M. In analogy with the finite indices case, we shall say that X satisfies the Morse – Smale property up to order k ∈ Z if W u (x) meets W s (y) transversally whenever m(x, V) − m(y, V) ≤ k. Let us study what happens when the Morse – Smale condition up to order 2 holds, and x, z are rest points with m(x, V) − m(z, V) = 2. Let W be a connected component of W u (x) ∩ W s (z). It is a two-dimensional manifold, and R acts freely on it. Therefore W/R is a connected one-dimensional manifold, that is it is either a circle or an interval. In the first case, it is easy to see that W = W ∪ {x, z} is a two-dimensional sphere, and the restriction of φ to W is topologically conjugated to the exponential flow on the Riemann sphere S 2 = C ∪ {∞}, R × S 2 (t, ζ) → et ζ ∈ S 2 . We shall be more interested in the second case, in which W is the union of W and two “broken orbits,” with exactly one intermediate rest point. More precisely, the situation is described by the following theorem. THEOREM 3.7. Assume that the Morse vector field X is complete, has a nondegenerate Lyapunov function f , and that (X, f ) satisfies (PS). Assume also that X satisfies (C1) – (C3) with respect to a subbundle V of T M, and has the Morse – Smale property up to order 2. Let x, y be rest points with m(x, V) − m(z, V) = 2, and let W be a connected component of W u (x) ∩ W s (z) such that W/R is an interval. Then restriction of the flow φ to W is topologically conjugated to the product of two shift flows on R: there exists a continuous surjective map h: R × R → W with the following properties:

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(i) φ(t, h(u, v)) = h(u + t, v + t) for every (u, v) ∈ R × R, t ∈ R; (ii) h(R2 ) = W, and there exist rest points y, y with m(y, V) = m(y , V) = m(x, V) − 1, and W1 , W2 , W1 , W2 connected components of W u (x) ∩ W s (y), W u (y) ∩ W s (z), W u (x) ∩ W s (y ), W u (y ) ∩ W s (z), respectively, such that W1 ∪ W2 W1 ∪ W2 , and h(R × {−∞}) = W1 , h({−∞} × R) = W1 ,

h({+∞} × R) = W2 , h(R × {+∞}) = W2 ;

(iii) the restrictions of h to R2 , to {±∞}×R, and to R×{±∞}, are diﬀeomorphisms; (iv) denoting by deg the Z-topological degree, referred to the orientations defined in Section 3.5, there holds deg h = − deg h|{−∞}×R · deg h|R×{+∞} = deg h|R×{−∞} · deg h|{+∞}×R . When y y , h is injective, so it is a conjugacy. When y = y , it may happen that W1 = W1 , or that W2 = W2 , but these identities cannot hold simultaneously. Statement (iv) expresses a form of coherence of the orientations defined in Section 3.5. Let us describe the main idea in the construction of h. By compactness and transversality, we can find a “broken orbit” in the closure of W, with exactly one intermediate rest point y of relative Morse index m(y, V) = m(x, V) − 1. Let W1 and W2 be the corresponding components of W u (x) ∩ W s (y) and W u (y) ∩ W s (z). Let p ∈ W1 , and let q ∈ W2 . Let A be a small hypersurface in W u (x) meeting W s (y) transversally at p, and let B be a small hypersurface in W s (z) meeting W u (y) transversally at q. Consider a neighborhood U of y of the form U = Eyu (r) × Eys (r), where r is so small that the local stable manifold of y is the graph of a θ-Lipschitz map σs : Eys (r) → Eyu (r), while the local unstable manifold of y is the graph of a θ-Lipschitz map σu : Eyu (r) → Eys (r), for some θ < 1. The forward evolution of A eventually intersects U in the graph of a θ-Lipschitz map from Eyu (r) to Eys (r): there is t0 ≥ 0 such that for every t ≥ t0 φ({t} × A) ∩ U = graph αt : Eyu (r) → Eys (r),

lip(αt ) ≤ θ,

and αt − σu ∞ → 0 for t → +∞. Similarly, for every t ≤ −t0 , φ({t} × B) ∩ U = graph βt : Eys (r) → Eyu (r),

lip(βt ) ≤ θ,

and βt − σs ∞ → 0 for t → −∞. Let u ≥ t0 and v ≤ −t0 . Since lip(αu ) ≤ θ < 1 and lip(βv ) ≤ θ < 1, the graphs of αu and of βv intersect in exactly one point, and we can define h(u, v) as h(u, v) := (graph αu ) ∩ (graph βv ).

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This defines h in a neighborhood of (+∞, −∞). See Abbondandolo and Majer (2003b, Section 11) for a complete proof. An analogous argument allows to prove the following result. PROPOSITION 3.8. Let x, y, z be rest points such that m(x, V) = m(y, V) + 1 = m(z, V) + 2, and let W1 , W2 be connected components of W u (x) ∩ W s (y), W u (y) ∩ W s (z), respectively. Then there exists a unique connected component W

of W u (x)∩W s (z) such that W1 ∪ W2 belongs to the closure of φ(R × {p}) p ∈ W with respect to the Hausdorﬀ distance.

3.8. THE MORSE COMPLEX

We now dispose of all the ingredients to build the Morse complex. The assumptions are that the Morse C 1 vector field X on the Hilbert manifold M is complete, satisfies (C1) – (C3) with respect to a subbundle V of T M, with a global presentation Q: M → N, that X satisfies the Morse – Smale condition up to order 2, has a nondegenerate Lyapunov function f ∈ C 2 (M), and that the pair (X, f ) satisfies (PS). For any k ∈ Z, denote by restk (X) the set of rest points x of X of relative Morse index m(x, V) = k, and let Ck (X) be the free Abelian group generated by restk (X). Assume the following finiteness condition: (C4) for every k ∈ Z, f is bounded below on restk (X). For every rest point x, we fix an orientation of the Fredholm pair (Tx W s (x),V(x)) in an arbitrary way. This choice induces an orientation of all the intersections W u (x) ∩ W s (y), for m(x, V) − m(y, V) ≤ 2. Let x, y be rest points of X with m(x, V) − m(y, V) = 1. Then W u (x) ∩ W s (y) is a 1-dimensional manifold with a free action of R, that is it is the union of the orbits of a discrete set of points. By Theorem 3.5 and by transversality, W u (x) ∩ W s (y) is compact: otherwise we could find a sequences of orbits in W u (x) ∩ W s (y) converging to a “broken orbit” from x to y, with at least one intermediate rest point, violating the Morse – Smale condition (up to order 0). Therefore, W u (x) ∩ W s (y) consists of finitely many orbits Wi , i = 1, . . . , h, each of which can be given a sign (Wi ) ∈ {+1, −1} depending on whether the direction of X agrees or does not agree with the orientation of Wi . In other words, if Wi = φ(R × {p}), (Wi ) is the degree of the map φ(·, p): R → Wi . We define the integer n(x, y) as n(x, y) =

h i=1

(Wi ).

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By assumption (C4), we can define a homomorphism ∂k : Ck (X) → Ck−1 (X) generatorwise, as n(x, y)y, ∀x ∈ restk (X). ∂k x = y∈restk−1 (X)

The results of Section 3.7 imply that these homomorphisms are boundary operators. PROPOSITION 3.9. For every k ∈ Z, ∂k−1 ∂k = 0. Proof. Let x and z be rest points with m(x, V) − m(z, V) = 2, and let S(x, z) be the set of “broken orbits” from x to z with exactly one intermediate rest point, necessarily of relative index m(z, V) + 1. By compactness and transversality, S(x, z) is a finite set. By Proposition 3.8, for every element W1 ∪ W2 of S(x, z) there is a unique connected component W of W u (x) ∩ W s (y) such that W1 ∪ W2 belongs to the closure of {φ(R × {p}) | p ∈ W} with respect to the Hausdorﬀ distance. By Theorem 3.7, the closure of W contains exactly one other element W1 ∪ W2 , diﬀerent from W1 ∪ W2 . So there is an involution W1 ∪ W2 → W1 ∪ W2 on S(x, z), without fixed points, and by Theorem 3.7 (iv), (W1 )(W2 ) = −(W1 )(W2 ).

(57)

If m(x, V) = k, the coeﬃcient of z in ∂k−1 ∂k x is the number n(x, y)n(y, z) = (W1 )(W2 ), y∈restk−1 (X)

which is zero by (57).

W1 ∪W2 ∈S(x,z)

Therefore, the Abelian groups Ck (X) and the homomorphisms ∂k , for k ∈ Z, are the data of a chain complex, called the Morse complex of X. The construction depends on the choice of the subbundle V, and on the choice of the orienta tions of T x W s (x), V(x) . Replacing the subbundle V by a compact perturbations produces a shift in the indices, equal to the relative dimension of the compact perturbation. A change of the orientations produces an isomorphic chain complex, the isomorphism being actually an involution. When the conditions (C1) – (C3) hold only with respect to a (0)-essential subbundle, there is no orientation theory available, and the above construction produces a chain complex of Z2 -vector spaces. Replacing the vector field X by another one (still satisfying conditions (C1) – (C3) with respect to the same subbundle V) having the same Lyapunov function f , produces an isomorphic Morse complex: the argument is analogous to the one used in the proof of Theorem 2.26. In particular, the homology of the Morse complex does not depend on the vector field, at it can be denoted by H∗ ( f ), the Morse homology of f .

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Although in this situation we should not expect the Morse homology H∗ ( f ) to be directly related to the singular homology of M, H∗ ( f ) is still considerably stable with respect to modifications of the function f . For instance, if (X0 , f0 ) and (X1 , f1 ) satisfy conditions (PS) and (C1) – (C4) (with respect to the same subbundle V), and if f1 − f0 is bounded, then the corresponding Morse homologies are isomorphic (see Abbondandolo and Majer, 2001, Theorem 1.8, but see also Theorem 1.10). This fact is a consequence of a more general functorial property of the Morse homology: Morse homology is a functor from the class of Morse functions which are Lyapunov functions of some vector field satisfying (PS) and (C1) – (C4), seen as a small category with the usual order relation, to the category of graded Abelian groups. In other words, to each inequality f0 ≥ f1 is associated a sequence of homomorphisms of Abelian groups φ f0 f1 : Hk ( f0 ) → Hk ( f1 ),

∀k ∈ Z,

such that φ f1 f2 φ f0 f1 = φ f0 f2 and φ f f = id. Actually, φθ◦ f f = id, if θ: R → R is a smooth function such that θ > 0 and θ(s) ≥ s. This fact is clearly useful in order to compute the Morse homology of a given function f : if one can squeeze f between two functions, f0 ≥ f ≥ f1 , the knowledge of the Morse homology of f0 and f1 and of the homomorphism φ f0 f1 allows to get information on the Morse homology of f . For instance, if φ f0 f1 is an isomorphism, then φ f0 f is injective and φ f f1 is surjective, hence the Morse homology of f is at least as rich as the Morse homology of f0 and f1 . The construction of the homomorphism φ f0 f1 involves the same idea used in the proof of Theorem 2.26: f0 and f1 can be used to build a new function f˜ on R × M, whose boundary operator ∂ is the cone of some homomorphism ψ f0 f1 from the Morse complex of f0 to the one of f1 . The ∂2 = 0 formula then implies that ψ f0 f1 is a chain map, so it induces a homomorphism φ f0 f1 in homology. Bibliographical note The Morse complex approach for compact manifolds When M is a compact manifold and X is the negative gradient flow of a smooth function, the relations (36) were proved by Morse (1925), see also Morse (1934; 1947). A classical reference for Morse theory is Milnor (1963). See also the review papers by Bott (1982; 1988). The dynamical system point of view arose after the seminal work of Smale, see Smale (1960; 1961) and the beautiful foundational paper Smale (1967), and it immediately had influences in topology, see Milnor’s book on the h-cobordism theorem (Milnor, 1965). In this framework, one can consider Morse – Smale flows, which are dynamical systems more general than gradient-like flows since they

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may have periodic orbits. The connection between Morse theory for Morse – Smale flows and the homotopy of the underlying manifold has been further clarified by Franks (1979), see also Franks (1980), and Cornea (2002a; 2002b). Interpreting the boundary homomorphism of a cellular filtration in terms of an algebraic count of the gradient flow lines connecting critical points of index diﬀerence 1, was already implicit in a paper by Thom (1949), who however did not clarify the conditions required on the gradient flow. This interpretation was pointed out by Witten (1982), where it is deduced quite indirectly from a relationship between Morse theory and certain deformations of the Laplace – Beltrami operator. The first explicit construction of the Morse complex is due to Floer (1989), see also Salamon (1990). Floer’s proof makes use of Conley index theory, a general and powerful method to decompose a dynamical system into simpler invariant sets, see Conley (1978), Conley and Zehnder (1984), and Salamon (1985). Weber (1993) contains a concise construction of the Morse complex, by dynamical systems techniques (see also Weber, 2005). A systematic study of the Morse complex of a function as a tool to build a homology theory which satisfies the Eilenberg – Steenrod axioms can be found in Schwarz (1993). Here the methods are closer to those used in Floer homology. The isomorphism with the singular homology is deduced by the fact that all the homology theories which satisfy the Eilenberg – Steenrod axioms are equivalent on compact CW-complexes. A more direct proof of this isomorphism, still in this spirit, can be obtained by interpreting singular homology theory in terms of pseudocycles, see Schwarz (1999). Banyaga and Hurtubise (2004) presents a self-contained exposition of Morse homology, adopting the dynamical system point of view and providing all the necessary tools from hyperbolic dynamics, as well as applications to Morse theory on Grassmannians and on Lie groups. The dynamical system point of view is at the basis of Harvey and Lawson’s approach to Morse theory in terms of the de Rham – Federer theory of currents (Harvey and Lawson, 2001). The idea is to construct a chain map from the complex of smooth diﬀerential forms to the complex of currents, by taking the limit for t → +∞ of the pullback of a diﬀerential form by the flow φ(t, ·). Such a chain map is chain homotopic to the inclusion, and it is a retraction onto the subcomplex of currents spanned by the stable manifolds of the flow. The cohomology of such a subcomplex is then isomorphic to the de Rham cohomology of the manifold, a result which implies the Morse relations (36). Infinite-dimensional Morse theory Morse theory for C 2 functions on Hilbert manifolds was developed by Palais (1963) and Smale (1964a; 1964b). The Palais – Smale condition was introduced in these papers. This version of Morse theory has been extensively used in the study of geodesics, see Klingenberg (1978; 1982). The first of these references contains also a description of the cellular complex approach to infinite-dimensional Morse

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theory, in the case of self-indexing functions. A complete presentation of infinitedimensional Morse theory including many applications to diﬀerential equations can be found in Mawhin and Willem (1989) and Chang (1993). Morse theory in the case of infinite Morse indices In simple situations, functions with critical points of infinite Morse index and co-index can be studied by taking finite-dimensional approximations. See, for instance, Chang, (1981; 1993), Conley and Zehnder, (1983; 1984), and Abbondandolo (2001). Another way of overcoming the lack of rigidity due to the presence of critical points of infinite Morse index and co-index is to restrict the class of admissible deformations to more rigid classes, as in Benci and Rabinowitz (1979) and Rabinowitz (1986). In the same spirit, a Morse theory for special classes of functions on a Hilbert space has been introduced by Szulkin (1992), and further refined by Abbondandolo (1997; 2000), Kryszewski and Szulkin (1997), Ge ba et al. (1999), and Izydorek (2001). The idea is to develop a generalized cohomology theory, which satisfies all the Eilenberg – Steenrod axioms except the dimension axiom. This axiom is replaced by the requirement that suitable infinite-dimensional spheres should have nontrivial cohomology. These generalized cohomologies will be functorial only with respect to restricted classes of continuous maps (the infinite-dimensional sphere is contractible), and it is possible to develop a Morse theory for functions whose gradient flow belongs to such a class. The idea of forgetting about the whole ambient space and looking only at the gradient flow lines connecting critical points is due to Floer, who applied it to a Cauchy – Riemann type equation which does not even produce a local flow (so the framework is quite diﬀerent from the setting of these notes). See Floer (1988a; 1988b; 1988c; 1989), and the expository paper Salamon (1999). Angenent and van der Vorst (1999) have used this approach to study the gradient flow of a function associated to a class of elliptic systems. A complete study of the Morse complex approach in the case of functions on a Hilbert space consisting of a compact perturbation of a nondegenerate quadratic form has been carried on by the authors in Abbondandolo and Majer (2001). The results of Abbondandolo and Majer (2003b) summarized in the third part of these notes, allow a much more general setting. There is a large literature about the Hilbert Grassmannian, and related constructions. In particular, the space of all compact perturbations of an infinitedimensional and -codimensional closed linear subspace is called restricted Grassmannian by some authors (although sometimes this term is reserved for Hilbert-Schmidt perturbations). See for instance Sato (1981), Segal and Wilson (1985), Pressley and Segal (1986), Guest (1997), and Arbarello (2002). The role of these objects in the homotopy theory underlying Floer homology is discussed in Cohen et al. (1995).

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References Abbondandolo, A. (1997) A new cohomology for the Morse theory of strongly indefinite functionals on Hilbert spaces, Topol. Methods Nonlinear Anal. 9, 325 – 382. Abbondandolo, A. (2000) Morse theory for asymptotically linear Hamiltonian systems, Nonlinear Anal. 39, 997 – 1049. Abbondandolo, A. (2001) Morse Theory for Hamiltonian Systems, Vol. 425 of Chapman Hall/CRC Res. Notes Math., Boca Raton, FL, Chapman & Hall/CRC. Abbondandolo, A. and Majer, P. (2001) Morse homology on Hilbert spaces, Comm. Pure Appl. Math. 54, 689 – 760. Abbondandolo, A. and Majer, P. (2003)a Infinite-dimensional Grassmannians, arXiv:math.AT/ 0307192. Abbondandolo, A. and Majer, P. (2003)b A Morse complex for infinite-dimensional manifolds. I, Adv. in Math., to appear; arXiv:math.DS/0309020. Abbondandolo, A. and Majer, P. (2003)c Ordinary diﬀerential operators on Hilbert spaces and Fredholm pairs, Math. Z. 243, 525 – 562. Abbondandolo, A. and Majer, P. (2004) When the Morse index is infinite, Int. Math. Res. Not. 71, 3839 – 3854. Abraham, R. and Robbin, J. (1967) Transversal Mappings and Flows, New York, W. A. Benjamin, Inc. Angenent, S. and van der Vorst, R. (1999) A superquadratic indefinite elliptic system and its Morse – Conley – Floer homology, Math. Z. 231, 203 – 248. Arbarello, E. (2002) Sketches of KdV, In A. Bertram, J. A. Carlson, and H. Kley (eds.), Symposium in Honor of C. H. Clemens, Vol. 312 of Contemp. Math., Salt Lake City, UT, 2000, p. 9 – 69, Providence, RI, Amer. Math. Soc. Banyaga, A. and Hurtubise, D. (2004) Lectures on Morse Homology, Vol. 29 of Kluwer Texts Math. Sci., Dordrecht, Springer. Benci, V. and Rabinowitz, P. H. (1979) Critical point theorems for indefinite functionals, Invent. Math. 52, 241 – 273. Bott, R. (1959) The stable homotopy of the classical groups, Ann. of Math. (2) 70, 313 – 317. Bott, R. (1982) Lectures on Morse theory, old and new, Bull. Amer. Math. Soc. (N.S.) 7, 331 – 358. ´ Bott, R. (1988) Morse theory indomitable, Inst. Hautes Etudes Sci. Publ. Math. 68, 99 – 114. Chang, K. C. (1981) Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34, 693 – 712. Chang, K. C. (1993) Infinite-dimensional Morse theory and multiple solution problems, Vol. 6 of Progr. Nonlinear Diﬀerential Equations Appl., Boston, MA, Birkh¨auser. Cohen, R. L., Jones, J. D. S., and Segal, G. B. (1995) Floer’s infinite-dimensional Morse theory and homotopy theory, In H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder (eds.), The Floer Memorial Volume, Vol. 133 of Progr. Math, p. 297 – 325, Basel, Birkh¨auser. Conley, C. (1978) Isolated invariant sets and the Morse index, Vol. 38 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc. Conley, C. and Zehnder, E. (1983) The Birkhoﬀ – Lewis fixed point theorem and a conjecture of V. I. Arnol d, Invent. Math. 73, 33 – 49. Conley, C. and Zehnder, E. (1984) Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37, 207 – 253. Cornea, O. (2002)a Homotopical dynamics. II. Hopf invariants, smoothing, and the Morse complex, ´ Ann. Sci. Ecole Norm. Sup. (4) 35, 549 – 573. Cornea, O. (2002)b Homotopical dynamics. IV. Hopf invariants, and Hamiltonian flows, Comm. Pure Appl. Math. 55, 1033 – 1088.

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Nemirovski˘ı, A. S. and Semenov, S. M. (1973) The polynomial approximation of functions in Hilbert spaces, Mat. Sb. (N.S.) 21 (92), 255 – 277, Russian. Palais, R. S. (1963) Morse theory on Hilbert manifolds, Topology 2, 299 – 340. Palais, R. S. (1965) On the homotopy type of certain groups of operators, Topology 3, 271 – 279. Palis, J. (1968) On Morse – Smale dynamical systems, Topology 8, 385 – 405. Palis, Jr., J. and de Melo, W. (1982) Geometric Theory of Dynamical Systems, New York, Springer. Po´zniak, M. (1991) The Morse complex, Novikov cohomology and Fredholm theory, preprint, University of Warwick. Pressley, A. and Segal, G. (1986) Loop Groups, Oxford Math. Monogr., Oxford, Oxford Univ. Press. Quillen, D. (1985) Determinants of Cauchy – Riemann operators over a Riemann surface, Funct. Anal. Appl. 19, 31 – 34. Rabinowitz, P. H. (1986) Minimax Methods in Critical Point Theory with Applications to Differential Equations, Vol. 65 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc. Salamon, D. (1985) Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291, 1 – 41. Salamon, D. (1990) Morse theory, the Conley index and Floer homology, Bull. Amer. Math. Soc. 22, 113 – 140. Salamon, D. (1999) Lectures on Floer homology, In Y. Eliashberg and L. Traynor (eds.), Symplectic Geometry and Topology, Vol. 7 of IAS/Park City Math. Ser., Park City, UT, 1997, p. 143 – 229, Providence, RI, Amer. Math. Soc. Sato, M. (1981) Soliton equations as dynamical systems on an infinite-dimensional Grassman manifold, S¯urikaisekikenky¯usho K¯oky¯uroku 439, 30 – 46. Schwarz, M. (1993) Morse Homology, Vol. 111 of Progr. Math, Basel, Birkh¨auser. Schwarz, M. (1999) Equivalence for Morse homology, In M. Barge and K. Kuperberg (eds.), Geometry and Topology in Dynamics, Vol. 246 of Contemp. Math., Winston-Salem, NC, 1998/San Antonio, TX, 1999, p. 197 – 216, Providence, RI, Amer. Math. Soc. ´ Segal, G. and Wilson, G. (1985) Loop groups and equations of KdV type, Inst. Hautes Etudes Sci. Publ. Math. 61, 5 – 65. Shub, M. (1987) Global Stability of Dynamical Systems, New York, Springer. Smale, S. (1960) Morse inequalities for a dynamical system, Bull. Amer. Math. Soc. 66, 43 – 49. Smale, S. (1961) On gradient dynamical systems, Ann. of Math. (2) 74, 199 – 206. Smale, S. (1964)a A generalized Morse theory, Bull. Amer. Math. Soc. 70, 165 – 172. Smale, S. (1964)b Morse theory and a nonlinear generalization of the Dirichlet problem, Ann. of Math. (2) 80, 382 – 396. Smale, S. (1965) An infinite-dimensional version of Sard’s theorem, Amer. J. Math. 87, 861 – 866. Smale, S. (1967) Diﬀerentiable dynamical systems, Bull. Amer. Math. Soc. 73, 747 – 817. Szulkin, A. (1992) Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209, 375 – 418. Thom, R. (1949) Sur une partition en cellules associ´ee a` une fonction sur une vari´et´e, C. R. Acad. Sci. Paris 228, 973 – 975. Weber, J. (1993) Der Morse – Witten Komplex, Master’s thesis, TU Berlin. Weber, J. (2005) The Morse – Witten complex via dynamical systems, Exposition. Math., to appear. Witten, E. (1982) Supersymmetry and Morse theory, J. Diﬀerential Geom. 17, 661 – 692.

NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY ALBERTO ABBONDANDOLO Universit`a di Pisa MATTHIAS SCHWARZ∗ Universit¨at Leipzig

Abstract. Given the cotangent bundle T ∗ Q of a smooth manifold with its canonical symplectic structure, and a Hamiltonian function on T ∗ Q which is fiberwise asymptotically quadratic, its well-defined Floer homology with the pair-of-pants ring structure is ring-isomorphic to the singular homology of the free loop space of Q endowed with its loop product. The analogous statement is true for the based loop space versions and the Pontrjagin product. This article gives an overview of the construction of this ring isomorphism which is based on Legendre duality and moduli spaces of flow trajectories of hybrid type, which are half Floer trajectories for the Hamiltonian problem and half Morse trajectories for the Lagrangian problem.

1. Introduction Initially, Floer homology had been invented in order to provide a proof of the nondegenerate Arnold conjecture for symplectic fixed points (Floer, 1988a, 1988b, 1989). Given a closed symplectic manifold (M 2n , ω) and a nondegenerate, timeperiodic Hamiltonian H: R/Z × M → R, the Arnold conjecture claims that there should be at least a minimal number of 1-periodic solutions x ∈ P1 (H) of the Hamiltonian diﬀerential equation x˙(t) = XH t, x(t) , that is, equivalently, fixed points of the time-1-map of the flow generated by the Hamiltonian vector field XH of H. This minimal number should be not smaller than the sum of the Betti numbers of M, that is rk H∗ (M). Floer’s key idea was to build formally a chain complex generated by the 1-periodic solutions P1 (H), a finite subset of the free loop space of M, and to use a Cauchy – Riemann type PDE for cylindrical maps u: R × R/Z → M connecting x, y ∈ P1 (H), in order to select and count particular trajectories in the free loop space in order to define a chain boundary operator. This Cauchy – Riemann type PDE makes use of Gromov’s theory of pseudoholomorphic curves (Gromov, 1985). The homology of the Floer chain complex turns out to be an invariant of the manifold M, independent of the Hamiltonian function H. In fact, this Floer homology HF∗ was proven to be isomorphic as a module over ∗

Work partially supported by DFG grant no. SCHW892/2-1.

75 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 75–108. © 2006 Springer. Printed in the Netherlands.

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some suitably chosen coeﬃcient ring to the standard homology of M, or in view of Poincar´e duality, to the cohomology of M (e.g., Hofer and Salamon, 1995). Subsequently, further algebraic structures were constructed on this Floer homology, in particular a natural ring structure, the so-called pair-of-pants ring structure (Schwarz, 1995). In view of the above isomorphism to H ∗ (M), it was shown in Piunikhin et al. (1996) that, in general, the pair-of-pants ring structure is a deformation of the cohomology ring of M known as its quantum cohomology. However, all these results did not expose any nontrivial topological structure concerning the free loop space of M. This turns out to be essentially due to the compactness assumption on the symplectic manifold M. As soon as this is dropped and as also the Hamiltonian function H deviates essentially from being compactly supported, its associated Floer homology, if well-defined, tends to be a truly infinite-dimensional homology theory. In this paper, we focus on the case of cotangent bundles (M, ω) = (T ∗ Q, ωLiouv ) with their canonical symplectic structure and their canonical Lagrangian fibration by the cotangent spaces. A result first proven by Viterbo (1996) shows that Floer homology for the fiberwise quadratic Hamiltonian on T ∗ Q given by a Riemannian metric g on Q is isomorphic as a graded Abelian group to the singular homology of the free loop space of the manifold Q. Another proof of this result was given by Salamon and Weber (2003). Whereas Viterbo uses a method based on generating functions, Salamon and Weber use an approach relating the Cauchy – Riemann type PDE of Floer theory to the parabolic heat equation for the L2 -gradient of the energy functional on the free loop space of Q. The present article gives an outline of again another approach (Abbondandolo and Schwarz, 2004, 2005). Our aim was to find a more immediate construction on the level of the Floer chain complex which could also be used to analyze the pairof-pants ring structure of Floer homology, which clearly must correspond to some ring structure on the homology of the free loop space of the manifold Q. It is worth mentioning that here, in the context of noncompact symplectic manifolds T ∗ Q, the symmetry between a cohomology and a homology theory is finally broken. In fact, it turns out that the pair-of-pants ring structure of Floer homology corresponds to the loop product known from the work by Chas and Sullivan (1999).1 In our approach, the concrete tool to represent the homology of the free loop space of Q is Morse homology (Schwarz, 1993; Abbondandolo and Majer, 2005). The philosophy of our construction of the chain complex isomorphism between the Floer chain complex for H and the Morse chain complex of the free loop space of Q is to build so-called hybrid moduli spaces of connecting trajectories which are half Morse trajectories for the negative gradient flow of a Lagrangian action functional on the free loop space of Q, parametrized by “flow-time” from minus infinity 1 Here, we thank Octav Cornea for pointing out to us the presence of this preprint in November 2002 when already our first version of the ring isomorphism had been established.

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to zero, and half Floer trajectories in the free loop space of T ∗ Q parametrized from time zero to infinity. Coupling these two diﬀerent types of half-trajectories is established via a Lagrangian boundary condition which uses crucially the fact that the Hamiltonian function on T ∗ Q on the Floer chain complex side is Legendre dual to the Lagrangian function on the tangent bundle T Q providing the Morse theory for the free loop space of Q. Counting such hybrid trajectories defines the chain complex homomorphism from the Morse chain complex into the Floer chain complex. It is intriguing to see that Legendre duality only allows such a construction in one direction, namely from the Lagrangian picture towards the Hamiltonian picture if using the negative gradient flow which corresponds to homology theories. Nevertheless, one can still show that this chain complex homomorphism has to be invertible. The result that Floer homology is isomorphic to the free loop space homology as a graded Abelian group already on the chain complex level has been proven for a large, fairly general class of Hamiltonian functions with fiberwise asymptotic quadratic upper and lower bound in Abbondandolo and Schwarz (2004). The proof that this isomorphism of homologies identifies the pair-of-pants product with the loop product of Chas-Sullivan is given in Abbondandolo and Schwarz (2005). Moreover, also the analogous versions for the based loop space are considered in both papers. Related results have been also achieved by Cohen (2005) and Ramirez (2005). In this article, we present a condensed outline of our results from Abbondandolo and Schwarz (2004; 2005). The main result is given in Theorem 2.7 below. The present article is based on a series of talks given by the second author during the workshop SMS 2004 in Montr´eal. Both authors wish to thank the organizers Paul Biran, Octav Cornea and Franc¸ois Lalonde for the very stimulating atmosphere of this workshop. We are also indebted to Ralph Cohen, Octav Cornea, Yong-Geun Oh, Dietmar Salamon, Dennis Sullivan, Claude Viterbo and Joa Weber for many fruitful discussions. 2. Main result 2.1. LOOP SPACE HOMOLOGY

Let us briefly recollect the classical and well-known facts about loop space homology which will be extended by the loop product ring structure in the setting of manifolds. Let (Q, q0 ) be a pointed topological space and denote its free and based loop space respectively by ΛQ = C 0 (T, Q), T = R/Z, ΩQ = {c ∈ ΛQ | c(0) = q0 }.

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We have the classical Serre fibration i

ev

ΩQ → ΛQ −→ Q,

ev(c) = c(0).

Without further remarks we will denote in the following H∗ for singular homology with integer coeﬃcients. We have the continuous concatenation injection #: ΩQ × ΩQ → ΩQ, (c, c ) → c # c , c(2t), 0 ≤ t ≤ 12 , c # c (t) = c (2t − 1), 12 ≤ t ≤ 1,

(1)

which induces the homomorphism #∗ : H∗ (ΩQ) ⊗ H∗ (ΩQ) → H∗ (ΩQ)

(2)

of graded groups and for which we have the classical result in the case of coeﬃcients in a field k (see for instance F´elix et al., 2001), THEOREM 2.1. (H∗ (ΩQ; k), #∗ ) is a graded Hopf algebra. From now on, let us assume that Q is a smooth, oriented, n-dimensional manifold. Moreover, for simplicity we will restrict our perspective throughout this article to compact manifolds without boundary. For analytical reasons we restrict ΛQ and ΩQ to dense subspaces which carry the structure of smooth Hilbert manifolds. In fact, given any smooth embedding Q ⊂ RN , the Hilbert space W 1,2 (T, RN ) of square integrable functions with weak derivatives in L2 defines a smooth structure of a Hilbert manifold on W 1,2 (T, Q) = {c ∈ W 1,2 (T, RN ) | c(T) ⊂ Q}. Using an auxiliary Riemannian structure g on Q, we obtain a well-defined Riemannian structure on W 1,2 (T, Q) as a Hilbert manifold by gc(t) v(t), w(t) + gc(t) (∇c˙ v, ∇c˙ w) dt. v, w = T

Clearly, ev−1 (q0 ) = {c ∈ W 1,2 (T, Q) | c(0) = q0 } is a smooth submanifold of codimension n and it is well-known that the dense inclusions W 1,2 (T, Q) → C 0 (T, Q) = ΛQ, ev−1 (q0 ) → ΩQ, are homotopy equivalences. We now use the notations ΛQ and ΩQ for the smooth Hilbert manifolds. Summing up, we observe

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LEMMA 2.2. ΩQ → ΛQ is a smooth embedding of codimension n with oriented ev normal bundle ν → ΩQ, and ΩQ → ΛQ −→ Q is a locally trivial fiber bundle. Proof. The first assertion follows from the fact that the map ev is a submersion onto an n-dimensional oriented manifold. Find open coverings Q = Uα = Vα , Uα ⊂⊂ Vα , with marked points xα ∈ Uα and smooth maps hα : Uα → Diﬀ Q in the sense that hα : Uα × Q → Q is smooth and hα (x, ·) is a diﬀeomorphism for all x ∈ Uα , with hα (x, xα ) = x and hα (x, Uα ) ⊂ Vα ∀ x ∈ Uα , hα (x, y) = y ∀x ∈ Q \ Vα . Then consider ΛQ|Uα = ev−1 (Uα ) = {c ∈ ΛQ | c(0) ∈ Uα } and define φα : ΛQ|Uα → ev−1 (xα ) × Uα , φα (c) = hα c(0) −1 ◦ c, c(0) . We have the fiber diﬀeomorphism −1 −1 φ−1 α (·, x): ev (xα ) → ev (x)

for all x ∈ Uα , and ev: ΛQ → Q is a locally trivial fiber bundle.

Note that the normal bundle ν of ΩQ = ev−1 (q0 ) can be identified with |ΩQ . Since ν is an oriented rank-n vector bundle, we have a well-defined Thom class u ∈ H n (ν(ΩQ), ν(ΩQ) \ OΩQ ) ev∗ T Q

such that the Thom isomorphism

∩ u: H∗ (ν, ν \ O) − → H∗−n (ΩQ) yields the definition of ∩: H∗ (ΛQ) → H∗−n (ΩQ)

(3)

by composing j∗

H∗ (ΛQ) −→ H∗ (ΛQ, ΛQ \ ΩQ) − → H∗ (ν, ν \ O) − → H∗−n (ΩQ), where j∗ is taken from the long exact sequence and the middle isomorphism is excision. Note that this definition of ∩: H∗ (ΛQ) → H∗−n (ΩQ) uses the manifold property of Q and its orientability in an essential way. Let us now consider in an analogous manner the codimension-n-submanifold X = {(c, d) ∈ ΛQ × ΛQ | c(0) = d(0)},

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consisting of figure-eight-loops. Similarly, the normal bundle is oriented and we can define ∩: H∗ (ΛQ × ΛQ) → H∗−n (X). (4) This allows now the following DEFINITION 2.3. The loop product o: H∗ (ΛQ) ⊗ H∗ (ΛQ) → H∗−n (ΛQ)

is given by the composition ×

∩

#∗

→ H∗ (ΛQ × ΛQ) − → H∗−n (X) −→ H∗−n (ΛQ). H∗ (ΛQ) ⊗ H∗ (ΛQ) − In view of the intersection product •: H∗ (Q) ⊗ H∗ (Q) → H∗−n (Q) which is well-defined for the oriented manifold Q, we have PROPOSITION 2.4. The following are ring homomorphisms, ∩: (H∗ (ΛQ), o) → (H∗−n (ΩQ), #∗ ), ev∗ : (H∗ (ΛQ), o) → (H∗ (Q), •). Our aim is to find precisely this triple of homological ring structures o, #∗ and • with ring homomorphisms isomorphically within the framework of Floer homology. We mention here again, that the loop product was already constructed in Chas and Sullivan (1999), considered in a similar context in Cohen and Jones (2002) and Cohen and Voronov (2005), and that similar results relating the ring structures to Floer theory are also contained in Cohen (2005) and Ramirez (2005). 2.2. FLOER HOMOLOGY FOR THE COTANGENT BUNDLE

Here we recall briefly the general setup of Floer homology for Hamiltonian systems before we apply it to the situation of cotangent bundles. Let (P2n , ω) be a symplectic manifold, H: T × P → R a smooth Hamiltonian, explicitly time-dependent and 1-periodic in time t. Let XH be the associated nonautonomous vector field defined by ω(XH , ·) = −dH and denote the set of 1-periodic solutions by PΛ (H) = {x ∈ ΛP | x˙(t) = XH t, x(t) ∀ t ∈ T}. Assume that H is nondegenerate, i.e., we have transverse intersection graph φ1H P ⊂ P × P for the time-1-map of the nonautonomous flow φtH of XH , see also

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(H0 )Λ below in Section 4. Let us also assume that we have an integral grading µΛ : PΛ (H) → Z by the Conley – Zehnder index. Given a t-dependent almost complex structure J: T × P → End(T P) on P, compatible with ω (i.e., ω(·, J(t)·) Riemannian metric ∀ t ∈ T) we define for x, y ∈ PΛ (H) the space of parametrized Floer trajectories

M x,y (J, H) = u: R × T → P lim u(s, ·) = x, lim u(s, ·) = y, s→−∞ s→+∞ u s − J(t, u) ut − XH (t, u) = 0 . Througout this article we make frequent use in an ad hoc manner of the symbol M denoting varying kinds of solution spaces. Note the choice of the sign in the (anti-)CR-operator which implies that u ∈ M x,y can be viewed as a negative flow line for the unregularized L2 -gradient of the Hamiltonian action functional. The first essential compactness result of Floer theory states that for generic #x,y = M x,y (J, H)$R of unparametrized trajectories divided by the J, the space M R-shift-action, for x y, is a manifold of dimension µΛ (x) − µΛ (y) − 1, and that it is compact in dimension 0 and compact up to simple splitting of trajectories in dimension 1, provided that (a) P is a closed symplectic manifold and the cohomology classes {ω}, c1 (T M, J) ∈ H 2 (M) satisfy suitable conditions with respect to π2 (M), or (b) P is noncompact, {ω}, c1 satisfy suitable conditions and J, H have a suitable asymptotic behaviour on the end of P. Below, the latter case will be made precise for cotangent bundles P = T ∗ Q. Next, we set for x, y ∈ PΛ (H) # , µΛ (x) − µΛ (y) = 1, # M x, y = alg x,y 0, else, where #alg denotes counting of trajectories with signs according to their coherent orientation (Floer and Hofer, 1993). We define the finitely generated, graded, free Abelian group CFΛ ∗ (H) = Λ P (H) ⊗ Z and the boundary operator Λ ∂: CFΛ ∗ (H) → CF∗−1 (H), ax x = a x x, yy. ∂ x

µΛ (y)=µΛ (x)−1

Note that this is the simplest possible version of the Floer chain complex without the use of Novikov rings (see Hofer and Salamon, 1995), which will be suﬃcient for our framework of cotangent bundles. The main theorem of Floer theory is

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THEOREM 2.5 (Floer et al.). (a) CF∗ (H), ∂ = ∂(J, H) is a chain complex, i.e., ∂2 = 0 and thus HF∗ (J, H) = ker ∂/ im ∂ is well-defined. (b) If P is a closed symplectic manifold (with suitable conditions on ω|π2 and c1|π2 ), then HF∗ (H, J) H n−∗ (P) are isomorphic in a natural way. Note the shift in the grading for the isomorphism of Floer homology with singular homology of P. This is due to the chosen normalizations for the Conley – Zehnder index. The choices are such that the grading µΛ on PΛ (H) and the Hamiltonian proportional on iteration action functional (see below) are positively (m) (m) Λ (−1) Λ = −µ (x) under time-reversal , x (t) = x(mt) and µ x towers x m∈Z which in view of (b) corresponds to Poincar´e duality H n−∗ (P) Hn+∗ (P). Let us now consider this concept of Floer homology for cotangent bundles of smooth n-dimensional manifolds Q which we assume (mainly for reasons of simplicity) to be closed and oriented. P = T ∗ Q carries the canonical symplectic structure ω = dθLiouv where the Liouville-1-form is canonically given by θLiouv =

n

pi dqi

i=1

in local coordinates (q1 , . . . , qn ) ∈ Q. Let again H: T × T ∗ Q → R be a nondegenerate Hamiltonian and consider x, y ∈ PΛ (H) and MΛx,y = M x,y (J, H) as above. Note that, for the exact symplectic form ω = dθLiouv we have the Hamiltonian action functional for paths or closed curves x in T ∗ Q defined by 1 ∗ AH (x) = x θLiouv − H t, x(t) dt . 0

There is also a version of Floer homology for the case of the based loop space ΩQ. Consider q0 ∈ Q and the Lagrangian submanifold T q∗0 Q ⊂ (T ∗ Q, ω). Then, let

x, y ∈ PΩ (H) = x: [0, 1] → T ∗ Q x˙(t) = XH t, x(t) , x(0), x(1) ∈ T q∗0 Q ,

∗ MΩ x,y (J, H) = u: R × [0, 1] → T Q u s = J(t, u) ut − XH (t, u) , u(s, 0), u(s, 1) ∈ T q∗0 Q ∀ s ∈ R, lim u(s, ·) = x, lim u(s, ·) = y . s→−∞

s→+∞

Here, we have to assume H to be nondegenerate with respect to q0 , that is for x ∈ PΩ (H) the linearized time-1-map Dφ1H x(0) maps T q∗0 Q = T x(0) (T q∗0 Q) transverse

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83

to T q∗0 Q = T x(1) (T q∗0 Q). We also have a Z-grading by the Maslov index µΩ for paths of Lagrangian subspaces, see (H0 )Ω below in Section 4. We now try to define Λ resp. Ω

∂: CF∗

Λ resp. Ω

(J, H) → CF∗−1

(J, H)

for nondegenerate H and generic J. The crucial new problem is to guarantee the #x,y of unparametrized trajectonecessary compactness properties of the spaces M ries. Since ω = dθ is exact, there is no obstruction by bubbling-oﬀ, but we need suitable asymptotic conditions on H and J in order to grant a priori C 0 -estimates for trajectories u ∈ M x,y (J, H). Pick any Riemannian metric g on Q. The associated Levi-Civita connection provides a canonical ω-compatible almost complex structure JLC by mapping vertical tangent spaces onto horizontal ones and reversely. The main assumption on the admissible class of Hamiltonian functions H: T × T ∗ Q → R is the following: For any coordinate chart U, (q1 , . . . , qn ) on Q (H1 ) there exist constants h0 > 0 and h1 ≥ 0 such that % ∂& dH(t, q, p) p − H(t, q, p) ≥ h0 |p|2 − h1 , ∂p (H2 ) there exists a constant h2 ≥ 0 such that |∂q H(t, q, p)| ≤ h2 (1 + |p|2 ),

|∂ p H(t, q, p)| ≤ h2 (1 + |p|),

for all t ∈ T, p ∈ T q∗ Q, q ∈ U. Here, p∂/∂p is the canonical Liouville vector field, ω(p∂/∂p, ·) = θLiouv . (H1 ) essentially states that H grows at least fiberwise quadratically like 12 h0 |p|2 − const and (H2 ) implies that H grows at most fiberwise quadratically like 12 h2 |p|2 + const, and it is radially convex in p for |p| large. Note that apart from the size of the concrete constants h0 , h1 , h2 , the form of the estimates (H1 ) and (H2 ) is invariant under coordinate changes on Q and independent of the choice of g. A typical example is given by physical Hamiltonians such as H(t, q, p) = 12 |p − A(t, q)|2 + V(t, q). The asymptotic condition on the ω-compatible almost complex structure J is that for a suitable constant j0 depending on g (J) J − JLC ∞ < j0 . The first result now states that we have well-defined Floer homology for the given class of Hamiltonian functions on the cotangent bundle.

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THEOREM 2.6 (Abbondandolo and Schwarz, 2004). Given H and a generic J on P = T ∗ Q satisfying (H1 ), (H2 ), (J) above, the Floer chain complexes Λ CF∗ (H), ∂(J, H) and CFΩ ∗ (H), ∂(J, H) are well-defined and for any generic J, Ω J satisfying (J) there exist quasi-isomorphisms providing HFΛ ∗ (H) and HF∗ (H) independent from J. Moreover, given two Hamiltonians H, H satisfying (H1 ), Ω (H2 ) there exist natural isomorphisms HFΛ → HFΛ → ∗ (H) − ∗ (H ) and HF∗ (H) − HFΩ ∗ (H ) satisfying the cocycle condition. The latter isomorphisms are called the continuation isomorphisms. The central result of the present article formulates the full context of the ring isomorphisms of the above Floer homology with loop space homology. THEOREM 2.7 (Abbondandolo and Schwarz, 2005). Given a Hamiltonian H on the cotangent bundle T ∗ Q satisfying above conditions (H1 ) and (H2 ), we have the following additional structures and relations for its Floer homology. Ω (a) HFΛ ∗ and HF∗ carry natural bilinear operations (r) (s) (r+s) ⊗ HFΩ → HFΩ , mΩ : HFΩ k H l H k+l H (r) (s) (r+s) ⊗ HFΛ → HFΛ , mΛ : HFΛ k H l H k+l−n H

for all k, l ∈ Z, r, s ∈ N with H (r) (t, ·) = rH(rt, ·). (b) There exist natural homomorphisms compatible with mΩ and mΛ , Λ Ω → HFΩ h: HFΛ ∗ (H), m ∗−n (H), m , Λ e: HFΛ → (H∗ (Q), •). ∗ (H), m (c) There exist natural Z-module isomorphisms compatible with the ring structures mΩ and o, respectively mΛ and #∗ , Λ − → (H∗ (ΛQ), o), ΦΛ : HFΛ ∗ (H), m Ω Ω Ω Φ : HF∗ (H), m − → (H∗ (ΩQ), #∗ ) such that the diagram (H∗ (Q), •) o

e

Λ Λ HF∗ , m

h

ΦΛ

(H∗ (Q), •) o commutes.

ev∗

(H∗ (ΛQ), o)

/ HFΩ , mΩ ∗−n ΦΩ

∩

/ (H∗−n (ΩQ), #∗ )

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85

The isomorphisms ΦΛ and ΦΩ as Z-module-isomorphisms are already given in Abbondandolo and Schwarz (2004). All statements about the ring structures are proven in Abbondandolo and Schwarz (2005). The following remarks should be made: (i) By naturality in (a) – (c) we mean two aspects: One is the compatibility with the canonical continuation isomorphism HF∗Λ/Ω (H) − → HF∗Λ/Ω (H ) associated to changes of the Hamiltonians H, H both satisfying (H1 ) and (H2 ). The other important aspect is that these homological structures are already defined on chain level. (ii) The homomorphisms e and ev∗ are endowed with natural right inverses Λ i: (H∗ (Q), • → HFΛ ∗ (H), m , i∗ : H∗ (Q), • → (H∗ (ΛQ), o), i.e., e ◦ i = idH∗ (Q) = ev∗ ◦i∗ , where i∗ is induced by the inclusion of constant loops, Q → ΛQ, c(t) = q, t ∈ T. The latter right-inverse property is seen in an analogous manner as Proposition 2.4 by standard algebraic topology. (iii) Not explicitly mentioned above is a natural R-filtration on HFΛ , HFΩ by the values of the Hamiltonian action functional. The “ring structures” mΛ , mΩ can be shown to be fully compatible with that filtration. (See (9) below.) (iv) It is an intriguing question, open to us, whether the Hopf algebra structure Ω Ω of (HFΩ ∗ , m ) implied by the isomorphism Φ can be seen immediately on the side of Floer homology. 3. Ring structures and ring-homomorphisms In this section we sketch the definitions of mΛ , mΩ , h, e, i and ΦΛ resp. Ω . Full details will be contained in Abbondandolo and Schwarz (2005). 3.1. THE PAIR-OF-PANTS PRODUCT

3.1.1. Based loop space From now on, the symbol # will denote concatenation in time without rescaling. We first construct mΩ for the based loop space case. Let Σ = R × [0, 1] ∪˙ R × [1, 2]/(s, 1−) ∼ (s, 1+) for s ≥ 0 i.e., the interior Σ˚ = R×(0, 2)\(−∞, 0)×{1} carries the standard complex structure as an open subset of C, and Σ is a noncompact Riemann surface with boundary √ where local conformal boundary coordinates at (0, 1) ∈ Σ are given by z → z ∈

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A. ABBONDANDOLO AND M. SCHWARZ

{z ≥ 0}. For s ≤ 0 we distinguish the diﬀerent boundary points of Σ by the notation (s, 1+) and (s, 1−). Let H: T × T ∗ Q → R and q0 ∈ Q be such that both

∗ ∗ PΩ 1 (H) = {x: [0, 1] → T Q | x(0), x(1) ∈ T q0 Q, x˙ = XH (t, x) , ∗ ∗ PΩ 2 (H) = {x: [0, 2] → T Q | x(0), x(2) ∈ T q0 Q, x˙ = XH (t, x)}

consist only of nondegenerate solutions. Observe that we have the canonical identification P2 (H) = P1 (H (2) ), H (2) (t, x) = 2H(2t, x) by rescaling. Moreover, we add the further, generic nondegeneracy assumption Ω Ω that x(1) T q∗0 Q for all x ∈ PΩ 2 (H). Then, given x, y ∈ P1 (H) and z ∈ P2 (H), we define the solution set Ω ∗ ∞ ˚ 0 M x,y;z = u: Σ → T Q ∈ C (Σ) ∩ C (Σ) ˚ u s − J(t, u) ut − XH (t, u) = 0, ∀ (s, t) ∈ Σ, x(t), 0 ≤ t ≤ 1−, lim u(s, t) = y(t − 1), 1+ ≤ t ≤ 2, s→−∞ lim u(s, t) = z(t), 0 ≤ t ≤ 2, s→∞ u(s, t) ∈ T q∗ Q, ∀ [s, t] ∈ ∂Σ = Σ \ Σ˚ . 0

Note that J in general has to be allowed to be dependent on t ∈ [0, 2] and J(t + 1, ·) J(t, ·), 0 ≤ t ≤ 1. PROPOSITION 3.1. For a generic choice of J(t, ·) t∈[0,2] and if x # y z, the Ω Ω Ω solution set MΩ x,y;z is a manifold of dimension µ (x) + µ (y) − µ (z). An important feature is the sharp energy estimate ∞ 2 |u s |2J ds dt = AH (x) + AH (y) − AH (z) ≥ 0. −∞

(5)

0

To allow for more general situations, we consider nondegenerate H: [0, 1] × T ∗ Q → R and K: [1, 2] × T ∗ Q → R with H(1, ·) = K(1, ·) and we let H # K: [0, 2] ×

Figure 1.

Σ, strip with a slit

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87

T ∗ Q be the concatenation which generically is again nondegenerate. Then, more generally, we will construct the operation Ω Ω mΩ : HFΩ ∗ (H) ⊗ HF∗ (K) → HF∗ (H # K),

(6)

where H # K is again viewed as rescaled over [0, 1] like above. By compatibility with the continuation isomorphisms we obtain also the operation mΩ for K = H and H # K = H (2) . PROPOSITION 3.2. Under the assumptions (H1 ) and (H2 ) for H, K and generic J satisfying (J) we have: (a) MΩ x,y;z is compact in dimension 0, (b) in dimension 1, MΩ x,y;z is compact up to simple splitting of Floer trajectories of relative index 1 at at most one end of Σ. As a consequence, we can define the operation mΩ on chain level Ω Ω mΩ : CFΩ k (H) ⊗ CFl (K) → CFk+l (H # K), #alg MΩ mΩ (x, y) = x,y;z (H # K, J) z,

(7)

µ(z)=µ(x)+µ(y)

and it satisfies ∂H#K ◦ mΩ (x, y) = mΩ (∂H x, y) ± mΩ (x, ∂K y).

(8)

Due to the energy estimate (5), mΩ is compatible with the R-filtration by the Hamiltonian action, a x x ∈ CF∗ (H) a x = 0 for AH (x) > a , (9) CFa∗ (H) = x∈P(H)

Ω

m : CF (H) ⊗ CFb (K) → CFa+b (H # K). a

3.1.2. Free loop space The construction of mΛ for HFΛ is carried out in analogous manner. Let Σ = R × [0, 1] ∪˙ R × [1, 2]/∼ with (s, 0) ∼ (s, 1−) and (s, 1+) ∼ (s, 2) (s, 1−) ∼ (s, 1+) and (s, 0) ∼ (s, 2)

for s ≤ 0, for s ≥ 0.

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Figure 2. The pair of pants

This endows Σ with the structure of a smooth, noncompact Riemann surface with an obvious holomorphic 2-sheeted branched cover p: Σ → R × T, [s, t] → (s, t mod 1), with a single branch point on (0, 0) ∼ (0, 1) ∼ (0, 2). As before, √ holomorphic coordinates near [0, 1] are provided by the z-map. Choose generic nondegenerate perturbations of H (r) and H (s) , again denoted by H, K, such that H(1) = K(0) and such that x # y PΛ (H # K) for any x ∈ P(H), y ∈ P(K) with x(1) = y(0). ∞ ∗ Again, after rescaling we identify PΛ 2 (H # K) ⊂ C (R/2Z, T Q) with a set of Λ Λ Λ 1-periodic loops. Given x ∈ P1 (H), y ∈ P1 (K) and z ∈ P2 (H # K) we set Λ ∗ M x,y;z = u: Σ → T Q smooth u s − J(t, u) ut − XH (t, u) = 0, ∀ (s, t) ∈ Σ, x(t), 0 ≤ t ≤ 1−, lim u(s, t) = y(t), 1+ ≤ t ≤ 2, s→−∞ lim u(s, t) = z(t), 0 ≤ t ≤ 2 . s→∞

Analogously to Proposition 3.1, for generic J, MΛx,y;z is a manifold of dimension dim MΛx,y;z = µΛ (x) + µΛ (y) − µΛ (z) − n. (10) We have also the analogous compactness result as in Proposition 3.2, and thus we can define Λ Λ mΛ : CFΛ k (H) ⊗ CFl (K) → CFk+l−n (H # K), with

Λ Λ Λ Λ ∂Λ H#K ◦ m = m ◦ ∂H ⊗ idK ± idH ⊗ ∂K .

(11)

Here, we point out that the above branched-cover definition of the pair-of-pants domain Σ is essential for the sharp energy estimate |∂ s u|2J ds dt ≥ 0, AH (x) + AK (y) − AH#K (z) = Σ

NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY

Figure 3.

89

The cylinder with slit

and it is crucial for the C 0 -estimate for solutions in the noncompact space T ∗ Q, where H and K have to satisfy (H1 ) and (H2 ). Also again, mΛ is compatible with the R-filtration of HF∗Λ . 3.2. THE RING HOMOMORPHISMS BETWEEN FREE LOOP SPACE FLOER HOMOLOGY AND BASED LOOP SPACE FLOER HOMOLOGY AND CLASSICAL HOMOLOGY

3.2.1. The homomorphism h Let us now give the construction of the homomorphism Λ Ω h: HFΛ → HFΩ ∗ ,m ∗−n , m . We define the underlying domain as Σ = R × [0, 1]/∼ with (s, 0) ∼ (s, 1) for s ≤ 0, i.e., the interior Σ˚ is given by the open subset R × T \ [0, ∞) × {0} of the cylinder R × R/Z with the standard complex structure. Again, Σ carries the structure of a Riemann surface with boundary, the holomorphic boundary chart at (0, 0) again √ given by the z-map. We now assume H: T × T ∗ Q → R to be simultaneously nondegenerate for Ω closed orbits PΛ 1 (H) as well as for the open orbits P1 (H) based at q0 ∈ Q. Moreover, we assume that x(0) T q∗0 Q for all x ∈ PΛ 1 (H). The corresponding Floer-trajectory solution set is defined as Λ→Ω ∞ ˚ ∗ 0 ∗ M x;y (J, H) = u ∈ C (Σ, T Q) ∩ C (Σ, T Q) u s − J ut − XH (t, u) = 0, lim u(s, ·) = x, s→−∞ ∗ lim u(s, ·) = y, u(s, 0), u(s, 1) ∈ T q0 Q, ∀ s ≥ 0 . s→∞

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PROPOSITION 3.3. In view of the nondegeneracy assumption on H and q0 , it holds that (a) for generic J, MΛ→Ω is a smooth manifold of dimension µΛ (x) − µΩ (y) − n, x;y (b) it is compact in dimension 0 and AH (x) ≥ AH (y) whenever it is nonempty, and (c) it is compact in dimension 1 up to simple splitting of Floer trajectories of type Λ respectively Ω at the respective end of Σ. As before in (7) and (11) we define h on the chain level by Ω h: CFΛ h(x) = x, yΛ→Ω y, ∗ → CF∗−n ,

(12)

y

with x, y

Λ→Ω

=

Λ Ω #alg MΛ→Ω x;y , µ (x) − µ (y) = n, 0, else,

Ω Ω Λ for x ∈ PΛ 1 (H), y ∈ P1 (H). By Proposition 3.3(c) we have ∂ ◦ h = h ◦∂ and hence, h is well-defined on homology level. We also have a sharp energy estimate Λ Λ→Ω ∅, so that compatibility which implies that AΩ H (y) ≤ AH (x) whenever M x;y with the R-filtration follows,

h: HFΛ,≤a (H) → HFΩ,≤a ∗−n (H). ∗

(13)

PROPOSITION 3.4. The two operators h ◦mΛ

and

mΩ ◦ (h ⊗ h): CFΛ,≤a (H) ⊗ CFΛ,≤b (K) → CFΩ,≤a+b k l k+l−2n (H # K)

for generic H, K satisfying the conditions for Proposition 3.2, (11) and Proposition 3.3 are chain-homotopy equivalent. Consequently, Λ Ω h: HFΛ → HFΩ ∗ ,m ∗−n , m is a homomorphism compatible with mΛ and mΩ . We briefly sketch the key ideas of the proof. Composing h with mΛ corresponds to a gluing result for the solutions spaces MΛx,y;z (H # K) and Λ→Ω (H # K) for x ∈ PΛ (H), y ∈ PΛ (K), z ∈ PΛ (H # K) and z ∈ PΩ (H # K) Mz;z 1 1 2 2 Λ→Ω is rescaled to width 2. This defines a 1 – 1 where the domain Σ for Mz;z correspondence of ˙ Λ→Ω MΛx,y;z × Mz,z µΛ (z)=k+l−n

NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY

−R

−R

R Figure 4.

The glued domain

91

R

ΣIR

with a set MIx,y;z (R) of solutions u: ΣIR → T ∗ Q,

u s − J(u) ut − XH (t, u) = 0,

on a domain ΣIR = R × [0, 1] ∪˙ R × [1, 2]/ ∼ with (s, 0) ∼ (s, 1−) and (s, 1+) ∼ (s, 2) (s, 0) ∼ (s, 2) and (s, 1−) ∼ (s, 1+) (s, 1−) ∼ (s, 1+)

for s ≤ −R, for − R ≤ s ≤ R, for s ≥ R,

satisfying u(s, 0), u(s, 2) ∈ T q∗0 Q for s ≥ R and x(t), 0 ≤ t ≤ 1−, and lim u(s, t) = y(t), 1+ ≤ t ≤ 2, s→−∞

lim u(s, t) = z(t), 0 ≤ t ≤ 2,

s→∞

provided that R ≥ R(x, y, z) is large enough. This latter solution set MIx,y;z (R) is again a 0-dimensional compact, oriented manifold, and, up to simple splitting of Λ- and Ω-Floer trajectories, it is compactly cobordant to a set MIIx,y;z (R ) of solutions u: ΣIIR → T ∗ Q, u s − J(u) ut − XH (t, u) = 0, on a domain ΣIIR = R × [0, 1] ∪˙ R × [1, 2]/∼ with (s, 0) ∼ (s, 1−) and (s, 1+) ∼ (s, 2) (s, 1−) ∼ (s, 1+)

for s ≤ −R , for s ≥ R ,

satisfying u(s, 0), u(s, 2) ∈ T q∗0 Q u(s, 1−), u(s, 1+) ∈ T q∗0 Q

for s ≥ −R for − R ≤ s ≤ R .

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−R

−R

R Figure 5.

R

The glued domain ΣIIR

The cobordism between MI and MII is obtained by a family of domains realizing a continuous deformation of ΣIR into ΣIIR . Again, for R ≥ R (x, y, z) large enough, the solutions in MIIx,y;z (R ) are in 1 – 1 correspondence with the disjoint union Λ

˙ Λ→Ω Λ→Ω Ω Λ M x;x (H) × My;y (K) × M x ,y ;z (H # K) µ (x ) = k − n, µ (y ) = l − n by a second gluing argument. The fact, that the cobordism between MIx,y;z (R) and MIIx,y;z (R ) is only compact up to simple splitting of Floer trajectories, is responsible for h ◦mΛ and mΩ ◦ (h ⊗ h) being only chain-homotopy equivalent instead of being equal. 3.2.2. Construction of e with right-inverse i In order to carry out all constructions on the chain level, we represent the classical homology of Q by means of Morse homology with auxiliary Morse functions f ∈ C ∞ (Q, R). Also the intersection product • is defined on chain level, namely by means of the trivalent-graph construction. This means that we define a chain complex (CM∗ ( f ), ∂) by CM∗ ( f ) = Crit∗ f ⊗ Z, #alg (M x,y ( f, g)/R)y, ∂x = i(y)=i(x)−1

where the set of nondegenerate critical points Crit∗ f is Z-graded by the Morse index i(·), and ( f, g) is a Morse – Smale pair of Morse function f and Riemannian metric g such that transversality holds, and thus

M x,y ( f, g) = γ: R → Q γ˙ (s) + ∇g f γ(s) = 0, γ(−∞) = x, γ(+∞) = y = W u (x) W s (y). is a manifold of dimension i(x) − i(y). We have ∂2 = 0 and def

H∗ (Q) HM∗ ( f ) = (CM∗ ( f ), ∂) = ker ∂/ im ∂.

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A homotopy of Morse functions f f provides a canonical continuation isomorphism H∗ ( f ) H∗ ( f ). For the definition of the intersection product, we choose a generic triple of Morse functions f , f , f such that Crit f ∩ Crit f = ∅. For x ∈ Critk f , y ∈ Critl f and z ∈ Critm f we set M x,y;z ( f, f , f , g) = γ, γ ∈ C ∞ (−∞, 0], Q , γ ∈ C ∞ ([0, ∞), Q) γ˙ + ∇g f (γ) = 0, γ˙ + ∇g f (γ ) = 0, γ˙ + ∇g f (γ ) = 0,

γ(−∞) = x, γ (−∞) = y, γ (+∞) = z, γ(0) = γ (0) = γ (0) . For a generic Riemannian metric g, M x,y;z ( f, f , f , g) is a manifold of dimension i(x) + i(y) − i(z) − n, compact in dimension 0 and compact up to simple splitting in dimension 1, such that counting elements of M x,y;z with orientation defines •: CMk ( f ) ⊗ CMl ( f ) → CMk+l−n ( f ),

(14)

which induces the intersection product of Q on homology level. Note that here, the orientability of Q is required for the existence of a coherent orientation of the solution sets M x,y;z . Let us now consider the half-cylinders Z + = [0, ∞) × T,

Z − = (−∞, 0] × T,

and define for x ∈ PΛ 1 (H) the space M−x (J, H) = u ∈ C ∞ (Z − , T ∗ Q) u s − J(u) ut − XH (t, u) = 0, lim u(s, t) = x(t), s→−∞

u(0, t) ∈ OQ ⊂ T ∗ Q, ∀ t ∈ T

(15)

for generic J as above under the generic assumption on H that no x ∈ PΛ 1 (H) lies completely within the zero section OQ . Since the zero section is a Lagrangian submanifold with vanishing Maslov class, µMaslov (c) = 0 for all c ∈ ΛQ, we obtain that M−x is a smooth manifold of dimension dim M−x (J, H) = µΛ (x). Similarly, we define for any q ∈ Q and x ∈ PΛ 1 (H) the set M+q;x (J, H) = u ∈ C ∞ (Z + , T ∗ Q) u s − J(u) ut − XH (t, u) = 0, lim u(s, t) = x(t), s→+∞

u(0, t) ∈ T q∗ Q, ∀ t ∈ T

(16)

which, if nonempty, represents a manifold of dimension µΛ (x) for J generically chosen for the fixed q ∈ Q, provided that no x ∈ PΛ 1 (H) lies entirely within the Lagrangian fiber T q∗ Q.

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Consider now y ∈ Crit f with W u (y), W s (y) the unstable and stable manifolds of the negative gradient flow of −∇g f on Q. We set M−x;y (H; f ) = {u ∈ M−x (H) | u(0, 0) ∈ W s (y)}, M+y;x ( f ; H) = {(q, u) | q ∈ W u (y), u ∈ M+q;x (H)}.

(17)

For generic J and g, these spaces are manifolds of dimensions dim M−x;y = µΛ (x) − i(y),

dim M+y;x = i(y) − µΛ (x).

They are compact in dimension 0 and compact, up to simple splitting of either ΛFloer trajectories on one end or Morse-trajectories on the other end, in dimension 1. Hence, we obtain on chain level well-defined group homomorphisms e: CFΛ (H) → CM ( f ), e(x) = #alg M−x;y (H, f ) y, k k i(y)=k

i: CMk ( f ) →

CFkΛ (H),

i(y) =

#alg M+y;x ( f, H) x.

(18)

µΛ (x)=k

It can now be shown that e ◦ i is chain-homotopy equivalent to idCM∗ ( f ) . The main argument is that the intermediate 0-dimensional solution space ' (R) = u: [−R, R] × T → T ∗ Q u − J(u)u − X (t, u) = 0, M y,y

s

t

H T q∗ Q, q

u({−R} × T) ⊂ ∈ W u (y),

u({+R} × T) ⊂ OQ , u(R, 0) ∈ W s (y ) is cobordant to W u (y)∩W s (y ) via a homotopy H we have for (−J)-holomorphic annuli

0. Namely, by Stokes’ theorem

u: [−R, R] × T → T ∗ Q, u s − J(u)ut = 0, u({−R} × T) ⊂ T q∗ Q, u({+R} × T) ⊂ OQ by compatibility of J with ω that R 1 2 2 1 E(u) = 2 (|u s | J + |ut | J ) ds dt = − u∗ ω −R 0 [−R,R]×T u(−R, ·)∗ θLiouv − u(+R, ·)∗ θLiouv = 0, = T

T

(19)

because θLiouv|OQ = 0 = θLiouv|T q∗ Q , and hence u = const. This proves that e ◦ i idCM∗ ( f ) . Similarly, one can establish a suitable cobordism between the solution space 'I (R) = u: Σ− → T ∗ Q u − J(u)u − X (t, u) = 0, M x,x ;y

R

s

t

H

u(−∞, ·)|[0,1−] = x, u(−∞, ·)|[1+,2] = x ,

u {R} × R/2Z ⊂ OQ , u(R, 0) ∈ W s (y; f ) ,

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where Σ−R = {(s, t) ∈ Σ | s ≤ R}, R > 0, for the pair-of-pants domain Σ for mΛ from Section 3.1.2, and the solution space 'II (R) = {(u, u , γ, γ ) | u ∈ M− (J, H), u ∈ M− (J, H ), M x x,x ;y x γ, γ : [−R, 0] → Q, γ˙ = −∇g f (γ), γ˙ = −∇g f (γ ), γ(−R) = u(0, 0), γ (−R) = u (0, 0), γ(0) = γ (0) ∈ W s (y; f )} for the Morse functions f , f , f from (14). This is the main argument for proving that on chain level, the compositions e ◦mΛ

Λ Λ • ◦ (e ⊗ e): CFΛ k (H) ⊗ CF l (K) → CFk+l−n ( f )

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are chain homotopy equivalent. From these considerations we obtain PROPOSITION 3.5. For the linear operators e: HFΛ ∗ → H∗ (Q) and i: H∗ (Q) → Λ : HFΛ ⊗ HFΛ → HFΛ and •: H (Q) ⊗ H (Q) HFΛ , and the bilinear operators m ∗ ∗ ∗ ∗ ∗ ∗−n → H∗−n (Q) we have (a) e ◦ i = idH∗ (Q) , (b) e mΛ (α, β) = e(α) • e(β), for all α, β ∈ HFΛ ∗. This completes the exposition of the arguments for statements (a) and (b) in Theorem 2.7 and the upper row in the commutative diagram. In the following section, we give details on the remaining vertical isomorphisms in that diagram which gives the main statement of the theorem, namely that Floer homology in the cotangent bundle encodes the well-known loop space homology. 4. Morse-homology on the loop spaces ΛQ and ΩQ, and the isomorphism In this section, we give a short account of the construction of the explicit isomorphisms ΦΛ and ΦΩ from Theorem 2.7. These isomorphisms have been established in Abbondandolo and Schwarz (2004), by representing the classical homology of the loop space in terms of Morse homology for the Lagrangian action functional. The central idea of the isomorphism is the Legendre transform as a duality between the Lagrangian and the Hamiltonian picture. Diﬀerent approaches to the same isomorphism of homology groups had been established first by Viterbo (1996) and by Salamon and Weber (2003). Let us consider the connected, compact, oriented smooth manifold Q with an auxiliary Riemannian metric g and an admissible Lagrangian function L: T × T Q → R such that the following assumptions hold: For any coordinate chart U, (q1 , . . . , qn ) on Q and the associated local trivialization of T Q|U there exist constants l0 > 0, l1 ≥ 0 such that we have in the coordinates (q1 , . . . , qn , v1 , . . . , vn )

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(L1 ) ∂2vv L(t, q, v) ≥ l0 as a quadratic form, (L2 ) |∂2qq L(t, q, v)| ≤ l1 (1 + |v|2 ), |∂2qv L(t, q, v)| ≤ l1 (1 + |v|), |∂2vv L(t, q, v)| ≤ l1 for all t ∈ T, q ∈ U, v ∈ T q Q. These assumptions are formally similar to (H1 ) and (H2 ). They imply that the Lagrangian L grows fiberwise at least quadratically and at most quadratically. By the strict convexity condition (L1 ) we have the Legendre transform

LL : T × T Q − → T × T ∗ Q, v TQ , (t, q, v) → (t, q, dL(t, q, v)|T (q,v)

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v T Q = ker Dτ = T (T Q) = T Q can be canonically identified, and where T (q,v) v q q τ: T Q → Q is the bundle projection. Using the inverse transform ∗ L−1 L : T × T Q → T × T Q, (t, q, p) → (t, q, v(t, q, p)),

we have the Hamiltonian function H(t, q, p) associated to L(t, q, v) by H(t, q, p) = max p, v − L(t, q, v) v∈T q Q

= p, v(t, q, p) − L(t, q, v(t, q, p)).

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The Legendre transform LL gives a diﬀeomorphism by which we can also pull back the symplectic structure ωLiouv from T ∗ Q to T Q, and the Hamiltonian vector field XH is mapped to the vector field YL on T Q whose integral curves are characterized by the second order ODE in the q-variables ∇t ∇v L(t, q(t), q(t) ˙ = ∇q L t, q(t), q(t) ˙ ,

(23)

where ∇t denotes the covariant derivation along q, and ∇v , ∇q denote the vertical and the horizontal part of the gradient of L with respect to the induced Riemannian metric from g and its Levi-Civita connection. (23) is also the Euler-Lagrange equation for the Lagrangian action functional

1

EL (q) =

L t, q(t), q(t) ˙ dt

0

viewed on ΛQ and, respectively, ΩQ. Its critical sets are the solution spaces PΛ (L) = {q ∈ ΛQ | q solves (23)} PΩ (L) = {q ∈ ΩQ | q solves (23)}.

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We use the same notation as in the Hamiltonian picture since by Legendre transform we have the canonical one-to-one correspondence 1:1

π: PΛ resp. Ω (H) −−→ PΛ resp. Ω (L), x = (q, p) → q. The nondegeneracy conditions for the Legendre picture read: (L0)Λ For every solution q ∈ PΛ (L) there exists no 1-periodic Jacobi vector field along q, (L0)Ω for every solution q ∈ PΩ (L) there exists no Jacobi vector field along q vanishing for t = 0 and t = 1. These conditions are equivalent to the nondegeneracy conditions for x = π−1 (q) ∈ PΛ resp. Ω (H) as solutions of the Legendre dual Hamiltonian problem, i.e., (H0)Λ the diﬀerential of the time-one integral map of XH at x(0) does not have the eigenvalue 1, (H0)Ω the diﬀerential of the time-one integral map of XH , T x(0) T ∗ Q → T x(1) T ∗ Q, maps the vertical subspace of x(0) onto a subspace having intersection {0} with the vertical subspace at x(1). The crucial observation for the isomorphism between Morse homology for EL in the Lagrangian picture and Floer homology for AH in the Hamiltonian picture is the LEMMA 4.1 (Main lemma). If x = (q, p): [0, 1] → T ∗ Q is continuous, with q of class W 1,2 , then AH (x) ≤ EL (q), v T Q , that is if and only the equality holding if and only if p(t) = dL(t, q(t), q(t))| ˙ T q(t) ˙ ˙ for every t ∈ [0, 1]. In particular, the Hamilif (t, q(t), p(t)) = LL (t, q(t), q(t)) tonian and the Lagrangian action functionals coincide on the solutions of the two systems. ∗ Q that Proof. From the definitions we have for x(t) = q(t), p(t) ∈ T q(t)

AH (x) =

1

p(t), q(t) ˙ − H t, q(t), p(t) dt

0 1

= ≤ 0

(p(t), q(t) ˙ − max[p(t), v − L(t, q(t), v)]) dt v

0 1

L t, q(t)q(t) ˙ dt = EL (q),

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by choosing pointwise v = q(t), ˙ with equality exactly if (t, q, p) = LL (t, q, q). ˙

On the canonically identified solution sets PΛ resp. Ω (H) = Crit AH and = Crit EL we have not only the same critical values for AH and EL by the previous lemma, we also can identify the Conley – Zehnder index µΛ and the Maslov index µΩ on the Hamiltonian side with the Morse indices iΛ and iΩ of EL on the Lagrangian side. PΛ resp. Ω (L)

THEOREM 4.2 (Duistermaat, 1976; Robbin and Salamon, 1993; Weber, 2002). Assuming that Q is orientable, let q ∈ PΛ (L), and let x ∈ PΛ (H), t, x(t) = LL (t, q(t), q(t)), ˙ be the corresponding 1-periodic solution of the Hamiltonian ∗ system on T Q. Then iΛ (q) = µΛ (x). Let q ∈ PΩ (L) and let x ∈ PΩ (H), t, x(t) = LL t, q(t), q(t) ˙ , be the corresponding solution of the Hamiltonian system on T ∗ Q. Then iΩ (q) = µΩ (x). The crucial property allowing the construction of Morse homology for EL is the Palais – Smale property which had been proven by V. Benci, THEOREM 4.3 (Benci, 1986). Consider standard W 1,2 -metrics on ΛQ and ΩQ, and let L satisfy the conditions (L1 ) and (L2 ). Then EL has the Palais – Smale property, i.e, for any sequence (qn ) ⊂ ΛQ respectively ΩQ with (EL (qn )) bounded and gradW 1,2 EL (qn ) → 0 there exists a convergent subsequence of (qn ). As a consequence, see for instance Abbondandolo and Majer (2005) in this volume, we have well-defined Morse homology for EL . That is, we have the DEFINITION 4.4. Given a, b ∈ Crit EL ⊂ ΛQ, we have

W u (a) = γ ∈ C ∞ (−∞, 0], ΛQ γ(−∞) = a, γ˙ = −∇EL (γ) ,

W s (b) = γ ∈ C ∞ ([0, ∞), ΛQ) γ(+∞) = b, γ˙ = −∇E (γ) , L

where ΛQ is endowed with a W 1,2 -structure and the gradient ∇EL is taken with respect to a Hilbert-metric on ΛQ as W 1,2 -Hilbert manifold. We have W u (a) ∩ W s (b) = MΛ a,b (EL ) = {γ: R → Λ | γ˙ + ∇EL (γ) = 0, γ(−∞) = a, γ(+∞) = b}, and completely analogously for the corresponding negative gradient flow on ΩQ.

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Again, one can show that for a generic perturbation of the Hilbert metric on ΛQ respectively ΩQ, we have transversality W u (a) W s (b), such that this trajectory space is a smooth, finite-dimensional manifold of dimension Λ resp. Ω

dim Ma,b

(EL ) = iΛ resp. Ω (a) − iΛ resp. Ω (b).

Moreover, due to the Palais – Smale property of EL , the trajectory spaces Ma,b have the required compactness properties up to splitting of trajectories. We then define CM∗ (EΛ resp. Ω ) = Crit EL ⊗ Z, which are Z-graded by iΛ respectively iΩ and endowed with the R-filtration CMa∗ (EΛ resp. Ω ) = {x ∈ Crit EL | EL (x) ≤ a} ⊗ Z. Using the concept of Morse homology for EL , we define chain boundary operators Λ resp. Ω ∂ on CM∗ (EL ) by counting Ma,b (El ) as before and we obtain the Morse homology HM∗ (EΛ resp. Ω ) = H∗ (CM∗ (EΛ resp. Ω ), ∂) and likewise for the filtered complex (CMa∗ , ∂). Recalling from Section 2.1 that the inclusions of the Hilbert manifolds W 1,2 (T, Q) and {c ∈ W 1,2 ([0, 1], Q) | c(0) = c(1) = q0 } into Λ0 Q = C 0 (T, Q) and Ω0 Q = C 0 ([0, 1], {0, 1}), (Q, q0 ) are homotopy equivalences, we obtain THEOREM 4.5. For all a ∈ R there exist canonical isomorphisms Λ resp. Ω

HMa∗ (EL

Λ resp. Ω

) H∗ ({EL

Λ resp. Ω HM∗ (EL )

≤ a}),

H∗ (Λ Q resp. Ω0 Q), 0

compatible with the induced homomorphisms from {E ≤ a} ⊂ {E ≤ b} for a ≤ b, and lim HMa∗ (EL ) = HM∗ (EL ). −→ We now show that this Morse homological representation by the Lagrangian action functional is directly isomorphic to the Floer homological representation based on the Legendre dual Hamiltonian action functional. THEOREM 4.6. Assume that L is nondegenerate and satisfies (L1 ), (L2 ) above and that H is the Legendre transform of L. Let J be a suﬃciently small generic perturbation of JLC for some Riemannian metric g0 on Q, and let g be a generic Riemannian structure on the Hilbert manifolds Λ1 Q and Ω1 close to ·, ·1,2 from g0 . Then, there exists a chain complex isomorphism for both loop space cases Λ and Ω, → CF∗ (H), ∂(H, J) Θ: CM∗ (EL ), ∂(L, g) −

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of the form Θ(q) =

d+ (q, x) · x

∀ q ∈ P(L)

x∈P(H),µ(x)=i(q)

with d+ (q, x) = 0 whenever EL (q) ≤ AH (x) except for d+ (q, x) = ±1 if π(x) = q. In particular, Θ is also an isomorphism between the restricted chain complexes CMa∗ (EL ) and CFa∗ (H) for a ∈ R. We remark that the conditions (H1 ) and (H2 ) are automatically satisfied for H = LL (L) if L satisfies (L1 ) and (L2 ). In the following we sketch the proof of Theorem 4.6 which is given in full terms in Abbondandolo and Schwarz (2004). For simplicity, we drop the notational distinction between ΛQ and ΩQ which are considered completely analogously. Proof. Recall first that the generators of CM∗ (EL ) and CF∗ (H) allow a canonical identification by P(L) = Crit EL q ↔ x ∈ Crit AH = P(H), via π(x) = q where π: T ∗ Q → Q is the cotangent bundle projection, and where EL (q) = AH (x) for those critical points, in view of the Legendre duality. Choose a numbering (qn )n∈N of Crit EL such that EL (qn ) is monotone nondecreasing, and let (xn ) be the corresponding sequence in Crit AH . The crucial issue is the observation that the operator Θ is represented in terms of this infinite basis by a matrix D = d+ (qi , x j ) i, j∈N which has the properties 1. D is lower-triangular, i.e., d+ (qi , x j ) = 0 for i < j, 2. dii+ = ±1 for all i ∈ N, 3. DE = A D, where E and A are the matrices representing the chain boundary operators ∂E : CM∗ → CM∗−1 and ∂A : CF∗ → CF∗−1 . It is an immediate consequence from → (CF∗ , ∂A ) is 1 and 2 that D is invertible and with 3 we obtain that Θ: (CM∗ , ∂E ) − a chain complex isomorphism. It remains to give the construction of d+ (q, x) for q ∈ P(L), x ∈ P(H) such that 1, 2 and 3 are satisfied. Let q ∈ Crit EL and x ∈ Crit AH and define ME→A = (γ, u) ∈ W u (q) × C ∞ ([0, ∞) × T, T ∗ Q) q;x u s − J ut − XH (t, u) = 0, u(+∞) = x,

π u(0, t) = γ(0)(t) ∀ t ∈ T . (24) Here, we use the loop γ(0) in the finite-dimensional unstable manifold as a 1∗ parameter family of Lagrangian subspaces T γ(0)(t) Q providing a t-dependent t∈T family of boundary conditions for the pseudoholomorphic half-cylinder u where

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101

the boundary condition identifies the loop parameter for u(0) with the family parameter. This is again an admissible nonlinear Fredholm problem, for which we can prove PROPOSITION 4.7. is a (a) For generic J and g as in Theorem 4.6, the mixed solution space ME→A q;x smooth manifold of dimension i(q) − µ(x). is compact in dimension 0 and compact, up to simply splitting of (b) ME→A q;x either Morse trajectories on the negative end or Floer trajectories on the positive end, in dimension 1. (c) We have ME→A ∅ only if EL (q) ≥ AH (x) with EL (q) = AH (x) if and only q;x

= (q, x) , i.e. γ(0) = q and u(0) = x in (24) above. if π(x) = q and ME→A q;x Defining now d+ (q, x) = #alg ME→A by counting with signs, which are q;x determined by a concept of coherent orientations described completely in (Abbondandolo and Schwarz, 2004), we obtain the properties 1 – 3 as claimed. Concerning this proposition, the essential new argument is based on Main Lemma 4.1. Namely, for any (γ, u) ∈ ME→A we have EL (γ(0)) ≥ AH (u(0)) q;x and thus EL (q) ≥ AH (x) with equality if and only if γ and u are constant with respect to the flow parameter. This implies an a priori energy bound required for the compactness statements in (b). Moreover, we see that d+ (qi , x j ) = 0 for i < j with respect to the above ordering of the generators. Note that the transversality property of (a) in case of the constant solution (q, x) ∈ ME→A cannot be achieved by perturbing J or g. In fact, we can prove q;x that for π(x) = q we have automatic transversality which follows from the second variation estimate d2 AH (x)[ζ, ζ] ≤ d2 EL (q)[Dπ(x)ζ, Dπ(x)ζ],

for all ζ ∈ T x Λ(T ∗ Q).

This estimate is again a consequence of Main Lemma 4.1. For details, see Abbondandolo and Schwarz (2004). Thus, the core argument is based on Legendre duality in an essential way. Another important remark is that there is no analogous construction of the inverse chain homomorphism Θ−1 : CF∗ → CM∗ via a correspondingly reversed mixed solution space. If we consider the problem = (u, γ) ∈ C ∞ (−∞, 0] × T, T ∗ Q × W s (q) u s − J(ut − XH (t, u)) = 0, MA→E x;q

u(−∞) = x, π u(0) = γ(0) ,

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there is no possibility to obtain an analogous statement as in Proposition 4.7. Both, the Fredholm property and the compactness property will fail in general, because dim W s (q) = ∞ and EL (γ(0) ≥ AH (u(0)) which allows no a priori energy bound. 5. Products in Morse-homology In this last section, we sketch the Morse-homological definitions of the product and intersection operations o, ∩ and # from Section 2.1. These are needed in order to prove that the explicit isomorphism Θ intertwines the product structures on both sides as claimed in Theorem 2.7. The finite-dimensional model of the homomorphism ∩ in Morse homology is given by the following construction. Let M m , N n be smooth, finite-dimensional manifolds, for simplicity closed. Let i: M → N be a smooth cooriented embedding and f M : M → R, f N : N → R be Morse functions such that Crit f N ∩ i(M) = ∅.

(25)

Then we obtain induced homomorphisms of Morse homology, i∗ : HM∗ (M, f M ) → HM∗ (N, f N ), i! : HM∗ (N, f N ) → HM∗−(n−m) (M, f M ), where i! equals ∩ from Section 2.1. The construction goes as follows on chain level. By assumption (25), which is generically fulfilled, we can find generic Riemannian metrics g M , gN on M and N such that we have the transverse intersections i(M) W s (x; f N , gN ) = γ: [0, ∞) → N γ˙ + ∇gN f N (γ) = 0,

γ(0) ∈ i(M), γ(∞) = x , i(M) W u (x; f N , gN ) = γ: (−∞, 0] → N γ˙ + ∇gN f N (γ) = 0,

γ(0) ∈ i(M), γ(−∞) = x , (26) for all x ∈ Crit f N , and ∗ Mix,y = W u (x; f M , g M ) i W s (y; f N , gN ) = (γ M , γ N ) ∈ C ∞ (−∞, 0], M × C ∞ ([0, ∞), N)

! Miy,x

γ˙ K + ∇gK f K (γ K ) = 0, K ∈ {M, N},

γ M (−∞) = x, γ N (+∞) = y, i γ M (0) = γ N (0) , = W u (y; f N , gN ) i W s (x; f M , g M ) = (γ N , γ M ) ∈ C ∞ ((−∞, 0], N) × C ∞ ([0, ∞), M) γ˙ K + ∇gK f K (γ K ) = 0, K ∈ {M, N}, γ N (−∞) = y, γ M (+∞) = x, i(γ M (0)) = γ N (0)

NOTES ON FLOER HOMOLOGY AND LOOP SPACE HOMOLOGY

= W u (y; f N , gN ) N i(M) M W s (x; f M , g M ),

103 (27)

with ∗ = i f M (x) − i f N (y), dim Mix,y

! dim Miy,x = i f N (y) − i f M (x) − (n − m),

for x ∈ Crit f M and y ∈ Crit f N . Here, M , N indicate the ambient manifold of the transversely intersecting submanifolds. Then, counting the solution of (27) in dimension 0 defines i∗ and i! on chain-level. Let us use this concept now in the infinite-dimensional Morse-homological framework of the codimension-n embedding i: Ωq0 Q → ΛQ of the Hilbert manifolds, where the Morse function is given by the Lagrangian action functional for some nondegenerate Lagrangian L satisfying (L1 ) and (L2 ). The transversality condition (25) is now given by the assumption that for all c ∈ PΛ (L) we have c(0) q0 , i.e., PΛ (L) ∩ i(Ωq0 Q) = ∅,

(28)

which is fulfilled generically for (L, q0 ). Choosing generic Riemannian metrics gΛ and gΩ on ΛQ and ΩQ we construct Λ i∗ : HM∗ (EΩ L ) → HM∗ (EL )

on chain level by

i∗ (a) =

#alg W u (a; EΩ , gΩ ) W s (b; EΛ , gΛ ) · b

b∈PΛ (L),i(b)=i(a)

for all a ∈ Crit EΩ L , where the intersection means counting γ: R → ΛQ such that γ˙ (s) = −∇gΩ EΩ (γ) and γ(s)(0) = q0 for all s ≤ 0 and γ˙ (s) = −∇gΛ EΛ (γ) for all s ≥ 0. Similarly, we define Ω i! : HM∗ (EΛ L ) → HM∗−n (EL )

on chain level by i! (a) =

s Ω Ω #alg W u (a; EΛ , gΛ ) ev−1 0 (q0 ) W (b; E , g ) · b

b∈PΩ (L),i(b)=i(a)−n

for a ∈ PΛ , where the double intersection means counting γ− : (−∞, 0] → ΛQ and γ+ : [0, ∞) → ΩQ such that γ˙ ± = −∇gΛ resp. Ω EΛ resp. Ω (γ± ), γ− (−∞) = a, γ+ (∞) = b, ev0 γ− (0) = q0 and γ− (0) = γ+ (0). One can show that, assuming (28), all intersections are transverse for generic gΛ , gΩ .

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Next we consider the codimension-n embedding, #: ΩQ × ΩQ → Ω2 Q ⊂ C 0 [0, 2], Q , c(t), 0 ≤ t ≤ 1, #(c, d)(t) = d(t), 1 ≤ t ≤ 2, which for convenience we view into the based loop space of period 2. The generic assumption on (L, q0 ) for transversality is now that c(1) q0

for all c ∈ PΩ 2 (L).

(29)

Then, for generic Riemannian metrics g1 , g2 on ΩQ on g3 on Ω2 Q we have # W u (a; EΩ , g1 ) × W u (b; EΩ , g2 ) W s (c; EΩ2 , g3 ) Ω for all a, b ∈ PΩ 1 (L), c ∈ P2 (L). Counting in dimension 0 with signs thus defines on chain level an operator which on homology gives Ω Ω 2 #∗ : HMk EΩ L ⊗ HMl EL → HMk+l EL .

Let us now turn to the codimension-n submanifold #

X = {(c, d) ∈ ΛQ × ΛQ | c(0) = d(0)} → Λ2 Q = W 1,2 (R/2Z, Q). In order to construct

Λ Λ i! : HMk EΛ L1 ⊗ HMl EL2 → HMk+l−n X; EL1 #L2

and

Λ → HM EL12#L2 #∗ : HMk+l−n X; EΛ k+l−n L1 #L2

we assume a(0) b(0) cX

for all (a, b) ∈ PΛ (L1 ) × PΛ (L2 ), for all c ∈ PΛ 2 (L1 # L2 ),

(30)

where L1 (1, ·) = L2 (0, ·), L = L1 # L2 : R/2Z × T Q → R, with L|[0,1] = L1 and L|[1,2] = L2 . By assumption (30) we can find generic Riemannian metrics g1 , g2 on ΛQ and g3 on Λ2 Q such that we have transverse intersections u Λ × W b; E X, , g , g Y = W u a; EΛ 1 2 L1 L2 2 Z(a, b; c) = #(Y) W s c; EΛ L1 #L2 , g3 for all (a, b) ∈ PΛ (L1 ) × PΛ (L2 ) and c ∈ PΛ2 (L1 #L2 ). We define Λ Λ → CM ⊗ CM E EL12#L2 , o: CMk EΛ l k+l−n L1 L2 aob= #alg Z(a, b; c) · c, i(c)=i(a)+i(b)−n

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γ1

γ3

105

u

γ2 0 Figure 6.

R The domain for MIa,b;x (R)

which on homology gives o = # ◦ i! , the Morse-homological definition of the loop product. 5.1. RING ISOMORPHISM BETWEEN MORSE HOMOLOGY AND FLOER HOMOLOGY

Finally, we sketch very briefly the proof of the ring homomorphism property of Ω Ω , # A , m Θ∗ : HM∗ EΩ − → HF ∗ ∗ L H in the based loop space case. Full details, also for the free loop space case, will be contained in Abbondandolo and Schwarz (2005). We show that on chain level, the compositions of Θ and # and mΩ and Θ, respectively, are chain homotopy equivalent, Θ◦#

Ω2 Ω mΩ ◦ (Θ ⊗ Θ): CMΩ ∗ (EL ) ⊗ CM∗ (EL ) → CF∗ (AH ).

(32)

Consider a generic triple of Riemannian metrics g1 , g2 on ΩQ and g3 on Ω2 Q as before, and J generic on T ∗ Q for AH on Λ2 (T ∗ Q). Then, by gluing, the composition (Θ ◦ #∗ )(a, b) is given by counting MIa,b;x (R) = (γ1 , γ2 , γ3 , u) γi : (−∞, 0] → ΩQ, i = 1, 2, γ3 : [0, R] → Ω2 Q, γ˙ i + ∇gi E(γi ) = 0, i = 1, 2, 3, γ1 (−∞) = a, γ2 (−∞) = b, γ3 (0) = γ1 (0) # γ2 (0), u: [R, ∞) × [0, 2] → T ∗ Q, u s − J(u) ut − XH (t, u) = 0, u(s, 0), u(s, 2) ∈ T q∗0 Q, R ≤ s < ∞,

u(+∞) = x, u(R, t) ∈ T γ∗3 (R)(t) Q, 0 ≤ t ≤ 2 for µΩ (x) = i(a) + i(b) and for R " 1 suﬃciently large depending on a, b.

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γ1 u

γ2 −r Figure 7.

0 The domain for MIIa,b;x (r)

For the other composition mΩ ◦ (Θ ⊗ Θ)(a, b) we have to count for r " 1 II Ma,b;x (r) = (γ1 , γ2 , u) γi : (−∞, 0] → ΩQ, i = 1, 2, γ˙ i + ∇gi E(γi ) = 0, i = 1, 2, γ1 (−∞) = a, γ2 (−∞) = b, u: Σr → T ∗ Q, u s − J(u)(ut − XH (t, u)) = 0, u(s, 0), u(s, 2) ∈ T q∗0 Q, for − r ≤ s < ∞, u(s, 1±) ∈ T q∗0 Q, for r ≤ s ≤ 0, ∗ ( T Q, 0 ≤ t ≤ 1−, u(+∞) = x, u(−r, t) ∈ γ∗1 (0)(t) T γ2 (0)(t−1) Q, 1+ ≤ t ≤ 2, Ω where x ∈ PΩ 2 (H) with µ (x) = i(a) + i(b), and Σr is the domain $ Σr = [−r, ∞) × [0, 1] ∪˙ [−r, ∞) × [1, 2] ∼,

with (s, 1−) ∼ (s, 1+) for all s ≥ 0. One can now show that both solution spaces MIa,b;x (R) and MIIa,b;x (r) are compactly, up to simple splitting, cobordant to 0 Ma,b;x = (γ1 , γ2 , u) γi : (−∞, 0] → ΩQ, i = 1, 2, γ˙ i + ∇gi E(γi ) = 0, i = 1, 2, γ1 (−∞) = a, γ2 (−∞) = b, u: [0, ∞) × [0, 2] → T ∗ Q, u s − J(u)(ut − XH (t, u)) = 0, u(s, 0), u(s, 2) ∈ T q∗0 Q, for 0 ≤ s < ∞, ∗ ( T Q, 0 ≤ t ≤ 1−, u(+∞) = x, u(0, t) ∈ γ∗1 (0)(t) . T γ2 (0)(t−1) Q, 1+ ≤ t ≤ 2, This shows that Θ ◦ # − mΩ ◦ (Θ ⊗ Θ) = ∂Ω (J, H) ◦ S ± S ◦ id ⊗ ∂Ω (EΩ , g2 ) ± ∂Ω (EΩ , g1 ) ⊗ id for some chain homotopy operator Ω Ω 2 S : CM∗ EΩ L ⊗ CM∗ EL → CF∗+1 AH ,

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γ1 γ2

Figure 8.

u

The intermediate domain for M0a,b;x

thus proving (32). References Abbondandolo, A. and Majer, P. (2005) Lectures on the Morse complex for infinite-dimensional manifolds, in this volume. Abbondandolo, A. and Schwarz, M. (2004) On the Floer homology of cotangent bundles, Comm. Pure Appl. Math., to appear; arXiv:math.SG/0408280. Abbondandolo, A. and Schwarz, M. (2005) Floer homology of cotangent bundles and the loop product, in preparation. Benci, V. (1986) Periodic solutions of Lagrangian systems on a compact manifold, J. Diﬀerential Equations 63, 135 – 161. Chas, M. and Sullivan, D. (1999) String topology, arXiv:math.GT/9911159. Cohen, R. (2005) Morse theory, graphs, and string topology, in this volume. Cohen, R. and Jones, J. D. S. (2002) A homotopy theoretic realization of string topology, Math. Ann. 324, 773 – 798. Cohen, R. L. and Voronov, A. A. (2005) Notes on string topology, arXiv:math.GT/0503625. Duistermaat, J. J. (1976) On the Morse index in variational calculus, Adv. in Math. 21, 173 – 195. Floer, A. (1988)a A relative Morse index for the symplectic action, Comm. Pure Appl. Math. 41, 393 – 407. Floer, A. (1988)b The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41, 775 – 813. Floer, A. (1989) Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120, 575 – 611. Floer, A. and Hofer, H. (1993) Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212, 13 – 38. F´elix, Y., Halperin, S., and Thomas, J.-C. (2001) Rational Homotopy Theory, Vol. 205 of Grad. Texts in Math., New York, Springer. Gromov, M. (1985) Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82, 307 – 347. Hofer, H. and Salamon, D. A. (1995) Floer homology and Novikov rings, In H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder (eds.), The Floer Memorial Volume, Vol. 133 of Progr. Math, pp. 483–524, Basel, Birkh¨auser. Piunikhin, S., Salamon, D., and Schwarz, M. (1996) Symplectic Floer – Donaldson theory and quantum cohomology, In C. B. Thomas (ed.), Contact and Symplectic Geometry, Vol. 8 of Publ. Newton Inst., Cambridge 1994, pp. 171–200, Cambridge, Cambridge Univ. Press. Ramirez, A. (2005), Ph.D. thesis, Stanford University, in preparation.

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Robbin, J. and Salamon, D. (1993) The Maslov index for paths, Topology 32, 827 – 844. Salamon, D. and Weber, J. (2003) Floer homology and the heat flow, arXiv:math.SG/0304383. Schwarz, M. (1993) Morse Homology, Vol. 111 of Progr. Math, Basel, Birkh¨auser. Schwarz, M. (1995) Cohomology operations from S 1 -cobordisms in Floer homology, Ph.D. thesis, ETH Zurich. Viterbo, C. (1996) Functors and computations in Floer homology with applications. II, preprint. Weber, J. (2002) Perturbed closed geodesics are periodic orbits: index and transversality, Math. Z. 241, 45 – 82.

HOMOTOPICAL DYNAMICS IN SYMPLECTIC TOPOLOGY JEAN-FRANC¸OIS BARRAUD Universit´e de Lille 1 OCTAV CORNEA Universit´e de Montr´eal

Abstract. This is mainly a survey of recent work on algebraic ways to “measure” moduli spaces of connecting trajectories in Morse and Floer theories as well as related applications to symplectic topology. The paper also contains some new results. In particular, we show that the methods of Barraud and Cornea (2003) continue to work in general symplectic manifolds (without any connectivity conditions) but under the bubbling threshold. Key words: Serre spectral sequence, Morse theory, Lagrangian submanifolds, Hopf invariants, Hofer distance 2000 Mathematical Subject Classification: Primary 53D40, 53D12; Secondary 37D15

1. Introduction The main purpose of this paper is to survey a number of Morse-theoretic results which show how to estimate algebraically the high-dimensional moduli spaces of Morse flow lines and to describe some of their recent applications to symplectic topology. We also deduce some new applications. The paper starts with a brief discussion of the various proofs showing that the diﬀerential in the Morse complex is indeed a diﬀerential. With this occasion we introduce the main concepts in Morse theory and, in particular, the notion of connecting manifold (or, equivalently, the moduli space of flow lines connecting two critical points) which is the main object of interest in our further constructions. Moreover, an extension of one of these proofs leads naturally to an important result of John Franks (1979) which describes the framed cobordism class of connecting manifolds between consecutive critical points as a certain relative attaching map. After describing Franks’ result, we proceed to a stronger result initially proved in Cornea (2002a) which computes a framed bordism class naturally associated to the same connecting manifolds in terms of certain Hopf invariants. While these results only apply to consecutive critical points we then describe a recent method to estimate general connecting manifolds by means of the Serre spectral sequence of the path-loop fibration having as base the ambient 109 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 109–148. © 2006 Springer. Printed in the Netherlands.

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manifold (Barraud and Cornea, 2003). Some interesting topological consequences of these results are briefly mentioned as well as some other methods used in the study of these problems. The third section discusses a number of symplectic applications. We start with some results which first appeared in Cornea (2002b). These use the nonvanishing of certain Hopf invariants to deduce the existence of bounded orbits of Hamiltonian flows (obviously, inside noncompact manifolds). This is a very “soft” type of result even if diﬃcult to prove. We then continue in Section 3.2 by describing how to use the Serre spectral sequence result to detect pseudo-holomorphic strips as well as some consequences of the existence of the strips. Most of the results of this part have first appeared in Barraud and Cornea (2003) but there are some that are new: we discuss explicitly the detection of pseudoholomorphic strips passing through some submanifold and we present a way to construct in a coherent fashion our theory for Lagrangians in general symplectic manifolds as long as we remain under a bubbling threshold. Notice that even the analogue of the classical Floer theory (which is a very particular case of our construction) has not been explicited in the literature in the Lagrangian case even if all the necessary ideas are present in some form — see Schwarz (1998) for the Hamiltonian case. The paper contains a number of open problems and ends with a conjecture which is supported by the results in Section 3.2 as well as by recent joint results of the second author with Franc¸ois Lalonde. 2. Elements of Morse theory Assume that M is a compact, smooth manifold without boundary of dimension n. Let f : M → R be a smooth Morse function and let γ: M × R → M be a negative gradient Morse – Smale flow associated of f . In particular, f is strictly decreasing along any nonconstant flow line of γ and the stable manifolds

W s (P) = x ∈ M : lim γt (x) = P t→∞

and the unstable manifolds

W u (Q) = x ∈ M : lim γt (x) = Q t→−∞

of any pair of critical points P and Q of f intersect transversally. One of the most useful and simple tools that can be defined in this context is the Morse complex C(γ) = (Z/2Crit( f ), d). Here Z/2S is the Z/2-vector space generated by the set S , the vector space Z/2Crit( f ) has a natural grading given by |P| = ind f (P), ∀P ∈ Crit( f ) and d is

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the diﬀerential of the complex which is defined by dx = ayx y |y|=|x|−1

so that the coeﬃcients ayx = # W u (x) ∩ W s (y) /R . This definition makes sense because the set W u (P) ∩ W s (Q) which consists of all the points situated on some flow line joining P to Q is invariant by the R-action given by the flow. Moreover, W u (P) and W s (P) are homeomorphic to open disks which implies that the set MQP = W u (P) ∩ W s (Q) /R has the structure of a smooth (in general, noncompact) manifold of dimension |P|−|Q|−1. We call this space the moduli space of flow lines joining P to Q. It is not diﬃcult to understand the reasons for the noncompactness of MQP when M is compact as in our setting: this is simply due to the fact that a family of flow lines joining P to Q may approach a third, intermediate, critical point R. For this to happen it is necessary (and suﬃcient — see Smale, 1967, or Franks, 1979) to have some flow line which joins P to R and some other joining R to Q. This implies that when |P| = |Q| + 1 the set MQP is compact and thus the sum above is finite. For further use let’s define also the unstable sphere of a critical point P as S au (P) = W u (P) ∩ f −1 (a) as well as the stable sphere S as (P) = W s (P) ∩ f −1 (a) where a is a regular value of f . It should be noted that this names are slightly abusive as these two sets are spheres, in general, only if a is suﬃciently close to f (P). In that case S au (P) is homeomorphic to a sphere of dimension |P| − 1 and S as (P) is homeomorphic to a sphere of dimension n−|P|−1. With this notations the moduli space MQP is homeomorphic to S au (P) ∩ S as (P) for any a ∈ f (Q), f (P) which is a regular value of f . The main properties of the object defined above are that: d2 = 0 and

H∗ C(γ) ≈ H∗ (M; Z/2).

We will sometimes denote this complex by C( f ) and will call it the (classical) Morse complex of f . The flow γ may be in fact even a pseudo- (negative) gradient flow of f . There also exists a version of this complex over Z in which the counting of the elements in Myx takes into account orientations. There are essentially four methods to prove these properties: ) y (i) Deducing the equation y ayx az = 0 (which is equivalent to d2 = 0) from the properties of the moduli spaces Mzx with |x| − |z| = 2. (ii) Comparing ayx with a certain relative attaching map. (iii) Expressing ayx in terms of a connection map in Conley’s index theory. (iv) A method based on a deformation of the de Rham complex (clearly, in this case the coeﬃcients are required to be in R).

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For the rest of this paper the two approaches that are of the most interest are (i) and (ii). Therefore we shall first say a few words on the other two methods and will then describe in more detail the first two. Method (iii) consists in regarding two critical points x, y so that |x| = |y| + 1 as an attractor-repellor pair and to apply the general Conley theory of Morse decompositions to this situation (Salamon, 1985). Method (iv) has been introduced by Witten (1982) and is based on a deformation of the diﬀerential of the de Rham complex which provides a new diﬀerential with respect to which the harmonic forms are in bijection with the critical points of f . Method (i) has been probably folklore for a long time but it first appeared explicitly in Witten’s paper. It is based on noticing that the moduli space MQP admits P

a compactification M Q which is a compact, topological manifold with boundary so that the boundary verifies the formula: P P R MR × M Q. (1) ∂M Q = R

There are two main ways to prove this formula. One is analytic and regards a flow line from P to Q as a solution of a diﬀerential equation x˙ = −∇ f (x) and studies the properties of such solutions (this method is presented in Schwarz, 1993). A second approach is more topological/dynamical in nature as is described in detail ) y by Weber (2004). Clearly, from formula (1) we immediately deduce y ayx az = 0 and hence d2 = 0. Just a little more work is needed to deduce from here the second property. Method (ii) was the one best known classically and it is essentially implicit in Milnor’s h-cobordism book (Milnor, 1965a). It is based on the observation that ayx can be viewed as follows. First, to simplify slightly the argument assume that the only critical points in f −1 ([ f (y), f (x)]) are x and y. It is well known that for a ∈ f (y), f (x) , there exists a deformation retract r: M(a) = f −1 (−∞, a] → M( f (y) − ) ∪φy D|y| = M where the attaching map φy : S uf (y)− (y) → M( f (y) − )

(2)

is just the inclusion and is small. This deformation retract follows the flow till reaching of U W u (y) ∪ M( f (y) − ) where U(W u (y) is a tubular neighborhood of W u (y) so that the flow is transverse to its boundary and then collapses this neighborhood to W u (y) by the canonical projection. Clearly, applying this remark to each critical point of f provides a CW-complex of the same homotopy type as that of M and with one cell x¯ for each element of x ∈ Crit( f ). To this cellular decomposition we may associate a cellular complex (C ( f ), d ) with the property ) that d x¯ = kyx¯¯ y¯ where kyx¯¯ is, by definition, the degree of the composition: φx

r

u

− M → − M /M( f (y) − ) ψyx : S au (x) −→ M(a) →

(3)

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with φ x : S au (x) → M(a) again the inclusion and where the last map, u, is the projection onto the respective topological quotient space (which is homeomorphic to the sphere S |y| ). Notice now that Myx ⊂ S au (x) is a finite union of points say P1 , . . . , Pk . Imagine a small disk Di ⊂ S au (x) around Pi . The key (but geometrically clear) remark is that the composition of the flow γ together with the retraction r transports Di (if it is chosen suﬃciently small) homeomorphically onto a neighborhood of y inside W u (y). Therefore, the degree of deg(ψyx ) = ayx and thus d = d which shows that d is a diﬀerential and that the homology it computes agrees with the homology of M. As we shall see further, the points of view reflected in the approaches at (i) and (ii) lead to interesting applications which go much beyond “classical” Morse theory. Method (iv), while striking and inspiring appears for now not to have been exploited eﬃciently. 2.1. CONNECTING MANIFOLDS

One way to look to the Morse complex is by viewing the coeﬃcients ayx of the differential as a measure of the 0-dimensional manifold Myx . The question we discuss here is in what way we can measure algebraically the similar higher dimensional moduli spaces. This is clearly a significant issue because, obviously, only a very superficial part of the dynamics of the negative gradient flow of f is encoded in the 0-dimensional moduli spaces of connecting flow lines. As a matter of terminology, the space MQP when viewed inside the unstable sphere S au (P) (with f (P) − a positive and very small) is also called (Franks, 1979) the connecting manifold of P and Q. It was mentioned above that, in general, a connecting manifold MQP is not closed. However, if the critical points P and Q are consecutive in the sense that R ∅, then M P is closed. there does not exist a critical point R so that MRP × MQ Q 2.1.1. Framed cobordism classes An important remark of John Franks (1979) is that connecting manifolds are canonically framed. First recall that a framed manifold V is a submanifold V → S n which has a trivial normal bundle together with a trivialization of this bundle. Two such trivializations are equivalent (and will generally be identified) if they are restrictions of a trivialization of the normal bundle of V × [0, 1] inside S n × [0, 1]. We also recall the Thom – Pontryagin construction in this context (Milnor, 1965b). Assuming V → S n is framed we define a map φV : S n → S codim(V) as follows: consider a tubular neighborhood U(V) of V, use the framing to define a homeomorphism ψ: U(V) → Dk × V where Dk is the closed disk of dimension k =

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p1

codim(V), consider the composition ψ : U(V) − → Dk × V −→ Dk → Dk /S k−1 = S k and define φV by extending ψ outside U(V) by sending each x ∈ S n \U(V) to the base point in Dk /S k−1 = S k . The homotopy class of this map is the same if two framings are equivalent. It is easy to see that two framed manifolds (of the same dimension) are cobordant iﬀ their associated Thom maps are homotopic. We return now to Franks’ remark and notice that the manifolds MQP are framed. First, we make the convention to view MQP as a submanifold of the unstable sphere of P, S au (P) (the other choice would have been to use S s (Q) as ambient manifold). Notice that we have MQP = S au (P) ∩ W s (Q) and this intersection is transversal. Clearly, as W s (Q) is homeomorphic to a disk, its normal bundle in M is trivial and any two trivializations of this bundle are equivalent. This implies that the normal bundle of MQP → S au (P) is also trivial and a trivialization of the normal bundle of W s (Q) provides a trivialization of this bundle which is unique up to equivalence. Recall that if P and Q are consecutive critical points, then MQP is closed. As we have seen that it is also framed we may associate to it a framed cobordism class |Q| *P ∈ π M |P|−1 (S ). Q

Moreover, it is easy to see that the function f may be perturbed without modifying the dynamics of the the negative gradient flow so that the cell attachments corresponding to the critical points Q and P are in succession. Therefore the map ψPQ : S |P|−1 → S |Q| defined as in formula (3) is still defined. The main result of Franks in Franks (1979) is: THEOREM 2.1 (Franks, 1979). Assume P and Q are consecutive critical points *P coincides with the homotopy class of ψP . of f . Up to sign M Q Q The idea of proof of this result is quite simple. All that is required is to make even more precise the constructions used in the approach (ii) used to show d2 = 0 for the Morse complex. For this we fix for W s (Q) a normal framing o which is invariant by translation along the flow γ and which at Q ∈ W s (Q) is given by a basis e of T Q W u (Q) (this is possible because W u (Q) and W s (Q) intersect transversally at Q). We also fix the tubular neighborhood U W u (Q) so that the projection r : U W u (Q) → W u (Q) has the property that (r )−1 (Q) = W s (Q) ∩ U(W u (Q)) and, for any point y ∈ (r )−1 (Q), we have (r )∗ (oy ) = e. Moreover, we may assume that the normal bundle of MQP in S au (P) is just the restriction of the

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normal bundle of W s (Q) (in fact, the two are, in general, only isomorphic and not equal but this is just a minor issue). Now, follow what happens with the framing of MQP along the composition u◦r. For this we write r = r ◦ r where r follows the flow till reaching U W u (Q) . Now pick a point in x ∈ MQP together with its normal frame o x at x. After applying r , the pair (x, o x ) is taken to a pair (x , o x ) with x ∈ ∂ (r )−1 (Q) . Applying now r , the image of (x , o x ) is (Q, e). Take now V a tubular neighborhood of MQP → S au (P) together with an identification V ≈ D|Q| × MQP which is provided by the framing o. The argument above implies that if the constant used to construct the map u: M( f (Q) − ) ∪ W u (Q) = M → M /M( f (Q) − ) is very small, then the composition u ◦ r ◦ r coincides with the relevant Thom – Pontryagin map. 2.1.2. Framed bordism classes and Hopf invariants It is natural to wonder whether, besides their framing, there are some other properties of the connecting manifolds which can be detected algebraically. A useful point of view in this respect turns out to be the following: imagine the elements of MQP as path or loops on M. The fact that they are paths is obvious (we parametrize them by the value of − f ; the negative sign gives the flow lines the orientation coherent with the negative gradient) but they can be transformed into loops very easily. Indeed, fix a simple path in M which joins all the critical points which has the of f and contract this to a point thus obtaining a quotient space M same homotopy type as M. Let q: M → M be the quotient map. We denote by ΩM the space of based loops on M and keep the notation q for the induced map This discussion shows that there are continuous maps ΩM → Ω M. jPQ : MQP → Ω M. These maps have first been defined and used in Cornea (2002a) and they have some interesting properties. For example, given such a map jPQ and assuming that P and Q are consecutive it is natural to ask whether the homology class [MQP ] ∈ H|P|−|Q|−1 (ΩM; Z) is computable (here [MQP ] is the image by jPQ of the fundamental class of MQP ). We shall see that quite a bit more is indeed possible: the full framed bordism class associated to jPQ and to the canonical framing on MQP can be expressed as a relative Hopf invariant. To explain this result first recall that if V → S n is framed and l: V → X is a continuous map with V a closed manifold we may construct a richer Thom – Pontryagin map as follows. We again consider a tubular neighborhood U(V) of V in S n together with an identification U(V) ≈ Dk × V where k = codim(V) which is provided by the framing. We now define a map j

id×l φ¯ V : U(V) → − Dk × V −−−→ Dk × X → (Dk × X)/(S k−1 × X)

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where j: U(V) = Dk × V → V is the projection and the last map is just the quo tient map (which identifies S k−1 × X to the base point). Notice that φ¯ V ∂U(V) ⊂ S k−1 × X. Therefore, we may extend the definition above to a map φ¯ V : S n → (Dk × X)/(S k−1 × X) by sending all the points in the complement of U(V) to the base point. It is well known (and a simple exercise of elementary homotopy theory) that there exists a (canonical) homotopy equivalence (Dk × X)/(S k−1 × X) Σk (X + ) where ΣX is the (reduced) suspension of X, Σi is the suspension iterated i-times and X + is the space X with an added disjoint point (notice also that Σk (X + ) = Σk X ∨ S k where ∨ denotes the wedge or the one point union of spaces). This allows us to view the map φ¯ V as a map with values in Σk (X + ). The framed bordism class of V is simply the homotopy class [φ¯ V ] ∈ πn Σk (X + ) . This is independent of the various choices made in the construction. Two pairs of data (framings included) (V, l) and (V , l ) admit an extension to a manifold W ⊂ S n × [0, 1] with ∂W = V × {0} ∪ V × {1} iﬀ φ¯ V φ¯ V . Notice also that to an element α ∈ πn (Σk X + ) we may associate a homology class [α] ∈ Hn−k (X) obtained by applying the Hurewicz homomorphism, desuspending k-times and projecting on the H∗ (X) term in H∗ (X + ). Returning now to our connecting manifolds MQP we again focus on the case when P and Q are consecutive. The map jPQ together with the canonical framing provide a homotopy class {MQP } ∈ π|P|−1 Σ|Q| (ΩM + ) (to simplify notation we have with M here — the two are homotopy equivalent). replaced M As indicated above, it turns out that this class can be computed in terms of a relative Hopf invariant. We shall now discuss how this invariant is defined. β

α

Assume that S q−1 − → X0 → X and S p−1 → − X → X are two successive cell attachments and that X is a subspace of some larger space X. In particular, X = X0 ∪α Dq , X = X ∪β D p . Let S ⊂ Dq be the q − 1-sphere of radius 12 . There is an important map called the coaction associated to α which is defined by the composition ∇: X → X /S ≈ S q ∨ X where the first map identifies all the points of S to a single one and the second is a homeomorphism (in practice it is convenient to also assume that all the maps and spaces involved are pointed and in that case we view Dq as the reduced cone over S q−1 ). We consider the composition β

∇

id∨ j

− X − → S q ∨ X −−−→ S q ∨ X → S q ∨ X ψ(β, α): S p−1 → and notice that if p2 : S q ∨ X → X is the projection on the second factor, then the composition p2 ◦ψ(β, α) is null-homotopic. This is due to the fact that this compoβ

sition is homotopic to S p−1 → − X → X → X which is clearly null-homotopic.

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We now consider the map p2 : S q ∨ X → X. It is well known in homotopy theory that any map may be transformed into a fibration. In our case this comes down to considering the free path fibration →X t: PX is the set of all continuous path in X parametrized by [0, 1], t(γ) = γ(0). where PX of the resulting We take the pull-back of this fibration over p2 . The total space E q fibration has the same homotopy type as S ∨ X and it is endowed with a canonical projection → X p: ˜ E which replaces p2 , p(z, ˜ γ) = γ(1). It is an exercise in homotopy theory to see that the fibre of the fibration p is homotopic to Σq (ΩX)+ and that, moreover, the inclusion of this fibre in the total space is injective in homotopy. As the composition p2 ◦ ψ(β, α) is homotopically trivial, the homotopy exact sequence of the → X implies that ψ(β, α) admits a lift to ψ(α, ¯ β): S p−1 → Σq (ΩX)+ fibration E whose class does not depend on the choice of lift. We let H(α, β) ∈ homotopy q + π p−1 Σ (ΩX) be equal to this homotopy class and we call it the relative Hopf invariant associated to the attaching maps α and β (for a discussion of the relations between this Hopf invariants and other variants see Chapters 6 and 7 in Cornea et al., 2003). To return to Morse theory, recall from (2) that passing through the two consecutive critical points Q and P leads to two successive attaching maps φQ : S |Q|−1 → M( f (Q) − ) and φP : S |P|−1 → M( f (P) − ) (we assume again — as we may — that the set f −1 ([ f (Q), f (P)]) does not contain any other critical points besides P and Q). Moreover, as we know the inclusion M = M( f (Q) − ) ∪ W u (Q) → M( f (P) − ) is a homotopy equivalence. Therefore, the construction above can be applied to φQ and φP and it leads to a relative Hopf invariant H(P, Q) ∈ π|P|−1 Σ|Q| (ΩM + ) . With these constructions our statement is: THEOREM 2.2 (Cornea, 2002a; 2002b). The homotopy class H(P, Q) coincides (up to sign) with the bordism class {MQP }. In particular, the homology class [MQP ] equals (up to sign and desuspension) the Hurewicz image of H(P, Q). The proof of this result can be found in Cornea (2002b) (a variant proved by a slightly diﬀerent method appears in Cornea, 2002a). The proof is considerably more complicated than the proof of Theorem 2.1 so we will only present a rough justification here. To simplify notation we let M0 = M( f (Q) − ). Let M1 = M( f (Q) − ) ∪ U W u (Q) . Recall, that the inclusions M → M1 → M( f (P) − ) are homotopy equivalences.

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Let P: ΩM → PM → M be the path-loop fibration (of total space the based paths on M and of fibre the space of based loops on M). We denote by E0 the total space of the pull-back of the fibration P over the inclusion M0 ⊂ M. Similarly, we let E1 be the total space of the pull-back of P over the inclusion M1 → M. The key remark is that the attaching map φP : S u (P) → M1 admits a natural lift 'P : S u (P) → E1 . Indeed, we assume here that all the critical points to a map φ are identified to the base point. The space E1 consists of the based paths in M that end at points in M1 . But each element of the image of φP corresponds to precisely such a path which is explicitly given by the corresponding flow line (we need to use here Moore paths and loops which are paths parametrized by arbitrary intervals [0, a] and not only the interval [0, 1]). Consider the inclusion E0 → E1 . It is not diﬃcult to see that the quotient topological space E1 /E0 admits a canonical homotopy equivalence η : E1 /E0 → Σ|Q| (ΩM + ). Therefore, we may consider the 'P . It is possible to show that this map η is homotopic to composition η = η ◦ φ 'P to M P coincides with H(P, Q). At the same time, we see that the restriction of φ Q 'P is homotopic jPQ . Moreover, by making explicit η it is also possible to see that φ P to the Thom – Pontryagin map associated to MQ . 2.1.3. Some topological applications We now describe a couple of topological applications of Theorem 2.2. The idea behind both of them is quite simple: the function − f is also a Morse function and the critical points Q, P are consecutive critical points for − f . Therefore, the connecting manifold MPQ is well-defined as well as its associated bordism homotopy class {MPQ }. Clearly, the underlying space for both MQP and MPQ is the same. The P map jQ P is diﬀerent from jQ just by reversing the direction of the loops. The two relevant framings may also be diﬀerent. The relation between them is somewhat less straightforward but it still may be understood by considering M embedded inside a high-dimensional Euclidean space and taking into account the twisting induced by the stable normal bundle. In all cases, this establishes a relationship between the two Hopf invariants H(P, Q) and H(Q, P).

A. The first application (Cornea, 2002a) concerns the construction of examples of nonsmoothable, simply-connected, Poincar´e duality spaces. The idea is as follows: we construct Poincar´e duality spaces which have a simple CW-decomposition and with the property that for certain two successively attached cells e, f the resulting Hopf invariant H and the Hopf invariant H associated to the dual cells f , e are not related in the way described above. If the respective Poincar´e duality space is smoothable, then the given cell decomposition can be viewed as associated to an appropriate Morse function and this leads to a contradiction. The obstructions to smoothability constructed in this way are obstructions to the lifting of the Spivak normal bundle to BO. This is an obstruction theory problem but one

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which can be very diﬃcult to solve directly in the presence of many cells. Thus, this approach is quite eﬃcient to construct examples. B. The second application (Cornea, 2004) concerns the detection of obstructions to the embedding of CW-complexes in Euclidean spaces in low codimension. The argument in this case goes roughly as follows. If the CW-complex X embeds in S n , then we may consider a neighborhood U(X) of X which is a smooth manifold with boundary. We consider a smooth Morse function f : U(X) → R which is constant, maximal and regular on the boundary of U(X). If P and Q are two consecutive critical points for this function we obtain that Σk H(P, Q) = Σk H(Q, P) for certain values of k and k which can be estimated explicitly — the main reason for this equality is that the Morse function in question is defined on the sphere so all the questions involving the stable normal bundle become irrelevant. If X admits some reasonably explicit cell-decomposition it is possible to express H(P, Q) as the Hopf invariant H of some successive attachment of two cells e, f and H(Q, P) as Σk H where H is another similar Hopf invariant. The obstructions to embedding appear because the low codimension condition translates to the fact that k + k > k. This can be viewed as an obstruction because it means that after k suspensions the homotopy class of H has to desuspend more than k-times. REMARK 2.3. The applications at A and B are purely of homotopical type. It is natural to expect that the Morse theoretical arguments that were used to establish these statements can be replaced by purely homotopical ones but this has not been done till now.

2.1.4. The Serre spectral sequence Theorem 2.2 provides considerable information on connecting manifolds for pairs of consecutive critical points. However, it does not shed any light on the case of nonconsecutive ones. Clearly, if the critical points are not consecutive the respective connecting manifold is not closed and thus no bordism or cobordism class can be directly associated to it. However, after compactification, the boundary of this connecting manifold has a special structure reflected by equation (1). As we shall see following Barraud and Cornea (2003), this structure is suﬃcient to construct an algebraic invariant which provides an eﬃcient “measure” of all connecting manifolds. This construction is based on the fact that the maps jPQ : MQP → ΩM are compatible with compactification and with the formula (1) in the following sense. Recall that here ΩM are the based Moore loops on M (these are loops parametrized by intervals [0, a]), the critical points of f have been identified to

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a single point and, moreover, in the definition of jPQ we use the parametrization of the flow lines by the values of − f . Recall that we have a product given by the concatenation of loops µ: ΩM × ΩM → ΩM. With these notations it is easy to see that we have the following formula: jPQ (u, v) = µ( jRP (u), jRQ (v)) P

R

(4)

P

where (u, v) ∈ M R × M Q ⊂ ∂M Q . We proceed with our construction. Let C∗ (X) be the (reduced) cubical complex of X with coeﬃcients in Z/2. Notice that there is a natural map Ck (X) ⊗ Ck (Y) → Ck+k (X × Y). x

A family of cubical chain syx ∈ C|x|−|y|−1| (M y ), x, y ∈ Crit( f ) is called a representing chain system for the moduli spaces Myx if for each pair of critical points x, z we have: (i) y dszx = syx ⊗ sz y

(ii) szx represents the fundamental class in H|x|−|z|−1 (Mzx , ∂Mzx ). It is easy to show by induction on the index diﬀerence |x| − |z| that such representing chain systems exist. We now fix such a representing chain system {syx } and we define ayx ∈ C|x|−|y|−1 (ΩM) by ayx = ( jyx )∗ (syx ). Notice that this definition extends the definition of these coeﬃcients in the usual Morse case when |x| − |y| − 1 = 0. We have a product map C∗ (µ)

·: Ck (ΩM) ⊗ Ck (ΩM) → Ck+k (ΩM × ΩM) −−−−→ Ck+k (ΩM) which makes C∗ (ΩM) into a diﬀerential ring. The discussion above shows that inside this ring we have the formula dazx = azx · azy . y

An elegant way to rephrase this formula is to group these coeﬃcients in a matrix A = (ayx ) and then we have (5) dA = A2 .

HOMOTOPICAL DYNAMICS IN SYMPLECTIC TOPOLOGY

We now define a new chain complex C( f ) associated to f by C( f ) = (C∗ (ΩM) ⊗ Z/2Crit( f ), d), dx = ayx ⊗ y.

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(6)

y

We shall call this complex the extended Morse complex of f . Here, C∗ (ΩM) ⊗ Z/2Crit( f ) is viewed as a graded C∗ (ΩM)-module and d respects this structure in the sense that it verifies d(a ⊗ x) = (da) ⊗ x + a(dx) (the grading on Crit( f ) is given, as before, by the Morse index). Choosing orientations on all the stable manifolds of all the critical points induces a co-orientation on all the unstable manifolds, and hence an orientations on the intersections W u (P) ∩ W s (Q) and finally on all the moduli spaces MQP : we may then use Z-coeﬃcients for this complex as well as, of course, for the classical Morse complex. In this case appropriate signs appear in the formulas above. Clearly, d2 = 0 due to (5). By definition, the coeﬃcients ayx represent the moduli spaces Myx . However, these coeﬃcients are not invariant with respect to the choices made in their construction. Therefore, it is remarkable that there is a natural construction which extracts from this complex a useful algebraic invariant which is not just the homology of the complex — as it happens, this homology is not too interesting as it coincides with that of a point. Consider the obvious diﬀerential filtration which is defined on this complex by F k C( f ) = C∗ (ΩM) ⊗ Z/2x ∈ Crit( f ) : ind f (x) ≤ k. Denote the associated spectral sequence by E( f ) = (E rp,q ( f ), dr ). THEOREM 2.4 (Barraud and Cornea, 2003). When M is simply connected and if r ≥ 2 the spectral sequence E( f ) coincides with the Serre spectral sequence of the path-loop fibration P: ΩM → PM → M. REMARK 2.5. (a) A similar result can be established even in the absence of the simple-connectivity condition which has been assumed here to avoid some technical complications. (b) The Serre spectral sequence of the path-loop fibration of a space X contains considerable information on the homotopy type of the space. In particular, there are spaces with the same cohomology and cup-product but which may be distinguished by their respective Serre spectral sequences.

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To outline the proof of the theorem we start by recalling the construction of the Serre spectral sequence in the form which will be of use here. We shall assume here that the Morse function f is self-indexed (in the sense that for each critical point x we have ind f (x) = k ⇒ f (x) = k) and that it has a single minimum denoted by m. Let Mk = f −1 (−∞, k + ] . We have Dky Mk = Mk−1 φy

where the union is taken over all the critical points y ∈ Critk ( f ) and φy : S u (y) → Mk−1 are the respective attaching maps. Denote by Ek the total space of the fibration induced by pull-back over the inclusion Mk → M from the fibration P. Consider the filtration of C∗ (PM) given by F k P = Im C∗ (Ek ) → C∗ (PM) . The spectral sequence induced by this filtration is invariant after the second page and is precisely the Serre spectral sequence (this spectral sequence may be constructed as above but by using an arbitrary skeletal filtration {Xk } of a space X which has the same homotopy type as that of M; in our case the particular filtration given by the sets Mk is a natural choice). For further use, we also notice that there is an obvious action of ΩM on PM and this action induces one on each Ek . Therefore, we may view C∗ (Ek ) as a C∗ (ΩM)-module. The first step in proving the theorem is to consider a certain compactification of the unstable manifolds of the critical points of f . Recall that f is self-indexed and that m is the unique minimum critical point of f . Fix x ∈ Crit( f ) and define x the following equivalence relation on the set M m × [0, f (x)]: (a, t) ∼ (a , t )

iﬀ

t = t and a(−τ) = a (−τ) ∀τ ≥ t.

x

Here the elements of M m are viewed as paths in M parametrized by the value of − f (so that f a(−τ) = τ). Denote by W(x) the resulting quotient topological space. Notice that, if x u y ∈ W (x), then there exists some a ∈ M m so that y is on the (possibly broken) flow line represented by a. Or, in other words, so that a(− f (y)) = y. This path a might not be unique. Indeed, inside W(x) there is precisely one equivalence class [a, f (y)] (with a − f (y) = y) for each (possibly) broken flow line joining x to y. Clearly, if y ∈ W u (x), then there is just one such flow line and so the natural surjection π: W(x) → W u (x), π([a, t]) = a(−t) as a is a homeomorphism when restricted to π−1 (W u (x)). Thus we may view W(x) u u special compactification of W (x) or as a desingularization of W (x). It is not difficult to believe (but harder to show and will not be proved here; see Barraud and

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Cornea, 2003) that W(x) is a topological manifold with boundary and moreover ∂W(x) = Myx × W(y). (7) y

We continue the proof of the Theorem 2.4 with the remark that there are obvious maps h x : W(x) → PM which associate to [a, t] the path in M which follows a from x to a(−t). These maps and the maps jyx are compatible with formula (7) in the sense that h x (a , [a , t]) = jyx (a ) · hy ([a , t])

(8)

and · represents the action of ΩM on PM. where (a , [a , t]) ∈ Myx × W(y) Now, we may obviously rewrite (7) as: x M y × W(y). ∂W(x) = y

Given the representing chain system {syx } it is easy to construct an associated representing chain system for W(x). This is a system of chains v(x) ∈ C|x| W(x) and we have the so that v(x) represents the fundamental class of C|x| W(x), ∂W(x) formula dv(x) = syx ⊗ v(y). y

Finally, we define a C∗ (ΩM)-module chain map α: C( f ) → C∗ (PM) by α(x) = (h x )∗ (v(x)). The formulas above show that we have x d (h x )∗ v(x) = ay · (hy )∗ v(y) y

and so α is a chain map. It is clear that the map α is filtration preserving and it is not diﬃcult to see that it induces an isomorphism at the E 2 level of the induced spectral sequences and this concludes the proof of Theorem 2.4. REMARK 2.6.

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(a) Another important but immediate property of W(x) is that it is a contractible x space. Indeed, all the points in M m × { f (x)} are in the same equivalence class. Moreover, each point [a, t] ∈ W(x) has the property that it is related by the path [a, τ], τ ∈ [t, f (x)] to ∗ = [a, f (x)]. The contraction of W(x) to ∗ is obtained by deforming W(x) along these paths. Given that W(x) is a contractible topological manifold with boundary, it is natural to suspect that W(x) is homeomorphic to a disk. This is indeed the case as is shown in Barraud and Cornea (2003) and is an interesting fact in itself because it implies that the union of the unstable manifolds of a self-indexed Morse – Smale function gives a CW-decomposition of M. The attaching map of the cell W(x) is simply the restriction of π to ∂W(x). (b) The Serre spectral sequence result above and the bordism result in Theorem 2.2 are obviously related via the central role of the maps jPQ . There is also a more explicit relation. Indeed, (a stable version of) the Hopf invariants appearing in Theorem 2.2 can be interpreted as diﬀerentials in the Atyiah – Hirzebruch – Serre spectral sequence of the path-loop fibration with coeﬃcients in the stable homotopy of ΩM. Moreover, the relation (1) can be understood as also keeping track of the framings. This leads to a type of extended Morse complex in which the coeﬃcients of the diﬀerential are stable Hopf invariants (Cornea, 2002a). All of this strongly suggests that the construction of the complex C( f ) can be enriched so as to include the framings of the connecting manifolds and, by the same method as above, that the whole Atyiah – Hirzebruch – Serre spectral sequence should be recovered from this construction. (c) Another interesting question, open even for consecutive critical points P, Q, is whether there are some additional constraints on the topology of the connecting manifolds MQP besides those imposed by Theorem 2.2. (d) Yet another open question is how this machinery can be adapted to the Morse – Bott situation and how it can be extended to general Morse – Smale flows (not only gradientlike ones). (e) It is natural to wonder what is the richest level of information that one can extract out of the moduli spaces of Morse flow lines. At a naive level, the union of all the points situated on the flow lines of f is precisely the whole underlying manifold M so we expect that there should exist some assembly process producing the manifold M out of these moduli spaces. Such a machine has been constructed by Cohen et al. (1995b; 1995a). They show that one can form a category out of the moduli spaces of connecting trajectories and that the classifying space of this category is of the homeomorphism type of the underlying manifold. In their construction an essential point is that the gluing of flow lines is associative. This approach is quite diﬀerent from

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the techniques above and does not imply the results concerning the extended Morse complex or the Hopf invariants that we have presented. The two points of view are, essentially, complementary. To end this section it is useful to make explicit a relation between Theorems 2.4 and 2.2 (we assume as above that M is simply-connected). PROPOSITION 2.7. Assume that there are q, p ∈ N so that in the Serre spectral 2 = 0 for q < k < p and there sequence of the path loop fibration of M we have Ek,s is an element a ∈ E 2p,0 so that d p−q a 0, then any Morse-Smale function on M has a pair of consecutive critical points P, Q of indexes at least q and at most p so that the homology class [MQP ] ∈ H|P|−|Q|−1 (ΩM) 0. Clearly, Theorem 2.4 directly implies that, even without any restriction on E 2 , if we have dr a 0 with a ∈ E rp,0 , then for any Morse – Smale function f there are critical points P and Q with |P| = p and |Q| = p − r so that MQP ∅. Indeed, if this would not be the case, then all the coeﬃcients ayx in the extended Morse complex of f are null whenever |x| = p, |y| = p − r. By the construction of the associated spectral sequence, this leads to a contradiction. However, the pair P, Q resulting from this argument might have a connecting manifold which is not closed so that its homology class is not even defined and, thus, Proposition 2.7 provides a stronger conclusion. The proof of the proposition is as follows. Recall that E 2s,r ≈ H s (M) ⊗ Hr (ΩM) and so H∗ (M) = 0 for q ≤ 0 ≤ p. If there are some points P, Q ∈ Crit( f ) with q ≤ |Q|, |P| ≤ p so that the diﬀerential of P in the classical Morse complex contains Q with a nontrivial coeﬃcient then this pair P, Q may be taken as the one we are looking for. If all such diﬀerentials in the classical Morse complex are trivial it follows that the critical points of index p and q are consecutive. In this case, the geometric arguments used in the proofs of either Theorem 2.2 or 2.4 imply that if for all pairs P, Q with |P| = p, |Q| = q we p−q would have [MQP ] = 0, then the diﬀerential d p−q would vanish on E p,0 . REMARK 2.8. Notice that the pair of critical points P and Q constructed in the proposition verify the property that |P| and |Q| are consecutive inside the set {ind f (x) : x ∈ Crit( f )}.

2.2. OPERATIONS

We discuss here a diﬀerent and, probably, more familiar approach to understanding connecting manifolds as well as other related Morse theoretic moduli spaces. This point of view has been used extensively by many authors — Fukaya (1993),

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Betz and Cohen (1994) being just a few of them. For this reason we shall review this technique very briefly. Given two consecutive critical points x, y notice that the set T yx = W u (x) ∩ W s (y) is homeomorphic to the unreduced suspension of Myx . Therefore, we may see this as an obvious inclusion iyx : ΣMyx → M and we may consider the homology class [T yx ] = (iyx )∗ (s[Myx ]) where s is suspension and [Myx ] is the fundamental class. There exists an obvious evaluation map e: ΣΩM → M which is induced by ΩM × [0, 1] → M, (β, t) → β(t) (the loops here are parametrized by the interval [0, 1] but this is a minor technical diﬃculty). It is easy to see, by the definition of this evaluation map, that [T yx ] = e∗ ( jyx )∗ ([Myx ]) . In general the map e∗ is not injective in homology. Clearly, the full bordism class {Myx } carries much more information than the homology class [T yx ]. Still, there is a direct way to determine [T yx ] without passing through a calculation of {Myx } and we will now describe it. Consider a second Morse – Smale function g: M → R so that its associated unstable and stable manifolds Wgu (−), Wgs (−) intersect transversally the stable and unstable manifolds of f and, except if they are of top dimension, they avoid the critical points of f . Fix x, y ∈ Crit( f ) and s ∈ Crit(g) so that |x| − |y| − indg (s) = 0. We may define k(x, y; s) = #(T yx ∩ Wgs (s) (where the counting takes into account the relevant orientations if we work over Z). We now put k¯ yx = k(x, y; s)s ∈ C(g). s

The essentially obvious claim is that: PROPOSITION 2.9. The chain k¯ yx is a cycle whose homology class is [T yx ]. ) Indeed we have s k(x, y; s)hzs = 0 where indg (z) = indg (s) − 1 and hzs are the coeﬃcients in the classical Morse complex of g. This equality is valid because we may consider the 1-dimensional space T yx ∩ Wgs (z). This is an open 1-dimensional manifold whose compactification is a 1-manifold whose boundary points are ) counted precisely by the formula s k(x, y; s)hzs = 0. By basic intersection theory it is immediate to see that the homology class represented by this cycle is [T yx ]. While this construction does not shed a lot of light on the properties of Myx its role is important once we use it to recover the various homological operations of

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M. To see how this is done from our perspective notice that the intersection T yx ∩ Wgs (s) = W uf (x) ∩ W sf (y) ∩ Wgs (s) can be viewed as a particular case of the following situation: assume that f1 , f2 , f3 are three Morse-Smale functions in general position and define x,y T z = W uf1 (x) ∩ W uf2 (y) ∩ W sf3 (z) . x,y

If we assume that |x| + |y| − |z| − n = 0 we may again count the points in T z with x,y x,y appropriate signs and we may define coeﬃcients tz = #T z . This leads to an operation (Betz and Cohen, 1994; Fukaya, 1993) C( f1 ) ⊗ C( f2 ) → C( f3 ) given as a linear extension of x⊗y→

x,y

tz z.

z

It is easy to see that this operation descends in homology and that it is in fact x,y the dual of the ∪-product. Moreover, the space T z may be viewed as obtained by considering a graph formed by three oriented edges meeting into a point with the first two entering the point and the other one exiting it and considering all the configurations obtained by mapping this graph into M so that to the first edge we associate a flow line of f1 which exits x, to the second edge a flow line of f2 which exits y and to the third a flow line of f3 which enters z. Clearly, this idea may be pushed further by considering other, more complicated graphs and understanding what are the operations they correspond to as was done by Betz and Cohen (1994). 3. Applications to symplectic topology We start with some applications that are rather “soft” even if diﬃcult to prove and we shall continue in the main part of the section,Section 3.2, with some others that go deeper. 3.1. BOUNDED ORBITS

We fix a symplectic manifold (M, ω) which is not compact. Assume that H: M → R is a smooth Hamiltonian whose associated Hamiltonian vector field is denoted by XH . One of the main questions in Hamiltonian dynamics is whether a given regular hypersurface A = H −1 (a) of H has any closed characteristics, or equivalently, whether the Hamiltonian flow of H has any periodic orbits in A. As M is not compact, from the point of view of dynamical systems, the first natural question

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is whether XH has any bounded orbits in A. Moreover, there is a remarkable result of Pugh and Robinson (1983), the C 1 -closing lemma, which shows that, for a generic choice of H, the presence of bounded orbits insures the existence of some periodic orbits. Therefore, we shall focus in this subsection on the detection of bounded orbits. It should be noted however that the detection of periodic orbits in this way is not very eﬀective because the periods of the orbits found can not be estimated. Moreover, there is no reasonable test to decide whether a given Hamiltonian belongs to the generic family to which the C 1 -closing lemma applies. Finally, it will be clear from the methods of proof described below that these results are also soft in the sense that they are not truly specific to Hamiltonian flows but rather they apply to many other flows. An example of a bounded orbit result is the following statement (Cornea, 2002b). THEOREM 3.1. Assume that H is a Morse – Smale function with respect to a Riemannian metric g on M so that M is metrically complete and there exists an and a compact set K ⊂ M so that ∇g H(x) ≥ for x K. Suppose that P and Q are two critical points of H so that |P| and |Q| are successive in the set {indH (x) : x ∈ Crit(H)}. If the stabilization [H(P, Q)] ∈ πS|P|−|Q|−1 (ΩM) of the Hopf invariant H(P, Q) is not trivial then there are regular values v ∈ H(Q), H(P) so that H −1 (v) contains bounded orbits of XH . Before describing the proof of this result let’s notice that the theorem is not diﬃcult to apply. Indeed, one simple way to verify that there are pairs P, Q as required is to use Proposition 2.7 together with Remark 2.8 with a minor adaptation required in a noncompact setting. This adaptation consists of replacing the Serre spectral sequence of the path loop fibration with the Serre spectral sequence of a relative fibration ΩM → (E1 , E0 ) → (N1 , N0 ) where N1 is an isolating neighborhood for the gradient flow of H which contains K and N0 is a (regular) exit set for this neighborhood (to see the precise definition of these Conley index theoretic notions see Salamon, 1985). The fibration is induced by pull-back from the path-loop fibration ΩM → PM → M over the inclusion (N1 , N0 ) → (M, M). In short, because the gradient of H is away from 0 outside of a compact set, pairs (N1 , N0 ) as above are easy to produce and if the pair (N1 , N0 ) has some interesting topology it is easy to deduce the existence of nonconstant bounded orbits. Here is a concrete example. COROLLARY 3.2. Assume that M is the cotangent bundle of some closed, simply-connected manifold N of dimension k ≥ 2 and ω is an arbitrary symplectic form. Assume that H: M → R is Morse and that outside of some compact set containing the 0 section, H restricts to each fibre of the bundle to a nondegenerate quadratic form. Then, XH has bounded, nonconstant orbits.

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Of course, this result is only interesting when there are no compact level hypersurfaces of H. This does happen if the quadratic form in question has an index which is neither 0 nor k. The proof of this result comes down to the fact that as N is closed and not a point there exists a lowest dimensional homology class u ∈ Ht (N) which is transgressive in the Serre spectral sequence (this means dt u 0). Using the structure of function quadratic at infinity of H it is easy to construct a pair (N1 , N0 ) where N1 is is homotopic to a disk bundle of base N and N0 is the associated sphere bundle. The spectral sequence associated to this pair can be related by the Thom isomorphism to the Serre spectral sequence of the path-loop fibration over N and the element u¯ ∈ H∗ (N1 , N0 ) which corresponds to u by the Thom isomorphism will have a nonvanishing diﬀerential. This means that Proposition 2.7 may be used to show the nontriviality of a homology class [MQP ] for P and Q as in Theorem 3.1. By Theorem 2.2, [MQP ] is the same up to sign as the homology class of the Hopf invariant H(P, Q) so Theorem 3.1 is applicable to detect bounded orbits. We now describe the proof of the theorem. The basic idea of the proof is simpler to present in the particular case when H −1 H(Q), H(P) does not contain any critical value. In this case let A = H −1 (a) where a ∈ H(Q), H(P) . We intend to show that A contains some bounded orbits of XH . To do this notice that the two sets S 1 = W u (P) ∩ A, S 2 = W s (Q) ∩ A are both diﬀeomorphic to spheres. We now assume that no bounded orbits exist and we consider a compact neighborhood U of S 1 ∪ S 2 . Assume that we let S 2 move along the flow XH . As this flow has no bounded orbits, each point of S 2 will leave U at some moment. Suppose that we are able to perturb the flow induced by XH to a new deformation η: M × R → M so that for some finite time T all the points in S 2 are taken simultaneously outside U (in other words ηT (S 2 ) ∩ U = ∅) and so that η leaves Q fixed. It is easy to see that this implies that S 1 ∩ S 2 is bordant to the empty set which, by Theorem 2.2, is impossible because H(P, Q) 0. This perturbation η is in fact not hard to construct by using some elements of Conley’s index theory and the fact that the maximal invariant set of XT inside U is the empty set (the main step here is to possibly modify also U so that it admits a regular exit region U0 ⊂ U and we then construct η so that it follows the flow lines of XT but stops when reaching U0 , this eliminates the problem of “bouncing” points which first exit U but later re-enter it). The case when there are critical points in H −1 H(Q), H(P) follows the same idea but is considerably more diﬃcult. The main diﬀerence comes from the fact that the sets S 1 and S 2 might not be closed manifolds. Even their closures S 1 and S 2 are not closed manifolds in general but might be singular sets. To be able to proceed in this case we first replace P and Q with a pair of critical points of the same index so that for any critical point Q ∈ H −1 H(Q), H(P) with indH (Q ) = indH (Q) we have [H(P, Q )] = 0. We then take a very close to H(Q) so that S 2

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at least is diﬀeomorphic to a sphere. We then first study the stratification of S¯1 : there is a top stratum of dimension |P| − 1 which is S 1 and a singular stratum S of dimension |Q| − 1 which is the union of the sets W u (Q ) ∩ A for all Q so that MQP ∅ and |Q | = |Q|. Notice that the way to construct the null-bordism of S 1 ∪ S 2 is to consider in A × [0, T ] the submanifold W = (ηt (S 2 ), t) and intersect it with W = S¯1 × [0, T ] — we assume here ηT (S 2 ) ∩ S 1 = ∅. Clearly, we need this intersection to be transverse and this can be easily achieved by a perturbation of η. The main technical diﬃculty is that L might intersect the singular part, S × [0, T ]. Indeed, dim(W) = n − q, dim(S ) = q − 1, dim(A) = n − 1 and so generically the intersection I between W and S ×[0, T ] is 0-dimensional and not necessarily void. By studying the geometry around each of the points of I it can be seen that S 1 ∩S 2 is bordant to the union of the MQP ’s where Q ∈ H −1 H(Q), H(P) (roughly, this follows by eliminating from the singular bordism W ∩ W a small closed, conelike neighborhood around each singular point and showing that the boundary of this conelike neighborhood is homeomorphic to a MQP ). We now use the fact that all the stable bordism classes of the MQP ’s vanish (because [H(P, Q )] = 0) and this leads us to a contradiction. Notice also that, at this point, we need to use stable Hopf invariants (or bordism classes) ∈ πS (ΩM) because, by contrast to the stable case, the unstable Thom – Pontryagin map associated to a disjoint union is not necessarily equal to the sum of the Thom – Pontryagin maps of the terms in the union and hence, unstably, even if we know H(P, Q ) = 0, ∀Q we still can not deduce H(P, Q) = 0. REMARK 3.3. It would be interesting to see whether, under some additional assumptions, a condition of the type [H(P, Q)] 0 implies the existence of periodic orbits and not only bounded ones.

3.2. DETECTION OF PSEUDOHOLOMORPHIC STRIPS AND HOFER’S NORM

In this subsection we shall again use the Morse theoretic techniques described in Section 2 and, in particular, Theorem 2.4 to study some symplectic phenomena by showing that Floer’s complex can be enriched in a way similar to the passage from the classical Morse complex to the extended one. 3.2.1. Elements of Floer’s theory We start by recalling very briefly some elements from Floer’s construction (for a more complete exposition see, for example, Salamon, 1999). We shall assume from now on that (M, ω) is a symplectic manifold — possibly noncompact but in that case convex at infinity — of dimension m = 2n. We also assume that L, L are closed (no boundary, compact) Lagrangian submanifolds of M which intersect transversally.

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To start the description of our applications it is simplest to assume for now that L, L are simply-connected and that ω|π2 (M) = c1 |π2 (M) = 0. Cotangent bundles of simply-connected manifolds oﬀer immediate examples of manifolds verifying these conditions. We fix a path η ∈ P(L, L ) = {γ ∈ C ∞ ([0, 1], M) : γ(0) ∈ L, γ(1) ∈ L } and let Pη (L, L ) be the path-component of P(L, L ) containing η. This path will be trivial homotopically in most cases, in particular, if L is Hamiltonian isotopic to L . We also fix an almost complex structure J on M compatible with ω in the sense that the bilinear form X, Y → ω(X, JY) = α(X, Y) is a Riemannian metric. The set of all the almost complex structures on M compatible with ω will be denoted by Jω . Moreover, we also consider a smooth 1-periodic Hamiltonian H: [0, 1] × M → R which is constant outside a compact set and its associated 1-periodic family of Hamiltonian vector fields XH determined by the equation t ω(XH , Y) = −dHt (Y),

∀Y.

we denote by φtH the associated Hamiltonian isotopy. We also assume that φ1H (L) intersects transversally L . In our setting, the action functional below is well-defined: 1 ∗ AL,L ,H : Pη (L, L ) → R, x → − x¯ ω + H t, x(t) dt (9) 0

where x¯(s, t): [0, 1] × [0, 1] → M is such that x¯(0, t) = η(t), x¯(1, t) = x(t), ∀t ∈ [0, 1], x([0, 1], 0) ⊂ L, x([0, 1], 1) ⊂ L . The critical points of A are the orbits of XH that start on L, end on L and which belong to Pη (L, L ). These orbits are in bijection with a subset of φ1H (L) ∩ L so they are finite in number. If H is constant these orbits coincide with the intersection points of L and L which are in the class of η. We denote the set of these orbits by I(L, L ; η, H) or shorted I(L, L ) if η and H are not in doubt. We now consider the solutions u of Floer’s equation: ∂u ∂u + J(u) + ∇H(t, u) = 0 ∂s ∂t

(10)

with u(s, t): R × [0, 1] → M,

u(R, 0) ⊂ L,

u(R, 1) ⊂ L.

When H is constant, these solutions are called pseudo-holomorphic strips. For any strip u ∈ S(L, L ) = {u ∈ C ∞ (R×[0, 1], M) : u(R, 0) ⊂ L, u(R, 1) ⊂ L } consider the energy ++ ++2 ++ ++2 ∂u + + ∂u 1 + t ++ ++ + ++ − XH (u)+++ ds dt. (11) E L,L ,H (u) = 2 R×[0,1] ∂s ∂t

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For a generic choice of J, the solutions u of (10) which are of finite energy, E L,L ,H (u) < ∞, behave like negative gradient flow lines of A. In particular, A decreases along such solutions. We consider the moduli space M = {u ∈ S(L, L ) : u verifies (10), E L,L ,H (u) < ∞}.

(12)

The translation u(s, t) → u(s + k, t) obviously induces an R action on M and we let M be the quotient space. For each u ∈ M there exist x, y ∈ I(L, L ; η, H) such that the (uniform) limits verify lim u(s, t) = x(t),

s→−∞

lim u(s, t) = y(t).

s→+∞

(13)

We let M (x, y) = {u ∈ M : u verifies (13)} and M(x, y) = M (x, y)/R so that M = x,y M(x, y). If needed, to indicate to which pair of Lagrangians, to what Hamiltonian and to what almost complex structure are associated these moduli spaces we shall add L and L , H, J as subscripts (for example, we may write ML,L ,H,J (x, y)). For x, y ∈ I(L, L ; η, H) we let S(x, y) = {u ∈ C ∞ ([0, 1] × [0, 1], M) : u([0, 1], 0) ⊂ L, u([0, 1], 1) ⊂ L , u(0, t) = x(t), u(1, t) = y(t)}. To each u ∈ S (x, y) we may associate its Maslov index µ(u) ∈ Z (Viterbo, 1987) and it can be seen that, in our setting, this number only depends on the points x, y. Thus, we let µ(x, y) = µ(u). Moreover, we have the formula µ(x, z) = µ(x, y) + µ(y, z).

(14)

According to these relations, the choice of an arbitrary intersection point x0 and the normalization |x0 | = 0, defines a grading | · | such that: µ(x, y) = |x| − |y|. There is a notion of regularity for the pairs of (H, J) so that, when regularity is assumed, the spaces M (x, y) are smooth manifolds (generally noncompact) of dimension µ(x, y) and in this case M(x, y) is also a smooth manifold of dimension µ(x, y) − 1. Regular pairs (H, J) are generic and, in fact, they are so even if L and L are not transversal (but in that case H can not be assumed to be constant), for example, when L = L . Floer’s construction is natural in the following sense. Let L = (φ1H )−1 (L ). Consider the map bH : P(L, L ) → P(L, L ) defined by bH (x) (t) = φtH x(t) . Let η ∈ P(L, L ) be such that η = bH (η ). Clearly, bH restricts to a map between Pη (L, L ) and Pη (L, L ) and it restricts to a bijection I(L, L ; η , 0) → I(L, L ; η, H).

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It is easy to also check AL,L ,H bH (x) = AL,L ,0 (x) and that the map bH identifies the geometry of the two action functionals. Indeed for u: R × [0, 1] → M with u(R, 0) ⊂ L, u(R, 1) ⊂ L , u˜ (s, t) = φt u(s, t) , J˜ = φ∗ J we have ∂u ∂˜u ∂u ∂˜u φ∗ + J˜ = +J − XH . ∂s ∂t ∂s ∂t Therefore, the map bH induces diﬀeomorphisms: bH : ML,L , J,0 ˜ (x, y) → ML,L ,J,H (x, y) where we have identified x, y ∈ L ∩ L with their orbits φtH (x) and φtH (y). Finally, the noncompactness of M(x, y) for x, y ∈ I(L, L ; η, H) is due to the fact that, as in the Morse-Smale case, a sequence of strips un ∈ M(x, y) might “converge” (in the sense of Gromov) to a broken strip. There are natural compactifications of the moduli spaces M(x, y) called Gromov compactifications and denoted by M(x, y) so that each of the spaces M(x, y) is a topological manifold with corners whose boundary verifies: ∂M(x, y) = M(x, z) × M(z, y). (15) z∈I(L,L ;η,H)

A complete proof of this fact can be found in Barraud and Cornea (2003) (when dim(Myx ) = 1 the proof is due to Floer and is now classical). 3.2.2. Pseudoholomorphic strips and the Serre spectral sequence We will now construct a complex C(L, L ; H, J) by a method that mirrors the construction of C( f ) in Section 2.1.4. This complex, called the extended Floer complex associated to L, L , H, J has the form: C(L, L ; H, J) = (C∗ (ΩL) ⊗ Z/2I(L, L ; η, H), D) where the cubical chains C∗ (ΩL) have, as before, Z/2-coeﬃcients. If needed, the moduli spaces M(x, y) can be endowed with orientations which are compatible with formula (15), and so we could as well use Z-coeﬃcients. To define the diﬀerential we first fix a simple path w in L which joins all the points γ(0), γ ∈ I(L, L ; H) and we identify all these points to a single one by collapsing this path to a single point. We shall continue to denote the resulting space by L to simplify notation. For each moduli space M(x, y) there is a continuous map lyx : M(x, y) → ΩL

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which is defined by associating to u ∈ M(x, y) the path u(R, 0) parametrized by the (negative) values of the action functional A. This is a continuous map and it is seen to be compatible with formula (15) in the same sense as in (8). We pick a representing chain system {kyx } for the moduli spaces M(x, y) and we let myx = (lyx )∗ (kyx ) ∈ C∗ (ΩL) and Dx =

myx ⊗ y.

(16)

y

As in the case of the extended Morse complex the fact that D2 = 0 is an immediate consequence of formula (15). REMARK 3.4. (a) There is an apparent asymmetry between the roles of L and L in the definition of this extended Floer complex. In fact, the coeﬃcients of this complex belong naturally to an even bigger and more symmetric ring than C∗ (ΩL). Indeed, consider the space T (L, L ) which is the homotopy-pullback of the two inclusions L → M, L → M. This space is homotopy equivalent to the space of all the continuous paths γ: [0, 1] → M so that γ(0) ∈ L, γ(1) ∈ L . By replacing both L and M by the respective spaces obtained by contracting the path w to a point, we see that there are continuous maps M(x, y) → Ω T (L, L ) . We may then use these maps to construct a com plex with coeﬃcients in C∗ (Ω T (L, L ) . Clearly, there is an obvious map T (L, L ) → L and it is precisely this map which, after looping, changes the coeﬃcients of this complex into those of the extended Floer complex. (b) At this point it is worth mentioning why using representing chain systems is useful in our constructions. Indeed, for the extended Morse complex representing chain systems are not really essential: the moduli spaces Myx are triangulable in a way compatible with the boundary formula and so, to represent this moduli space inside the loop space ΩM, we could use instead of the chain ayx a chain given by the sum of the top dimensional simplexes in such a triangulation. This is obviously, a simpler and more natural approach but it has the disadvantage that it does not extend directly to the Floer case. The reason is that it is not known whether the Floer moduli spaces M(x, y) admit coherent triangulations (even if this is likely to be the case). The chain complex C(L, L ; H) admits a natural degree filtration which is given by

F k C(L, L ; H, J) = C∗ (L) ⊗ Z/2x ∈ I(L, L ; η, H) : |x| ≤ k.

(17)

It is clear that this filtration is diﬀerential. Therefore, there is an induced spectral sequence which will be denoted by E(L, L ; H, J) = (Erp,q , Dr ). We write

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E(L, L ; J) = E(L, L ; 0, J). For convenience we have omitted η from this notation (the relevant components of the paths spaces P(L, L ) will be clear below). Here is the main result concerning this spectral sequence. THEOREM 3.5. For any two regular pairs (H, J), (H , J ), the spectral sequences E(L, L ; H, J) and E(L, L ; H , J ) are isomorphic up to translation for r ≥ 2. Moreover, if φ is a Hamiltonian diﬀeomorphism, then E(L, L ; J) and E(L, φ(L ); J ) are also isomorphic up to translation for r ≥ 2 (whenever defined). The second term of this spectral sequence is E2 (L, L ; H, J) ≈ H∗ (ΩL)⊗HF∗ (L, L ) where HF∗ (−, −) is the Floer homology. Finally, if L and L are Hamiltonian isotopic, then E(L, L ; J) is isomorphic up to translation to the Serre spectral sequence of the path-loop fibration ΩL → PL → L. Isomorphism up to translation of two spectral sequences E rp,q , F rp,q means that there exists a k ∈ Z and chain isomorphisms φr : E rp,q → F rp+k,q . This notion appears naturally here because the choice of the element x0 ∈ I(L, L ; H) with |x0 | = 0 is arbitrary. A diﬀerent choice will simply lead to a translated spectral sequence. As follows from the discussion in Section 3.2.3B, it is possible to replace this choice of grading with one that only depends on the path η. However, this might make the absolute degrees fractionary and, as the choice of η is not canonical, the resulting spectral sequence will still be invariant only up to translation. The outline of the proof of this theorem is as follows (see Barraud and Cornea, 2003, for details). First, in view of the naturality properties of Floer’s construction, it is easy to see that the second invariance claim in the statement is implied by the first one. Now, we consider a homotopy G between H and H as well as a one-parameter family of almost complex structures J¯ relating J to J . For x ∈ I(L, L ; H, J) and y ∈ I(L, L ; H , J ) we define moduli spaces N(x, y) which are solutions of an equation similar to (10) but replaces H with G, J with J¯ (and takes into account the additional parameter — this is a standard construction in Floer theory). These moduli spaces have properties similar to the M(x, y)’s. In particular they admit compactifications which are manifolds with boundary so that the following formula is valid ∂N(x, y) = M(x, z) × N(z, y) ∪ N(x, z ) × M(z , y). z∈I(L,L ;H)

z ∈I(L,L ;H )

The representing chain idea can again be used in this context and it leads to coeﬃcients nyx ∈ C∗ (ΩL). If we group these coeﬃcients in a matrix B and we group the coeﬃcients of the diﬀerential of C(L, L ; H, J) in a matrix A and the coeﬃcients of C(L, L ; H , J ) in a matrix A , then the relation above implies that we have: ∂B = A · B + B · A . (18)

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It follows that the module morphism φG, J¯: C(L, L ; H, J) → C(L, L ; H, J) which is the unique extension of ryx ⊗ y, φG, J¯(x) =

∀x ∈ I(L, L ; H)

y

is a chain morphism. Moreover, the chain morphism constructed above preserves filtrations (of course, to for this it is required that the choices for the point x0 with |x0 | = 0 for our two sets of data be coherent — this is why the isomorphisms are “up to translation”). After verifying that E2 ≈ H∗ (ΩL) ⊗ FH∗ (L) for both spectral sequences it is not diﬃcult to see that φG induces an isomorphism at the E2 -level of these spectral sequences (Barraud and Cornea, 2003). Hence it also induces an isomorphism for r > 2. For the last point of the theorem we use Floer’s reduction of the moduli spaces M J,L,L (x, y) of pseudoholomorphic strips to moduli spaces of Morse flow lines Myx ( f ). In short, this shows (Floer, 1988; 1989b) that for certain choices of J, f and L which is Hamiltonian isotopic to L we have homeomorphisms ψ x,y : M(x, y) → Myx which are compatible with the compactification and with the boundary formulas. This means that with these choices we have an isomorphism C(L, L ) → C( f ) and it is easy to see that this preserves the filtrations of these two chain complexes. By Theorem 2.4 this completes the outline of proof. REMARK 3.6. It is shown in Barraud and Cornea (2003) that the E1 -term of this spectral sequence has also some interesting invariance properties.

3.2.3. Applications We will discuss here a number of direct corollaries of Theorem 3.5 most (but not all) of which appear in Barraud and Cornea (2003). A. Localization and Hofer’s metric An immediate adaptation of Theorem 2.4 provides a statement which is much more flexible. This is a “change of coeﬃcients” or “localization” phenomenon that we now describe. Assume that f : L → X is a smooth map. Then we can consider the induced map Ω f : ΩL → ΩX and we may use this map to change the coeﬃcients of C(L, L ; H, J) thus getting a new complex CX (L, L ; H, J) = (C∗ (ΩX) ⊗ Z/2I(L, L ; H), DX ) ) so that DX (x) = y (Ω f )∗ (myx ) ⊗ y where myx are the coeﬃcients in the diﬀerential D of C(L, L ; H, J) (compare with (16)). This complex behaves very much like the

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one studied in Theorem 3.5. In particular, this complex admits a similar filtration and the resulting spectral sequence, EX (L, L ), has the same invariance properties as those in the theorem and, moreover, for L, L Hamiltonian isotopic this spectral sequence coincides with the Serre spectral sequence of the fibration ΩX → E → L which is obtained from the path-loop fibration ΩX → PX → X by pull-back over the map f . In particular, the homology of this complex coincides with the singular homology of E. If X is just one point, &, it is easy to see that the complex C& (L, L ; H, J) coincides with the Floer complex. The complex CX (L, L ; H, J) may also be viewed as a sort of localization in the following sense. Assume that we are interested to see what pseudo-holomorphic strips pass through a region A ⊂ L. Then we may consider the closure C of the complement of this region and the space L/C obtained by contracting C to a point. There is the obvious projection map L → L/C which can be used in place of f above. Now, if some nonvanishing diﬀerentials appear in EL/C (L, L ) for r ≥ 2, then it means that there are some coeﬃcients myx so that |(myx )| > 0 and (Ω f )∗ (myx ) 0. This means that the map lyx

Ωf

M(x, y) − → ΩL −−→ Ω(C/L) carries the representing chain of M(x, y) to a nonvanishing chain in C∗ (Ω(L/C)). But this means that the intersection M (x, y) ∩ A is of dimension equal to µ(x, y). The typical choice of region A is a tubular neighborhood of some submanifold V → L. In that case L/C is the associated Thom space. Let ∇(L, L ) = inf max H(x, t) − min H(x, t) . H,φ1H (L)=L

x,t

x,t

be the Hofer distance between Lagrangians. It has been shown to be nondegenerate by Chekanov (2000) for symplectic manifolds with geometry bounded at infinity. COROLLARY 3.7. Let a ∈ Hk (L) be a nontrivial homology class. If a closed submanifold V → L represents the class a, then for any generic J and any L Hamiltonian isotopic to L, there exists a pseudoholomorphic strip u of Maslov index at most n − k which passes through V and which verifies: u∗ ω ≤ ∇(L, L ). In view of the discussion above, the proof is simple (we are using here a variant of that used in Barraud and Cornea, 2003).

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We start with a simple topological remark. Take A to be a tubular neighborhood of V. Then L/C = T V is the associated Thom space. In the Serre spectral sequence of Ω(T V) → P(T V) → T V we have that dn−k (τ) 0 where τ ∈ Hn−k (T V) is the Thom class of V. By Poincar´e duality, there is a class b ∈ Hn−k (T V) which is taken to τ by the projection map L → T V (b corresponds to the Poincar´e dual a∗ of a via the isomorphism H n−k (L) ≈ Hk (L)). This means that Dn−k (b) is not zero in ET V (L, L ; H) (for any Hamiltonian H). To proceed with the proof notice that, by the naturality properties of the Floer moduli spaces, it is suﬃcient to show that for any Hamiltonian H (and any generic J) so that φ1H (L) = L there exists an element u ∈ M L,L,J,H which is of Maslov index (n − k) and so that u(R, 0) ∩ A ∅ and E L,L,H (u) ≤ ||H|| where H = max x,t H(x, t) − min x,t H(x, t). We may assume that min x,t H(x, t) = 0 and we let K = max x,t H(x, t). We consider a Morse function f : L → R which is very small in C 2 norm and we extend it to a function (also denoted by f ) which is defined on M and remains C 2 small. In particular, we suppose min x f (x) = 0 and max x f (x) < . We denote f = f − and f¯ = f + K. It follows that we may construct monotone homotopies G : f¯ H and G : H f . Consider the following action filtration of CT V (L, L; H) Fv CT V (L, L; J, H) = C∗ (ΩT V) ⊗ Z/2x ∈ I(L, L; H) : AL,L,H (x) ≤ v and similarly on the complexes CT V (L, L; f¯) and CT V (L, L; f ). It is obvious that this is a diﬀerential filtration and, if the choice of path η (used to define the action functional, see (9) is the same for all the three Hamiltonians involved, these monotone homotopies preserve these filtrations. We now denote C = F K+ CT V (L, L ; f¯)/F− CT V (L, L ; f¯), C = F K+ CT V (L, L ; H)/F− CT V (L, L ; H), C = F K+ CT V (L, L ; f )/F− CT V (L, L ; f ) These three complexes inherit degree filtrations and there are associated spectral sequences E(C), E(C ), E(C ). We have induced morphisms φG : C → C and φG : C → C which also induce morphisms among these spectral sequences. Moreover, as C and C both coincide with CT V ( f ) (because f is very C 2 -small and 0 ≤ f (x) < ), the composition φG ◦ φG induces an isomorphism of spectral sequences for r ≥ 2 (here CT V ( f ) is the extended Morse complex obtained from C( f ) by changing the coeﬃcients by the map L → T V). But, as the class b has the property that its Dn−k diﬀerential is not trivial in E(C), this implies that Dn−k φG (b) 0 which is seen to immediately imply that there is some moduli space M (x, y) of dimension n − k with AL,L ;H (x), AL,L ;H (y) ∈ [−, K + ] and M (x, y) ∩ A ∅. Therefore there are J-strips passing through A which have

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Maslov index n − k and area less than H + 2. By letting A tend to V and → 0, H → ∇(L, L ), these strips converge to strips with the properties desired. We may apply this even to 1 ∈ H0 (L) and Corollary 3.7 shows in this case that through each point of L passes a strip of Maslov index at most n (again, for J generic) and of area at most ∇(L, L ). The case V = pt was discussed explicitly in Barraud and Cornea (2003). REMARK 3.8. (a) It is clear that the strips detected in this corollary actually have a symplectic area which is no larger than c(b; H) − c(1; H) where c(x; H) is the spectral value of the homology class x relative to H,

c(x; H) = inf v ∈ R : x ∈ Im H∗ (FC≤v )(L, L; H) → FH∗ (L, L; H) where FC≤v (L, L; H) is the Floer complex of L, L, H generated by all the elements of I(L, L; η, H) of action smaller or equal than v; FH∗ (L, L; H) is the Floer homology. Under our assumptions we have a canonical isomorphism (up to translation) between HF∗ (L, L, H) and H∗ (L) so we may view b ∈ HF∗ (L, L; H). (b) Clearly, in view of Gromov compactness our result also implies that for any J (even nonregular) and for any L Hamiltonian isotopic to L and for any x ∈ L\L there exists a J-holomorphic strip passing through x which has area less than ∇(L, L ). This result (without the area estimate) also follows from independent work of Floer (1989a) and Hofer (1988). Another method has been mentioned to us by Dietmar Salamon. It is based on starting with disks with their boundary on L and which are very close to be constant maps. Therefore, an appropriate evaluation defined on the moduli space of these disks is of degree 1. Each of these disks is made out of two semi-disks which are pseudo-holomorphic and which are joined by a short semi-tube verifying the nonhomogeneous Floer equation for some given Hamiltonian H. This middle region is then allowed to expand till, at some point, it will necessarily produce a semi-tube belonging to some M H (x, y). It is also possible to use the pair of pants product to produce Floer orbits joining the “top and bottom classes” (Schwarz, 2000). Still, having simultaneous area and Maslov index estimates appears to be more diﬃcult by methods diﬀerent from ours. Of course, detecting strips of lower Maslov index so that they meet some fixed submanifold is harder yet. Corollary 3.7 has a nice geometric application. COROLLARY 3.9. Assume that, as before, L and L are Hamiltonian isotopic. For any symplectic embedding e: (Br , ω0 ) → M so that e−1 (L) = Rn ∩ B(r) and e(Br ) ∩ L = ∅ we have πr2 /2 ≤ ∇(L, L ).

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This is proved (see Barraud and Cornea, 2003) by using a variant of the standard isoperimetric inequality: a J0 -pseudoholomorphic surface in the standard ball (Br , ω0 ) of radius r whose boundary is on ∂Br ∪ Rn has area at least πr2 /2. Clearly, this implies the nondegeneracy result of Chekanov that was mentioned before under the connectivity conditions that we have always assumed till this point. B. Relaxing the connectivity conditions assumption that

We have worked till now under the

L, L are simply-connected and ω|π2 (M) = c1 |π2 (M) = 0.

(19)

These requirements were used in a few important places: in the definition of the action functional, the definition of the Maslov index, the boundary product formula (because they forbid bubbling). Of these, only the bubbling issue is in fact essential: the boundary formula is precisely the reason why d2 = 0 as well as the cause of the invariance of the resulting homology. We proceed below to extend the corollaries and techniques discussed above to the case when all the connectivity conditions are dropped but we assume that L and L are Hamiltonian isotopic and only work below the minimal energy that could produce some bubbling (this is similar to the last section of Barraud and Cornea (2003) but goes beyond the cases treated there). First, for a time dependent almost complex structure Jt , t ∈ [0, 1], we define δL,L (J) as the infimum of the symplectic areas of the following three types of objects: − the Jt -pseudoholomorphic spheres in M (for t ∈ [0, 1]). − the J0 -pseudoholomorphic disks with their boundary on L. − the J1 -pseudoholomorphic disks with their boundary on L . By Gromov compactness this number is strictly positive. We will proceed with the construction in the case when L = L and in the presence of a Hamiltonian H. We shall assume that the pair (H, J) is regular in the sense that the moduli spaces of strips defined below, M(x, y), are regular. We take the fixed reference path η to be a constant point in L (see (9)). Denote P0 (L, L) = Pη (L, L) and consider in this space the base point given by η. Notice that there is a morphism ω: π1 P0 (L, L) → R obtained by integrating ω over the disk represented by the element z ∈ π1 P0 (L, L) (such an element can be viewed as a disk with boundary in L). Similarly, let µ: π1 P0 (L, L) → Z be the Maslov morphism. Let K be the kernel of the morphism ω × µ: π1 P0 (L, L) → R × Z.

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The group π = π1 (P0 (L, L))/K is an abelian group (as it is a subgroup of R × Z) and is of finite rank. Let’s also notice that this group is the quotient of π2 (M, L) by the equivalence relation a ∼ b iﬀ ω(a) = ω(b), µ(a) = µ(b) (with this definition this group is also known as the Novikov group). This is a simple homotopical result. First P(L, L) is the homotopy pull-back of the map L → M over the map L → M. But this means that we have a fibre sequence F → P(L, L) → L with F the homotopy fibre of L → M and that this fibre sequence admits a canonical section. This implies that π1 P0 (L, L) ≈ π1 (F) × π1 (L). But π1 (F) = π2 (M, L). As both ω and µ are trivial on π1 (L) the claim follows. It might not be clear at first sight why µ is null on π1 (L) here. The reason is that the term π1 (L) in the product above is the image of the map induced in homotopy by jL : L → P0 (L, L). This map associates to a point in L the constant path. Consider a loop γ(s) in L. Then jL ◦ γ is a loop in P0 (L, L) which at each moment s is a constant path. We now need to view this loop as the image of a disk and µ([ jL (γ)]) is the Maslov index of this disk. But this disk is null homotopic so µ([ jL (γ)]) = 0. Consider the regular covering p: P 0 (L, L) → P0 (L, L) which is associated to the group K. We fix an element η0 ∈ P 0 (L, L) so that p(η0 ) = η. Clearly, the action functional A L,L,H : P 0 (L, L) → R may be defined by essentially the same formula as in (9): 1 ∗ AL,L,H (x) = − (p ◦ u) ω + H t, (p ◦ x)(t) dt 0

P 0 (L, L)

is such that u(0) = η0 , u(1) = x and is now wellwhere u: [0, 1] → defined. Let I (L, L, H) = p−1 I(L, L, H) . For x, y ∈ I (L, L, H) we may define µ(x, y) = µ(p ◦ u) where u: [0, 1] → P 0 (L, L) is a path that joins x to y. This is again welldefined. For each x ∈ I (L, L, H) we consider a path v x : [0, 1] → P 0 (L, L) so that v x (0) = η0 and v x (1) = x. The composition p ◦ v x can be viewed as a “semidisk” whose boundary is resting on the orbit p(x) and on L. Therefore, we may associate to it a Maslov index µ(v x ) (Robbin and Salamon, 1993) and it is easy to see that this only depends on x. Thus we define µ(x) = µ(v x ) and we have µ(x, y) = µ(x) − µ(y) for all x, y ∈ I (L, L, H). To summarize what has been done till now: once the choices of η and η0 are made, both the action functional A : P 0 (L, L, H) → R and the “absolute” Maslov index µ(−): I (L, L, H) → Z are well-defined.

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Fix an almost complex structure J. Consider two elements x, y ∈ I (L, L, H). We may consider the moduli space which consists of all paths u: R → P 0 (L, L) which join x to y and are so that p ◦ u satisfies Floer’s equation (10) modulo the R-action. If regularity is achieved, the dimension of this moduli space is precisely µ(x, y) − 1. The action functional A decreases along such a solution u and the energy of u (which is defined as the energy of p◦u) verifies, as in the standard case, E(u) = A (x)−A (y). Bubbling might of course be present in the compactification of these moduli spaces. As we only intend to work below the minimal bubbling energy δL,L (J) we artificially put: M(x, y) = ∅

if A (x) − A (y) ≥ δL,L (J)

and, of course, for A (x) − A (y) < δL,L (J), M(x, y) consists of the elements u mentioned above. We only require these moduli spaces to be regular. With this convention, for all x, y ∈ I (L, L, H) so that M(x, y) is not void we have the usual boundary formula (15). Notice at the same time that this formula is false for general pairs x, y (and so there is no way to define a Floer type complex at this stage). Now consider a map f : L → X so that X is simply-connected (the only reason to require this is to insure that the Serre spectral sequence does not require local coeﬃcients). We consider the group: C(L, L, H; X) = C∗ (ΩX) ⊗ Z/2I (L, L, H). = {x ∈ I (L, L, H) : w ≥ A (x) ≥ v} and we define For w ≥ v ∈ R, we denote Iv,w the subgroup Cw,v (L, L, H; X) = C∗ (ΩX) ⊗ Z/2Iv,w .

Suppose that w − v ≤ δL,L (J) − . We claim that in this case we may define a diﬀerential on Cv,w (L, L, H; X) by the usual procedure. Consider representing chain systems for all the moduli spaces M(x, y) and let the image of these chains inside C∗ (ΩX) be respectively myx . Let D be the linear extension of the map given by Dx = myx ⊗ y. y∈Iv,w

PROPOSITION 3.10. The linear map D is a diﬀerential. A generic monotone homotopy G between two Hamiltonians H and H φGX : Cv,w (L, L, H, J; X) → Cv,w (L, L, H , J; X). A monotone homotopy between monotone homotopies G and G induces a chain homotopy between φGX and φGX so that H∗ (φGX ) = H∗ (φGX ).

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) ) y Now, D2 x = z ( y myx ·mz +∂mzx )⊗z. In this formula we have A (x)−A (z) ≤ δL,L (J) and, because the usual boundary formula (15) is valid in this range, all the terms vanish so that D2 (x) = 0. The same idea may be applied to a monotone homotopy as well as to a monotone homotopy between monotone homotopies and it implies the claim. REMARK 3.11. (a) If we take for the space X a single point ∗ we get a chain complex whose diﬀerential only takes into account the 0-dimensional moduli spaces and which is a truncated version of Floer homology. (b) The complex Cv,w (L, L, H, J; X) admits a degree filtration which is perfectly similar to the one given by (17). Let ECv,w (L, L, H, J; X) be the resulting spectral sequence. Then, under the restrictions in the Proposition 3.10, a monotone homotopy G induces a morphism of spectral sequences EX (φG ) and two such homotopies G, G which are monotonously homotopic have the property that they induce the same morphism ErX (φG ) = ErX (φG ) for r ≥ 2. This last fact follows from Proposition 3.10 by computing E2X (φG ) = ∗ ) = E2 (φ ) where φ∗ : C (L, L, H; ∗) → C (L, L, H ; ∗). idH∗ (ΩX) ⊗ H∗ (φG v,w v,w X G G Naturally, the next step is to compare our construction with its Morse theoretical analogue. Consider the map jL : L → P0 (L, L) and consider p: L˜ → L the regular covering obtained by pull-back from P 0 (L, L) → P0 (L, L). Notice that, because both compositions ω ◦ π1 ( jL ) and µ ◦ π1 ( jL ) are trivial, it follows that the covering L˜ → L is trivial. Let f¯: L˜ → R be defined by f¯ = f ◦ p and consider C( f¯; X) the extended Morse complex of f¯ with coeﬃcients changed by the map ΩL˜ → ΩX. Notice that, in general, the group π acts on I (L, L, H) and we have the formula: A (gy) = ω(g) + A (y), µ(gy) = µ(g) + µ(y),

∀y ∈ I (L, L, H), ∀g ∈ Π.

In our particular case, when H = f , we have I (L, L, H) = Crit( f¯). For each point x ∈ Crit( f ) let x¯ ∈ Crit( f¯) be the element of p−1 (x) which belongs to the component of L˜ which also contains η0 . We then have A ( x¯) = f (x) and µ( x¯) = ind f¯( x¯). The extended Morse complex C( f¯; X) is therefore isomorphic to C( f ; X)⊗ Z[π] and the action filtration is determined by writing A (x ⊗ g) = f (x) + ω(g). The degree filtration induces, as usual, a spectral sequence which will be denoted by EC( f ; X). The remarks above together with Theorem 3.5 show that this spectral sequence consists of copies the Serre spectral sequence of ΩX → E → L: one copy for each ˜ We denote by C0 ( f ; X) and EC0 ( f ; X) the copies of the connected component of L. extended complex and of the spectral sequence that correspond to the connected component L0 of L˜ which contains η0 .

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PROPOSITION 3.12. Suppose H0 < δ J (L, L). There exists a chain morphisms φ: C0 ( f ; X) → C0,H (L, L, H, J; X) and ψ: C0,H (L, L, H, J; X) → C0 ( f ; X) which preserve the respective degree filtrations and so that ψ◦φ induces an isomorphism at the E 2 level of the respective spectral sequences. To prove this proposition we shall use a diﬀerent method than the one used in Corollary 3.7. The comparison maps φ, ψ will be constructed by the method introduced in Piunikhin et al. (1996) and later used in Schwarz (1998; 2000). Compared to Proposition 3.10 this is particularly eﬃcient because it avoids the need to control the bubbling threshold along deformations of J. We fix as before the Morse function f as well as the pair H, J. To simplify the notation we shall assume that inf H(x, t) = 0. The construction of φ is based on defining certain moduli spaces W(x, y) with x ∈ Crit( f¯) and y ∈ I (L, L, H). They consist of pairs (u, γ) where u: R → P 0 (L, L), γ: (−∞, 0] → L˜ and if we put u = p(u), γ = p(γ) (p: P 0 (L, L) → P0 (L, L) is the covering projection) then we have: u (R × {0, 1}) ⊂ L,

∂ s (u ) + J(u )∂t (u ) + β(s)∇H(u , t) = 0,

u(+∞) = y

and

dγ = −∇g f (γ ), γ(−∞) = x, γ(0) = u(−∞). dt Here g is a Riemannian metric so that ( f, g) is Morse – Smale and β is a smooth cut-oﬀ function which is increasing and vanishes for s ≤ 12 and equals 1 for s ≥ 1. It is useful to view an element (u, γ) as before as a semi-tube connecting x to y. Under usual regularity assumptions these moduli spaces are manifolds of dimension µ(x) − µ(y). , The energy of such an element (u, γ) is defined in the obvious way by E(u, γ) = ∂ s u 2 ds dt. A simple computation shows that: E(u, γ) = I(u) + where I(u) =

, R×[0,1]

∗

R×[0,1]

(u ) ω −

1

H y(t) dt

0

β (s)H(u (s), t) ds dt. If x ∈ L0 , then the energy verifies

E(u, γ) = I(u) − A (y) ≤ sup(H) − A (y). As before, we only want to work here under the bubbling threshold and we are only interested in the critical points x ∈ L0 so we put W(x, y) = ∅ for all those pairs (x, y) with either y ∈ I (L, L, H) so that A (y) [0, H] or with x L0 . This means that there is no bubbling in our moduli spaces. Thus we may apply the usual procedure: compactification, representing chain systems, representation in the loop space (for this step we need to choose a convenient way to parametrize

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the paths represented by the elements (u, γ)). Notice that the boundary of W(x, y) x is the union of two types of pieces M z × W(z, y) and W(x, z) × M(z, y). We then ) x define φ(x) = wy ⊗ y where wyx is a cubical chain representing the moduli space W(x, y). This is a chain map as desired. We now proceed to construct the map ψ. The construction is perfectly similar: we define moduli spaces W (y, x), y ∈ I (L, L, J), x ∈ Crit( f¯) except that the pairs (u, γ) considered here, start as semi-tubes and end as flow lines of f . The equation verified by u is similar to the one before but instead of the cut-oﬀ function β we use the cut-oﬀ function −β. For x ∈ L0 the energy estimate in this case gives E(u, γ) ≤ A (y). By the same method as above we define W (y, x) to be void ) y y whenever x L0 or A (y) > H and we define ψ(y) = w x ⊗ x where w x is a cubical chain representing the moduli space W (y, x). Notice that because E(u, γ) ≤ A (y) this map ψ does in fact vanish on C0 (L, L, H, J; X) and so it induces a chain map (also denoted by ψ) as desired. The next step is to notice that the composition ψ ◦ φ induces an isomorphism at E 2 . This is equivalent to showing that H∗ (ψ ◦ φ) is an isomorphism for X = ∗. In turn, this fact follows by now standard deformation arguments as in Piunikhin et al. (1996). COROLLARY 3.13. Assume that L and L are Hamiltonian isotopic and suppose that J is generic. If ∇(L, L ) < δL,L (J), then the statement of Corollary 3.7 remains true (for J) without the connectivity assumption (19). Notice that if H is a Hamiltonian so that φ1H (L) = L and JH = (φH )∗ (J) then, by the naturality described in Section 3.2, we have: δL,L (JH ) = δL,L (J). This implies, again by this same naturality argument, that the problem reduces to finding appropriate semi-tubes whose detection comes down to showing the nonvanishing of certain diﬀerentials in EC0,w (L, L, H; T V) for some well-chosen w < δL,L (JH ). But this immediately follows from Proposition 3.12 by the same topological argument as the one used in the proof of Corollary 3.7 We formulate the geometric consequence which corresponds to Corollary 3.9. For two Lagrangians L and L the following number has been introduced in Barraud and Cornea (2003): B(L, L ) is the supremum of the numbers r ≥ 0 so that there exists a symplectic embedding e: (B(r), ω0 ) → (M, ω) so that e−1 (L) = Rn ∩ B(r) and Im(e) ∩ L = ∅. COROLLARY 3.14. There exists an almost complex structure J so that we have the inequality: π ∇(L, L ) ≥ min δL,L (J), B(L, L )2 . 2

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Clearly, this implies that ∇(−, −) is nondegenerate in full generality (and recovers, in particular, the fact that the usual Hofer norm for Hamiltonians is nondegenerate). It is useful to also notice as in Barraud and Cornea (2003) that this result is a Lagrangian version of the usual capacity — displacement energy inequality (Lalonde and McDuﬀ, 1995). Indeed, this inequality (with the factor 12 ) is implied by the following statement which has been conjectured to hold for any two compact Lagrangians in a symplectic manifold (Barraud and Cornea, 2003): ∇(L, L ) ≥

π B(L, L )2 . 2

(20)

This remains open. An even stronger conjecture is the following: CONJECTURE 3.15. For any two Hamiltonian isotopic closed Lagrangians L, L ⊂ (M, ω) and for any almost complex structure J which compatible with ω and any point x ∈ L\L there exists a pseudoholomorphic curve u which is either a strip resting on L and , on L or a pseudoholomorphic disk with boundary in L so ∗ that x ∈ Im(u) and u ω ≤ ∇(L, L ). By the isoperimetric inequality used earlier in this paper, it follows that this statement implies (20). There is a substantial amount of evidence in favor of this conjecture: − as explained in this paper, in the absence of any pseudoholomorphic disks (that is, when ω|π2 (M,L) = 0) it was proved in Barraud and Cornea (2003). − the statement in Corollary 3.14 shows that the area estimate is not unreasonable. − one striking consequence of Conjecture 3.15 is that if the disjunction energy of the Lagrangian L is equal to E0 < ∞, then, for any J as in the statement and any x ∈ L there is a pseudoholomorpic disk of area at most E0 which passes through x. Under certain conditions, this is indeed true and follows from recent work of the second author joint with Franc¸ois Lalonde. By the same geometric argument as above we deduce a nice consequence. Define the relative (or real) Gromov radius of L, Gr(L), to be the supremum of the positive numbers r so that there exists a symplectic embedding e: (B(r), ω0 ) → (M, ω) with the property that e−1 (L) = Rn ∩ B(r), then π Gr(L) 2 /2 ≤ E0 (where E0 , as before, is the disjunction energy of L). It is also useful to note that if L is the zero section of a cotangent bundle, then Gr(L) = ∞. There are numerous other interesting consequences of Conjecture 3.15 besides (20). To conclude, Conjecture 3.15 appears to be a statement worth investigating.

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References

Barraud, J.-F. and Cornea, O. (2003) Lagrangian intersections and the Serre spectral sequence, prepint. Betz, M. and Cohen, R. L. (1994) Graph moduli spaces of graphs and cohomology operations, Turkish J. Math. 18, 23 – 41. Chekanov, Y. (2000) Invariant Finsler metrics on the space of Lagrangian embeddings, Math. Z. 234, 605 – 619. Cohen, R. L., Jones, J. D. S., and Segal, G. B. (1995)a Floer’s infinite-dimensional Morse theory and homotopy theory, In H. Hofer, C. H. Taubes, A. Weinstein, and E. Zehnder (eds.), The Floer Memorial Volume, Vol. 133 of Progr. Math, pp. 297–325, Basel, Birkh¨auser. Cohen, R. L., Jones, J. D. S., and Segal, G. B. (1995)b Morse theory and classifying spaces, preprint. Cornea, O. (2002)a Homotopical dynamics. II. Hopf invariants, smoothing, and the Morse complex, ´ Ann. Sci. Ecole Norm. Sup. (4) 35, 549 – 573. Cornea, O. (2002)b Homotopical dynamics. IV. Hopf invariants, and Hamiltonian flows, Comm. Pure Appl. Math. 55, 1033 – 1088. Cornea, O. (2004) New obstructions to the thickening of CW-complexes, Proc. Amer. Math. Soc. 132, 2769 – 2781. Cornea, O., Lupton, G., Oprea, J., and Tanr´e, D. (2003) Lusternik – Schnirelmann category, Vol. 103 of Math. Surveys Monogr., Providence, RI, Amer. Math. Soc. Floer, A. (1988) Morse theory for Lagrangian intersections, J. Diﬀerential Geom. 28, 513 – 547. Floer, A. (1989)a Cuplength estimates on Lagrangian intersections, Comm. Pure Appl. Math. 42, 335 – 356. Floer, A. (1989)b Witten’s complex and infinite-dimensional Morse theory, J. Diﬀerential Geom. 30, 207 – 221. Franks, J. M. (1979) Morse – Smale flows and homotopy theory, Topology 18, 199 – 215. Fukaya, K. (1993) Morse homotopy, A∞ -category, and Floer homologies, In H.-J. Kim (ed.), Proceedings of GARC Workshop on Geometry and Topology ’93, Vol. 18 of Lecture Notes Ser., Seoul, 1993, pp. 1–102, Seoul, Seoul National Univ. Hofer, H. (1988) Lusternik – Schnirelman-theory for Lagrangian intersections, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 5, 465 – 499. Lalonde, F. and McDuﬀ, D. (1995) The geometry of symplectic energy, Ann. of Math. (2) 141, 349 – 371. Milnor, J. (1965)a Lectures on the h-cobordism theorem, Princeton, NJ, Princeton Univ. Press. Milnor, J. W. (1965)b Topology from the Diﬀerentiable Viewpoint, Charlottesville, VA, The University Press of Virginia. Piunikhin, S., Salamon, D., and Schwarz, M. (1996) Symplectic Floer – Donaldson theory and quantum cohomology, In Contact and Symplectic Geometry, Vol. 8 of Publ. Newton Inst., Cambridge, 1994, pp. 171–200, Cambridge, Cambridge Univ. Press. Pugh, C. C. and Robinson, C. (1983) The C 1 closing lemma, including Hamiltonians, Ergodic Theory Dynam. Systems 3, 261 – 313. Robbin, J. and Salamon, D. (1993) The Maslov index for paths, Topology 32, 827 – 844. Salamon, D. (1985) Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291, 1 – 41. Salamon, D. (1999) Lectures on Floer homology, In Y. Eliashberg and L. Traynor (eds.), Symplectic Geometry and Topology, Vol. 7 of IAS/Park City Math. Ser., Park City, UT, 1997, pp. 143–229, Providence, RI, Amer. Math. Soc. Schwarz, M. (1993) Morse Homology, Vol. 111 of Progr. Math, Basel, Birkh¨auser.

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Schwarz, M. (1998) A quantum cup-length estimate for symplectic fixed points, Invent. Math. 133, 353 – 397. Schwarz, M. (2000) On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193, 419 – 461. Smale, S. (1967) Diﬀerentiable dynamical systems, Bull. Amer. Math. Soc. 73, 747 – 817. Viterbo, C. (1987) Intersection de sous-vari´et´es lagrangiennes, fonctionnelles d’action et indice des syst`emes hamiltoniens, Bull. Soc. Math. France 115, 361 – 390. Weber, J. (2004) The Morse – Witten complex via dynamical systems, preprint. Witten, E. (1982) Supersymmetry and Morse theory, J. Diﬀerential Geom. 17, 661 – 692.

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY RALPH L. COHEN∗ Stanford University

Abstract. In these lecture notes we discuss a body of work in which Morse theory is used to construct various homology and cohomology operations. In the classical setting of algebraic topology this is done by constructing a moduli space of graph flows, using homotopy theoretic methods to construct a virtual fundamental class, and evaluating cohomology classes on this fundamental class. By using similar constructions based on “fat” or ribbon graphs, we describe how to construct string topology operations on the loop space of a manifold, using Morse theoretic techniques. Finally, we discuss how to relate these string topology operations to the counting of J-holomorphic curves in the cotangent bundle. We end with speculations about the relationship between the absolute and relative Gromov – Witten theory of the cotangent bundle, and the open-closed string topology of the underlying manifold.

Introduction Recently, an intersection theory has been developed for spaces of paths and loops in compact, oriented, manifolds. This theory, which goes under the name of “string topology,” was initiated in the seminal work of Chas and Sullivan (1999), and has been expanded by several authors (Cohen and Jones, 2002; Cohen and Godin, 2004; Voronov, 2005; Chataur, 2003; Klein, 2003; Hu, 2004). Various operad and field theoretic properties of this theory are now known (Voronov, 2005; Cohen and Godin, 2004). In particular, it was shown in Cohen and Godin (2004) how, given a surface Σ viewed as a cobordism between p circles and q-circles, there is an associated operation in homology (as well as generalized homologies) µΣ : H∗ (LM)⊗p → H∗ (LM)⊗q . These operations share many formal properties with Gromov – Witten theory for symplectic manifolds, and it is the goal of an ongoing project of the author to understand this relationship. The possibilities for relating these theories became more compelling with the recent work of Viterbo (1996), Salamon and Weber (2004), and Abbondandolo and Schwarz (2004), which proves that the Floer homology of the cotangent ∗

Partially supported by a grant from the NSF.

149 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 149–184. © 2006 Springer. Printed in the Netherlands.

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bundle of a closed, oriented manifold is isomorphic to the homology of the loop space, HF∗ (T ∗ M) H∗ (LM). Here the Floer homology is defined with respect to a Hamiltonian HV on T ∗ (M) defined in terms of a potential function on the manifold, V: R/Z × M → R. Because Floer homology is a Morse homology based on the symplectic action functional, in order to compare Floer theory and Gromov – Witten theory to string topology, a first step is to devise a Morse theoretic approach to string topology. This will be the main topic of these lecture notes. The basic strategy is to expand and generalize a theory of graph flows of Betz and Cohen (1994) for constructing classical (co)homology invariants in algebraic topology. Generalizing it in the appropriate way is the subject of joint work with P. Norbury, which we will report on in this paper. The basic idea is to make “toy model” of Gromov – Witten theory in which the role of surfaces are replaced by finite graphs. More specifically, we develop a theory in which we make the following replacements from classical Gromov – Witten theory. 1. A smooth surface F is replaced by a finite, oriented graph Γ. 2. The role of the genus of F is replaced by the first Betti number, b = b1 (Γ). 3. The role of marked points in F is replaced by the univalent vertices (“leaves”) of Γ. 4. The role of a complex structure Σ on F is replace by a metric on Γ. 5. The notion of a J-holomorphic map to a symplectic manifold with compatible almost complex structure, Σ → (N, ω, J), is replaced by the notion of a “graph flow” γ: Γ → M which, when restricted to each edge is a gradient trajectory of a Morse function on M. More specifically, each edge of the graph is labeled by a Morse function, fi : M → R, and the restriction of γ to the ith edge is a gradient trajectory of fi . We describe the moduli space of graph flows M(Γ, M), and then define, using homotopy theory, virtual fundamental classes. These are integral homology classes in our case, and indeed we define these fundamental classes in any generalized homology theory that supports an orientation of M. In particular we can therefore define integral Gromov – Witten type invariants in this toy model situation. These are operations Σ Σ qΓ : H∗ BAut(Γ) ⊗ H∗ p (M p ) → H∗ q (M q ). Here H∗Σn (M n ) is the Σn -equivariant homology of the n-fold Cartesian product M n , where the symmetric group Σn acts by permuting the coordinates. Aut(Γ) is the finite group of combinatorial automorphisms of the graph Γ, and BAut(G) is its classifying space. The cohomology of this automorphism group is playing the

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role in this theory, of the cohomology of mapping class group (or equivalently the moduli space of Riemann surfaces) in Gromov – Witten theory. We review the results of Betz and Cohen (1994) and of Cohen and Norbury (2005) to show how one obtains classical cohomology operations such as cup product, intersection product, Steenrod squares, and Stiefel – Whitney classes from these Morse-graph constructions. We also discuss certain field theoretic (or gluing) properties of these operations, as well as a kind of homotopy invariance. These properties are proved in Cohen and Norbury (2005). We then indicate how these properties can be used to prove the classical relations, such as the Cartan and Adem relations, for these operations. Our next step is to move to the loop space. Here we need to use “fat” or “ribbon graphs” instead of all finite graphs. Spaces of fat graphs have been used by many authors to study moduli spaces of Riemann surfaces (Harer, 1985; Penner, 1987; Strebel, 1984; Kontsevich, 1992). We describe work of Chas and Sullivan (1999), Cohen and Jones (2002), and Cohen and Godin (2004), that show how to use fat graphs to produce operations on the homology (or generalized homology) of loop spaces, µg,n : H∗ (Mg,n ) ⊗ H∗ (LM)⊗p → H∗ (LM)⊗q . Here Mg,n is the moduli space of genus g-Riemann surfaces with n = p + q parameterized boundary circles. These operations are constructed using classical techniques of algebraic and diﬀerential topology, such as transversal intersections of chains, and the Thom – Pontrjagin construction. We also describe work of Sullivan (2004) and Ramirez (2005) that gives an “open-closed” version of these string topology operations. This is motivated by open string theory in physics, where one studies configurations of paths in a manifold that take boundary values in “D-branes.” Next we adapt the Morse theoretic techniques described above to the loop space. We describe work of Cohen (2005) and Ramirez (2005) that describe how to construct the string topology operations using Morse theoretic techniques similar to those used to construct classical (co)homology operations on finitedimensional manifolds. Finally we review the work of Salamon and Weber (2004) on the Floer homology of the cotangent bundle of a closed, oriented manifold, with its canonical symplectic structure. They establish a relationship between moduli spaces of J-holomorphic cylinders in the cotangent bundle, and gradient flow lines of perturbed energy functionals on the loop space. In both cases, they are using perturbations by a potential function, V: R/Z × M → R. In the first case they use this to define a Hamiltonian on the cotangent bundle, HV : R/Z × T ∗ M → R, and in the second case to define a Morse function S V : LM → R. We use this correspondence to relate the space of gradient graph flows to the loop space, using a fat graph Γ, to the space of “cylindrical J-holomorphic curves” from a surface built from

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the graph Γ to the cotangent bundle, T ∗ M. In the case when the graph is a Figure 2, we show how recent work of Abbondandolo and Schwarz (2005) implies, when one inputs our Morse theoretic interpretation of string topology (Cohen, 2005; Ramirez, 2005), that the pair of pants product in the Floer homology of the cotangent bundle is equal to the Chas – Sullivan loop product on H∗ (LM). The organization of the paper is the following. In Section 1 we recall results from Betz and Cohen (1994) and describe results from Cohen and Norbury (2005) concerning the classical cohomology operations obtained using Morse theory, from our “toy model” of Gromov – Witten theory. In Section 2 we recall some of the theory of fat graphs, and describe the basic constructions in string topology from Chas and Sullivan (1999), Cohen and Jones (2002), Cohen and Godin (2004), Sullivan (2004), and Ramirez (2005). In Section 3 we apply the methodology of studying graph flows to describe a Morse theoretic description of the (closed) string topology operations. In Section 4 we recall the results of Salamon and Weber (2004) and describe how they might be applied to get descriptions of string topology operations using “cylindrical J-holomorphic maps” in the cotangent bundle. We also speculate about the further relationships between the Gromov – Witten theory (including the relative theory) of the cotangent bundle and the string topology of the manifold. 1. Graphs, Morse theory, and cohomology operations In this section we describe a generalization of the constructions and results of Betz and Cohen (1994) that is joint work with P. Norbury (Cohen and Norbury, 2005). As described in the introduction, our goal is to make a toy model of Gromov – Witten theory, where surfaces are replaced by graphs, and so on. The analogue of Teichm¨uller space in this theory is the space of graphs with metrics. To describe this we use ideas of Culler and Vogtmann (1986), and modifications of them due to Igusa (2002), and to Godin (2004). We define this space of structures in terms of a category of graphs. DEFINITION 1.1. Define Cb,p+q to be the category of oriented graphs of first Betti number b, with p+q leaves. More specifically, the objects of Cb,p+q are finite graphs (one-dimensional CW-complexes) Γ, with the following properties: 1. Each edge of the graph Γ has an orientation. 2. Γ has p + q univalent vertices, or “leaves.” p of these are vertices of edges whose orientation points away from the vertex (toward the body of the graph). These are called “incoming” leaves. The remaining q leaves are on edges whose orientation points toward the vertex (away from the body of the graph). These are called “outgoing” leaves.

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

Figure 1.

153

An object Γ in C2,2+2

3. Γ comes equipped with a “basepoint,” which is a nonunivalent vertex. For set theoretic reasons we also assume that the objects in this category (the graphs) are subspaces of a fixed infinite-dimensional Euclidean space, R∞ . A morphism between objects φ: Γ1 → Γ2 is combinatorial map of graphs (cellular map) that satisfies: 1. φ preserves the orientations of each edge. 2. The inverse image of each vertex is a tree (i.e., a contractible subgraph). 3. The inverse image of each open edge is an open edge. 4. φ preserves the basepoints. Notice these conditions on morphisms is equivalent to saying that φ: Γ1 → Γ2 is a combinatorial map of graphs that is orientation preserving on edges and is a basepoint preserving homotopy equivalence on the geometric realizations of the graphs. Given a graph Γ ∈ Cb,p+q , we define the automorphism group Aut(Γ) to be the group of invertible morphisms from Γ to itself in this category. Notice that Aut(Γ) is a finite group, as it is a subgroup of the group of permutations of the the edges. We now fix a graph Γ (an object in Cb,p+q ), and we discuss the category of “graphs over Γ,” CΓ . As we will see below, this category will be viewed as the space of metrics on (subdivisions) of Γ. DEFINITION 1.2. Define CΓ to be the category whose objects are morphisms in Cb,p+q with target Γ: φ: Γ0 → Γ. A morphism from φ0 : Γ0 → Γ to φ1 : Γ1 → Γ is a morphism ψ: Γ0 → Γ1 in Cb,p+q with the property that φ0 = φ1 ◦ ψ: Γ0 → Γ1 → Γ. We notice that the identity map id: Γ → Γ is a terminal object in CΓ . That is, every object φ: Γ0 → Γ has a unique morphism to id: Γ → Γ. This implies that the geometric realization of the category, |CΓ | is contractible. (|CΓ | is essentially the cone on the vertex represented by the terminal object.) But notice that the category CΓ has a free right action of the automorphism group, Aut(Γ), given on the objects by composition: Objects(CΓ ) × Aut(Γ) → Objects(CΓ ) g φ (φ: Γ0 → Γ) · g → g ◦ φ: Γ0 −→Γ−→Γ

(1)

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Figure 2.

A 2-simplex in |CΓ |

This induces an action on the geometric realization CΓ . We therefore have: PROPOSITION 1.3. The orbit space is homotopy equivalent to the classifying space, |CΓ |/ Aut(Γ) BAut(Γ). As mentioned above, we can think of |CΓ | as the space of “metrics on subdivisions of Γ.” The following idea of Igusa (2002) associates to a point in |CΓ | a metric on a graph over Γ. Recall that ψ1 ψk ψk−1 φ |CΓ | = ∆k × {Γk −→ Γk−1 −−−→ Γk−2 → · · · −→ Γ0 − → Γ}/ ∼ k

where the identifications come from the face and degeneracy operations. ) be a point in |CΓ |, where t = (t0 , t1 , . . . , tk ) is a vector of positive numLet (t, ψ is a sequence of k-composable morphisms in bers whose sum equals one, and ψ CΓ . Recall that a morphism φi : Γi → Γi−1 can only collapse trees, or perhaps compose such a collapse with an automorphism. So in a sense, given a composition of morphisms, : Γk → · · · → Γ0 → Γ ψ Γk is a (generalized) subdivision of Γ, and in particular Γ is obtained from Γk by collapsing various edges. We use the coordinates t of the simplex ∆k to define a metric on Γk as follows. For each edge E of Γk , define k + 1 numbers,

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

Figure 3.

155

A 2-simplex of metrics

λ0 (E), . . . , λk (E) given by λi (E) =

in Γi , 0 if E is collapsed by ψ 1 if E is not collapsed in Γi .

We then define the length of the edge E to be (E) =

k

ti λi (E).

(2)

i=0

Notice also that the orientation on the edges and the metric determine parameterizations (isometries) of standard intervals to the edges of the graph Γk over Γ, θE : [0, (E)] − →E (3) ) ∈ |CΓ | determines a metric on a graph Γk living over Γ, as Thus a point (t, ψ well as a parameterization of its edges.

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We now fix a target closed manifold M of dimension d. Our goal is describe a moduli space of maps from the space of graphs over Γ to M, which, when restricted to each edge, satisfy a gradient flow equation of a function on M. This moduli space of maps (which we call “graph flows”) will be our analogue in this model, of the moduli space of J-holomorphic curves in a symplectic manifold. For these purposes, we will need a bit more structure on an object, φ0 : Γ0 → Γ. Namely, we want to label the edges of Γ0 by distinct functions, fi : M → R. To say this more precisely, we first introduce some notation. Let V be a real vector space, and let F(V, m) ⊂ V m be the configuration space of m- distinct vectors in V. It is an open, dense subset of the m-fold Cartesian product, V m . We note that if V is infinite-dimensional, F(V, m) is contractible. Consider the functor µ: CΓ → Spaces which assigns to a graph over Γ, φ0 : Γ0 → Γ, the configuration space F C ∞ (M), e(Γ0 ) , where C ∞ (M) is the vector space of smooth, real valued functions on M, and e(Γ0 ) is the number of edges of Γ0 . Given a morphism ψ: Γ1 → Γ0 , which collapses certain edges and perhaps permutes others, there is an obvious induced map, µ(ψ): F C ∞ (M), e(Γ1 ) → F C ∞ (M), e(Γ0 ) . This map projects oﬀ of the coordinates corresponding to edges collapsed by ψ, and permutes coordinates corresponding to the permutation of edges induced by ψ. We can now do a homotopy theoretic construction, called the homotopy colimit (see for example Bousfield and Kan, 1972). DEFINITION 1.4. We define the space of metric structures and Morse labelings on G, S(Γ), to be the homology colimit, S(Γ) = hocolim(µ: CΓ → Spaces). The homotopy colimit construction is a simplicial space whose k simplices ), where ψ : Γk → Γk−1 → · · · Γ0 → Γ is a k-tuple of composconsist of pairs, ( f, ψ able morphisms in CΓ , and f ∈ µ(Γk ). That is, f is a labeling of the edges of Γk by distinct functions fE : M → R, for E an edge of Γk . We refer to f as a “M-Morse labeling” of the edges of Γk . So we can think of a point σ ∈ S(Γ) as consisting of a metric on a graph over Γ, together with an M-Morse labeling of its edges. We now make the following observation. LEMMA 1.5. The space of structures S(Γ) is contractible with a free Aut(Γ) action.

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157

Proof. The contractibility follows from standard facts about the homotopy colimit construction, considering the fact that both |CΓ | and F(C ∞ (M), m) are contractible. The free action of Aut(Γ) on |CΓ | extends to an action on S(Γ), since Aut(Γ) acts by permuting the edges of Γ, and therefore permutes the labels accordingly. We now define our moduli space of structures. DEFINITION 1.6. The moduli space of metric structures and M-Morse labelings on G, M(Γ), is defined to be the quotient, M(Γ) = S(Γ)/ Aut(Γ). We therefore have the following. COROLLARY 1.7. The moduli space is a classifying space of the automorphism group, M(Γ) BAut(Γ). We now define the moduli space of “graph flows” in M. Let σ ∈ S(Γ). As mentioned above, σ can be thought of as a graph Γk over Γ, with a metric and parameterizations of its edges, and a labeling of its edges by functions on M. Let γ: Γk → M be a continuous map which is smooth on the open edges. If we restrict γ to the edge E, and use the parameterization induced by the metric, this defines a smooth map γE : [0, (E)] → M for each edge E of Γk . ' M), to DEFINITION 1.8. Define the structure space of “graph flows” on Γ, M(Γ, be ' M) = (σ, γ) : σ ∈ S(Γ), γ: Γk → M is such that γE : [0, (E)] → M M(Γ, dγE satisfies the gradient flow equation + ∇ fE γ(t) = 0, dt for each edge E of Γk . We define the moduli space of “graph flows” on Γ M(Γ, M) to be the quotient, ' M)/ Aut(Γ). M(Γ, M) = M(Γ, We discuss the topology of the moduli space of graph flows in Cohen and Norbury (2005). In any case the topology can be deduced from the following two results.

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THEOREM 1.9. Suppose a graph Γ in Cb,p+q is a tree. Then there is a homeomorphism,

Ψ: M(Γ, M) − → M(Γ) × M

BAut(Γ) × M

(σ, γ) − → σ × γ(v) where v is the fixed vertex of the graph Γk over Γ determined by the structure σ. Proof. This follows from the existence and uniqueness theorem for solutions of ODE’s on compact manifolds. The point is that the values of γ on the edges emanating from v are completely determined by γ(v) ∈ M, since one has a unique flow line through that point for any of the functions labeling these edges. The value of γ on these edges determines the value of γ on coincident edges (i.e., edges that share a vertex) for the same reason. The theorem then follows. For general graphs Γ, we analyze the topology of M(Γ, M) in the following way. Let σ ∈ S(Γ). A tree flow of Γ with respect to the structure σ is a collection γ = {γT } where γT : T → M is a graph flow on a maximal subtree T ⊂ Γk . The collection ranges over all maximal subtrees T ⊂ Γk , and is subject only to the condition that the values at the basepoint are the same: γT 1 (v) = γT 2 (v) for any two maximal trees T 1 , T 2 ⊂ Γk . We define Mtree (Γ, M) = {(σ, γ) : σ ∈ S(Γ), and γ = {γT } is a tree flow of Γ with respect to σ}/ Aut(Γ). (4) We now have the following generalization of Theorem 1.9. THEOREM 1.10. 1. For any graph Γ ∈ Cb,p+q there is a homeomorphism

Ψ: Mtree (Γ, M) − → M(Γ) × M

BAut(Γ) × M

(σ, γ) − → σ × γ(v) where v is the fixed vertex of the graph Γk over Γ determined by the structure σ. 2. Let ρ: M(Γ, M) → Mtree (Γ, M) be the map that sends a graph flow γ to the tree flow obtained by restricting γ to each maximal tree. Then ρ is a codimension b1 (Γ) · d embedding.

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159

Idea of proof. The first part of this theorem follows from the existence and uniqueness theorem, just like Theorem 1.9. For the second part, we will show in (Cohen and Norbury, 2005) that there exists a Serre fibration over the Cartesian product, ev: Mtree (Γ, M) → M 2b where b = b1 (Γ), so that the restriction to the diagonal embedding ∆b : M b → (M 2 )b is equal to M(Γ, M). That is, there is a pullback square of fibrations, M(Γ, M) ev

Mb

ρ →

/ Mtree (Γ, M)

→

∆b

ev

(5)

/ (M 2 )b

To give an idea of the map ev: Mtree (Γ, M) → M 2b , assume for simplicity that b1 (Γ) = 1. Let σ ∈ S(Γ) be a structure. Then any maximal tree in Γk is obtained by removing a single edge from Γk . Say T E is a maximal tree obtained by removing the edge E of Γk . Let γT E : T E → M be a graph flow on T E with respect to σ. E is an oriented edge, so we can identify its vertices as a source vertex v0 and a target vertex v1 . Let ev1 (γT E ) ∈ M be the evaluation γT E (v1 ). Now γT E is not defined on E, but we can extend γT E to the edge E by considering the unique gradient flow line αE : E → M of the function fE : M → R labeling E given by the structure σ, that has the property that αE (v0 ) = γE (v0 ). We then define ev2 (γE ) ∈ M to be the evaluation αE (v1 ). Taking the pair ev1 × ev2 defines an element ev(γT E ) = γT E (v1 ), αE (v1 ) ∈ M 2 . Clearly the graph flow γT E on the tree T E is the restriction of a graph flow on all of Γk if and only if γE (v1 ) = αE (v1 ). This is the basic ingredient in the construction of the fibration ev: Mtree (Γ, M) → (M 2 )b , and verifying the pullback property (5). Details will appear in Cohen and Norbury (2005). This result will be used to define virtual fundamental classes for these moduli spaces. This will depend on the following homotopy theoretic construction described in Hatcher and Quinn (1974), Cohen and Jones (2002), and in full generality in Klein (2005). PROPOSITION 1.11. Let ι: P → N be a smooth embedding of closed manifolds with normal bundle ν(ι). Let p: E → N be a Serre fibration, ι∗ (E) → P the restriction to P. Let ι∗ (E) ν(ι) be the Thom space of the pullback of the normal

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bundle via the projection map p: ι∗ (E) → P. Then there is a “Thom collapse map” τ: E → ι∗ (E) ν(ι) compatible with the usual Thom collapse of the embedding, N → Pν(ι) . Notice that there is no smoothness requirement about the fibration E → N. In particular this proposition implies that if h∗ is any (generalized) homology theory for which the normal bundle ν(ι) is orientable then one can define an “umkehr map,” ι! : hq (E) → hq ι∗ (E) ν(ι) − → hq−k ι∗ (E) , where k is the codimension of the embedding ι, and the second map in this composition is the Thom isomorphism. (The existence of the Thom isomorphism is exactly what is meant by saying the normal bundle is orientable with respect to the homology theory h∗ .) Equivalently, (or dually), one gets a “push-forward map” in cohomology, ι! : hr ι∗ (E) → hr+k (E). So by Theorem 1.10, and in particular the structure displayed in Diagram 5, we can therefore construct a Thom collapse τ: M(Γ) × M → (M(Γ, M))ν where ν is the pullback along the evaluation map ev: M(Γ, M) → M b of the normal bundle to the diagonal embedding ∆b : M b → (M 2 )b , which is isomorphic to the exterior product of the tangent bundle, ν = (T M)b → M b . Thus if h∗ is any homology theory that supports an orientation of M (i.e., the tangent bundle T M), there is an umkehr map, ρ! : h∗ (BAut(Γ) × M) h∗ (M(Γ) × M) → h∗−b·dim(M) (M(Γ, M)). Now notice that since Aut(Γ) is a finite group, its classifying space BAut(Γ) is infinite-dimensional. Thus by Theorem 1.10 the moduli space M(Γ, M) is infinitedimensional. So in order to construct a fundamental class, we need to “cut down” the moduli space M(Γ) BAut(Γ) by a homology class. When we do that we can make the following definition. DEFINITION 1.12. Let h∗ be a generalized homology theory that supports an orientation of M. Given a homology class α in hk BAut(Γ) = hk M(Γ) , we can define a virtual fundamental class [Mα (Γ, M)] ∈ hk+χ(Γ)·dim(M) (M(Γ, M))

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

by the formula

161

[Mα (Γ, M)] = ρ! (α × [M])

where [M] ∈ hdim(M) (M) is the fundamental class. Although using generalized homology theories such as K-theory and bordism theory are very useful, and will be pursued in future work, for the rest of this paper we restrict to ordinary homology with integral coeﬃcients, or perhaps with Z/2 coeﬃcients if the manifolds in question are not orientable. GEOMETRIC EXPLANATION

The idea for defining this virtual fundamental class is the following. Suppose that the homology class α ∈ Hk M(Γ) is represented by a manifold Nα ⊂ M(Γ). Let α ⊂ M(Γ) be the induced Aut(Γ)-covering. We can then define a subspace N ' M) 'Nα (Γ, M) ⊂ M(Γ, M ˜ We then define the “cut down” moduli to consist of those pairs {(σ, γ) : σ ∈ N}. space 'Nα (Γ, M)/ Aut(Γ). MNα (Γ, M) = M Notice that the embedding ρ restricts to give a codimension b · d embedding, ρα : MNα (Γ, M) → Nα × M. Now if Nα could be chosen in such a way that MNα (Γ, M) is a smooth, oriented submanifold, which is compact, or could be compactified, then Poincar´e duality would imply that its fundamental class would be ρ! ([Nα ] × [M]). This class would be independent of the particular manifold Nα we chose to represent α, so long as the smoothness and compactness properties hold. Verifying that such manifolds can be chosen to satisfy these properties, would be a technical analytical problem. By defining our virtual fundamental classes using the algebraic topological methods represented in Theorems 1.10 and 1.9, we avoid the analytical issues of smoothness and compactness in our construction. ' M), we can evaluate γ on the p + q univalent Given a graph flow (σ, γ) ∈ M(Γ, vertices (leaves) to define a map ' M) → (M p ) × (M q ). ev: M(Γ,

(6)

Notice furthermore that if we restrict an automorphism in Aut(Γ) to the incoming and outgoing leaves, there is a group homomorphism r = rin × rout : Aut(Γ) → Σ p × Σq .

(7)

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The evaluation map is clearly equivariant with respect to this homomorphism. This will allow us to pass to the (homotopy) orbit space to define an evaluation map ' M)/ Aut(Γ) → (EΣ p ×Σ M p ) × (EΣq ×Σ M q ). ev: M(Γ, M) = M(Γ, p q

(8)

Here EΣk is a contractible space with a free Σk action. This type of passage to homotopy orbit spaces will be discussed in detail in Cohen and Norbury (2005). By pulling back equivariant cohomology classes of (M p ) and (M q ) to ∗ H (M(Γ, M)) and then evaluating on the virtual fundamental classes defined above, then for each α ∈ H∗ BAut(Γ) we obtain an operation qΓ (α): HΣ∗ p (M p ) ⊗ HΣ∗ q (M q ) → k

x ⊗ y → ev∗ (x × y), [Mα (Γ, M)],

(9)

where k is any ground field taken as the coeﬃcients of the cohomology groups. Using the “umkehr” maps described above, in Cohen and Norbury (2005) we actually study the corresponding adjoint operations in equivariant homology, Σ Σ qΓ : H∗ BAut(Γ) ⊗ H∗ p (M p ) → H∗ q (M q ). (10) In order to describe the properties of these operations, consider the following constructions. 1. Suppose Γ and Γ are two graphs (objects) in the category Cb,p+q , and φ: Γ → Γ a morphism. By Definition 1.1 this morphism induces a homomorphism between their automorphism groups and the associated classifying spaces, φ∗ : Aut Γ → Aut Γ

Bφ: BAut Γ → BAut Γ .

2. Let Γ1 be a graph in Cb,p+q , and Γ2 a graph in Cb ,q+r . Then let Γ1 # Γ2 be the graph in Cb+b +q−1,p+r obtained by gluing the q outgoing leaves of Γ1 to the q incoming leaves of Γ2 . 3. Let Γ1 and Γ2 be as above. Consider the homomorphisms ρout : Aut(Γ1 ) → Σq

ρin : Aut(Γ2 ) → Σq

defined by the induced permutations of the outgoing and incoming leaves, respectively. Let Aut(Γ1 ) ×Σq Aut(Γ2 ) be the fiber product of these homomorphisms. That is, Aut(Γ1 ) ×Σq Aut(Γ2 ) ⊂ Aut(Γ1 ) × Aut(Γ2 ) is the subgroup consisting of those (g1 , g2 ) with ρout (g1 ) = ρin (g1 ). Let p1 : Aut(Γ1 ) ×Σq Aut(Γ2 ) → Aut(Γ1 ),

p2 : Aut(Γ1 ) ×Σq Aut(Γ2 ) → Aut(Γ2 )

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

Figure 4.

163

Γ1 # Γ2

be the projection maps. There is also an obvious inclusion as a subgroup of the automorphism group of the glued graph, ι: Aut(Γ1 ) ×Σq Aut(Γ2 ) → Aut(Γ1 # Γ2 ) which realizes Aut(Γ1 ) ×Σq Aut(Γ2 ) as the subgroup of Aut(Γ1 # Γ2 ) consisting of automorphisms that preserve the subgraphs, Γ1 and Γ2 . In Cohen and Norbury (2005) we prove the following theorem. THEOREM 1.13. The operations Σ Σ qΓ : H∗ BAut(Γ) ⊗ H∗ p (M p ) → H∗ q (M q ) satisfy the following properties. 1. (Homotopy invariance) Let φ: Γ → Γ be a morphism in Cb,p+q . Then the following diagram commutes: Σ H∗ BAut(Γ) ⊗ H∗ p (M p ) qΓ Σ

φ∗ ⊗1

H∗ q (M q )

/ H BAut(Γ ) ⊗ H Σ p (M p ) ∗ ∗

=

qΓ

/ H Σq (M q ) ∗

2. (Gluing formula) Let Γ1 ∈ Cb,p+q , and Γ2 ∈ Cb ,q+r . Let α ∈ H∗ B Aut(Γ1 ) ×Σq Aut(Γ2 ) , and Σ

x ∈ H∗ p (M p ). Then Σ qΓ1 #Γ2 (ι∗ (α) ⊗ x) = qΓ2 (p2 )∗ (α) ⊗ qΓ1 ((p1 )∗ (α) ⊗ x) ∈ H∗ q (M q ).

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Figure 5.

The “Y-graph”

Examples. 1. Consider the graph Γ in Figure 5. Γ has one incoming and two outgoing leaves. The automorphism group Aut(Γ) = Z/2. So the operation is a map qΓ : H∗ (BZ/2) ⊗ H∗ (M) → H∗Σ2 (M × M). It is easy to see that this map is induced by Steenrod’s “equivariant diagonal” BZ/2 × M → EZ/2 ×Z/2 (M × M) given by applying the construction EZ/2×Z/2 (−) to the diagonal map ∆: M → M × M. Now work with Z/2 coeﬃcients. The Steenrod squares are defined if we take the dual map in cohomology. Namely, if α ∈ H k (M), then ∗

(qΓ ) (α ⊗ α) =

k

ak ⊗ S qk−i (α).

i=0

Here a ∈ H 1 (BZ/2; Z/2) Z/2 is the generator. It will be shown in Cohen and Norbury (2005) that the Cartan and Adem relations for the Steenrod squares follow from the homotopy invariance and gluing formulas in Theorem 1.13. 2. Now consider the graph Γ given in Figure 2 below. In this case there is only one incoming leaf, and the automorphism group is also Z/2. So the invariant (with Z/2-coeﬃcients) is a map qΓ : H∗ (BZ/2) ⊗ H∗ (M) → Z/2,

Figure 6.

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

165

or equivalently, qΓ ∈ H ∗ (BZ/2) ⊗ H ∗ (M). In Cohen and Norbury (2005) we will show that d qΓ = ai ⊗ wd−i (M) i=0

where d = dim(M), and w j (M) ∈ H j (M; Z/2) is the jth Stiefel – Whitney class of the tangent bundle.

2. String topology In this section we review the basic constructions of “string topology.” This is an intersection theory for loop spaces and path spaces in manifolds, invented by Chas and Sullivan (1999). Their theory was constructed by studying chains in a manifold and in the loop space. In our discussion we shall follow the approach used in Cohen and Jones (2002) and Cohen and Godin (2004) based on the “Thom collapse map.” This will allow us to identify a Morse theoretic approach to string topology based on graph operations, similar to those described in the last section. This, in turn, will allow us to relate string topology to the counting of holomorphic curves in the cotangent bundle of the loop space. To motivate the constructions of string topology, we first recall how classical intersection is done in the diﬀerential topology of compact manifolds. Let e: P p → N n be an embedding of smooth, closed, oriented manifolds. p and n are the dimensions of P and N respectively, and let k be the codimension, k = n − p. The idea in intersection theory is to take a q-cycle in N, which is transverse to P in an appropriate sense, and take the intersection to produce a q − k-cycle in P. Homologically, one can make this rigorous by using Poincar´e duality, to define the intersection map, e! : Hq (N) → Hq−k (P) by the composition e∗

e! : Hq (N) H n−q (N) −→ H n−q (P) Hq−k (P) where the first and last isomorphisms are given by Poincar´e duality. Intersection theory can also be realized by the “Thom collapse” map. Namely, extend the embedding e to a tubular neighborhood, P ⊂ ηe ⊂ N, and consider the projection map, τe : N → N/(N − ηe ). By the tubular neighborhood theorem, this space is homeomorphic to the Thom space of the normal bundle, N/(N−ηe ) Pηe . So the Thom collapse map can be viewed as a map, τe : N → Pηe .

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Then the homology intersection map e! is equal to the composition, (τe )∗

e! : Hq (N) −−−→ Hq (Pηe ) Hq−k (P)

(11)

where the last isomorphism is given by the Thom isomorphism theorem. In fact this description of the intersection (or “umkehr”) map e! shows that it can be defined in any generalized homology theory, for which there exists a Thom isomorphism for the normal bundle. This is an orientation condition. In these notes we will restrict our attention to ordinary homology, but intersection theories in such (co)homology theories as K-theory and cobordism theory are very important as well. We remark that the “intersection product” is the example induced by the diagonal embedding, ∆: M → M × M. This induces a product, ∆! : H p (M) ⊗ Hq (M) → H p+q−d (M),

(12)

where d is the dimension of M. The Chas – Sullivan “loop product” in the homology of the free loop space of a closed oriented n-manifold, µ: H p (LM) ⊗ Hq (LM) → H p+q−d (LM)

(13)

is defined as follows. Let Map(8, M) be the mapping space from the Figure 2 (i.e., the wedge of two circles) to the manifold M. The maps are required to be piecewise smooth (see Cohen and Jones, 2002). Notice that Map(8, M) is the subspace of LM × LM consisting of those pairs of loops that agree at the basepoint 1 ∈ S 1 . In other words, there is a pullback square Map(8, M) ev

e

M

/ LM × LM

∆

ev × ev

/M×M

where ev: LM → M is the fibration given by evaluating a loop at 1 ∈ S 1 . The map ev: Map(8, M) → M evaluates the map at the crossing point of the Figure 2. Since ev × ev is a fibration, e: Map(8, M) → LM × LM can be viewed as a codimension d embedding, with normal bundle ev∗ (η∆ ) ev∗ (T M). As was done in Cohen and Jones (2002) this diagram allows us to define a Thom-collapse map ∗ (T M)

τe : LM × LM → Map(8, M)ev

,

and therefore an intersection map, e! : H∗ (LM × LM) → H∗−d Map(8, M) .

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

Figure 7.

167

The “pair of pants” surface P

Now by going around the outside of the Figure 2, there is a map γ: Map(8, M) → LM, and the Chas – Sullivan pairing is the composition, e! γ∗ µ∗ : H p (LM) ⊗ Hq (LM) −→ H p+q−d Map(8, M) −→ H p+q−d (LM)

which defines an associative, commutative algebra structure. One can think of this structure in the following way. Consider the “pair of pants” surface P, viewed as a cobordism from two circles to one circle (see Figure 2). Consider the smooth mapping space, Map(P, M). Then there are restriction maps to the incoming and outgoing boundary circles, ρin : Map(P, M) → LM × LM,

ρout : Map(P, M) → LM.

Notice that the Figure 2 is homotopy equivalent to the surface P, with respect to which the restriction map ρin : Map(P, M) → LM × LM is homotopic to the embedding e: Map(8, M) → LM × LM. Also restriction to the outgoing boundary, ρout : Map(P, M) → LM is homotopic to γ: Map(8, M) → LM. So the Chas – Sullivan product can be thought of as a composition, (ρin )! (ρout )∗ µ∗ : H p (LM) ⊗ Hq (LM) −−−→ H p+q−d Map(P, M) −−−−→ H p+q−d (LM).

The role of the Figure 2 can therefore be viewed as just a technical one, that allows us to define the umkehr map e! = (ρin )! . More generally, consider a surface F, viewed as a cobordism from p-circles to q-circles. See Figure 2 below. We can consider the mapping space, Map(F, M), and the resulting restriction maps, ρin ρout (LM)q ←−− Map(F, M) −−→ (LM) p . (14) The idea in Cohen and Godin (2004) is to construct an “umkehr map” (ρin )! : h∗ (LM) p → h∗+χ(F)·d Map(F, M)

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Figure 8. The surface F

where χ(F) is the Euler characteristic of the surface F and d = dim(M). Like above, h∗ is any generalized homology theory that supports an orientation of M. This then allows the definition of a string topology operation (ρin )! (ρin )∗ µF : h∗ ((LM) p ) −−−→ h∗+χ(F)·d Map(F, M) −−−−→ h∗+χ(F)·d (LM)q .

It was proved in Cohen and Godin (2004) that the operations µF respect gluing of surfaces. That is, if Fg,p+q is a cobordism between p circles and q-circles of genus g, and Fh,q+r is a cobordism between q-circles and r circles of genus h, and Fg+h+q−1,p+r is the glued cobordism between p circles and r circles, then there is a relation µFg+h+q−1,p+r = µFh,q+r ◦ µFg,p+q . This can be described in field theoretic language. Notice in this case the number of outgoing boundary components must be positive. In Cohen and Godin (2004) this was condition was referred to as a “positive boundary” condition, and the following was proved. THEOREM 2.1. If h∗ is a homology theory with respect to which the closed d-manifold M is oriented, then the string topology operations which assigns to a circle S 1 the homology h∗ (LM), and to a cobordism F, the operation µF : h∗ (LM) p → h∗ (LM)q is a two-dimensional, positive boundary, topological quantum field theory. Clearly the diﬃcult part in defining the string topology operations µF is the definition of the umkehr map (ρin )! . To do this, Cohen and Godin used the Chas – Sullivan idea of representing the pair of pants surface P by a Figure 2, and realized the surface F by a “fat graph” (or ribbon graph). Fat graphs have been used to represent surfaces for many years, and to great success. See for example the following important works: Harer (1985), Strebel (1984), Penner (1987), Kontsevich (1992). We recall the definition.

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

Figure 9.

169

Thickenings of two fat graphs

DEFINITION 2.2. A fat graph is a finite graph with the following properties: 1. Each vertex is at least trivalent 2. Each vertex comes equipped with a cyclic order of the half edges emanating from it. We observe that the cyclic order of the half edges is quite important in this structure. It allows for the graph to be “thickened” to a surface with boundary. This thickening, which will be defined carefully below, can be thought of as assigning a “width” to the ink used in drawing a fat graph. Thus one is actually drawing a two-dimensional space, and it is not hard to see that it is homeomorphic to a smooth surface. Consider the following two examples (Figure 2) of fat graphs which consist of the same underlying graph, but have diﬀerent cyclic orderings at the top vertex. The orderings of the edges are induced by the counterclockwise orientation of the plane. Notice that Γ1 thickens to a surface of genus zero with four boundary components. Γ2 thickens to a surface of genus 1 with two boundary components. Of course these surfaces are homotopy equivalent, since they are each homotopy equivalent to the same underlying graph. But their diﬀeomorphism types are diﬀerent, and that is encoded by the cyclic ordering of the vertices.

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Figure 10.

The fat graph Γ2

These examples make it clear that we need to study the combinatorics of fat graphs more carefully. For this purpose, for a fat graph Γ, let E(Γ) be the set of edges, and let E(Γ) be the set of oriented edges. E(Γ) is a 2-fold cover of E(G). It has an involution E → E which represents changing the orientation. The in the following way. cyclic orderings at the vertices determines a partition of E(Γ) Consider the example illustrated in Figure 2. As above, the cyclic orderings at the vertices are determined by the counterclockwise orientation of the plane. To obtain the partition, notice that an oriented edge has well-defined source and target vertices. Start with an oriented edge, and follow it to its target vertex. The next edge in the partition is the next oriented edge in the cyclic ordering at that vertex. Continue in this way until one is back at the original oriented edge. This will be the first cycle in the partition. Then continue with this process until one exhausts all the oriented edges. The resulting cycles in the partition will be called “boundary cycles” as they reflect the boundary circles of the thickened surface. In the case of Γ2 illustrated in Figure 2, the partition into boundary cycles are given by: ¯ D, E, B, D, C, E). Boundary cycles of Γ2 : (A, B, C) (A, So one can compute combinatorially the number of boundary components in the thickened surface of a fat graph. Furthermore the graph and the surface have the same homotopy type, so one can compute the Euler characteristic of the surface directly from the graph. Then using the formula χ(F) = 2 − 2g − n, where n is the number of boundary components, we can solve for the genus directly in terms of the graph. The main theorem about spaces of fat graphs is the following (see Penner, 1987; Strebel, 1984).

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171

THEOREM 2.3. For g ≥ 2, the space of metric fat graphs Fatg,n of genus g and n boundary cycles is homotopy equivalent to the moduli space Mng of closed Riemann surfaces of genus g with n marked points. Notice that the boundary cycles of a metric fat graph Γ nearly determines a parameterization of the boundary of the thickened surface. For example, the boundary cycle (A, B, C) of the graph Γ2 can be represented by a map S 1 → Γ2 where the circle is of circumference equal to the sum of the lengths of sides A, B, and C. The ambiguity of the parameterization is the choice of where to send the basepoint 1 ∈ S 1 . In her thesis (Godin, 2004), Godin described the notion of a “marked” fat graph, and proved the following analogue of Theorem 2.3 THEOREM 2.4. Let Fat∗g,n be the space of marked metric fat graphs of genus g and n boundary components. Then there is a homotopy equivalence Fat∗g,n

Mg,n

where Mg,n is the moduli space of Riemann surfaces of genus g having n parameterized boundary components. In Cohen and Godin (2004) the umkehr map ρin : h∗ (LM) p → h∗+χ(F)·d Map(F, M) was constructed as follows. Let Γ be a marked fat graph representing a surface F. Assume p of the boundary cycles of Γ have been distinguished as “incoming,” and the remaining q have been distinguished as “outgoing.” Assume furthermore that Γ satisfies the following technical condition: DEFINITION 2.5. A fat graph Γ is called a “Sullivan chord diagram” if it satisfies the following property. An oriented edge E is contained in an incoming boundary cycle of Γ if and only if E is contained in an outgoing boundary cycle. It is easy to see that every surface F is represented by a marked chord diagram Γ. In this case the map ρin : Map(F, M) → (LM) p is homotopic to a map ρin : Map(Γ, M) → (LM) p which is obtained by restricting a map from Γ to its p incoming boundary cycles, using the parameterizations determined by the markings. Furthermore it was shown in Cohen and Godin (2004) that this map is a codimension χ(F) · d embedding, and that a Thom collapse map could be defined, τF : (LM) p → Map(Γ, M)ν

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where ν is the normal bundle of ρin . This bundle was computed explicitly in Cohen and Godin (2004). This allows for the definition of the umkehr map, as was discussed in the last section. This in turn, allowed for the definition of the string topology operation µF described above. We end this section with a discussion of “open-closed,” or perhaps a better term is “relative” string topology. In this setting our background manifold comes equipped with a collection of submanifolds, B = {Di ⊂ M}. Such a collection is referred to as a set of “D-branes,” which in string theory supplies boundary conditions for open strings. In string topology, this is reflected by considering the path spaces P M (Di , D j ) = {γ: [0, 1] → M : γ(0) ∈ Di , γ(1) ∈ D j }. Following Segal (2001), in a theory with D-branes, one associates to a connected, oriented compact one-manifold S whose boundary components are labeled by Dbranes, a vector space VS . In the case of string topology, if S is topologically a circle, the vector space VS = h∗ (LM). If S is an interval with boundary points labeled by Di and D j , then VS = h∗ P M (Di , D j ) . As is usual in field theories, to a disjoint union of such compact one manifolds, one associates the tensor product of the above vector spaces. Now to an appropriate cobordism, one needs to associate an operator between the vector spaces associated to the incoming and outgoing parts of the boundary. In the presence of D-branes these cobordisms are cobordisms of manifolds with boundary. More precisely, in a theory with D-branes, the boundary of a cobordism F is partitioned into three parts: 1. incoming circles and intervals, written ∂in (F), 2. outgoing circles and intervals, written ∂out (F), 3. the “free part” of the boundary, written ∂ f (F), each component of which is labeled by a D-brane. Furthermore ∂ f (F) is a cobordism from the boundary of the incoming one manifold to the boundary of the outgoing one manifold. This cobordism respects the labeling. We will call such a cobordism an “open-closed cobordism” (see Figure 2). In a theory with D-branes, associated to such an open-closed cobordism F is an operator, φF : V∂in (F) → V∂out (F) . Of course such a theory must respect gluing of open-closed cobordisms. Such a theory with D-branes has been put into the categorical language of PROPs by Ramirez (2005) extending notions of Segal and Moore (Segal, 2001). He called such a field theory a B-topological quantum field theory.

MORSE THEORY, GRAPHS, AND STRING TOPOLOGY

Figure 11.

173

Open-closed cobordism

In the setting of string topology, operators φF were defined by Sullivan (2004) using transversal intersections of chains. They were defined via Thom-collapse maps in Ramirez (2005) where he proved the following. THEOREM 2.6. Given a set of D-branes B in a manifold M and a generalized homology theory h∗ that supports orientations of M and all the submanifolds of B, then the open-closed string topology operations define a positive boundary B-topological quantum field theory.

3. A Morse theoretic view of string topology The goal of this section is to apply the methods for constructing homology operations using graphs described in Section 1, to the loop space LM, and to thereby recover the string topology operations from this Morse theoretic perspective. This is an exposition of the work to be contained in Cohen (2005). In order to do this, we need a plentiful supply of Morse functions on LM. Inspired by the work of Salamon and Weber (2004) we take as our Morse functions certain classical energy functions. These are defined as follows. Endow our closed d-dimensional manifold with a Riemannian metric g. Consider a potential function on M, defined to be a smooth map V: R/Z × M → R.

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We can then define the classical energy functional SV : LM → R 1 2 1 dγ − V t, γ(t) dt. γ → 0 2 dt

(15)

For a generic choice of V, SV is a Morse function (Weber, 2002). Its critical points are those γ ∈ LM satisfying the ODE ∇t

dγ = −∇Vt (x) dt

(16)

where ∇Vt (x) is the gradient of the function Vt (x) = V(t, x), and ∇t dγ/dt is the Levi-Civita covariant derivative. By perturbing SV in a precise way as in Salamon and Weber (2004, Section 2) , it is possible to assume that SV satisfies the Morse – Smale transversality condition, and one obtains a Morse chain complex, C∗V (LM), for computing the homology of the loop space, H∗ (LM): ∂

∂

V → · · · → CqV (LM) → − Cq−1 (LM) → − ···

(17)

where, as usual, the boundary map is computed by counting gradient trajectories connecting critical points. Now as in Section 1, we wish to study graph flows, but now the target is the loop space, rather than a compact manifold. Recall that a graph flow is made of gradient trajectories of diﬀerent Morse functions, that fit together according to the combinatorics and the metric of a graph. In the case of the loop space, a gradient trajectory, being a curve in the loop space, may be thought of as a map of a cylinder to M. In order to fit these cylinders together, we use the combinatorics of a fat graph, as described in Section 2. This is done by the following construction. Let Γ be a metric marked chord diagram as described in Definition 2.5. Recall this means that the boundary cycles of Γ are partitioned into p incoming and q outgoing cycles, and there are parameterizations determined by the markings, α− : S 1 → Γ, α+ : S 1 → Γ. p

q

By taking the circles to have circumference equal to the sum of the lengths of the edges making up the boundary cycle it parameterizes, each component of α+ and α− is a local isometry. Define the surface ΣΓ to be the mapping cylinder of these parameterizations, ΣΓ = S 1 × (−∞, 0] ( S 1 × [0, +∞) ∪ Γ/ ∼ (18) p

q

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175

Figure 12. ΣΓ

where (t, 0) ∈ S 1 × (−∞, 0] ∼ α− (t) ∈ Γ, and (t, 0) ∈ S 1 × [0, +∞) ∼ α+ (t) ∈ Γ Notice that the Figure 2 is a fat graph representing a surface of genus g = 0 and 3 boundary components. This graph has two edges, say A and B, and has ¯ B). If we let (A) and (B) be the incoming cycles boundary cycles (A), (B), (A, ¯ and (A, B) the outgoing cycle, then the Figure 2 graph becomes a chord diagram. Figure 3 is a picture of the surface ΣΓ , for Γ equal to the Figure 2. Notice that a map φ: ΣΓ → M is a collection of p curves φi : (−∞, 0] → LM and q-curves, φ j : [0, +∞) → LM, that have an intersection property at t = 0 determined by the combinatorics of the fat graph Γ. In particular we think of the mapping cylinder ΣΓ shown in Figure 3 as the analogue of the “Y-graph” (see Figure 5) with the vertex “blown up” via the graph Γ. Now in Section 1, a structure on a graph consisted of a metric and a Morse labeling, which was a labeling of the edges by Morse functions. For example, the Y-graph would have three distinct functions labeling its three edges. The analogue of the edges of the Y-graph in our situation are the boundary cylinders of ΣΓ . So we need to label these cylinders by functions on the loop space LM → R. We choose to restrict our attention to the energy functionals, SV : LM → R defined by a potential V: R/Z × M → R. This leads to the following definition. DEFINITION 3.1. Given a marked chord diagram Γ with p-incoming and qoutgoing boundary cycles, we define an LM-Morse structure σ on Γ to be a metric on Γ together with a labeling of each boundary cylinder of the surface ΣΓ by a distinct potential function V: R/Z× M → R. (Equivalently, the potential functions label the boundary cycles of the fat graph Γ.) This leads to the following definition of the moduli space of cylindrical flows. Compare to Definition 1.8.

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DEFINITION 3.2. Let Γ be a marked chord diagram as above. Let σ be a LMMorse structure on Γ. Suppose φ: ΣΓ → M is a continuous map, smooth on the cylinders. Let φi : S 1 × (−∞, 0] → M be the restriction of φ to the ith incoming cylinder, i = 1, . . . , p, and φ j : S 1 × [0, +∞) → M be the restriction to the jth outgoing cylinder, j = 1, . . . , q. We consider the φi ’s and φ j ’s as curves in the loop space, LM. Then the moduli space of cylindrical flows is defined to be MσΓ (LM)

dφ j dφi + ∇S Vi φi (t) = 0 and + ∇S V j φi (t) = 0 = φ: ΣΓ → M : dt dt for i = 1, . . . , p and j = 1, . . . , q .

Let φ ∈ MσΓ (LM). For i = 1, . . . , p, let φi,−1 : S 1 → M be the restriction of φi : S 1 × (−∞, 0] → M to S 1 × {−1}. Similarly, for j = 1, . . . , q, let φ j,1 : S 1 → M be the restriction of φ j to S 1 × {1}. These restrictions define the following maps (compare with the restriction and evaluation maps (6) and (14)). ρout

ρin

(LM)q ←−− MσΓ (LM) −−→ (LM) p .

(19)

In Cohen (2005) it is shown that one can define a Thom collapse map, τΓ : (LM) p → (MσΓ (LM)ν where ν is a certain vector bundle of dimension −χ(Γ)·d. This can be thought of as a normal bundle in an appropriate sense. This allows the definition of an “umkehr map” (ρin )! : h∗ ((LM) p ) → h∗+χ(Γ)·d MσΓ (LM) for any homology theory h∗ supporting an orientation of M. One can then define an operation (ρin )! (ρout )∗ qmorse : h∗ ((LM) p ) −−−→ h∗+χ(Γ)·d MσΓ (LM) −−−−→ h∗+χ(Γ)·d (LM)q . Γ

(20)

The definition of the Thom collapse map τG , the induced umkehr map (ρin )! is a consequence of the following technical result, proved in Cohen (2005). Consider the map ψ: MσΓ (LM) → Map(Γ, M) defined by restricting a cylindrical flow φ: ΣΓ → M to the graph Γ ⊂ ΣΓ . THEOREM 3.3. For a generic choice of LM-Morse structure σ on a marked chord diagram Γ, the map ψ: MσΓ (LM) → Map(Γ, M) is a homotopy equivalence.

Map(ΣΓ , M)

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177

This result then allows the construction of the Thom collapse map τΓ so that the induced umkehr map (ρin )! is equal to the umkehr map (14) via the homotopy equivalence ψ. As a result we have the following, proved in Cohen (2005). THEOREM 3.4. For any marked chord diagram Γ, the Morse theoretic operation qmorse : h∗ ((LM) p ) → h∗+χ(Γ)·d (LM)q Γ given in (20) is equal to the string topology operation qΓ : h∗ ((LM) p ) → h∗+χ(Γ)·d (LM)q defined in Theorem 2.1. This Morse theoretic viewpoint of the string topology operations has another, more geometric due to Ramirez (2005). It is a direct analogue of the perspective on the graph operations in Betz and Cohen (1994). As above, let Γ be a marked chord diagram. In Ramirez’s setting, an LMMorse structure on Γ can involve a labeling of the boundary cycles of Γ (or equivalently the cylinders of ΣΓ ) by any distinct Morse functions on LM that are bounded below, and satisfy the Palais – Smale condition as well as the Morse – Smale transversality condition. Let σ be an LM-Morse structure on Γ in this sense. Let ( f1 , . . . , f p+q ) be the Morse functions on LM labeling the p + q cylinders of ΣΓ . As above, the first p of these cylinders are incoming, and the remaining q are outgoing. Let a = (a1 , . . . , a p+q ) be a sequence of loops such that ai ∈ LM is a critical point of fi : LM → R. Let W u (ai ) and W s (ai ) be the unstable and stable manifolds of these critical points. Then define MσΓ (LM, a) = φ: ΣΓ → M that satisfy the following two conditions: dφi 1. + ∇ fi φi (t) = 0 for i = 1, . . . , p + q dt 2. φi ∈ W u (ai ) for i = 1, . . . , p, s and φ j ∈ W (a j ) for j = p + 1, . . . , p + q . Ramirez then proved that under suﬃcient transversality conditions described in Ramirez (2005) then MσΓ (LM, a) is a smooth manifold of dimension p+q p Ind(ai ) − Ind(a j ) + χ(Γ) · d. dim MσΓ (LM, a) = i=1

(21)

j=p+1

Moreover, an orientation on M induces an orientation on MσΓ (LM, a). Furthermore compactness issues are addressed, and it is shown that if dim(MσΓ (LM, a) =

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0 then it is compact. This leads to the following definition. For fi one of the f labeling Morse functions, let C∗i (LM) be the Morse chain complex for computing H∗ (LM), and let C ∗fi (LM) be the corresponding cochain complex. Consider the chain qmorse (LM) Γ

= #MσΓ (LM, a) σ dim(MΓ (LM,a))=0

· [a] ∈

p "

C ∗fi (LM)

i=1

⊗

p+q "

f

C∗ j (LM) (22)

j=p+1

We remark that the (co)chain complexes C ∗fi (LM) are generated by critical points, so this large tensor product of chain complexes is generated by vectors of critical points [a]. It is shown in Ramirez (2005) that this chain is a cycle and if one uses (arbitrary) field coeﬃcients this defines a class qmorse (LM) ∈ H ∗ (LM) ⊗p ⊗ H∗ (LM) ⊗q Γ = Hom H∗ (LM) ⊗p , H∗ (LM) ⊗q .

(23)

Ramirez then proved that these operations are the same as those defined by (20), and hence by Theorem 3.4 is equal to the string topology operation. In the case when Γ is the Figure 2, then this operation is the same as that defined by Abbondandolo and Schwarz (2004) in the Morse homology of the loop space. 4. Cylindrical holomorphic curves in the cotangent bundle This is a somewhat speculative section. Its goal is to indicate possible relations between string topology operations and holomorphic curves in the cotangent bundles. It is motivated by the work ofSalamon and Weber (2004). As before, we let M be a d-dimensional, closed oriented manifold, and T ∗ M its cotangent bundle. This is a 2d-dimensional open manifold with a canonical symplectic form ω defined as follows. Let p: T ∗ M → M be the projection map. Let x ∈ M and u ∈ T x∗ M. Consider the composition Dp u α(x, u): T (x,u) T ∗ (M) −−→ T x M → − R where T (x,u) T ∗ (M) is the tangent space of T ∗ (M) at (x, u), and Dp is the derivative of p. Notice that α is a one form, α ∈ Ω1 T ∗ (M) , and we define ω = dα ∈ Ω2 T ∗ (M) . It is well known that ω is a nondegenerate symplectic form on T ∗ (M). Now given → T ∗ M, one gets a corresponding almost a Riemannian metric on M, g: T M − complex structure Jg on T ∗ (M) defined as follows.

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The Levi-Civita connection defines a splitting of the tangent bundle of the T ∗ (M), T T ∗ (M) p∗ (T M) ⊕ p∗ T ∗ (M) . With respect to this splitting, Jg : T T ∗ (M) → T T ∗ (M) is defined by the matrix, 0 −g−1 Jg = . g 0 The induced metric on T ∗ (M) is defined by g 0 Gg = . 0 g−1 The symplectic action functional is defined on the loop space of the cotangent bundle, L T ∗ (M) . Such a loop is given by a pair, (γ, η), where γ: S 1 → M, and ∗ M. The symplectic action has the formula η(t) ∈ T γ(t) A: L(T ∗ M) → R 1 dγ (γ, η) → η(t), (t) dt. dt 0

(24)

As done by Viterbo (1996) and Salamon and Weber (2004), one can do Floer theory on T ∗ M, by perturbing the symplectic action functional by a Hamiltonian induced by a potential function V: R/Z × M → R in the following way. Given such a potential V, define HV : R/Z × T ∗ (M) → R by the formula HV t, (x, u) = 12 |u|2 + V(t, x).

(25)

Then one has a perturbed symplectic action AV : L(T ∗ M) → R

1

(γ, η) → A(γ, η) −

H t, γ(t), η(t) dt.

(26)

0

As observed in Salamon and Weber (2004), via the Legendre transform one sees that the critical points of AV are loops (γ, η), where γ ∈ LM is a critical point of the energy functional SV : LM → R, and η is determined by the derivative dγ/dt via the metric, η(v) = v, dγ/dt. Thus the critical points of AV and those of SV are in bijective correspondence. These generate the Floer complex, CF∗V (T ∗ M) and the Morse complex, C∗V (LM) respectively. The following result is stated in a form proved by Salamon and Weber (2004), but the conclusion of the theorem was first proved by Viterbo (1996).

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THEOREM 4.1. The Floer chain complex CF∗V (T ∗ M) and the Morse complex C∗V (LM) are chain homotopy equivalent. There is a resulting isomorphism of the Floer homology of the cotangent bundle with the homology of the loop space, HF∗V (T ∗ M) H∗ (LM). This result was also proved using somewhat diﬀerent methods by Abbondandolo and Schwarz (2004). The Salamon – Weber argument involved scaling the metric on M, g → (1/)g, which scales the almost complex structure J → J , and the metric on T ∗ M, 1 g 0 G → G = . 0 g−1 Notice that in this metric, the “vertical” distance in the cotangent space is scaled by . Now the boundary operator in the Floer complex CF∗V (T ∗ M) is defined by counting gradient flow lines of AV , which are curves (u, v): R → L(T ∗ M), or equivalently, (u, v) : R × S 1 → T ∗ M that satisfy the perturbed Cauchy Riemann equations, ∂ s u − ∇t v − ∇Vt (u) = 0 and

∇ s v + ∂t u − v = 0.

(27)

We refer to these maps as holomorphic cylinders in T ∗ M with respect to the almost complex structure J and the Hamiltonian HV . Salamon and Weber proved that there is an 0 > 0 so that for < 0 , the set of these holomorphic cylinders defined with respect to the metric G , that connect critical points (a1 , b1 ) and (a2 , b2 ) of relative Conley – Zehnder index one, is in bijective correspondence with the set of gradient trajectories of the energy functional SV : LM → R defined with respect to the metric (1/)g that connect a1 to a2 . Theorem 4.1 is then a consequence. The Salamon – Weber construction inspires the following idea. Let Γ be a marked chord diagram as in the previous section. Let ΣG be the cylindrical surface, and let σ be an LM - Morse structure on Γ. Recall that this means that the graph Γ has a metric, and hence the cylinders S 1 × [0, +∞) and S 1 × (−∞, 0] making up ΣΓ have well-defined widths, and hence complex structures. Recall that part of the data of a LM-Morse structure σ is a labeling of these boundary cylinders by distinct potentials, Vi : R/Z × M → R. Given a Riemannian metric g on M as above, and an > 0, we define the moduli space of “cylindrical holomorphic curves” in the cotangent bundle T ∗ (M) as follows. ∗ DEFINITION 4.2. We define Mhol (Γ,σ,) (T M) to be the space of continuous maps ∗ φ: ΣΓ → T (M) such that the restrictions to the cylinders,

φi : (−∞, 0] × S c1i → T ∗ M

and

φ j : [0, +∞) × S c1 j → T ∗ M

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are holomorphic with respect to the almost complex structure J and the Hamiltonians HVi and HV j respectively. Here the circles S c1 are round with circumference c j determined by the metric given by the structure σ. Like in the last section we have restriction maps (compare (19))

q

ρin ρout ∗ ←−− Mhol −→ L(T ∗ M) p . (28) (Γ,σ,) (T M) − .p ρin is defined by sending a cylindrical flow φ to i=1 φi,−1 : {−1} × S 1 → T ∗ M and . p+q ρout sends φ to j=p+1 φ j,1 : {1} × S 1 → T ∗ M. We conjecture the following analogue of the existence of the string topology operations, and their field theoretic properties.

L(T ∗ M)

CONJECTURE 4.3. For every marked chord diagram Γ, there is an umkehr map ∗ (ρin )! : HF∗ (T ∗ M) ⊗p → H∗+χ(Γ)·d Mhol (Γ,σ,) (T M) and a homomorphism ⊗q ∗ ∗ (ρout )∗ : H∗ Mhol (Γ,σ,) (T M) → HF ∗ (T M) so that the operations θΓ = (ρout )∗ ◦ (ρin )! : HF∗ (T ∗ M) ⊗p → HF∗ (T ∗ M) ⊗q satisfy the following properties: 1. The maps θ fit together to define a positive boundary, topological field theory. 2. With respect to the Salamon – Weber isomorphism HF∗ (T ∗ M) H∗ (LM) (Theorem 4.1) the Floer theory operations θΓ equal the string topology operations qΓ studied in the last two sections. Remark. The existence of a field theory structure on the Floer homology of a closed symplectic manifold was established by Lalonde (2004). The above conjecture should be directly related to Lalonde’s constructions. A possible way to approach this conjecture is to prove the following, which can be viewed as a generalization of the Salamon – Weber result relating the gradient trajectories of the Floer functional AV on L(T ∗ M) (i.e., J-holomorphic cylinders in T ∗ M) with the gradient trajectories of the energy functional S V on LM. CONJECTURE 4.4. There is a natural map induced by the Legendre transform, ∗ : MσΓ (LM) → Mhol (Γ,σ,) (T M)

which is a homotopy equivalence for suﬃciently small.

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In view of Theorem 3.4, this conjecture would imply Conjecture 4.3. Now one might also take the more geometric approach to the construction of these Floer theoretic operations, analogous to Ramirez’s geometrically defined Morse theoretic constructions of string topology operations. This would involve the study of the space of cylindrical holomorphic curves in T ∗ M, with boundary ∗ conditions in stable and unstable manifolds of critical points, Mhol a). (Γ,σ,) (T M, Smoothness and compactness properties need to be established for these moduli spaces. In particular, in a generic situation their dimensions should be given by the formula p+q p hol ∗ dim M(Γ,σ,) (T M, a) = Ind(ai ) − Ind(a j ) + χ(Γ) · d i=1

j=p+1

where Ind(ai ) denotes the Conley – Zehnder index. We remark that in the case of the Figure 2, this analysis has all been worked out by Abbondandolo and Schwarz (2005). In this case ΣΓ is a Riemann surface structure on the pair of pants. They proved the existence of a “pair of pants” algebra structure on HF∗V (LM) and with respect to their isomorphism, HF∗V (LM) H∗ (LM) it is isomorphic to a product structure they construct on the Morse homology of LM, which is the same as we defined in (23) in the case when the graph Γ is a Figure 2. In view of Theorem 3.4 one has the following consequence of Abbondandolo and Schwarz’s work: THEOREM 4.5. With respect to the isomorphism HF∗V (T ∗ M) H∗ (LM), the pair of pants product in the Floer homology of the cotangent bundle corresponds to the Chas – Sullivan string topology product. Another aspect of the relationship between the symplectic structure of the cotangent bundle and the string topology of the manifold, has to do with the relationship between the moduli space of J-holomorphic curves with cylindrical boundaries, Mg,n (T ∗ M), and moduli space of cylindrical holomorphic curves, ∗ Mhol (Γ,σ,) (T M), where we now let Γ and σ vary over the appropriate space of metric graphs. These moduli spaces should be related as a parameterized version of the relationship between the moduli space of Riemann surfaces and the space of metric fat graphs (Theorem 2.3). Once established, this relationship would give a direct relationship between Gromov – Witten invariants of the cotangent bundle, and the string topology of the underlying manifold. In this setting the Gromov – Witten invariants would be defined using moduli spaces of curves with cylindrical ends rather than marked points, so that the invariants would be defined in terms of the homology of the loop space (or, equivalently, the Floer homology of the cotangent bundle), rather than the homology of the manifold.

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One might also speculate about the relative invariants. As described in Section 2, there is an “open-closed” version of string topology, defined in the presence of D-branes, which are submanifolds Di ⊂ M. Recall that on the cotangent level, the conormal bundles, Conorm(Di ) ⊂ T ∗ M are Lagrangian submanifolds. It is interesting to speculate about the open-closed string topology invariants, defined on the homology of the space of paths, H∗ P M (Di , D j ) and how they may be related to the relative Gromov – Witten invariants of the conormal bundles in the cotangent space, or the similar (more general) invariants of the cotangent bundle coming from the symplectic field theory of Eliashberg et al. (2000). We believe that the relationship between the symplectic topology of the cotangent bundle and the string topology of the underlying manifold is very rich. Acknowledgements This paper is a survey of constructions and results of other papers, some of which have yet to appear. I would like to thank my collaborators and students, J. Jones, V. Godin, P. Norbury, and A. Ramirez, as well as my colleague Y. Eliashberg, for many stimulating conversations concerning this and related material. I would also like to thank O. Cornea , F. Lalonde, and the staﬀ of the mathematics department at the Universit´e de Montr´eal for organizing such a lively and informative workshop.

References Abbondandolo, A. and Schwarz, M. (2004) On the Floer homology of cotangent bundles, Comm. Pure Appl. Math., to appear; arXiv:math.SG/0408280. Abbondandolo, A. and Schwarz, M. (2005), in preparation. Betz, M. and Cohen, R. L. (1994) Graph moduli spaces of graphs and cohomology operations, Turkish J. Math. 18, 23 – 41. Bousfield, A. K. and Kan, D. M. (1972) Homotopy Limits, Completions, and Localizations, Vol. 304 of Lecture Notes in Math., New York, Springer. Chas, M. and Sullivan, D. (1999) String topology, Ann. of Math. (2), to appear; arXiv:math.GT/ 9911159. Chataur, D. (2003) A bordism approach to string topology, arXiv:math.AT/0306080. Cohen, R. L. (2005) String topology and Morse theory on the loop space, in preparation. Cohen, R. L. and Godin, V. (2004) A polarized view of string topology, In Topology, Geometry, and Quantum Field Theory, Vol. 308 of London Math. Soc. Lecture Note Ser., p. 127 – 154, Cambridge, Cambridge Univ. Press. Cohen, R. L. and Jones, J. D. S. (2002) A homotopy theoretic realization of string topology, Math. Ann. 324, 773 – 798. Cohen, R. L. and Norbury, P. (2005) Morse field theory, in preparation.

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Culler, M. and Vogtmann, K. (1986) Moduli of graphs and automorophisms of free groups, Invent. Math. 84, 91 – 119. Eliashberg, Y., Givental, A., and Hofer, H. (2000) Introduction to symplectic field theory, Geom. Funct. Anal. pp. 560–673. Godin, V. (2004) A Category of Bordered Fat Graphs and the Mapping Class Group of a Bordered Surfaces, Ph.D. thesis, Stanford University. Harer, J. L. (1985) Stability of the homology of the mapping class groups of orientable surfaces, Ann. of Math. (2) 121, 215 – 249. Hatcher, A. and Quinn, F. (1974) Bordism invariants of intersections of submanifolds, Trans. Amer. Math. Soc. 200, 327 – 344. Hu, P. (2004) Higher string topology on general spaces, arXiv:math.AT/0401081. Igusa, K. (2002) Higher Franz – Reidemeister Torsion, Vol. 31 of AMS/IP Stud. Adv. Math., Providence, RI, Amer. Math. Soc. & Somerville, MA, International Press. Klein, J. R. (2003) Fiber products, Poincar´e duality, and A∞ -ring specta, arXiv:math.AT/0306350. Klein, J. R. (2005), private communication. Kontsevich, M. (1992) Intersection theory on the moduli space of curves and the matrix Airy function, Comm. Math. Phys. 147, 1 – 23. Lalonde, F. (2004) A field theory for symplectic fibrations over surfaces, Geom. Topol. 8, 1189 – 1226. Penner, R. (1987) The decorated Teichm¨uller space of punctured surfaces, Comm. Math. Phys. 113, 299 – 339. Ramirez, A. (2005) ?TITLE?, Ph.D. thesis, Stanford University, in preparation. Salamon, D. and Weber, J. (2004) Floer homology and the heat flow, arXiv:math.SG/0304383. Segal, G. (2001) Topological structures in string theory, Philos. Trans. Roy. Soc. London Ser. A 359, 1389 – 1398. Strebel, K. (1984) Quadratic Diﬀerentials, Vol. 5 of Ergeb. Math. Grenzgeb. (3), Berlin, Springer. Sullivan, D. (2004) Open and closed string field theory interpreted in classical algebraic topology, In Topology, Geometry, and Quantum Field Theory, Vol. 308 of London Math. Soc. Lecture Note Ser., p. 344 – 357, Cambridge, Cambridge Univ. Press. Viterbo, C. (1996) Functors and computations in Floer homology, II, preprint. Voronov, A. (2005) Notes on universal algebra, In M. Lyubich and L. Takhtajan (eds.), Graphs and Patterns in Mathematics and Theoretical Physics, Vol. 73 of Proc. Sympos. Pure Math., Stony Brook, NY, 2001, Providence, RI, Amer. Math. Soc. Weber, J. (2002) Perturbed closed geodesics are periodic orbits: index and transversality, Math. Z. 241, 45 – 82.

TOPOLOGY OF ROBOT MOTION PLANNING MICHAEL FARBER University of Durham

Abstract. In this paper we discuss topological problems inspired by robotics. We study in detail the robot motion planning problem. With any path-connected topological space X we associate a numerical invariant TC(X) measuring the “complexity of the problem of navigation in X.” We examine how the number TC(X) determines the structure of motion planning algorithms, both deterministic and random. We compute the invariant TC(X) in many interesting examples. In the case of the real projective space RPn (where n 1, 3, 7) the number TC(RPn ) − 1 equals the minimal dimension of the Euclidean space into which RPn can be immersed. Key words: Robot motion planning algorithms, navigational complexity of configuration spaces, collision avoiding, cohomological lower bound, immersions of projective spaces 2000 Mathematics Subject Classification: Primary 68T40, 93C85; Secondary 55Mxx

1. Introduction This paper represents a slightly extended version of a mini-course consisting of four lectures delivered at the Universit´e de Montr´eal in June 2004. The main goal was to give an introduction to the topological robotics and in particular to describe a topological approach to the robot motion planning problem. This new theory appears to be useful both in robotics and in topology. In robotics, it explains how the instabilities in the robot motion planning algorithms depend on the homotopy properties of the robot’s configuration space. In topology, if one regards the topological spaces as configuration spaces of mechanical systems, one discovers a new interesting homotopy invariant TC(X) which measures the “navigational complexity” of X. The invariant TC(X) is similar in spirit to the Lusternik – Schnirelmann category cat(X) although in fact TC(X) and cat(X) are independent, as simple examples (given below) show. The topological approach to the robot motion planning problem was initiated by the author in Farber (2003; 2004). It was inspired by the earlier well-known work of Smale (1987) and Vassil iev (1988) on the theory of topological complexity of algorithms of solving polynomial equations. The approach of Farber (2003; 2004) was also based on the general theory of robot motion planning algorithms described in the book of J.-C. Latombe (1991). It is my pleasure 185 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 185–230. © 2006 Springer. Printed in the Netherlands.

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to acknowledge the importance of discussions of the initial versions of Farber (2003; 2004) with Dan Halperin and Micha Sharir in summer 2001 in Tel Aviv. The theory of a genus of a fiber space developed by Schwarz (1966) plays a technically important role in our approach as well as in the work of Smale and Vassiliev. Further developments and applications of the theory of topological complexity of robot motion planning of Farber (2003; 2004) are also mentioned in these notes. They include the study of collision free motion planning algorithms in Euclidean spaces (Farber and Yuzvinsky, 2004) and on graphs (Farber, 2005) and also applications to the immersion problem for the real projective spaces (Farber et al., 2003). These notes also include some new material. We explain how one may construct an explicit motion planning algorithm in the configuration space of n distinct particles in Rm having topological complexity ≤ n2 . Such an algorithm may have some practical applications. We also analyze the complexity of controlling many objects simultaneously. Finally we mention some interesting open problems. The plan of the lectures in Montreal was as follows: Lecture 1 Introduction. Interesting topological spaces provided by robotics and questions about the standard topological spaces one asks after encounters with robotics. Lecture 2 The notion of topological complexity of the motion planning problem. The Schwarz genus. Computations of the topological complexity in basic examples. Lecture 3 Topological complexity of collision free motion planning of many particles in Euclidean spaces and on graphs. Lecture 4 Motion planning in projective spaces. Relation with the immersion problem for the real projective spaces. Discussion of open problems. 2. First examples of configuration spaces The ultimate goal of robotics is creating of autonomous robots (Latombe, 1991). Such robots should be able to accept high-level descriptions of tasks and execute them without further human intervention. The input description specifies what should be done and the robot decides how to do it and performs the task. One expects robots to have sensors and actuators. A few words about history of robotics. The idea of robots goes back to ancient times. The word robot was first used in 1921 by Karel Capek in his play “Possum’s Universal Robots.” The word robotics was coined by Isaac Asimov in 1940 in his book “I, robot.”

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What is common to robotics and topology? Topology enters robotics through the notion of configuration space. Any mechanical system R determines the variety of all its possible states X which is called the configuration space of R. Usually a state of the system is fully determined by finitely many real parameters; in this case the configuration space X can be viewed as a subset of the Euclidean space Rk . Each point of X represents a state of the system and diﬀerent points represent diﬀerent states. The configuration spaces X comes with the natural topology (inherited from Rk ) which reflects the technical limitations of the system. Many problems of control theory can be solved knowing only the configuration space of the system. Peculiarities in the behavior of the system can often be explained by topological properties of the system’s configuration space. We will discuss this in more detail in the case of the motion planning problem. We will see how one may predict the character of instabilities of the behavior of the robot knowing the cohomology algebra of its configuration space. If the configuration space of the system is known one may often forget about the system and study instead the configuration space viewed with its topology and with some other geometric structures, e.g., with the Riemannian metric. EXAMPLE 2.1 (Piano movers’ problem; Schwartz and Sharir, 1983). In Figure 1 the large rectangles represent the obstacles and the black figures represent diﬀerent states of the “piano.” We assume that the picture is planar, i.e., the objects move in the horizontal plane only. Of course in practice the obstacles may have much more involved geometry than it is shown on the picture. One has to move the piano from one state to another avoiding the obstacles. The configuration space in this example is 3-dimensional having complicated geometry. The state of the piano is determined by the coordinates of the center and by the orientation. EXAMPLE 2.2 (The robot arm; Latombe, 1991). Schematically, the robot arm consists of several bars connected by revolving joins (Figure 2). One distinguishes

Figure 1.

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Figure 2.

the spacial case and the planar case (when the bars lie in a single 2-dimensional plane). The configuration space in this example is X = S1 × S1 × ··· × S1 (the n-dimensional torus) in the planar case and it is X = S2 × S2 × ··· × S2 in the spacial case. We allow the self-intersection of the arm. The space of all configurations of the planar robot arm with no self-intersections is topologically very much diﬀerent: it is homotopy equivalent to a circle, see the recent work of Connelly et al. (2003). EXAMPLE 2.3 (The “usual” configuration spaces). Let Y be a topological space and let X = F(Y, n) denote the subset of the Cartesian product Y × Y × · · · × Y (n times) containing the n-tuples (y1 , y2 , . . . , yn ) with the property that yi y j for i j (Figure 3). X = F(Y, n) is the configuration space of a system of n particles moving in the space Y avoiding collisions. The most interesting special cases are Y = Rm (the Euclidean space) and when Y is a connected graph. The configuration spaces F(Rm , n) were introduced by Fadell and Neuwirth (1962). Nowadays they are standard objects of topology. The configuration spaces F(R2 , n) and F(R2 , n)/Σn appear in the theory of braids. In 1968 V. Arnol d used information about cohomology of the configuration spaces to study algebraic functions.

Figure 3.

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In robotics it is natural to study the configuration spaces F(Γ, n) where Γ is a graph. Such spaces describe several objects moving along a prescribed net Γ (say, the factory floor) avoiding collisions, see Section 27.

3. Varieties of polygonal linkages In this section we consider the configuration spaces of polygonal linkages. These are remarkable manifolds which describe shapes of closed polygonal chains in robotics; they also appear in many areas of mathematics. The varieties of polygonal linkages carry a set of geometric structures; for example they are K¨ahler and support several Hamiltonian circle actions. These varieties were studied by Thurston (1987), Walker (1985), Klyachko (1994), Kapovich and Millson (1996), and Hausmann and Knutson (1998). Our exposition mainly follows the work of Klyachko (1994). We describe some basic facts about these varieties referring the reader to the articles mentioned above for more complete information and for proofs. Fix a vector a ∈ Rm + , a = (a1 , . . . , am ) consisting of m positive real numbers ai > 0. Define the variety M(a) as follows m / ai zi = 0 SO3 . M(a) = (z1 , . . . , zm ); zi ∈ S 2 , i=1

Here SO3 acts diagonally on the product S 2 × . . . × S 2 . M(a) is the variety of all polygonal shapes in R3 having the given side lengths (Figure 4). The first question is whether M(a) in nonempty.

Figure 4.

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Figure 5.

LEMMA 3.1. M(a) ∅ if and only if for any i = 1, . . . , m one has ai ≤ |a|/2, where |a| = a1 + a2 + · · · + am . This follows from the triangle inequality. DEFINITION 3.2. Vector a ∈ Rm + is generic if the equation solutions with i = ±1.

)m

i=1 i ai

= 0 has no

Equivalently, a is generic if the variety M(a) contains no lined configurations, i.e., configurations with all the edges lying on a single line (Figure 5). LEMMA 3.3. If a is generic then M(a) is a closed smooth manifold of dimension 2(m − 3). 3.1. SHORT AND LONG SUBSETS

How does the variety M(a) depend on the vector a? To answer this question we need to introduce the notions of short and long subsets. A subset J ⊂ {1, 2, . . . , m} is called short iﬀ ai ≤ aj i∈J

jJ

One element subsets J = {i} are always short assuming that M(a) is nonempty. S (a) will denote the set of all short subsets. S (a) is a partially ordered set, it is determined by its maximal elements (since a subset of a short subset is short). EXAMPLE 3.4. Let a = (1, 1, 1, 2). The maximal elements of S (a) are: {(12), (13), (23), {4}}. LEMMA 3.5. Assume that the vectors a, a ∈ Rm + are generic and such that the posets S (a) and S (a ) are isomorphic. Then M(a) and M(a ) are diﬀeomorphic. See Hausmann and Knutson (1998) for a proof. 3.2. POINCARE´ POLYNOMIAL OF M(A)

Klyachko (1994) found a remarkable formula for the Poincar´e polynomial of M(a).

TOPOLOGY OF ROBOT MOTION PLANNING

THEOREM 3.6. The Poincar´e polynomial of M(a) equals 1 (1 + t2 )m−1 − t2|J| P(t) = 2 2 t (t − 1) J∈S (a)

191

(1)

The proof uses the Morse theory, we shall sketch its main points. Fix a pair of indices i, j ∈ {1, . . . , m} and consider the smooth function H: M(a) → R given by H = −ai zi + a j z j 2 . One may assume without loss of generality that j = i + 1. Then H is the negative square of the length of a diagonal of a polygon (Figure 6). The critical points of H are of types (I), (II), (III) described in Figure 7. In the case of critical points of type (I) one has zi = z j . For the type (II) zi = −z j . In the case (III) all the sides of the polygon except zi and z j are lined up. The length of the base of the triangle equals k ak . (2) ki, j

Clearly the critical points of types (I) and (II) form critical submanifolds which are diﬀeomorphic to varieties of polygonal linkages with lower number of edges. Critical points of type (III) are isolated. The following lemma describes the Morse indices: LEMMA 3.7. The Morse – Bott index of the critical submanifold (I) is 0. The Morse – Bott index of the critical submanifold (II) is 2. The Morse index of any critical point of type (III) equals twice the number of minus signs k appearing in (2). We refer to Klyachko (1994) and to Kapovich and Millson (1996) for a proof. We only mention that the first statement concerning the type (I) critical points is

Figure 6.

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(I)

(II)

(III) Figure 7.

obvious: the diagonal is then the longest ai + a j and therefore the function H has minimum. Figure 8 shows why each minus sign k in the sum (2) gives a two-dimensional family of deformations of the shape of the polygon decreasing (quadratically) the value of the function H. LEMMA 3.8. H: M(a) → R is a perfect Morse – Bott function. The proof of this statement uses the existence of a symplectic structure on the

Figure 8.

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Figure 9.

manifold M(a) such that the function H generates a Hamiltonian circle action. The latter can be described geometrically as in Figure 9. One bends the polygon along the specified diagonal. The fixed points of this action are precisely the critical points of H, i.e., the polygons of types (I), (II), (III). THEOREM 3.9 (Klyachko). Assume that the vector a ∈ Rm + is generic. Then M(a) admits a symplectic structure such that the circle action described above is Hamiltonian with the function H as the Hamiltonian. Perfectness of the function H leads to the following recurrence relation for the Poincar´e polynomials. Denote a+ = (a1 , . . . , aˆ i , . . . , aˆ j , . . . , am , ai + a j ) ∈ Rm−1 + , a− = (a1 , . . . , aˆ i , . . . , aˆ j , . . . , am , |ai − a j |) ∈ Rm−1 + . Here the hat above a symbol means that this symbol should be skipped. We obtain (using the perfectness of H) the following recurrence equation: t2n , P M(a) (t) = P M(a+ ) (t) + t2 P M(a− ) (t) + )

|ai −a j |<

ki, j k ak
where k = ±1 and n is the number of negative k . This relation leads eventually to formula (1) for the Poincar´e polynomial. The reader may find the continuation of this beautiful story in (Klyachko, 1994).

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4. Universality theorems for configuration spaces How special are configuration spaces of the mechanisms? In other words, we ask if there exist specific topological properties which characterize the configuration spaces among the topological spaces? Universality theorems for configuration spaces claim (roughly) that all “reasonable” topological spaces are configuration spaces of linkages. Many theorems of this type are known. Lebesgue (1950) gave an account of several results including Kempe’s universality theorem, not for the configuration space of the mechanism itself but for the orbit of one of its vertices: “Toute courbe alg´ebrique peut eˆ tre trac´ee a` l’aide d’un syst`eme articul´e.” A theorem of Jordan and Steiner (1999) states: THEOREM 4.1. Any compact real algebraic variety V ⊂ Rn is homeomorphic to a union of components of the configuration space of a mechanical linkage. Kapovich and Millson (2002) prove the following statement: THEOREM 4.2. For any smooth compact manifold M there exists a linkage whose moduli space is diﬀeomorphic to a disjoint union of a number of copies of M. Let us explain the terms used here. An abstract linkage is a triple L = (L, , W) where L is a graph, W ⊂ V(L) is an ordered subset of vertices of L, and : E(L) → R+ is a function on the set of edges of L. Here W are the fixed vertices of L and is a “metric” (length function) on L. A planar realization of L is a map φ: V(L) → R2 such that if the vertices v, w ∈ V(L) are joined by an edge e ∈ E(L) in L then |φ(v) − φ(w)| = (e). Let W = (v1 , . . . , vn ) and let Z = (z1 , . . . , zn ) be an ordered set of n points zi ∈ R2 . A planar realization of L relative to Z is a realization φ: V(L) → R2 as above satisfying an additional requirement that φ(v j ) = z j for all j = 1, . . . , n. The set C(L, Z) of all relative planar realizations of L is called the relative configuration space of L. Elements of C(L, Z) are all planar realizations of L such that the vertices of W stay in the prescribed positions Z. The linkages which we studied in Section 3 are special cases when the graph L is homeomorphic to the circle.

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Kapovich and Millson (2002) observe that the configuration space of any planar linkage admits an involution (induced by a reflection of the plane) and this involution is nontrivial if the graph L is connected and the configuration space is not a point. Hence if M n is a closed manifold n > 0 which does not admit a nontrivial involution (such manifolds exist) then M is not homeomorphic to the moduli space of a planar linkage. 5. A remark about configuration spaces in robotics The notion of configuration space may seem obvious and even trivial for a topologist. But for people in robotics it is not so. In fact in some problems of robotics this notion appears to be even controversial. For a system of great complexity it is unrealistic to assume that its configuration space can be described completely; more reasonably to think that at any particular moment the topology and the geometry of the configuration space are known only partially or approximately. We want to emphasize that we do not question the existence of the configuration spaces. However in some particular cases it may happen to be too expensive to learn the topology of the configuration space entirely. Then one has to solve the control problems “on-line” and to learn the underlying configuration space at the same time. It seems plausible that there may exist a better mathematical notion of a configuration space describing a “partially known” topological space whose geometry is being gradually revealed. 6. The motion planning problem In this section we start studying the robot motion planning problem which is the main topic of these lectures. Imagine that you get into your advanced car and say “Go home!” and the car takes you home, automatically, obeying the traﬃc rules. Such a car must have a GPS (finding its current location) and a computer program suggesting a specific route from any initial state to any desired state. Computer programs of this kind are based on motion planning algorithms. In general, given a mechanical system, a motion planning algorithm is a function which assigns to any pair of states of the system (i.e., the initial state and the desired state) a continuous motion of the system starting at the initial state and ending at the desired state. A recent survey of algorithmic motion planning can be found in Sharir (1997); see also Latombe (1991). Papers Farber (2003; 2004) emphasize the topological nature of the robot motion planning problem. They show that the navigational complexity of configuration spaces, TC(X), is a homotopy invariant quantity which can be studied

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Figure 10.

using the algebraic topology tools. This theory explains how knowing the cohomology algebra of configuration space of a robot one may predict instabilities of its behavior. Below in this article we describe the results of Farber (2003; 2004) adding some more recent developments. Let X denote the configuration space of the mechanical system. Continuous motions of the system are represented by continuous paths γ: [0, 1] → X. Here the point A = γ(0) represents the initial state and γ(1) = B represents the final state of the system (Figure 10). Assume that X is path-connected. Practically this means that one may fully control the system and bring it to an arbitrary state from any given state. Denote by PX the space of all continuous paths γ: [0, 1] → X. The space PX is supplied with the compact-open topology. Let π: PX → X × X be the map which assigns to a path γ the pair γ(0), γ(1) ∈ X×X of the initial-final configurations. π is a fibration in the sense of Serre. DEFINITION 6.1. A motion planning algorithm is a section s: X × X → PX

(3)

π ◦ s = 1X×X .

(4)

of fibration π, i.e.,

The first question to ask is whether there exist motion planning algorithms which are continuous? Continuity of a motion planning algorithm s means that the suggested route s(A, B) of going from A to B depends continuously on the states A and B. LEMMA 6.2. A continuous motion planning algorithm in X exists if and only if the space X is contractible.1 1

This result was first observed in Farber (2003).

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Figure 11.

Proof. Let s: X × X → PX be a continuous MP algorithm. Here for A, B ∈ X the image s(A, B) is a path starting at A and ending at B. Fix B = B0 ∈ X. Define F(x, t) = s(x, B0 )(t). Here F: X × [0, 1] → X is a continuous deformation with F(x, 0) = x and F(x, 1) = B0 for any x ∈ X. This shows that X must be contractible. Conversely, let X be contractible. Then there exists a deformation F: X × [0, 1] → X collapsing X to a point B0 ∈ X. One may connect any two given points A and B by the concatenation of the path F(A, t) and the inverse path to F(B, t).

COROLLARY 6.3. For a system with noncontractible configuration space any motion planning algorithm must be discontinuous. We see that the motion planning algorithms appearing in real situations are most likely discontinuous. Our main goal is to study the discontinuities in these algorithms. Having this goal in mind, with any path-connected topological space X we associate a numerical invariant TC(X) measuring the complexity of the problem of navigation in X. We give four diﬀerent descriptions of how the number TC(X) influences the structure of the motion planning algorithms in X. One of these descriptions identifies TC(X) with the minimal number of “continuous rules” which are needed to describe a motion planning algorithm in X. On the other hand the number TC(X) equals the minimal “order of instability” of motion planning algorithms in X. We will also recover the number TC(X) while dealing with the random motion planning algorithms in X. Formally we act in a diﬀerent way: we define four a priori distinct notions of navigational complexity of topological spaces (which we denote TC j (X), where j = 1, 2, 3, 4) and we show that TCi (X) = TC j (X) for “good” spaces X (for example, for polyhedrons).

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7. Tame motion planning algorithms DEFINITION 7.1. A motion planning algorithm s: X × X → PX is called tame if X × X can be split into finitely many sets X × X = F1 ∪ F2 ∪ F3 ∪ · · · ∪ Fk

(5)

such that 1. s|Fi : Fi → PX is continuous, i = 1, . . . , k, 2. Fi ∩ F j = ∅, where i j, 3. Each Fi is an Euclidean Neighborhood Retract (ENR).2 For a fixed pair of points (A, B) ∈ Fi , the curve produced by the algorithm t → s(A, B)(t) ∈ X is a continuous curve in X which starts at point A ∈ X and ends at point B ∈ X. This curve depends continuously on (A, B) assuming that the pair of points (A, B) varies in the set Fi . Recall the definition of ENR: DEFINITION 7.2. A topological space X is called an ENR if it can be embedded into an Euclidean space X ⊂ Rk such that for some open neighborhood X ⊂ U ⊂ Rk there exists a retraction r: U → X, r|X = 1X . All motion planning algorithms which appear in practice are tame. The configuration space X is usually a semi-algebraic set and the sets F j ⊂ X × X are given by equations and inequalities involving real algebraic functions; thus they are semi-algebraic as well. In practical situations the functions s|F j : F j → PX are real algebraic and hence they are continuous. DEFINITION 7.3. The topological complexity of a tame motion planning algorithm (3) is the minimal number of domains of continuity k in a representation of type (5). DEFINITION 7.4. The topological complexity TC1 (X) of a path-connected topological space X is the minimal topological complexity of motion planning algorithms in X. Observation. TC1 (X) = 1 if and only if X is an ENR and it is contractible. We set TC1 (X) = ∞ if X admits no tame motion planning algorithms. 2

An equivalent concept was introduced in Farber (2004) under the name “motion planner.”

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Figure 12.

Figure 13.

EXAMPLE 7.5. Let us show that TC1 (S n ) = 2 for n odd and TC1 (S n ) ≤ 3 for n even. Let F1 ⊂ S n × S n be the set of all pairs (A, B) such that A −B. We may construct a continuous section s1 : F1 → PS n by moving A toward B along the shortest geodesic arc. Consider now the set F2 ⊂ S n × S n consisting of all pairs antipodal (A, −A). If n is odd we may construct a continuous section s2 : F2 → PS n as follows. Fix a nonvanishing tangent vector field v on S n . Move A toward the antipodal point −A along the semi-circle tangent to vector v(A). In the case when n is even find a tangent vector field v with a single zero A0 ∈ S n . Define F2 = {(A, −A); A A0 } and define s2 : F2 → PS n as above. The set F3 = {(A0 , −A0 )} consists of a single pair; define s3 : F3 → PS n by choosing an arbitrary path from A0 to −A0 .

8. The Schwarz genus Let p: E → B be a fibration. Its Schwarz genus is defined as the minimal number k such that there exists an open cover of the base B = U1 ∪ U2 ∪ · · · ∪ Uk with the property that over each set U j ⊂ B there exists a continuous section s j : U j → E of E → B. This notion was introduced by A. S. Schwarz in 1958. In 1987 – 1988 S. Smale and V. A. Vassiliev applied the notion of Schwarz genus to study complexity of algorithms of solving polynomial equations.

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The genus of a fibration equals 1 if and only if it admits a continuous section. The genus of the Serre fibration P0 X → X coincides with the Lusternik – Schnirelmann category cat(X) of X, see Cornea et al. (2003). Here P0 (X) is the space of paths in X which start at the base point x0 ∈ X. For the motion planning problem we need to study a diﬀerent fibration π: PX → X × X. 9. The second notion of topological complexity The invariant TC1 (X) introduced above seems to be quite natural from the robotics point of view. However a more convenient topological invariant uses open covers instead of decompositions into ENR’s. DEFINITION 9.1. Let X be a path-connected topological space. The number TC2 (X) is defined as the Schwartz genus of the fibration π: PX → X × X. This notion coincides with the original definition of the topological complexity of the robot motion planning problem given in Farber (2003). Explicitly, TC2 (X) is the minimal number k such that there exists an open cover X × X = U1 ∪ U2 ∪ · · · ∪ Uk with the property that π admits a continuous section s j : U j → PX over each U j ⊂ X × X. Note that the inclusion U j → X × X may be not null-homotopic. For example, if X is a polyhedron, there always exists a continuous section over a small neighborhood of the diagonal X ⊂ X × X. We know that TC2 (X) = 1 if and only if X is contractible. LEMMA 9.2. One has cat(X) ≤ TC2 (X) ≤ cat(X × X). Proof. We shall use two following simple properties of the Schwartz genus. Consider a fibration E → B. 1. Let B ⊂ B be a subset, E = p−1 (B ). Then the genus of E → B is less than or equal to the genus of E → B. 2. The genus of E → B is less than or equal to cat(B). To probe the lemma, apply 1 to the fibration PX → X × X and to the subset X × x0 ⊂ X × X. Note that π−1 (X × x0 ) = P0 X. We find TC2 (X) ≥ cat(X). 2 gives TC2 (X) ≤ cat(X × X). Exercise. Let G be a connected Lie group. Then TC2 (G) = cat(G).

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EXAMPLE 9.3. TC2 SO(3) = cat SO(3) = cat(RP3 ) = 4. This example is important for robotics since SO(3) is the configuration space of a rigid body in R3 fixed at a point. 10. Homotopy invariance THEOREM 10.1. The number TC2 (X) is a homotopy invariant of X. See Farber (2003) for a proof. 11. Order of instability of a motion planning algorithm Let s: X × X → PX be a tame motion planning algorithm and let X × X = F1 ∪ F2 ∪ . . . ∪ Fk

(6)

be a decomposition into domains of continuity as in Definition 7.1. Here Fi ∩ F j = ∅ and each F j is an ENR. DEFINITION 11.1. The order of instability of the decomposition (6) is the maximal r so that for some sequence of indices 1 ≤ i1 < i2 < · · · < ir ≤ k the intersection F i1 ∩ F i2 ∩ . . . ∩ F ir ∅ in not empty. The order of instability of a motion planning algorithm3 s is the minimal order of instability of decompositions (6) for s. The order of instability is an important functional characteristic of a motion planning algorithm. If the order of instability equals r then for any > 0 there exist r pairs of initial-final configurations (A1 , B1 ),

(A2 , B2 ),

...,

(Ar , Br )

which are within distance < from one another and which lie in distinct sets Fi . This means that small perturbations of the input data (A, B) may lead to r essentially distinct motions suggested by the motion planning algorithm. DEFINITION 11.2. Let TC3 (X) be defined as the minimal order of instability of all tame motion planning algorithms in X. 3

This notion was introduced and studied in Farber (2004).

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Obviously one has: TC3 (X) ≤ TC1 (X).

(7)

12. Random motion planning algorithms In this section we analyze complexity of random motion planning algorithms, following Farber (2005). Let X be a path-connected topological space. A random n-valued path σ in X starting at A ∈ X and ending at B ∈ X is given by an ordered sequence of paths γ1 , . . . , γn ∈ PX and an ordered sequence of real numbers p1 , . . . , pn ∈ [0, 1] such that each γ j : [0, 1] → X is a continuous path in X starting at A = γ j (0) and ending at B = γ j (1), and p j ≥ 0,

p1 + p2 + · · · + pn = 1.

(8)

One thinks of the paths γ1 , . . . , γn as being diﬀerent states of σ (Figure 15). The number p j is the probability that the random path σ is in state γ j . Random path σ as above will be written as a formal linear combination σ = p1 γ1 + p2 γ2 + · · · + pn γn . Equality between n-valued random paths is understood as follows: the random path σ = p1 γ1 + p2 γ2 + · · · + pn γn . is equal to σ = p 1 γ1 + p 2 γ2 + · · · + p n γn iﬀ p j = p j for all j = 1, . . . , n and, besides, γ j = γ j for all indices j with p j 0. In other words, the path γ j which appears with the zero probability p j = 0 could be replaced by any other path starting at A and ending at B.

Figure 14.

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We denote by Pn X the set of all n-valued random paths in X. The set Pn X has a natural topology: It is a factor-space of a subspace of the Cartesian product of n copies of PX × [0, 1]. The canonical map π: Pn X → X × X (9) assigning to a random path its initial and end points is continuous. DEFINITION 12.1. An n-valued random motion planning algorithm is defined as a continuous section s: X × X → Pn X. (10) of fibration (9). Given a pair (A, B) ∈ X × X (an input), the output of the random motion planning algorithm (10) is an ordered probability distribution s(A, B) = p1 γ1 + · · · + pn γn

(11)

supported on n paths between A and B. In other words, the algorithm s produces the motion γ j with probability p j where j = 1, . . . , n. Now we come to yet another notion of complexity of path-connected topological spaces: DEFINITION 12.2. Let TC4 (X) be defined as the minimal integer n such that there exists an n-valued random motion planning algorithm s: X × X → Pn X. 13. Equality theorem THEOREM 13.1. Let X be a simplicial polyhedron. Then four notions of topological complexity introduced above coincide, i.e., one has TC1 (X) = TC2 (X) = TC3 (X) = TC4 (X).

Figure 15.

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Proof. Let k denote TC1 (X) and let s: X × X → PX be a tame motion planning algorithm (as in Definition 7.1). We have a splitting X × X = F1 ∪ · · · ∪ Fk into disjoint ENRs such that the restriction s|F j is continuous. Let us show that one may enlarge each set F j to an open set U j ⊃ F j such that the section s|F j extends to a continuous section s j defined on U j . This would prove that TC2 (X) ≤ TC1 (X).

(13)

We will use the following property of the ENRs: If F ⊂ X and both spaces F and X are ENRs then there is an open neighborhood U ⊂ X of F and a retraction r: U → F such that the inclusion j: U → X is homotopic to i ◦ r, where i: F → X denotes the inclusion. See Dold, (1972, Chapter 4, Section 8) for a proof. Using the fact that the sets Fi and X × X are ENRs, we find that there exists an open neighborhood Ui ⊂ X × X of the set Fi and a continuous homotopy hiτ : Ui → X × X, where τ ∈ [0, 1], such that hi0 : Ui → X × X is the inclusion and hi1 is a retraction of Ui onto Fi . We will describe now a continuous map s i : Ui → PX with E ◦ s i = 1Ui . Given a pair (A, B) ∈ Ui , the path hiτ (A, B) in X × X is a pair of paths (γ, δ), where γ is a path in X starting at the point γ(0) = A and ending at a point γ(1), and δ is a path in X starting at B = δ(0) and ending at δ(1). Note that the pair γ(1), δ(1) belongs to Fi ; therefore the motion planner si : Fi → PX defines a path ξ = si γ(1), δ(1) ∈ PX connecting the points γ(1) and δ(1). Now we set s i (A, B) to be the concatenation of γ, ξ, and δ−1 (the reverse path of δ): s i (A, B) = γ · ξ · δ−1 . Now we want to show that X always admits a tame motion planning algorithm (see Definition 7.1) with the number of local domains F j equal to = TC2 (X). This will show that TC1 (X) ≤ TC2 (X). (14) Let U1 ∪ U2 ∪ · · · ∪ U = X × X,

where = TC2 (X),

(15)

be an open cover such that for any i = 1, . . . , there exists a continuous motion planning map si : Ui → PX with π ◦ si = 1Ui . Find a piecewise linear partition of unity { f1 , . . . , f } subordinate to the cover (15). Here fi : X × X → [0, 1] is a piecewise linear function with support in Ui and such that for any pair (A, B) ∈ X × X, it holds that f1 (A, B) + f2 (A, B) + · · · + f (A, B) = 1.

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Fix numbers 0 < ci < 1 where i = 1, . . . , with c1 + · · · + c = 1. Let a subset Vi ⊂ X × X, where i = 1, . . . , , be defined by the following system of inequalities f j (A, B) < c j for all j < i, fi (A, B) ≥ ci . Then: (a) each Vi is an ENR; (b) Vi is contained in Ui ; therefore, the section si : Ui → PX restricts onto Vi and defines a continuous section over Vi ; (c) the sets Vi are pairwise disjoint, Vi ∩ V j = ∅ for i j; (d) V1 ∪ V2 ∪ · · · ∪ Vk = X × X. Hence we see that the sets Vi and the sections si |Vi define a tame motion planning algorithm in the sense of Definition 7.1 with = TC2 (X) local domains. Now we prove that TC3 (X) ≤ TC2 (X). (16) Suppose that s: X × X → PX is a tame motion planning algorithm with domains of continuity F1 , . . . , Fk ⊂ X×X. Denote the order of instability of the decomposition X × X = F1 ∪ · · · ∪ Fk by r ≤ k. Then any intersection of the form F i1 ∩ · · · ∩ F ir+1 = ∅,

(17)

is empty, where 1 ≤ i1 < i2 < · · · < ir+1 ≤ k. For any index i = 1, . . . , k fix a continuous function fi : X × X → [0, 1] such that fi (A, B) = 1 if and only if the pair (A, B) belongs to F i and such that the support supp( fi ) retracts onto Fi . Let φ: X × X → R be the maximum of (finitely many) functions of the form fi1 + fi2 + · · · + fir+1 for all increasing sequences 1 ≤ i1 < i2 < · · · < ir+1 ≤ k of length r + 1. We have: φ(A, B) < r + 1 for any pair (A, B) ∈ X × X, as follows from (17). Let Ui ⊂ X × X denote the set of all (A, B) such that (r + 1) · fi (A, B) > φ(A, B). Then Ui is open and contains F i , and hence the sets U1 , . . . , Uk form an open cover of X × X. On the other hand, any intersection Ui1 ∩ Ui2 ∩ · · · ∩ Uir+1 = ∅ is empty. As above we may assume that the sets U1 , . . . , Uk are small enough so that over each Ui there exists a continuous motion planning section (here we use

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the assumption that each Fi is an ENR). Applying Lemma 13.2 (see below) we conclude that TC2 (X) ≤ r. Combining inequalities (7), (13), (14), (16) we obtain TC1 (X) = TC2 (X) = TC3 (X). Next we show that TC2 (X) = TC4 (X). This last argument is an adjustment of the proof of Schwarz (1966, Proposition 2). Assume that there exists an n-valued random motion planning algorithm s: X× X → Pn X in X. The right-hand side of formula (11) defines continuous real valued functions p j : X × X → [0, 1], where j = 1, . . . , n. Let U j denote the open set p−1 j (0, 1] ⊂ X × X. The sets U 1 , . . . , U n form an open covering of X × X. Setting s j (A, B) = γ j , one gets a continuous map s j : U j → PX with π ◦ si = 1U j . Hence, n ≥ TC2 (X) according to the definition of TC2 (X). Conversely, setting k = TC2 (X), we obtain that there exists an open cover U1 , . . . , Uk ⊂ X×X and a sequence of continuous maps si : Ui → PX where π◦si = 1Ui , i = 1, . . . , k. Extend si to an arbitrary (possibly discontinuous) mapping S i : X × X → PX satisfying π ◦ S i = 1X×X . This can be done without any diﬃculty; it amounts in making a choice of a connecting path for any pair of points (A, B) ∈ X×X−Ui . One may find a continuous partition of unity subordinate to the open cover U1 , . . . , Uk . It is a sequence of continuous functions p1 , . . . , pk : X × X → [0, 1] such that for any pair (A, B) ∈ X × X one has p1 (A, B) + p2 (A, B) + · · · + pk (A, B) = 1 and the closure of the set p−1 i (0, 1] is contained in U i . We obtain a continuous kvalued random motion planning algorithm s: X × X → Pn X given by the following explicit formula s(A, B) = p1 (A, B)S 1 (A, B) + · · · + pk (A, B)S k (A, B).

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The continuity of s follows from the continuity of the maps S i restricted to the domains p−1 i (0, 1]. This completes the proof. LEMMA 13.2. Let X be a path-connected metric space. Consider an open cover X × X = U1 ∪ U2 ∪ · · · ∪ U such that for any i = 1, . . . , there exists a continuous map si : Ui → PX with π◦si = 1Ui . Suppose that for some integer r any intersection Ui1 ∩ Ui2 ∩ . . . ∩ Uir = ∅ is empty where 1 ≤ i1 < i2 < · · · < ir ≤ . Then TC2 (X) < r. A proof of Lemma 13.2 can be found in Farber (2004).

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Although the numbers TC j (X) (where j = 1, 2, 3, 4) coincide when X is a simplicial polyhedron, they do not coincide when X is a general topological space. The most convenient notion topologically is TC2 (X). Notation. In what follows we will use the notation TC(X) = TC2 (X).

14. An upper bound for TC(X) THEOREM 14.1. For any path-connected paracompact locally contractible space X one has TC(X) ≤ 2 dim X + 1, (19) where dim X denotes the covering dimension of X. Proof. We know that TC(X) ≤ cat(X × X). Combine this with cat(X × X) ≤ dim(X × X) + 1 = 2 dim X + 1. This result can be improved assuming that X is highly connected: THEOREM 14.2. If X is an r-connected CW-complex then TC(X) <

2 · dim X + 1 + 1. r+1

(20)

See Farber (2004) for a proof. 15. A cohomological lower bound for TC(X) In this section we describe a result from Farber (2003). Let be a field. The cohomology H ∗ (X; ) = H ∗ (X) is a graded -algebra with the multiplication ∪: H ∗ (X) ⊗ H ∗ (X) → H ∗ (X) given by the cup-product. The tensor product H ∗ (X) ⊗ H ∗ (X) is again a graded -algebra with the multiplication (u1 ⊗ v1 ) · (u2 ⊗ v2 ) = (−1)|v1 |·|u2 | u1 u2 ⊗ v1 v2 . Here |v1 | and |u2 | denote the degrees of cohomology classes v1 and u2 correspondingly. The cup-product ∪ is an algebra homomorphism. DEFINITION 15.1. The kernel of the homomorphism ∪: H ∗ (X) ⊗ H ∗ (X) → H ∗ (X) is called the ideal of the zero-divisors of H ∗ (X). The zero-divisors-cup-length of H ∗ (X) is the length of the longest nontrivial product in the ideal of the zero-divisors of H ∗ (X).

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THEOREM 15.2. TC(X) is greater than the zero-divisors-cup-length of H ∗ (X). See Farber (2003) for a proof. 16. Examples EXAMPLE 16.1. Consider the case X = S n . Let u ∈ H n (S n ) be the fundamental class, and let 1 ∈ H 0 (S n ) be the unit. Then a = 1 ⊗ u − u ⊗ 1 ∈ H ∗ (S n ) ⊗ H ∗ (S n ) is a zero-divisor. Another zero-divisor is b = u ⊗ u. Computing a2 = a · a we find a2 = (−1)n−1 − 1 · u ⊗ u. Hence a2 = −2b for n even and a2 = 0 for n odd. We conclude: the zero-divisors-cup-length of H ∗ (S n ; Q) equals 1 for n odd and 2 for n even. Applying Theorem 15.2 we find that TC(S n ) ≥ 2 for n odd and TC(S n ) ≥ 3 for n even. In Section 7 we constructed explicit motion planning algorithms having topological complexity 2 for n odd and 3 for n even. Hence, 2, if n is odd, n TC(S ) = 3, if n is even. EXAMPLE 16.2. Here we calculate the number TC(X) when X is a graph. THEOREM 16.3. If X is a connected finite graph then   1, if b1 (X) = 0,    2, if b1 (X) = 1, TC(X) =    3, if b (X) > 1. 1

Proof. If b1 (X) = 0 then X is contractible and hence TC(X) = 1. If b1 (X) = 1 then X is homotopy equivalent to the circle and therefore TC(X) = TC(S 1 ) = 2, see above. Assume now that b1 (X) > 1. Then there exist two linearly independent classes u1 , u2 ∈ H 1 (X). Thus 1 ⊗ ui − ui ⊗ 1,

i = 1, 2

are zero-divisors and their product equals u2 ⊗ u1 − u1 ⊗ u2 0 which implies TC(X) ≥ 3. On the other hand, we know that TC(X) ≤ 3 by Theorem 14.1. This completes the proof. EXAMPLE 16.4. Let X = Σg be a compact orientable surface of genus g. Then 3, if g ≤ 1, TC(X) = 5, if g > 1. We leave the proof as an exercise.

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17. Simultaneous control of many systems Suppose that we have to control two systems simultaneously. We assume that the systems do not interact, i.e., the admissible states of one of the systems do not depend on the state of the other. Let X and Y be the corresponding configuration spaces. If we view these two systems as a new single system then the configuration space is the product X × Y. For the topological complexity of the product one has the inequality: THEOREM 17.1. TC(X × Y) ≤ TC(X) + TC(Y) − 1. A proof can be found in Farber (2003). Suppose now that one has to control simultaneously n systems having configuration spaces X1 , . . . , Xn . The total configuration space is the Cartesian product Y n = X 1 × X 2 × . . . × Xn . (21) We ask: What is the asymptotics of the topological complexity TC(Yn ) for large n? We shall assume that the topological complexity of the space Xn is bounded, i.e., there exists a constant M ≥ 1 such that TC(Xn ) ≤ M for all n. Applying Theorem 17.1 one obtains the inequality TC(Yn ) ≤ n · [M − 1] + 1.

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This shows that the sequence TC(Yn ) growths at most linearly. Let us assume additionally that each space Xn is path-connected and homologically nontrivial, i.e., H ∗ (Xn ) H ∗ (pt). Then one has TC(Yn ) ≥ n + 1. Proof. Let ur ∈ H ir (Xr ) be a nontrivial class, where ir > 0. Denote wr = 1 × 1 × · · · × ur × 1 × · · · × 1 ∈ H ir (Yn ) (here ur stands on the r-th place). Then n 0

w j ∈ H ∗ (Yn )

j=1

is a nonzero class. The class w j = w j ⊗ 1 − 1 ⊗ w j,

j = 1, . . . , n

is a zero-divisor and the product n 0 j=1

wj =

n 0 j=1

wj ⊗ 1 ± . . . 0

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is nonzero. This proves (22) as follows from the cohomological lower bound. Combining the inequalities (22) and (23) one obtains: COROLLARY 17.2. Assume that each space Xr is path-connected and homologically nontrivial and the topological complexity TC(Xr ) is bounded above. Then the topological complexity of the product Yn (see (21)) (viewed as a function of n) has a linear growth. In particular, for any finite-dimensional path-connected and homologically nontrivial polyhedron X the sequence TC(X n ) as a function of n has a linear growth. This result has an important implication in the control theory: THEOREM 17.3. A centralized control by n identical independent systems has topological complexity which is linear in n (more precisely, the inequalities (22) and (23) are satisfied). The distributed control, i.e., when each of the objects is controlled independently of the others, has an exponential topological complexity TC(X)n . We see that in practical situations the centralized control by many independent objects could be organized so that its “much more stable” than the distributed control. 18. Another inequality relating TC(X) to the usual category The result of this section was inspired by a discussion with H.-J. Baues. Consider the fibration π: PX → X × X, cf. Definition 6.1. LEMMA 18.1. Let U ⊂ X × X be a subset. There exists a continuous section s: U → PX, π ◦ s = 1U of π over U if and only if the inclusion U → X × X is homotopic to a map with values in the diagonal ∆X ⊂ X × X. Proof. Let s: U → PX be a section. Here s(A, B)(t) ∈ X is a continuous function of A, B, t (where (A, B) ∈ U and t ∈ [0, 1]) such that s(A, B)(0) = A and s(A, B)(1) = B. Define σ: U × [0, 1] → X × X by σ(A, B)(t) = (s(A, B)(t), B). Then one has σ(A, B)(0) = (A, B) and σ(A, B)(1) = (B, B) takes values on the diagonal ∆X. Hence σ is a homotopy between the inclusion U → X × X and a map with values on the diagonal. Conversely, suppose that σt : U → X × X is a homotopy from the inclusion to a map with values on the diagonal. Then t → σt (A, B) is a path in X × X which starts at (A, B) and ends at a point (C, C). In other words, σt (A, B) is a pair (γ1 , γ2 )

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of paths in X where γ1 starts at A, γ2 starts at B, and the end points of these paths coincide. Hence the path s = γ1 γ2−1 ∈ PX is well-defined, continuously depends on A and B and starts at A and ends at B. We obtain a continuous section of π over U. COROLLARY 18.2. The topological complexity TC(X) is the smallest k such that X × X can be covered by k open subsets U1 ∪ U2 ∪ · · · ∪ Uk = X × X such that each U j → X × X is homotopic to a map with values in the diagonal ∆X ⊂ X × X. The following inequality complements Lemma 9.2. LEMMA 18.3. If X is an ENR then TC(X) ≥ cat (X × X)/∆X − 1. Proof. Let X × X = U1 ∪ U2 ∪ · · · ∪ Uk and each Ui → X × X is homotopic to a map with values in ∆(X). Let U j = U j − ∆(X) and U j ⊂ (X × X)/∆(X) be the image of U j under the canonical map X × X → X × X/∆X. Then U j is nullhomotopic and these sets cover the whole X × X/∆X except the base point of the factor-space. Hence, adding a contractible neighborhood of the base point gives a categorical cover of the factor-space. Existence of such neighborhood follows from the ENR assumption. This completes the proof.

19. Topological complexity of bouquets It is quite obvious that TC(X ∨ Y) ≥ max{TC(X), TC(Y)}.

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We shall prove the following: THEOREM 19.1. Let X and Y be two polyhedrons. Then TC(X ∨ Y) is less than or equal to max{TC(X), TC(Y), cat(X) + cat(Y) − 1}. (25) Proof. The product (X ∨ Y) × (X ∨ Y) is a union of four spaces X × X,

Y × Y,

X × Y,

Y×X

and any two of these spaces intersect at a single point (p, p) where p is the join point of the wedge X ∨ Y. Over each of these sets one may construct a motion planning algorithm having respectively TC(X),

TC(Y),

cat(X) + cat(Y) − 1,

cat(X) + cat(Y) − 1

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domains of continuity. For example, the algorithm over X × Y takes pairs (x, y) ∈ X × Y as an input and finds a path α in X connecting x with p, a path β in Y connecting y with p and finally produces the path αβ−1 as the output. To make the choice of α continuous one has to split X into cat(X) pieces; to make the choice of β continuous one splits Y into cat(Y) pieces. Similarly to the proof of the product inequality (see Farber, 2003, Theorem 11) one may rearrange the totality of cat(X) × cat(Y) products into cat(X) + cat(Y) − 1 sets (the checkerboard trick) such that the algorithm is continuous over each of them. The remaining arguments of the proof are similar (compare with the next section), we leave them as an exercise for the reader.

20. A general recipe to construct a motion planning algorithm Let X be a path-connected polyhedron and let U = {U1 , U2 , . . . , Un } be a nice open cover of X with the property that each inclusion Ui → X is null-homotopic. The word “nice” means that the Main assumption (see below) is satisfied. Our goal is to construct a motion planning algorithm in X with 2m − 1 local domains where m is the multiplicity of the covering U, i.e., the maximal number of distinct domains U j having a nonempty intersection. Introduce subsets V1 , V2 , . . . , Vm where Vr ⊂ X denotes the set of points x ∈ X which are covered by precisely r sets U j . Main assumption. Each Vi is an ENR. For any multi-index α = (1 ≤ i1 < i2 < · · · < ir ≤ n) denote Uα =

r

Uik .

k=1

Then Vr =

|α|=r

Uα −

Uα .

|α|=r+1

Note that Vr = ∅ for r > m. LEMMA 20.1. (A) Each set Wα = Uα ∩ Vr (where |α| = r) is closed and open in Vr . (B) The sets Wα and Wβ are disjoint for α β, |α| = r = |β|.

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Proof. Clearly x ∈ Uα ∩ Uβ implies that x ∈ Uα∪β . This implies statement (B). Now we want to show slightly more, namely, that the sets W α and Wβ are disjoint for |α| = r = |β|, α β. Indeed, if x ∈ W α ∩ Wβ then x = lim xn where xn ∈ Wα . Since x lies in Wβ one has xn ∈ Wβ for all large n and hence xn ∈ Uα∪β Vr — a contradiction. These two statements imply that W α ∩ Vr = Wα i.e., Wα is closed in Vr . As follows from the definition Wα is also open in Vr . LEMMA 20.2. One has Vr ⊂

Vk .

k≤r

Proof. It follows directly from (26).

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LEMMA 20.3. Over each set Vr × Vr ⊂ X × X one may construct an explicit continuous section of the fibration π: PX → X × X. Proof. We know that Vr = |α|=k Wα and each Wα is open and closed in Vr . Hence it is enough to construct a continuous section over each Wα × Wβ . Let i and j be the smallest indices appearing in the multi-indices α and β correspondingly. j Then Wα ⊂ Ui and Wβ ⊂ U j . Let Hti : Ui × I → X and Ht : U j × I → X be the homotopies contracting Ui and U j to the base point x0 ∈ X. Then, given a pair (x, y) ∈ Wα × Wβ one constructs a path connecting them as follows: it is concatenation of the path Hti (x) leading from x to the base point and then follows j the reverse path to Ht (y). Denote Ak =

Vr × Vr ⊂ X × X,

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r+r =k+1

where k = 1, 2, . . . , 2m − 1. These sets are ENR’s (by the assumption) and cover X × X. LEMMA 20.4. Each product Vr × Vr , where r + r = k + 1, is closed and open in Ak . Proof. It follows from (20.2). Hence the described above local sections over each Vr × Vr combine into a continuous section over Ak . In total, we have 2m − 1 local sections. 21. How diﬃcult is to avoid collisions in R m? In this section we start discussing the problem of finding the topological complex ity TC F(Rm , n) of the configuration space F(Rm , n) of n distinct points in the

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Euclidean space Rm . A motion planning algorithm in F(Rm , n) takes as an input two configurations of n distinct points in Rm and produces n continuous curves A1 (t), . . . , An (t) ∈ Rm , where t ∈ [0, 1], such that Ai (t) A j (t) for all t ∈ [0, 1], i j and A1 (0), . . . , An (0) and A1 (1), . . . , An (1) are the first and the second given configurations. In other words, a motion planning algorithm in F(Rm , n) moves one of the given configurations into another avoiding collisions. The following theorem was obtained in Farber and Yuzvinsky (2004). THEOREM 21.1. One has TC(F(Rm , n)) =

2n − 1 2n − 2

for any odd m, for m = 2.

At the moment we do not know the answer for the case m ≥ 4 even. We know that in this case the number TC(F(Rm , n)) is either 2n − 1 or 2n − 2. Conjecture. For any m even one has TC(F(Rm , n)) = 2n − 2. We will give here some ideas of the proof of Theorem 21.1 referring the reader to Farber and Yuzvinsky (2004) for details. We will also discuss the possible approaches to construct explicit motion planning algorithms in F(Rm , n). Such algorithms could be useful in situations when a large number of objects must be moved automatically (without human intervention) from one position to another avoiding collisions. Consider the set Hi j = {(y1 , . . . , yn ); yi ∈ Rm , yi = y j } ⊂ Rnm . Here i, j ∈ {1, 2, . . . , n}, i < j. The set Hi j is a linear subspace of Rnm of codimension m. The system of subspaces {Hi j }i< j is an arrangement of linear subspaces of codimension m. Our approach to the problem is to view the union H= Hi j i< j

as the set of obstacles: F(Rm , n) = Rnm − H. 22. The case m = 2 Assume first that m = 2. This means that we are dealing with n distinct particles on the plane. Then Hi j ⊂ Cn is a complex subspace of codimension 1.

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Consider a slightly more general situation. Let A = {H} be a finite set of hyperplanes in an aﬃne complex space Cn . Denote by M(A) the complement H. M(A) = Cn − H∈A

We will study the motion planning problem in M(A). We may say that we live in Cn and the union of hyperplanes H represent our obstacles. Recall some terminology from the theory of arrangements (Orlik and Terao, 1 1992). If H∈A H ∅ then A is called central, and up to change of coordinates the hyperplanes can be assumed linear. Suppose that A is linear. For each H ∈ A one can fix a linear functional αH (unique up to a non-zero multiplicative constant) such that H = {αH = 0}. A set of hyperplanes Hi ∈ A is called linear independent if the corresponding functionals αHi are linearly independent. The rank of {αH }, i.e., the cardinality of a maximal independent subset, is called the rank of A and denoted by rk(A). Clearly rk(A) ≤ n and the equality occurs if and only 1 if H H = 0. If A is not central we define its rank as the rank of a maximal central subarrangement of A. While dealing with the arrangement complements we will need the following nontrivial result (Orlik and Terao, 1992): if A is an arbitrary arrangement of rank r then the complement M(A) has homotopy type of a simplicial complex of dimension r. Note that the rank of the braid arrangement {Hi j }i< j in Cn equals n − 1. COROLLARY 22.1. The configuration space F(C, n) has homotopy type of a simplicial complex of dimension n − 1. Combining this with Theorems 10.1 and 14.1 we obtain: COROLLARY 22.2. TC F(C, n) ≤ 2n − 1. This result can be improved: THEOREM 22.3. Let A be a central complex hyperplane arrangement of rank r. Then the topological complexity of the complement M(A) is less or equal than 2r. In particular one has TC F(C, n) ≤ 2n − 2. Proof. Let A be {H1 , . . . , H } ⊂ Cn . Let H1∗ be a parallel copy of H1 which is disjoint from H1 . Then the intersections Hi ∩ H1∗ ,

i = 2, 3, . . . ,

form a (in general, non-central) hyperplane arrangement A∗ in H1∗ Cn−1 of rank r − 1. There is a principal C∗ -fibration M(A) → M(A∗ ). The inclusion M(A∗ ) ⊂

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M(A) is a section of it. Hence the fibration is trivial M(A) using the product inequality (see Theorem 17.1) we find

M(A∗ ) × C∗ and

TC(M(A)) ≤ TC(M(A∗ )) + TC(C∗ ) − 1 ≤ [2(r − 1) + 1] + 2 − 1 = 2r.

The opposite inequality requires an additional geometric property of the arrangement: THEOREM 22.4. Let A be a complex central hyperplane arrangement of rank r. Assume that there exist 2r − 1 hyperplanes H1 , H2 , . . . , H2r−1 such that H1 , H2 , . . . , Hr are independent and for any 1 ≤ j ≤ r the hyperplanes H j , Hr+1 , Hr+2 , . . . , H2r−1 are independent. Then one has TC M(A) ≥ 2r. The proof (see Farber and Yuzvinsky, 2004) uses the cohomological lower bound for the topological complexity and combinatorics of Orlik – Solomon algebras. EXAMPLE 22.5. Consider the braid arrangement {Hi j }i< j ⊂ Cn . Here r = n − 1 and 2r − 1 = 2n − 3. We have 2n − 3 hyperplanes: H12 , H13 , . . . , H1n , H23 , H24 , . . . , H2n satisfying the condition of the above theorem. COROLLARY 22.6. One has TC F(R2 , n) = 2n − 2. 23. TC F(R m, n) in the case m ≥ 3 odd Assume that m ≥ 3 is odd. Then F(Rm , n) is (m − 2)-connected and in particular it is simply connected. Its cohomology algebra is generated by the cohomology classes ei j ∈ H m−1 F(Rm , n) , i j which arise as follows. Consider the map φi j : F(Rm , n) → S m−1 , Then

(y1 , y2 , . . . , yn ) →

yi − y j ∈ S m−1 . |yi − y j |

ei j = φ∗i j [S m−1 ]

where [S m−1 ] is the fundamental class of the sphere S m−1 .

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The cohomology classes ei j satisfy the following relations: e2i j = 0,

ei j e jk + e jk eki + eki ei j = 0

and

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for any triple i, j, k. It follows that a product ei1 j1 ei2 j2 · · · eik jk is nonzero if and only if the subgraph of the full graph on vertices {1, 2, . . . , n} having the edges (ir , jr ) contains no cycles. Hence for m ≥ 3 the configuration space F(Rm , n) has homotopy type of a polyhedron of dimension ≤ (n − 1)(m − 1). Since it is (m − 2)-connected we may use inequality (20) of Theorem 14.2 to find TC(F(Rm , n)) <

1 2(n − 1)(m − 1) + 1 + 1 = 2n − 1 + . m−1 m−1

We obtain: COROLLARY 23.1. TC F(Rm , n) ≤ 2n − 1. We want to show that an equality holds in Corollary!23.1. We shall use the cohomological lower bound (see Theorem 15.2). Set e¯ i j = 1 ⊗ ei j − ei j ⊗ 1. It is a zero-divisor of the cohomology algebra. Note that (¯ei j )2 = −2 · ei j ⊗ ei j 0. Here we use the assumption that m is odd. Consider the following product π=

n 0 (¯e1i )2 ∈ A ⊗ A. i=2

We find π = (−2)n−1 m ⊗ m, where m=

n 0

e1i .

i=2

The monomial m 0 is nonzero and hence the product π is nonzero. Using the cohomological lower bound for the topological complexity we obtain the opposite inequality TC(M) ≥ 2n − 1. This completes the proof of Theorem 21.1 in the case m ≥ 3 odd. 24. Shade Let X ⊂ Rn be a closed subset with connected complement Rn − X. Our purpose is to find (or to estimate) the number TC(Rn − X). Our main motivation is the special case when X = ∪H is the union of finitely many aﬃne subspaces.

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M. FARBER

DEFINITION 24.1. Let v ∈ S n−1 be a unit vector. The shade of X in the direction of v is defined as Shadev (X) = {x + λv; x ∈ X, λ ∈ R+ }. (30) In other words we assume that the light illuminating the space arrives from direction of vector v and that X is not transparent. Then Shadev (X) is precisely the shaded parts of the space. Assume that X ⊂ Rn satisfies the following condition: For any point p ∈ Rn and for any generic unit vector v ∈ S n−1 the distance dist(p − λv, X) → +∞

(31)

tends to +∞ as λ tends to +∞. This condition is satisfied in two cases which are particularly important for us: when either X is compact or X is a union of finitely many aﬃne subspaces. If X is a union of finitely many aﬃne subspaces then the condition above is satisfied assuming that the vector v is not parallel to any of the subspaces. LEMMA 24.2. If (31) is satisfied then for a generic nonzero v ∈ Rn the distance dist p − λv, Shadev (X) tends to +∞ as λ → +∞. LEMMA 24.3. If (31) is satisfied then for a generic nonzero vector v the complement of the shade Rn − Shadev (X) is contractible. Proof. We will show that any compact set K ⊂ Rn − Shadev (X) is nullhomotopic in the complement Rn − Shadev (X). Assume that K is contained in a ball with center p ∈ Rn and radius A > 0. Using Lemma 24.2 find λA such that the distance between Shadev (X) and p − λA v is greater than A. The homotopy ht : K → Rn − Shadev (X), t ∈ [0, 1], where ht (x) = x − λtv, takes K into the ball with center p − λA v of radius A which is disjoint from Shadev (X) and hence the image h1 (K) can be contracted to a point in this ball.

Figure 16.

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219

DEFINITION 24.4. Let X ⊂ Rn be a closed subset satisfying (31). The shading dimension of X is defined as the smallest r such that there exist unit vectors v1 , . . . , vr+1 such that the intersection r+1

Shadevi (X) = X

(32)

i=1

equals X. Equivalently, r + 1 is the minimal number of projectors (placed at infinity) needed to illuminate the space Rn − X. EXAMPLE 24.5. Let X ⊂ Rn be a finite set X = {p1 , . . . , pm } then its shading dimension is 1. Indeed, choose a generic unit vector u ∈ S n−1 such that no line through pi and p j has direction u. Then u and −u are two directions such that the intersection of their shades equals X. Any line in Rn in the direction of u intersects X in at most one point and hence the unit vectors u and −u illuminate the whole complement Rn − X. THEOREM 24.6. If X ⊂ Rn is closed subset satisfying (31) then for the topological complexity of the complement TC(Rn − X) one has TC(Rn − X) ≤ 2r + 1 where r is the shading dimension of X. Moreover, using the discussion of Section 20 one obtains an explicit motion planning algorithm in Rn − X with ≤ 2r + 1 local rules. Proof. It follows from the results described above, since Rn − X =

r+1

Rn − Shadevi (X)

i=1

and each term Rn − Shadevi (X) is contractible.

25. Illuminating the complement of the braid arrangement Consider n particles in Rm which are disjoint from each other. In this case the obstacle set X ∈ Rm × Rm × · · · × Rm = Rmn is Hi j X= i< j

where Hi j is the linear subspace zi = z j , i.e., the particle number i collides with the particle number j.

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M. FARBER

Let e ∈ Rm be a fixed unit vector. Let v = (0, e, 2e, . . . , (n − 1)e) ∈ Rnm . We will consider the shade of X in the direction of v. First note that v is generic, i.e., it is not parallel to any of the subspaces Hi j . Let z ∈ Rm be a point. Its tail is defined as the set T (z) = {z − λe; λ ≥ 0}. LEMMA 25.1. The shade Shadev (X) ⊂ Rnm coincides with the set of all configurations (z1 , z2 , . . . , zn ) ∈ Rnm , where zi ∈ Rm , such that zi ∈ T (z j ) for at least one pair i < j. (33) Proof. Consider a configuration (z1 , z2 , . . . , zn ) ∈ X. Assume that it lies in Hi j , i.e., zi = z j where i < j. Then the current configuration of the shade is (z 1 , z 2 , . . . , z n ) where z i = zi + (i − 1)λe. We see that z j − z i = ( j − i)λe which means that z i lies in the tail of z j , i.e., z i ∈ T (z j ). Conversely, suppose now that we are given a configuration z = (z1 , z2 , . . . , zn ) such that zi ∈ T (z j ) for some i < j. Then z j = zi + ( j − i)λe for some λ > 0. We see that the configuration z = (z 1 , z 2 , . . . , z n ) where z r = zr − (r − 1)λe, lies in Hi j and hence z lies in the shade of z in the direction of vector v. Note that the complement of the described set in the configuration space is indeed contractible (in accordance with Lemma 24.3). Since we have zi T (z j )

for all i < j,

one may first move the point zn far enough in the direction of vector −e, there will be no obstacles. Then one moves the point zn−1 again in the direction of −e also far, but closer than zn . And so on: each next point is moved not that far so that the points after the motion lie in diﬀerent slices of Rm (no interactions). 26. A quadratic motion planning algorithm in F(R m, n) Combining the general recipe for constructing motion planning algorithms described in Section 20 with Theorem 24.6 and the results of Section 25, one may

Figure 17.

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221

construct an explicit motion planning algorithm with ≤ n2 local rules where n is the number of particles. In this section we briefly explain how such algorithm can be built. Fix distinct unit vectors e1 , e2 , . . . , eN ∈ Rm , where N=

n(n − 1) + 1. 2

Then for any configuration z = (z1 , . . . , zn ) ∈ F(Rm , n) where zi z j , zi ∈ Rm , there exists 1 ≤ r ≤ N such that the vector er (one of the N fixed unit vectors) is distinct from all vectors zi − z j , for all i < j. |zi − z j | Therefore the configuration z lies in the complement of the shade Rnm − Shadeer (X). Hence N contractible sets Rnm − Shadeer (X), where r = 1, . . . , N, cover the complement Rnm −X. By the construction of Section 20 this leads to a motion planning algorithm with 2N − 1 = n2 − n + 1 local rules. 27. Configuration spaces of graphs Here we will discuss the configuration spaces F(Γ, n) where Γ is a connected graph. These spaces were studied by Ghrist (2001), Ghrist and Koditschek (2002) ´ and Abrams (2002); see also Gal (2001), Swia tkowski (2001). To illustrate the importance of these configuration spaces for robotics one may mention the control problems where a number of automated guided vehicles (AGV) have to move along a network of floor wires. The motion of the vehicles must be safe: it should be organized so that the collisions do not occur. If n is the number of AGV then the natural configuration space of this problem is F(Γ, n) where Γ is a graph. The first question to ask is whether the configuration space F(Γ, n) is connected. Clearly F(Γ, n) is disconnected if Γ = [0, 1] is a closed interval (and n ≥ 2) or if Γ = S 1 is the circle and n ≥ 3. These are the only examples of this kind as the following simple lemma claims: LEMMA 27.1. Let Γ be a connected finite graph having at least one essential vertex. Then the configuration space F(Γ, n) is connected. An essential vertex is a vertex which is incident to 3 or more edges.

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M. FARBER

Figure 18.

THEOREM 27.2. Let Γ be a connected graph having an essential vertex. Then the topological complexity of F(Γ, n) satisfies TC F(Γ, n) ≤ 2m(Γ) + 1, (34) where m(Γ) denotes the number of essential vertices in Γ. A proof can be found in Farber (2005). THEOREM 27.3. Let Γ be a tree having an essential vertex. Let n be an integer satisfying n ≥ 2m(Γ) where m(Γ) denotes the number of essential vertices of Γ. In the case n = 2 we will additionally assume that the tree Γ is not homeomorphic to the letter Y viewed as a subset of the plane R2 . Then the upper bound (34) is exact, i.e., TC F(Γ, n) = 2m(Γ) + 1. (35) Farber (2005) contains a sketch of the proof and also an explicit description of a motion planning algorithm in F(Γ, n) (assuming that Γ is a tree) having precisely 2m(Γ) + 1 domains of continuity. If Γ is homeomorphic to the letter Y then m(Γ) = 1 and F(Γ, 2) is homotopy equivalent to the circle S 1 . Hence in this case TC(F(Γ, 2)) = 2. The equality (35) fails in this case. For any tree Γ one has TC F(Γ, 2) = 3 assuming that Γ is not homeomorphic to the letter Y. This shows that the assumption n ≥ 2m(Γ) of Theorem 27.3 cannot be removed: if Γ is a tree with m(Γ) ≥ 2 then the inequality above would give TC F(Γ, 2) = 2m(Γ) + 1 ≥ 5. Here are more examples. For the graphs K5 and K3,3 (Figure 18) one has TC F(K5 , 2) = TC F(K3,3 , 2) = 5. (36) In these examples the equality (35) is violated. 28. Motion planning in projective spaces Next we consider the problem of computing the topological complexity of the real projective spaces. We will follow Farber et al. (2003) which shows that the

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223

Figure 19.

problem of computing the number TC(RPn ) is equivalent to a classical problem of manifold topology which asks what is the minimal dimension of the Euclidean space N such that there exists an immersion RPn → RN . The immersion problem for the real projective spaces was studied by many people and a variety of important results was obtained. However at the moment the immersion dimension of RPn as a function of n is not known. We refer to a recent survey (Davis, 1993). The problem of finding motion planning algorithms in the projective space n RP can be viewed as an elementary problem of topological robotics. Indeed, points of RPn represent lines through the origin in the Euclidean space Rn+1 and hence a motion planning algorithm in RPn should describe how a given line A in Rn+1 should be moved to another prescribed position B. Lines through the origin in R3 may represent metallic bars fixed at the fixed point by a revolving joint; this situation is common in the practical robotics. If the angle between the lines A and B is acute then one may rotate A toward B in the two-dimensional plane spanned by A and B such that A sweeps the acute angle. Hence the problem reduces immediately to the special case when the lines A and B are orthogonal. In this case, if the intention is to use simple rotations, one needs a continuous choice of the direction of rotation in the plane spanned by A and B. Note that the Lusternik – Schnirelmann category of the real projective spaces is well known and easy to compute: cat(RPn ) = n+1. Using the general properties of the topological complexity mentioned above we may write n + 1 ≤ TC(RPn ) ≤ 2n + 1. We shall see below (see Corollary 30.4) that in fact TC(RPn ) ≤ 2n for all n; the equality holds if n is a power of 2. The answer in the complex case is much simpler: LEMMA 28.1. TC(CPn ) = 2n + 1. More generally, for any simply connected symplectic manifold M one has TC(M) = dim M + 1.

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Proof. Let u ∈ H 2 (M) be the class of the symplectic form. We have a zerodivisor u ⊗ 1 − 1 ⊗ u satisfying 2n n (u ⊗ 1 − 1 ⊗ u)2n = (−1)n u ⊗ un n where 2n = dim M. The cohomological lower bound gives TC(M) ≥ 2n + 1. The cohomological upper bound of Farber (2004) (using the assumption that M is simply connected) gives the opposite inequality TC(M) ≤ 2n + 1. THEOREM 28.2. If n ≥ 2r−1 then TC(RPn ) ≥ 2r . Proof. Let α ∈ H 1 (RPn ; Z2 ) be the generator. The class α × 1 + 1 × α is a zero-divisor. Consider the power (α × 1 + 1 × α)2 −1 . r

Assuming that 2r−1 ≤ n < 2r it contains the nonzero term r 2 −1 k α ⊗ αn n where k = 2r − 1 − n < n. Applying the cohomological lower bound the result follows.

29. Nonsingular maps The main result concerning TC(RPn ) (see Theorem 29.2) uses the following classical notion: DEFINITION 29.1. A continuous map f : Rn × Rn → Rk

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is called nonsingular if: (a) f (λu, µv) = λµ f (u, v) for all u, v ∈ Rn , λ, µ ∈ R, and (b) f (u, v) = 0 implies that either u = 0, or v = 0. In the mathematical literature there exist several variations of the notion of a nonsingular map. We refer to Lam (1967) and Milgram (1967) where nonsingular maps (of a diﬀerent type) were used to construct immersions of real projective spaces into the Euclidean space. Problem. Given n find the smallest k such that there exists a nonsingular map f : Rn × Rn → Rk .

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Let us show that for any n there exists a nonsingular map f : Rn × Rn → R2n−1 . Fix a sequence α1 , α2 , . . . , α2n−1 : Rn → R of linear functionals such that any n of them are linearly independent. For u, v ∈ Rn the value f (u, v) ∈ R2n−1 is defined as the vector whose jth coordinate equals the product α j (u)α j (v), where j = 1, 2, . . . , 2n − 1. If u 0 then at least n among the numbers α1 (u), . . . , α2n−1 (u) are nonzero. Hence if u 0 and v 0 there exists j such that α j (u)α j (v) 0 and thus f (u, v) 0 ∈ R2n−1 . Remarks. 1. For k < n there exist no nonsingular maps f : Rn × Rn → Rk (as follows from the Borsuk – Ulam theorem). 2. For n = 1, 2, 4, 8 there exist nonsingular maps f : Rn × Rn → Rn having an additional property that for any u ∈ Rn , u 0 the first coordinate of f (u, u) is positive. These maps use the multiplication of the real numbers, the complex numbers, the quaternions, and the Cayley numbers, correspondingly. 3. For n distinct from 1, 2, 4, 8 there exist no nonsingular maps f : Rn × Rn → Rn (as follows from the famous theorem of J. F. Adams). Here is the main theorem of Farber et al. (2003): THEOREM 29.2. The number TC(RPn ) coincides with the smallest integer k such that there exists a nonsingular map Rn+1 × Rn+1 → Rk . We refer to Farber et al. (2003) for the proof. Here we will only explain (following Farber et al., 2003) how one uses the nonsingular maps to construct motion planning algorithms. PROPOSITION 29.3. If there exists a nonsingular map Rn+1 × Rn+1 → Rk with n + 1 < k then RPn admits a motion planner with k local rules, i.e., TC(RPn ) ≤ k. Proof. Let Φ: Rn+1 × Rn+1 → R be a scalar continuous map such that φ(λu, µv) = λµφ(u, v) for all u, v ∈ V and λ, µ ∈ R. Let Uφ ⊂ RPn × RPn denote the set of all pairs (A, B) of lines in Rn+1 such that A B and φ(u, v) 0 for some points u ∈ A and v ∈ B. It is clear that Uφ is open. There exists a continuous map s defined on Uφ with values in the space of continuous paths in the projective space RPn such that for any pair (A, B) ∈ Uφ the path s(A, B)(t), t ∈ [0, 1], starts at A and ends at B. One may find unit vectors u ∈ A and v ∈ B such that φ(u, v) > 0. Such pair u, v is not unique: instead of u, v we may take −u, −v. Note that both pairs u, v and −u, −v determine the same orientation of the plane spanned by A, B. The desired map s consists in rotating A toward B in this plane, in the positive direction determined by the orientation.

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Assume now additionally that φ: Rn+1 × Rn+1 → R is positive in the following sense: for any u ∈ Rn+1 , u 0, one has φ(u, u) > 0. Then instead of Uφ we may take a slightly larger set Uφ ⊂ RPn × RPn , which is defined as the set of all pairs of lines (A, B) in Rn+1 such that φ(u, v) 0 for some u ∈ A and v ∈ B. Now all pairs of lines of the form (A, A) belong to Uφ . For A B the path from A to B is defined as above (rotating A toward B in the plane, spanned by A and B, in the positive direction determined by the orientation), and for A = B we choose the constant path at A. Then continuity is not violated. A vector-valued nonsingular map f : Rn+1 × Rn+1 → Rk determines k scalar maps φ1 , . . . , φk : Rn+1 × Rn+1 → R (the coordinates) and the described above neighborhoods Uφi cover the product RPn × RPn minus the diagonal. Since n + 1 < k one may replace the initial nonsingular map by such an f that for any u ∈ Rn+1 , u 0, the first coordinate φ1 (u, u) of f (u, u) is positive. Now, the open sets Uφ 1 , Uφ2 , . . . , Uφk cover RPn × RPn . We have described explicit motion planning strategies over each of these sets. Therefore TC(RPn ) ≤ k.

30. TC(RP n) and the immersion problem THEOREM 30.1. For any n 1, 3, 7 the number TC(RPn ) equals the smallest k such that the projective space RPn admits an immersion into Rk−1 . The proof (see Farber et al., 2003) uses Theorem 29.2 and the following theorem of Adem et al. (1972): THEOREM 30.2. There exists an immersion RPn → Rk (where k > n) if and only if there exists a nonsingular map Rn+1 × Rn+1 → Rk+1 . We will give here a direct construction of a motion planning algorithm in RPn starting from an immersion RPn → Rk . THEOREM 30.3. Suppose that the projective space RPn can be immersed into Rk . Then TC(RPn ) ≤ k + 1. Proof. Imagine RPn being immersed into Rk . Fix a frame in Rk and extend it, by parallel translation, to a continuous field of frames. Projecting orthogonally onto RPn , we find k continuous tangent vector fields v1 , v2 , . . . , vk on RPn such that the vectors vi (p) (where i = 1, 2, . . . , k) span the tangent space T p (RPn ) for any p ∈ RPn . A nonzero tangent vector v to the projective space RPn at a point A (which we understand as a line in Rn+1 ) determines a line vˆ in Rn+1 , which is orthogonal to A, i.e., vˆ ⊥ A. The vector v also determines an orientation of the two-dimensional plane spanned by the lines A and vˆ , see Figure 20.

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Figure 20.

For i = 1, 2, . . . , k let Ui ⊂ RPn × RPn denote the open set of all pairs of lines (A, B) in Rn+1 such that the vector vi (A) is nonzero and the line B makes an acute n n angle with the line v2 i (A). Let U 0 ⊂ RP × RP denote the set of pairs of lines (A, B) in Rn+1 making an acute angle. The sets U0 , U1 , . . . , Uk cover RPn × RPn . Indeed, given a pair (A, B), there exist indices 1 ≤ i1 < · · · < in ≤ k such that the vectors vir (A), where r = 1, . . . , n, span the tangent space T A (RPn ). Then the lines A, v i1 (A), . . . , v in (A) span the Euclidean space Rn+1 and therefore the line B makes an acute angle with one of these lines. Hence, (A, B) belongs to one of the sets U0 , Ui1 , . . . , Uik . We may describe a continuous motion planning strategy over each set Ui , where i = 0, 1, . . . , k. First define it over U0 . Given a pair (A, B) ∈ U0 , rotate A toward B with constant velocity in the two-dimensional plane spanned by A and B so that A sweeps the acute angle. This defines a continuous motion planning section s0 : U0 → P(RPn ). The continuous motion planning strategy si : Ui → P(RPn ), where i = 1, 2, . . . , k, is a composition of two motions: first we rotate line A toward the line v2 i (A) in the in the 2-dimensional plane spanned by A and v2 i (A) in the direction determined by the orientation of this plane (see above). On the second step rotate the line v2 i (A) toward B along the acute angle similarly to the action of s0 . COROLLARY 30.4. One has TC(RPn ) ≤ 2n. Proof. The case n = 1 is trivial. For n > 1 by the Whitney immersion theorem there exists an immersion RPn → R2n−1 . The result now follows from Theorem 30.3. Below is the table of the values TC(RPn ) for n ≤ 23, see Farber et al. (2003). It is obtained by combining the results mentioned above with the information on the immersion problem available in the literature.

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M. FARBER TABLE I. n TC(RPn )

1 2

2 4

3 4

4 8

5 8

6 8

7 8

8 16

9 16

10 17

11 17

n TC(RPn )

13 23

14 23

15 23

16 32

17 32

18 33

19 33

20 35

21 39

22 39

23 39

12 19

As explained in Farber et al. (2003) explicit motion planning algorithms in with n ≤ 7 could be constructed using the multiplication of the complex numbers, the quaternions, and the Cayley numbers. RPn

31. Some open problems Finally we mention several open problems concerning the homotopy invariant TC(X). 1. Rational version of TC(X). It can be “formally” defined as TC(XQ ). One should be able to express this number in terms of Sullivan’s minimal model. This result may give stronger (more sophisticated) lower bounds than the cohomological lower bound mentioned above. The rational version of the LS category was introduced by Felix and Halperin (1982). 2. Symmetric motion planning. One may decide to impose on the motion planning algorithms s: X × X → PX two additional (quite natural) conditions: (a) The path s(A, A) is a constant path at point A; (b) For A B one has s(A, B)(t) = s(B, A)(1 − t). In other words, the motion from B to A goes along the same route as the motion from A to B but in the reverse order. The appropriate numerical invariant TCS (X) measuring the topological complexity is defined as one plus the Schwartz genus of the fibration (P X)/Z2 → (X × X − ∆)/Z2 . Here P X is the set of paths γ: [0, 1] → X with γ(0) γ(1). It has the following properties: (A) TCS (X) ≥ TC(X); (B) In some examples TCS (X) > TC(X); (C) The number TCS (X) is not a homotopy invariant of X. Problem. Find a cohomological lower bound for TCS (X). 3. Motion planning in aspherical spaces. The problem is to compute TC(X) in the case when the polyhedron X is aspherical, i.e., πi (X) = 0 for all i > 1. The

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homotopy type of an aspherical space X depends only on the fundamental group π = π1 (X). Hence in this case the number TC(X) depends only on the group π viewed as a discrete group. One should be able to express the number TC(X) in terms of the algebraic properties of the group π1 (X). A similar question for the Lusternik – Schnirelmann category was solved by Eilenberg and Ganea (1957). Their theorem states: If X is aspherical then cat(X) − 1 = dim π = geom dim π

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except 3 special low-dimensional cases. Here dim π is the least n such that H q (π; A) = 0 for any module A and for any q > n. The symbol geom dim π denotes the smallest dimension of a K(π, 1)-complex. References Abrams, A. (2002) Configuration spaces of colored graphs, Geom. Dedicata 92, 185 – 194. Adem, J., Gitler, S., and James, I. M. (1972) On axial maps of a certain type, Bol. Soc. Mat. Mexicana (2) 17, 59 – 62. Connelly, R., Demaine, E. D., and Rote, G. (2003) Straightening polygonal arcs and convexifying polygonal cycles, Discrete Comput. Geom. 30, 205 – 239. Cornea, O., Lupton, G., Oprea, J., and Tanr´e, D. (2003) Lusternik – Schnirelmann category, Vol. 103 of Math. Surveys Monogr., Providence, RI, Amer. Math. Soc. Davis, D. M. (1993) Immersions of projective spaces: a historical survey, In M. C. Tangora (ed.), Algebraic Topology, Vol. 146 of Contemp. Math., Oaxtepec, 1991, pp. 31–37, Providence, RI, Amer. Math. Soc. Dold, A. (1972) Lectures on Algebraic Topology, Vol. 200 of Grundlehren Math. Wiss., Berlin, Springer. Eilenberg, S. and Ganea, T. (1957) On the Lusternik – Schnirelmann category of abstract groups, Ann. of Math. (2) 65, 517 – 518. Fadell, E. and Neuwirth, L. (1962) Configuration spaces, Math. Scand. 10, 111 – 118. Farber, M. (2003) Topological complexity of motion planning, Discrete Comput. Geom. 29, 211 – 221. Farber, M. (2004) Instabilities of robot motion, Topology Appl. 140, 245 – 266. Farber, M. (2005) Collision free motion planning on graphs, In Algorithmic Foundations of Robotics. VI, Utrecht/Zeist, 2004, Berlin, Springer, to appear. Farber, M., Tabachnikov, S., and Yuzvinsky, S. (2003) Topological robotics: motion planning in projective spaces, Int. Math. Res. Not. 34, 1853 – 1870. Farber, M. and Yuzvinsky, S. (2004) Topological robotics: subspace arrangements and collision free motion planning, In V. M. Buchstaber and I. M. Krichever (eds.), Geometry, Topology, and Mathematical Physics, Vol. 212 of Amer. Math. Soc. Transl. Ser. 2, Moscow, 2002 – 2003, pp. 145–156, Providence, RI, Amer. Math. Soc. Felix, Y. and Halperin, S. (1982) Rational LS category and its applications, Trans. Amer. Math. Soc. 273, 1 – 38. ´ (2001) Euler characteristic of the configuration space of a complex, Colloq. Math. 89, Gal, S. 61 – 67. Ghrist, R. (2001) Configuration spaces and braid groups on graphs in robotics, In J. Gilman, W. W. Menasco, and X.-S. Lin (eds.), Knots, Braids, and Mapping Class Groups — Papers Dedicated

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to Joan S. Birman, Vol. 24 of AMS/IP Stud. Adv. Math., New York, 1998, pp. 29–40, Providence, RI, Amer. Math. Soc. & Somerville, MA, International Press. Ghrist, R. W. and Koditschek, D. E. (2002) Safe cooperative robot dynamics on graphs, SIAM J. Control Optim. 40, 1556 – 1575. Hausmann, J.-C. and Knutson, A. (1998) Cohomology rings of polygon spaces, Ann. Inst. Fourier (Grenoble) 48, 281 – 321. Jordan, D. and Steiner, M. (1999) Configuration spaces of mechamical linkages, Discrete Comput. Geom. 22, 297 – 315. Kapovich, M. and Millson, J. J. (1996) The symplectic geometry of polygons in Euclidean space, J. Diﬀerential Geom. 44, 479 – 513. Kapovich, M. and Millson, J. J. (2002) Universality theorems for configuration spaces of planar linkages, Topology 41, 1051 – 1107. Klyachko, A. A. (1994) Spatial polygons and stable configurations of points in the projective line, In Algebraic Geometry and its Applications, Vol. E25 of Aspects Math., Yaroslavl , 1992, pp. 67–84, Braunschweig, Vieweg. Lam, K. Y. (1967) Construction of nonsingular bilinear maps, Topology 6, 423 – 426. Latombe, J.-C. (1991) Robot Motion Planning, Dordrecht, Kluwer Acad. Publ. Lebesgue, H. (1950) Le¸cons sur les constructions g´eom´etriques, Paris, Gauthier-Villars. Milgram, R. J. (1967) Immersing projective spaces, Ann. of Math. (2) 85, 473 – 482. Orlik, P. and Terao, H. (1992) Arrangements of Hyperplanes, Vol. 300 of Grundlehren Math. Wiss., Berlin, Springer. Schwartz, J. T. and Sharir, M. (1983) On the “piano movers” problem. II. General techniques for computing topological properties of real algebraic manifolds, Adv. in Appl. Math. 4, 298 – 351. Schwarz, A. S. (1966) The genus of a fiber space, Amer. Math. Soc. Transl. (2) 55, 49–140. Sharir, M. (1997) Algorithmic motion planning, In J. E. Goodman and J. O’Rourke (eds.), Handbook of Discrete and Computational Geometry, CRC Press Ser. Discrete Math. Appl., Boca Raton, FL, CRC Press, p. 733 – 754. Smale, S. (1987) On the topology of algorithms. I, J. Complexity 3, 81 – 89. ´ Swia tkowski, J. (2001) Estimates for the homological dimension of configuration spaces of graphs, Colloq. Math. 89, 69 – 79. Thurston, W. (1987) Shapes of polyhedra, preprint. Vassil iev, V. A. (1988) Cohomology of braid groups and complexity of algorithms, Funktsional. Anal. i Prilozhen. 22, 15 – 24, (Russian); English transl. in Funct. Anal. Appl. 22, 182 – 190. Walker, K. (1985) Configuration spaces of linkages, Undergraduate thesis, Princeton University.

APPLICATION OF FLOER HOMOLOGY OF LANGRANGIAN SUBMANIFOLDS TO SYMPLECTIC TOPOLOGY KENJI FUKAYA Kyoto University

1. Introduction The purpose of this article is twofold. The first is to review the theory of Floer homology of Lagrangian submanifolds developed in Fukaya et al. (2000). The construction described in the present article is slightly diﬀerent from the one in Fukaya et al. (2000). Namely, in this article, we use de Rham theory while in Fukaya et al. (2000) we used a version of singular homology. In this article we emphasize applications to symplectic topology, while in other earlier survey articles (Fukaya, 2001, 2003; Ohta, 2001) of Fukaya et al. (2000), more emphasis is put on the applications to mirror symmetry. Some results that were not in the 2000 preprint version of Fukaya et al. (2000) is included here and will be included in the revised version of Fukaya et al. (2000) as well. The other purpose of this article is to describe a technique of using topology of loop space with the moduli space of pseudo-holomorphic disks, as well as some applications of this technique to symplectic topology. Especially we explain a classification of prime 3 manifolds which can be embedded in C3 as Lagrangian submanifolds. The author would like to thank the organizers of the school “Morse theoretic methods in nonlinear analysis and symplectic topology” for the opportunity to write this article. He would also like to thank his collaborators Y.-G. Oh, H. Ohta, K. Ono and acknowledge that many of the results discussed in this article are our joint work. The author would also like to thank D.Sullivan for a discussion which was quite useful for the author in developing the idea on how Floer theory and loop space homology are related. The author also thanks P. Biran who pointed out some errors in the earlier version of the manuscript. 231 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 231–276. © 2006 Springer. Printed in the Netherlands.

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2. Lagrangian submanifold of C n In this section we extract some arguments from Floer’s theory and use them to study Lagrangian submanifolds of Cn . We do not introduce Floer homology in √ this section yet. Let xi + −1yi , i = 1, . . . , n be the canonical coordinates of Cn . ) We put the standard symplectic structure ω = dxi ∧ dyi on Cn . Let J be the standard complex structure J(∂/∂xi ) = ∂/∂yi , J(∂/∂yi ) = −∂/∂xi . DEFINITION 2.1. An n-dimensional submanifold L of Cn is said to be a Lagrangian submanifold if ω|L = 0. In this article we always assume that a Lagrangian submanifold is compact. For a Lagrangian submanifold L ⊂ Cn , the energy function E L : π1 (L) → R is defined by ∗ E L (γ) = γ xi dyi = u∗ ω, (1) i

S1

D2

where u: D2 → Cn is a map with u(∂D2 ) ⊆ L and [u(∂D2 )] = γ. REMARK 2.2. For a general Lagrangian submanifold L ⊂ M, we can define E L : H2 (M, L) → R in the same way. The following result due to Gromov is one of the most basic results in symplectic topology. THEOREM 2.3 (Gromov, 1985). E L 0 for any embedded Lagrangian submanifold L ⊂ Cn . Theorem 2.3 in particular implies that H1 (L; Q) 0 for any Lagrangian submanifold L of Cn . To state the next result we need to define Maslov index of Lagrangian submanifold. DEFINITION 2.4. The set of all real n-dimensional R-linear subspaces V of Cn with ω|V = 0 is called Lagrangian Grassmannian. We write it as GLagn . Let Grn be the set of all real n-dimensional R-linear subspaces V of Cn , i.e., the Grassmannian manifold. LEMMA 2.5. Let n ≥ 2. Then π1 (GLagn ) = Z, and the map GLagn → Grn induces a surjective homomorphism Z = π1 (GLagn ) → π1 (Grn ) = Z2 . See for example Arnol d and Givental (1985) for the proof. In case L ⊂ Cn is a Lagrangian submanifold, we define a map TiL : L → GLagn by TiL (p) = T p L ∈ GLagn .

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DEFINITION 2.6. The Maslov index ηL : π1 (L) → Z = π1 (GLagn ) is the homomorphism induced by TiL . REMARK 2.7. For general Lagrangian submanifold L ⊂ M, we can define ηL : π2 (M, L) → Z in a similar way. The composition of π2 (M) → π2 (M, L) and ηL coincides with β → 2β ∩ c1 (M), where c1 (M) is the first Chern class. Lemma 2.5 implies that, if L is oriented, then the image of ηL is in 2Z. We recall that a spin manifold is an oriented manifold whose second Stiefel – Whitney class vanishes. THEOREM 2.8 (Fukaya et al., 2000). Let L ⊆ Cn be a Lagrangian submanifold. If H 2 (L; Q) = 0 and L is spin, then ηL 0. REMARK 2.9. When H 2 (L; Z2 ) = 0, we can show that ηL 0 without assuming that L is oriented. See Fukaya et al. (2000). As example, let us consider L = S 1 × S n . The assumption of Theorem 2.8 is satisfied when n > 2. For n = 1, L = S 1 × S 1 , it was proved by Viterbo (1990) and Polterovich (1991a), that there exists γ ∈ π1 (L) such that E(γ) > 0 and ηL (γ) = 2. For general n we can prove the following. PROPOSITION 2.10 (Oh, 1996; Fukaya et al., 2000). Let L = S 1 × S n ⊂ Cn+1 be a Lagrangian submanifold with n ≥ 2. We choose the generator γ ∈ π1 (L) = Z, such that E(γ) > 0. If n is odd then there exists a positive integer such that ηL (γ) = n + 1. If n is even then either ηL (γ) = 2 or there exists a positive integer such that ηL (γ) = 2 − n. REMARK 2.11. For the case ηL (γ) > 0, Proposition 2.10 follows from Oh (1996). Although not explicitly stated in the 2000 version of Fukaya et al. (2000), the proposition directly follows from results therein, as will be shown in Section 5. In fact, when n is even, we can show ηL (γ) = 2, i.e. the second possibility can be excluded. See Section 13. Before proving the results stated above we describe a construction of Lagrangian submanifolds. For details, see Audin et al. (1994). We say i: L → Cn is a Lagrangian immersion if i is an immersion, dim L = n and i∗ ω = 0. The map TiL : L → GLagn is defined in the same way. Two Lagrangian immersions i0 , i1 are said to be regular homotopic to each other if there exists a family of Lagrangian immersions i s , s ∈ [0, 1], joining them.

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THEOREM 2.12 (Gromov, 1986; Lees, 1976). 1. There exists a Lagrangian immersion i: L → Cn if and only if TL ⊗C is a trivial bundle. 2. Assume that TL ⊗C is trivial, then i → [T iL ] induces a bijection from the set of regular homotopy classes of Lagrangian immersions i: L → Cn to the set of the homotopy classes of maps L → GLagn . The proof is based on Gromov’s h-principle. See (Eliashberg and Mishachev, 2002). Theorem 2.12 implies that every element of πn (GLagn ) is realized by some Lagrangian immersion i: S n → Cn , while Theorem 2.3 implies that none of them is realized by a Lagrangian embedding. THEOREM 2.13 (Audin et al., 1994). If iL : L → Cn is a Lagrangian immersion and iL : L → Cm is a Lagrangian embedding then there exists a Lagrangian embedding i: L × L → Cn+m such that TiL×L : L × L → GLagn+m is homotopic to sum ◦(TiL ⊗ TiL ). Where sum: GLagn × GLagm → GLagn+m is given by (V, W) → V ⊕ W. Proof. Let us take f : L → Rm ⊂ Cm such that I = (iL , f ): L → Cn+m is an embedding. Clearly I ∗ ω = 0. Then Weinstein’s theorem (see Arnol d and Givental , 1985; Eliashberg and Mishachev, 2002) implies that there exists a neighborhood U of I(L) in Cn+m and a symplectic embedding I : U ⊂ T ∗ L × Cm such that I ◦ I is an embedding q → ((q, 0), 0). Then, for small , the map I+ : L × L → T ∗ L × Cm , I+ (q, Q) = (q, 0), iL (Q) sends L × L to I (U). We put iL,L = (I )−1 ◦ I+ : L × L → Cn+m . Now consider L = S n and L = S 1 . There is an obvious Lagrangian embedding S 1 → C whose Maslov index is ηS 1 [S 1 ] = 2. Hence we have an embedding S 1 × S n → Cn+1 such that ηS 1 ×S n ([S 1 ]) = 2. Note that the set of regular homotopy classes of Lagrangian immersion of S 1 × S n can be identified with π1 (GLagn+1 ) × πn (GLagn+1 ). The construction above realizes any element of the form (2, α) ∈ π1 (GLagn+1 ) × πn (GLagn+1 ) as a Lagrangian embedding. Polterovich (1991a) constructed a Lagrangian embedding S 1 × S 2n−1 ⊂ C2n such that ηL (γ) = 2n,1 where γ ∈ π1 (L) is a generator with E(γ) > 0. Let S 2n−1 = {x ∈ R2n | |x| = 1} and define L = {θx ∈ C2n | x ∈ S 2n−1 , θ ∈ C, |θ| = 1}. It can be shown that L is diﬀeomorphic to S 1 × S 2n−1 and is a Lagrangian submanifold with ηL (γ) = 2n. When k = 2, L = S 1 × S 3 , then ηL (γ) is 2 or 4 according to Proposition 2.10. They both actually occur. When k = 3, L = S 1 × S 5 , Proposition 2.10 implies 1 The author thanks Polterovich who pointed out his example and corrected an error author made during author’s lecture in Montreal.

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ηL (γ) is 2 or 6. They again both occur. When k = 4, L = S 1 × S 7 , Proposition 2.10 implies ηL (γ) is either 2, 4 or 8. The cases 2 and 8 occur, while example with ηL (γ) = 4 does not seem to be known. PROBLEM 2.14. Are there any Lagrangian submanifold L ⊂ C2n diﬀeomorphic to S 1 × S 2n−1 such that ηL (γ) 2, 2n? As for the the other factor π2n−1 (GLag2n ), there is no restriction in the case of ηS 1 ×S 2n−1 (γ) = 2. However, in other cases, especially in the case of ηS 1 ×S 2n−1 (γ) = 2n the following seems to be open. PROBLEM 2.15. For which homotopy class α ∈ π2n−1 (GLag2n ), the class (2n, α) ∈ π1 (GLag2n ) × π2n−1 (GLag2n ) is realized by a Lagrangian embedding? The first case to study seems to be the case of S 1 × S 3 . We remark that Theorem 13.1 determines when the regular homotopy class of Lagrangian immersion S 1 × S 2m → C2m+1 is realized by a Lagrangian embedding. We will discuss more results on Lagrangian submanifolds in Cn in Sections 11, 12 and 13. 3. Perturbing Cauchy – Riemann equation Let us start the proof of Theorems 2.3 and 2.8. We use the moduli space of holomorphic disks with boundary on L, as defined below. Let β ∈ π2 (Cn , L) π1 (L). ' β) be the set of all the maps ϕ: D2 → Cn with DEFINITION 3.1. Let Int M(L; the following properties: 1. ϕ is holomorphic. 2. ϕ(∂D2 ) ⊂ L. 3. The homotopy type of ϕ is β. A topology can be defined on it (see Fukaya et al., 2000) and the resulting space is ' β). The group PSL(2; R) can be identified with the group still denoted by Int M(L; of biholomorphic maps D2 → D2 . Let Aut(D2 , 1) = {g ∈ PSL(2; R) | g(1) = 1}, where we identify D2 = {z ∈ C | |z| < 1}, PSL(2; R) = Aut(D2 ), and Aut denotes the group of biholomorphic automorphisms. We define # β) = Int M(L;

' β) Int M(L; , Aut(D2 , 1)

Int M(L; β) =

' β) Int M(L; . PSL(2; R)

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# β) and Int M(L; β) can be compactified by inThe moduli spaces Int M(L; cluding stable maps (from open Riemann surface of genus 0). (See Fukaya et al., # β) and 2000, for the precise definition.) We denote the compactifications by M(L; M(L; β) respectively. In Section 5, these moduli spaces will be used to define and study Floer homology of Lagrangian submanifolds. In this section we use them more directly to prove Theorems 2.3 and 2.8. For this purpose, we consider a similar but diﬀerent moduli space using perturbation of the Cauchy – Riemann equation (whose solution ϕ is pseudo-holomorphic) by Hamiltonian vector field. Some notations are needed. DEFINITION 3.2. For each positive R we consider a smooth function χR : R → [0, 1] such that 1. χR (t) = 0, if |t| > R, 2. χR (t) = 1, if |t| < R − 1, 3. The C k norm of χR is bounded uniformly with respect to R. Take a compactly supported smooth function H: Cn → R of and let XH = J grad H

(3)

be the Hamiltonian vector field generated by it. DEFINITION 3.3. Let N(R) be the set of all maps ϕ = ϕ(τ, t): R × [0, 1] with the following properties: ∂ϕ ∂ϕ (τ, t) = J (τ, t) − χR (τ)XH (ϕ(τ, t)) , ∂τ ∂t ϕ(τ, 0), ϕ(τ, 1) ∈ L, ϕ∗ ω < ∞. R×[0,1]

→C

(4a) (4b) (4c)

It follows from (4a) that ϕ is holomorphic outside [−R, R] × [0, 1]. Therefore (4c) and removable singularity theorem of Gromov (1985) and Oh (1992) imply the following. LEMMA 3.4. There exists p+∞ , p−∞ ∈ L such that limτ→±∞ ϕ(τ, t) = p±∞ . We remark that D2 \ {1, −1} is biholomorphic to R × [0, 1]. Hence element ϕ of N(R) may be regarded as a map ϕ: (D2 , ∂D2 ) → (Cn , L). DEFINITION 3.5. Let N(R; β) be the set of all ϕ ∈ N(R) such that the homotopy class of ϕ is β. Fix p0 ∈ L and let N(R; β; p0 ) (respectively N(R; p0 )) be the set of all ϕ ∈ N(R; β) (respectively N(R)) such that limτ→+∞ ϕ(τ, t) = p0 .

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Let exptXH : Cn → Cn be the one parameter group of transformations generated by XH . ASSUMPTION 3.6. exp1XH (L) ∩ L = ∅. Since L is compact we can always find H satisfying Assumption 3.6. The following three propositions are basic points of the proof of Theorems 2.3 and 2.8. PROPOSITION 3.7. There exists C depending only on H and L (and independent of R) such that if N(R; β) ∅ then β ∩ ω > −C.

(5)

The proof is based on well established argument and is omitted. We only comment that Definition 3.2 3 is essential for the proof of Proposition 3.7. PROPOSITION 3.8. If Assumption 3.6 is satisfied, then there exists R0 depending only on H and L such that N(R) = ∅ for R > R0 . Proof (Sketch). If not, then there exists ϕi ∈ N(Ri ) with Ri → ∞. Proposition 3.7 and elliptic regularity imply that there exist τi ∈ [−Ri + 2, Ri − 2], δ > 0 such that the C ∞ norm of ϕi on [τi − δ, τi + δ] × [0, 1] is bounded and that ϕ∗i ω = 0. (6) lim i→∞

[τi −δ,τi +δ]×[0,1]

With (4a) and elliptic regularity, (6) implies that ++ ++ ∂ϕi lim sup +++ (τi , t) − XH +++ = 0. i→∞ t∈[0,1] ∂t It follows that

lim dist exp1XH ϕi (τi , 0) , ϕi (τi , 1) = 0.

i→∞

By (4b), this contradicts Assumption 3.6.

For the next proposition we need the following assumption. ASSUMPTION 3.9. If E(β) > 0 then ηL (β) ≤ 0. Let β0 = 0 ∈ π2 (Cn , L). PROPOSITION 3.10. Under Assumption 3.9, we may choose p0 such that N(0; β; p0 ) = ∅ for all β β0 .

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Proof (Sketch). Since for β β0 , dim M(L; β) = n + ηL (β) − 3,

(7)

(see Fukaya et al., 2000), we have by Assumption 3.9: dim ϕ(S 1 ) ≤ n + ηL (β) − 3 + 1 ≤ n − 2. ϕ∈M(L;β)

Therefore we may take p0 which is not contained in the union of ϕ∈M(L;β) ϕ(S 1 ) for various β. Because the equation (4a) is the Cauchy – Riemann equation when R is zero, the proposition holds. LEMMA 3.11. N(0; β0 ; p0 ) is one point. The lemma follows immediately from the fact that each element of N(0; β0 ) is a constant map. Let N(R; β; p0 ) × {R}. (8) N(β; p0 ) = R≥0

We can perturb our moduli spaces so that N(β; p0 ) is a manifold with boundary. Now we are in the position to prove Theorem 2.3. The main point is the following. LEMMA 3.12. If E: π2 (Cn , L) → R is zero then N(β; p0 ) is compact. Proof (Sketch). We prove by contradiction. Let ϕi ∈ N(Ri ; β0 ; p0 ) be a divergent sequence. We may assume by Proposition 3.8 that Ri converges to R. Then there exists pi = (τi , ti ) ∈ R × [0, 1] such that |d pi ϕi | = sup{|d x ϕi | | x ∈ R × [0, 1]}.

(9)

Let i = 1/Di . We consider the following three cases separately. Case 1 |d pi ϕi | = Di diverges. Di dist pi , ∂(R × [0, 1]) = Ci → ∞. Case 2 |d pi ϕi | = Di diverges. Di dist pi , ∂(R × [0, 1]) is bounded. Case 3 |d pi ϕi | = Di is bounded. |τi | diverges. For Case 1, we define hi : D2 (Ci ) → Cn by hi (z) = ϕi (i z+ pi )−ϕi (pi ). It follows from (9) that dhi is uniformly bounded. Moreover hi (0) = 0 and |d0 hi | = 1. Hence, by elliptic regularity, hi converges to h: C → Cn . Since |d0 h| = 1, h is not a constant map. On the other hand, by (4a), ∂h ∂hi i −J (z) z ∈ D2 (Ci ) < iC sup ∂τ ∂t

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where C is a constant , independent of i. Therefore h is holomorphic. , Moreover by (4c) the integral D2 (R ) h∗i ω is uniformly bounded. It follows that D2 (R ) h∗ ω is i i finite. Hence the holomorphicity of h implies that h is a constant map. This is a contradiction. For Case 2, we consider a similar limit and obtain a map h: h = {z ∈ C | Im , z≥ n 0} → C which is nonconstant and holomorphic. Moreover h(R) ⊂ L and h h∗ ω is finite. By removable singularity theorem of Gromov (1985) and Oh (1992), h can be extended to h+ : D2 → Cn with h+ (∂D2 ) ⊂ L, where we identify D2 \{1} = h with h+ = h on h. Since E = 0 by assumption, it follows that ∗ h ω= h+∗ ω = 0. h

D2

Holomorphicity then implies that h is a constant map. This is a contradiction. For Case 3, we can show that Di is bounded away from zero by using the fact ϕi is a divergent sequence. We define hi : R × [0, 1] → Cn by hi (z) = ϕi z − (τi , 0) . Since |τi | diverges it follows that hi , after taking a subsequence if necessary, converges to a holomorphic map h: R × [0, 1] → Cn which is nonconstant. Moreover h(R × {0, 1}) ⊂ L. Furthermore: ∗ h ω ≤ lim sup ϕ∗i ω < ∞. R×[0,1]

,

R×[0,1]

By Lemma 3.4 and E = 0, we find that R×[0,1] h∗ ω = 0 and h is a constant map again by holomorphicity. This is a contradiction. The proof of Lemma 3.12 is now complete. The rest of the proof of Theorem 2.3 goes as follows. Similar to (7) we can show that dim N(R; β) = n + ηL (β). (10) Then dim N(R; β; p0 ) = ηL (β)

(11)

dim N(β; p0 ) = ηL (β) + 1.

(12)

and hence In particular ∂CN(β0 ; p0 ) is a one-dimensional manifold. It is compact by Lemma 3.12. Its boundary is one point by Proposition 3.10, Lemma 3.11. This is a contradiction and Theorem 2.3 is proved. REMARK 3.13. To make the above argument rigorous, we need to find a perturbation by which various moduli spaces have fundamental chains. We omit the arguments on transversality needed at various places of this article. However,

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a novel argument on transversality will be explained in Section 6. We will not discuss the issue of sign (orientation) either. Fukaya et al. (2000) contains detailed discussion for those points. (The details of the new argument on transversality in Section 6 will be in Fukaya, 2005.) 4.

Maslov index of Lagrangian submanifold with vanishing second Betti number

In this section, we continue the argument of the previous section and sketch the proof of Theorem 2.8. We assume here that ηL : π2 (Cn , L) → Z is zero. It follows from (7), (11) and (12) that dim N(β; p0 ) = 1, dim N(R; β; p0 ) = 0 and dim M(L; β) = n − 3. We need the following result: THEOREM 4.1 (Fukaya et al., 2000; Silva, 1997). A spin structure of L determines an orientation of M(L; β), N(β; p0 ), etc. REMARK 4.2. In case of (oriented) Lagrangian submanifold L in a symplectic manifold M, the orientability of the moduli space of pseudo-holomorphic disks is a consequence of relative spin structure. Here L ⊂ M is said to be relatively spin if the second Stiefel – Whitney class w2 (L) is in the image of H 2 (M; Z2 ). We remark that a spin manifold is always relatively spin. Hereafter we assume that L is spin and use the orientation obtained by Theorem 4.1. Since we do not assume E = 0, Lemma 3.12 does not imply compactness of N(β; p0 ). We are going to study its compactification. We use our assumption ηL = 0 in the next lemma. LEMMA 4.3. If ηL = 0, we may choose p0 such that p0 ϕ(S 1 ) for all ϕ ∈ M(L; β). Proof. Since dim M(L; β) = n − 3, it follows that the union of all ϕ(S 1 ) for ϕ ∈ M(L; β) has dimension n − 2. The lemma follows. Using Lemma 4.3, we can describe compactification of N(β; p0 ) and of N(R; β; p0 ). In the rest of this section, set A = (R × {0, 1}) ∪ {−∞}. Define ev: N(R; β; p0 ) × A → L by ev(ϕ, (τ, ±1)) = ϕ(τ, ±1), # β) → L by and ev: M(L;

ev(ϕ, −∞) = lim ϕ(τ, t). τ→−∞

ev([ϕ]) = lim ϕ(τ, t). τ→+∞

Then we have

(13) (14)

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Figure 1.

PROPOSITION 4.4. Suppose ηL = 0. Then N(R; β; p0 ) and N(β0 ; p0 ) have compactifications CN(R; β; p0 ) and CN(β0 ; p0 ) respectively, which have Kuranishi structures with boundary. The boundaries are identified as # β1 ) ×L (N(R; β2 ; p0 ) × A), β β0 M(L; ∂CN(R; β; p0 ) = β1 +β2 =β

∂CN(β0 ; p0 ) \ N(0; β0 ; p0 ) =

# β1 ) ×L (N(β2 ; p0 ) × A), M(L;

β0 = 0.

β1 +β2 =β0

REMARK 4.5. We would not explain the definition of “Kuranishi structure with boundary” here. See Fukaya and Ono (1999). If the reader is not familiar with it, he may read the statement as “there is a perturbation so that the virtual fundamental chain satisfies the equality . . . .” Proof (Sketch). The proof of Proposition 4.4 goes similarly as the proof of Lemma 3.12. We consider a divergent sequence ϕi ∈ N(Ri ; β0 ; p0 ). Ri is bounded by Proposition 3.8 and hence we may assume Ri → R. We choose pi = (τi , ti ) ∈ R × [0, 1] satisfying (9). Then we consider Cases 1, 2, and 3 in the proof of Lemma 3.12. Case 1 does not occur by the same reason as the proof of Lemma 3.12 and we only need to consider Cases 2 and 3. If lim τi = τ ∈ R and lim ti = t∞ ∈ {0, 1}, the limit corresponds to an element # β1 ) ×L (N(R; β2 ; p0 ) × A), [h+ ], ϕ∞ , (τ, t∞ ) ∈ M(L; where h+ : D2 → Cn is a bubble at (τ, t∞ ). If τi → −∞ we have an element # β1 ) ×L (N(R; β2 ; p0 ) × A). [h+ ], ϕ∞ , (−∞) ∈ M(L;

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Figure 2.

Figure 3.

Using Lemma 4.3 we may assume that τi → +∞ does not occur. (See Figure 4.) Now consider Case 3. For the same reason, τi → +∞ does not occur. When τi → −∞, we again get an element # β1 ) ×L (N(R; β2 ; p0 ) × A). [h+ ], ϕ∞ , (−∞) ∈ M(L; This implies Proposition 4.4.

The next lemma is a consequence of Proposition 3.7 and Gromov compactness. LEMMA 4.6. There exists only finitely many β ∈ π1 (Cn , L) such that N(β; p0 ) ∅ and E(β) < 0. First we consider the case where the following additional assumption is satisfied. ASSUMPTION 4.7. There exists a unique β ∈ π1 (Cn , L) such that N(β; p0 ) ∅ # β ) = ∅ for all β ∈ π1 (Cn , L) with E(β ) > 0, and E(β) < 0. Furthermore, M(L; −E(β) > E(β ) > 0.

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This assumption is rather restrictive. Later in the next section, we will provide an argument to remove it. Now we assume ηL = 0, L is spin and Assumption 4.7. By Proposition 4.4, we have # β1 ) ×L (N(β2 ; p0 ) × A). (15) ∂CN(β0 ; p0 ) \ N(0; β0 ; p0 ) = M(L; β1 +β2 =β0

# β1 ) ∅ and E(β0 ) = 0 it follows from (15) and Because E(β1 ) > 0 if M(L; Assumption 4.7 that # −β) ×L (N(β; p0 ) × A). ∂CN(β0 ; p0 ) \ N(0; β0 ; p0 ) = M(L;

(16)

Note that CN(β0 ; p0 ) is an one-dimensional oriented chain and N(0; β0 ; p0 ) is a point. Hence the right-hand side of (16) is an oriented zero-dimensional manifold whose order counted with sign is 1. # −β) = Int M(L; # −β) By the second half of Assumption 4.7, we see that M(L; and defines a cycle of dimension is n − 2. Hence # −β)]) ∈ Hn−2 (L; Z). ev∗ ([M(L; Similarly we find ev∗ ([N(0; β; p0 )]) ∈ H2 (L; Z). We now have

# −β)]) · ev∗ ([N(0; β; p0 )]) = 1, ev∗ ([M(L;

since it equals to the right-hand side of (16). (Here · is the intersection pairing.) This implies H 2 (L; Q) 0. Theorem 2.8 is thus proved under additional hypothesis Assumption 4.7. 5. Floer homology and a spectral sequence We now introduce Floer cohomology of Lagrangian submanifold and explain how it can be used to study Lagrangian submanifold of, say Cn . Let L be a compact Lagrangian submanifold of a symplectic manifold M. (In case M is noncompact we assume that M is convex at infinity. See Eliashberg and Gromov, 1991.) Let us define a universal Novikov ring Λ by λi Λ= ai T ai ∈ R, λi → +∞, λi ∈ R . (17) Λ is in fact a field. We also set Λ0 = ai T λi ∈ Λ λi ≥ 0 ,

Λ+ =

ai T λi ∈ Λ λi > 0 .

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Λ0 is a local ring and Λ+ is its maximal ideal. Let ηL : π2 (M, L) → Z be the Maslov index and E: π2 (M, L) → R is defined by integrating the symplectic form. We say that L is relatively spin if there exists st ∈ H 2 (M; Z2 ) which is sent to the second Stiefel – ;Whitney class of L. (If L is spin then it is relatively spin.) We say that L ⊂ M is rational if the image of E: π2 (M, L) → R is in λQ for some λ. THEOREM 5.1 (Fukaya et al., 2000). We assume that L is relatively spin. Then there exists a series of elements βi ∈ π2 (M, L) with E(βi ) > 0, E(βi ) ≤ E(βi+1 ) and cohomology classes oβi ∈ H 2−ηL (βi ) (L; Q), such that oβ = 0 if 2 − ηL (β) = 0, n or if E(β) ≤ 0. The classes also have the following properties. If oβi = 0 for all βi then there exists a Λ module HF(L, L) and a spectral sequence E∗ , which satisfies 1, 2, 4, 5, 6 below. In the case where (M, L) is rational it satisfies 3 as well. p,q

1. E2 H(L; Q) ⊗ T qλ Λ0 /T (q+1)λ Λ0 . p,q

2. There exists Fq HF p (L, L) such that E∞ Fq HF p (L, L)/Fq+1 HF p (L, L). 3. The diﬀerential di = T E(βi ) dβi is a homomorphism induced by dβi : H k (L; Q) → H k+1−ηL (βi ) (L; Q). 4. dβ ([1]) = 0. Here [1] ∈ H 0 (L; Q) is a generator. 5. The fundamental cocycle [L] ∈ H n (L; Q) is not contained in the image of dβ . 6. If Φ: M → M is a Hamiltonian diﬀeomorphism and L is transversal to Φ(L), then dimΛ HF(L, L) ≤ (L ∩ Φ(L)). The statement of Theorem 5.1 looks rather complicated. However it seems to be impossible to find a simpler way to describe Floer cohomology of Lagrangian submanifolds in general, by the reasons that we are going to explain below. The ideal statements one might expect for Floer cohomology HF(L, L ) would be the following. (A) The Floer cohomology HF(L, L ) between two Lagrangian submanifolds L, L is always well-defined. (B) We have HF(L, L) H(L), where the right-hand side is the usual cohomology group. (C) Floer cohomology HF(L, L ) is invariant of Hamiltonian isotopies of L and of L . (D) dim HF(L, L ) ≤ (L∩L ) holds for any transversal pair (L, L ) of Lagrangian submanifolds.

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Floer (1988) constructed such HF(L, L ) under the assumption that π2 (M, L) = π2 (M, L ) = 0. Under this assumption (A), (B), (C), (D) hold as stated above. Oh (1993) generalized it under milder assumptions. However it is too much to expect general Floer cohomology to have all of the above properties (A), (B), (C), (D). In fact, (B), (C), (D) would imply that L should always intersect with Φ(L) where Φ is a Hamiltonian isotopy. But this is not the case, especially in the case of Lagrangian submanifold in Cn . So we need to modify (A), (B), (C), (D) above to have a correct Floer theory in the general situation. The series of obstruction classes oβ = 0 in Theorem 5.1 precisely describes when (A) holds. 2, 3, 4, 5 of Theorem 5.1 above is related to (B) and shows how H(L) is deformed to HF(L, L). (6) of Theorem 5.1 is a modified version of (D). REMARK 5.2. In Theorem 5.1 we considered only one Lagrangian submanifold L and HF(L) = HF(L, L). There is a version for two Lagrangian submanifolds L, L and the Floer cohomology HF(L, L ) between them. See Fukaya et al. (2000). REMARK 5.3. Floer cohomology HF(L, L) is not an invariant of symplectic diﬀeomorphism type of (M, L). Instead, it depends on an element of a moduli space M(L) of bounding chains b. In Fukaya et al. (2000), we introduced the notion of bounding chains and their moduli space M(L). In this article, we will not discuss bounding chains, since we use only the properties stated in Theorem 5.1. Any choice of bounding chain is good for the applications in this article. For other applications, especially for applications to mirror symmetry, the space M(L) itself plays a crucial role. REMARK 5.4. The boundary operators of the spectral sequence in Theorem 5.1 do not preserve degree. Namely its degree is 1 − ηL (βi ) which is not 1 in general. In (Fukaya et al., 2000) we add one extra generator (denoted by e there) to Λ to avoid this strange phenomenon. There is no essential diﬀerence between the presentation of the results in Fukaya et al. (2000) and in Theorem 5.1. For the readers who want to know the relation of this article with Fukaya et al. (2000), we remark that the Floer homology in Theorem 5.1 is called ‘weakly unobstructed after infinitesimal deformation’ in the terminology of the revised version of Fukaya et al. (2000). For the purpose of the present article the readers do not need to know what it means. A sketch of the proof of Theorem 5.1 is given just after Definition 8.6. Although Theorem 5.1 looks rather complicated, we can apply it to various situations as we see below. We first show: LEMMA 5.5. Theorem 5.1 implies Theorem 2.8.

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Proof. We prove it by contradiction. Let L ⊂ Cn be a spin Lagrangian submanifold, with H 2 (L; Q) = 0 and ηL = 0. Since oβ ∈ H 2−η(β) (L; Q) = H 2 (L; Q) it follows that we can define HF(L, L). By 6, we have HF(L, L) = 0. Hence by 1 and 2, [L] ∈ H n (L) does not survive in E∞ . Since [L] is not in the image of dβ by 5, it follows that dβ ([L]) 0 for some β. However this is impossible since dβ ([L]) ∈ H n+1+ηL (β) (L; Q) = 0. We next apply Theorem 5.1 to the case of (M, L) = (Cn+1 , S 1 × S n ) and prove Proposition 2.10. Namely we prove the following two lemmas. LEMMA 5.6 (Oh, 1996). Let L = S 1 × S n be a Lagrangian submanifold of Cn+1 and n is odd. We choose the generator β ∈ π2 (Cn+1 , L) so that E(β) > 0. Then ηL (β) is positive and divides n + 1. Proof. We first remark that E 0 by Theorem 2.3. Hence there is a unique generator β with E(β) > 0. Since ηL (β) is even, it follows that oβ ∈ H even (L; Q). Since n is odd, the cohomology group H even (L; Q) is nonzero for H 0 (L) and H n+1 (L) only. In that case oβ = 0 by Theorem 5.1. Therefore HF(L, L) is well-defined. Since L ⊆ Cn+1 , it follows that HF(L, L) 0. By 4 and 5 there exists k, k such that dkβ ([1]) 0, and [L] = dk β u for some u ∈ H(L). Since dkβ is of odd degree, it follows that either dkβ ([1]) = c[S 1 ] or dkβ = c[S n ] (c ∈ Q). Because E(β) > 0, it follows that k > 0. For dkβ ([1]) = [S 1 ], we have ηL (kβ) = 2. Hence k = 1 and ηL (β) = 2. For dkβ ([1]) = [S n ], we have ηL (kβ) = n + 1. Hence ηL (β) divides n + 1. LEMMA 5.7 (Fukaya et al., 2000; Oh, 1996). Let L = S 1 × S n be a Lagrangian submanifold of Cn+1 and n is even. We choose the generator β ∈ π2 (Cn+1 , L) so that E(β) > 0. Then either ηL (β) = 2 or it is nonpositive and divides 2 − n. Proof. If okβ 0 then, since deg okβ is nonzero and even, it follows that okβ ∈ n H (L; Q). It implies 2 − ηL (kβ) = n. Hence ηL (β) is negative and divides 2 − n. If okβ are all zero, then Floer cohomology HF(L, L) is well defined and is zero. Hence dkβ ([1]) 0 for some k. Since [L] is not in the image of dkβ , it follows that dkβ ([1]) = c[S 1 ]. Hence ηL (kβ) = 2. Therefore k = 1 and ηL (β) = 2. REMARK 5.8. The second possibility of ηL (β) < 0 in Lemma 5.7 will be eliminated in Section 13 (Theorem 13.1).

6. Homology of loop space and Chas – Sullivan bracket In Sections 6, 7, and 8, we explain a construction of a filtered A∞ structure on the cohomology group H(L) of Lagrangian submanifold L. We take a slightly diﬀerent approach than Fukaya et al. (2000) and use the de Rham cohomology.

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Figure 4.

REMARK 6.1. In Fukaya et al. (2000) a variant of singular chain complex was used. The approach in Fukaya et al. (2000) has an advantage that we can work over Z coeﬃcient at least when L is semi-positive.2 The approach here has an advantage that we can keep more symmetry. Especially, cyclic symmetry is established in Section 9. Let L be a compact smooth manifold. We denote its free loop space by L(L). Namely L(L) = {: S 1 → L | is piecewise smooth} (18) Chas and Sullivan (1999) introduced a structure of graded Lie algebra on the homology of L(L). Let us recall it here. We identify S 1 = R/Z. Let fi : Pi → L(L) be cycles and write Pˆ i = (Pi , fi ). We put

(19) P1 ∗ P2 = (x, y, t) ∈ P1 × P2 × S 1 f1 (x) (0) = f2 (y) (t) and define f1 ∗ f2 : P1 ∗ P2 → L(L) by the following formula.   f2 (y) (2s) 2s ≤ t,    (x) (2s − t) t ≤ 2s ≤ t + 1, f ( f1 ∗ f2 )(x, y, t) (s) =  1    f (x)(2s − 1) t + 1 ≤ 2s.

(20)

2

We then define

1 ∗ P 2 = (P1 ∗ P2 , f1 ∗ f2 ). P

(21) If P1 , P2 are cycles, of dimension k1 , k2 respectively, then, under an appropri1 ∗ P 2 is a cycle of dimension k1 + k2 − n + 1, where ate transversality condition, P n = dim L. Therefore ∗ defines a map ∗: Hk1 L(L) ⊗ Hk2 L(L) → Hk1 +k2 −n+1 L(L) . (22) 2

Actually we can do it in general using the method of (Fukaya and Ono, 2001).

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DEFINITION 6.2 (Chas and Sullivan, 1999). We define loop bracket {·, ·} by 2 ]} = [P 1 ∗ P 2 ] + (−1)(deg P1 +1)(deg P2 +1) [P 2 ∗ P 1 ]. 1 ], [P {[P THEOREM 6.3 (Chas and Sullivan, 1999). Loop bracket satisfies Jacobi identity. Namely it defines a structure of graded Lie algebra on H∗ L(L) . Actually we can work in chain level and construct an L∞ algebra, which is a homotopic version of graded Lie algebra. There are various approaches of working out transversality problem to justify Chas – Sullivan’s construction. Here we use the following one which works best with our construction of virtual fundamental chain of the moduli space of pseudoholomorphic disks. DEFINITION 6.4. (P, f, ω) is said to be an approximate de Rham chain of L(L) if the following holds. 1. P is an oriented smooth manifold of finite dimension (with or without boundary). ω is a smooth diﬀerential form on P with compact support. 2. f : P → L(L) is a smooth map.

3. The map ev: P → L defined by ev(x) = f (x) (0) is a submersion. When ∂P is nonempty we assume that ev: ∂P → L is also a submersion. We say (P, f, ω) is an approximate de Rham cycle if P does not have a boundary and dω = 0. We define the degree of approximate de Rham chain by deg(P, f, ω) = dim P − deg ω. We put ∂(P, f, ω) = (−1)deg ω (∂P, f, ω) + (−1)deg ω (P, f, dω)

(23)

An approximate de Rham cycle (P, f, ω) of degree k determines an element in Hk (L(L); R) as follows. Let Hc∗ (P; R) be the compactly supported de Rham cohomology group and PD: Hcdim P−k (P; R) → Hk (P; R) be the Poincar´e duality map. Then we associate f∗ (PD([ω])) ∈ Hk (L(L); R) to (P, f, ω). It is easy to see that any element of H∗ (L(L); R) can be realized by an approximate de Rham cycle. An approximate de Rham chain (P, f, ω) determines a smooth diﬀerential form evL (P, f, ω) on L of degree k = deg(P, f, ω) as following: evL (P, f, ω) = ev! (ω) ∈ Ωn−k (L).

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Here ev! is integration along fiber. It is well defined since ω is of compact support and ev: P → L is a submersion.

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We can easily check for an approximate de Rham cycle (P, f, ω) that PD([evL (P, f, ω)]) = f∗ (PD([ω])) The following lemma follows easily from Stokes’ theorem. LEMMA 6.5. d evL (P, f, ω) = evL ∂(P, f, ω) . Now we go back to the loop bracket. Let (Pi , fi , ωi ) be approximate de Rham chains of degree ki for i = 1, 2. LEMMA 6.6. (P1 ∗ P2 , f1 ∗ f2 , ω1 × ω2 ) is an approximate de Rham chain of degree k1 + k2 − n + 1. Proof. Notice that the map P1 × P2 × S 1 → L × L: (x, y, t) → f1 (x) (0), f2 (y) (t) is transversal to the diagonal by the condition 3 of Definition 6.4. The rest of the proof is straightforward. Thus ∗ as well as the loop bracket can be defined on chain level using approximate de Rham chains. By the argument of Chas and Sullivan (1999), loop bracket satisfies graded chain level Jacobi identity modulo parameterization of the loop. We can easily show that it induces an L∞ structure. (The argument we need is the same as the argument used to show that the (based) loop space is an A∞ space. See Stasheﬀ, 1963; Sugawara, 1957). We want to use moduli space of pseudo-holomorphic disks to construct an (approximate de Rham) chain in the loop space. Let M be a symplectic manifold and L be its Lagrangian submanifold. We assume that L is compact and relatively spin and M is convex at infinity (in case M is noncompact). For each β ∈ π2 (M; L) ' β), M(L; # β) and M(L; β) are defined as in Definition 3.1. Since the spaces M(L; n in general M C , we need to use sphere bubbles to compactify them. We will not discuss the argument which handles sphere bubbles since it is the same as in the case of the moduli space of pseudo-holomorphic map from Riemann surface without boundary (see Fukaya and Ono, 1999), which is a codimension 2 phenomenon. As for the compactification M(L; β), there is one new point we need to deal with, which we mention briefly in Section 14 (see Theorem 14.4 and its proof) and will be discussed in detail in the revised version of Fukaya et al. (2000). # β). Recall that Consider M(L; ' # β) = M(L; β) M(L; Aut(D2 , 1)

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and the group Aut(D2 , 1) is contractible. Hence we fix a lifting # β) → M(L; ' β). Liftβ : M(L; ' β) → L(L) by ev(ϕ) = ϕ|∂D2 , and consider ev ◦ Liftβ : We define the map ev: M(L; # # β), ev ◦ Liftβ ) to construct Floer M(L; β) → L(L). We want to use the chain (M(L; cohomology, etc. # β), ev ◦ Liftβ ) needs to be replaced by For the purpose of transversality, (M(L; an appropriate approximate de Rham chain. To describe the procedure precisely we need to use the notion of Kuranishi structure more systematically. We will do it in Fukaya (2005). Here we sketch the argument by simplifying the situation. Let us consider a Banach manifold B together with a Banach bundle E → B and a section s: B → E which is not necessarily transversal to zero. We assume that the diﬀerential of s is Fredholm. Let f˜: B → L(L) be a smooth map. (In our application B = Map((D2 , ∂D2 ); (M, L)), and s = 0 gives the equation for ϕ ∈ B to be pseudo-holomorphic.) We assume that s−1 (0) is compact. Then we can find a finite-dimensional space W and a family of perturbations sw of s parameterized by w ∈ W having the following properties: 1. P = {(x, w) ∈ B × W | sw (x) = 0}. is a smooth manifold. 2. The projection πW : P → W is smooth and proper. 3. If we put f (x, w) = f˜(x) then f : P → L(L) is a smooth map. 4. The composition ev ◦ f : P → L is a submersion. It is possible to achieve 4 if we take W to have very big dimension. Now let ωW be a smooth,form on W of top dimension. We assume that it is compactly supported with W ωW = 1. Let ω = π∗W ωW , then (P, f, ω) is an approximate de Rham cycle. In our actual application, we have locally an orbibundle Eα → Uα on an # orbifold Uα together with its sections sα such that α s−1 α (0) = M(L; β). Moreover (Eα , Uα , sα ) are glued in an appropriate sense. More precisely we have a Kuranishi # β). We can use a smooth family of multi-sections (see Fukaya structure on M(L; and Ono, 1999) in a similar way as above to obtain an approximate de Rham chain # β). We denote it by the same symbol M(L; # β) for simplicity. for each M(L; THEOREM 6.7 (Fukaya, 2005). We can choose Liftβ and approximate de Rham # β) such that chain M(L; # β) + ∂M(L;

1 # # β2 )} = 0. {M(L; β1 ), M(L; 2 β=β +β 1

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2

The proof is similar to the proof of Proposition 4.4 and details will be in Fukaya (2005).

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REMARK 6.8. The method to realize virtual fundamental chain using approximate de Rham chain is somewhat similar to Ruan’s approach (Ruan, 1999).

7. Iterated integral and Gerstenhaber bracket In Section 6, we studied homology of the loop space L(L). The relation of homology of L and of L(L) is classical. Especially there is a construction by Chen (1973), which we review here. Let L be a smooth manifold and (Ω(L), d, ∧) be its de Rham complex. We put Ω(L)[1]m = Ωm+1 (L), Bk (Ω(L)[1]) = Ω(L)[1] ⊗ · · · ⊗ Ω(L)[1] 3456 k B(Ω(L)[1]) = Bk (Ω(L)[1]) k

and define mk : Bk (Ω(L)[1]) → Ω(L)[1] of degree +1 by m1 (u) = (−1)deg u du,

m2 (u, v) = (−1)deg u(deg v+1) u ∧ v

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and mk = 0 for k 1, 2. It defines a structure of A∞ algebra, which will be described later. We now define ˆ B(Ω(L)[1]) → B(Ω(L)[1]), d: by ˆ 1 ⊗ · · · ⊗ uk ) = d(u

(−1)∗i u1 ⊗ · · · ⊗ m1 (ui ) ⊗ · · · ⊗ uk

i

+

(−1)∗i u1 ⊗ · · · ⊗ m2 (ui , ui+1 ) ⊗ · · · ⊗ uk

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i

where ∗i = deg u1 + · · · + deg ui−1 + i − 1, here deg is the degree as a diﬀerential ˆ is a cochain complex. form. It is easy to see that dˆ2 = 0 and hence (B(Ω(L)[1]), d) To define iterated integral we modify the above complex slightly. Fix a base point p0 of L and define Ω0 (L) as following: Ωk0 (L) = Ωk (L), Ω00 (L)

k 0, ∞

= { f ∈ C (L) | f (p0 ) = 0}

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and define B(Ω0 (L)[1]) as before. It is easy to see that dˆ preserves B(Ω0 (L)[1]) ˆ is a cochain complex as well. and hence (B(Ω0 (L)[1]), d) We denote by L0 (L) the based loop space, i.e., L0 (L) = {: S 1 → L | (0) = p0 }. Chen (1973) defined a cochain homomorphism ˆ → Ω(L0 (L), d). Ich: (B(Ω0 (L)[1]), d)

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Since L0 (L) is infinite-dimensional we need to be careful in defining de Rham complex Ω(L0 (L), d). We do not discuss this point here. See Chen (1973). Instead we take a smooth chain (P, f ) of L0 (L) and define an integration of Ich(u1 ⊗ · · · ⊗ uk ) over (P, f ) as follows. Let Ck = {(t1 , . . . , tk ) ∈ [0, 1]k |0 ≤ t1 ≤ · · · ≤ tk ≤ 1} (29) and define the map ev: L0 (L) × Ck → Lk by ev , (t1 , . . . , tk ) = (t1 ), . . . , (tk ) .

(30)

then we define ∗ f Ich(u1 ⊗ · · · ⊗ uk ) =

(31)

P

ev ◦( f × id) ∗ (u1 ∧ · · · ∧ uk ). P×Ck

THEOREM 7.1 (Chen, 1973). Ich is a cochain homomorphism. Proof. It suﬃces to show ∗ ˆ 1 ⊗ · · · ⊗ uk ) . f Ich(u1 ⊗ · · · ⊗ uk ) = f ∗ Ich d(u ∂P

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P

by studying the boundary of P × Ck . (We omit the details.)

We define a free loop space version of the homomorphism of Theorem 7.1. Consider 0 Hom(B(Ω(L)[1]), Ω(L)[1]) = Hom(Bk (Ω(L)[1]), Ω(L)[1]), k

and define a boundary operator δ as following. Let ϕ ∈ Hom(B(Ω(L)[1]), Ω(L)[1]) then ˆ 1 ⊗ · · · ⊗ ak ) (δϕ)(a1 ⊗ · · · ⊗ ak ) = m1 ϕ(a1 ⊗ · · · ⊗ ak ) − (−1)deg ϕ (ϕ ◦ d)(a (deg a1 +1) deg ϕ +(−1) m2 a1 , ϕ(a2 ⊗ · · · ⊗ ak ) +m2 (ϕ(a1 ⊗ · · · ⊗ ak−1 ), ak ). (33)

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It is easy to check that δ2 = 0. We denote by S D L(L) the set of all approximate de Rham chains on L(L). It is a chain complex. We next define a chain homomorphism Ich∗ : S D L(L) → Hom(B(Ω(L)[1]), Ω(L)[1]) First define the map ev+ = (ev, ev0 ): L(L) × Ck → Lk+1 by

ev+ , (t1 , . . . , tk ) = (t1 ), . . . , (tk ), (0) .

Then, we define Ich∗ by Ich∗ (P, f, ω) (u1 ⊗ · · · ⊗ uk ) = (ev0 ◦π)! ω ∧ ev ◦( f × id) ∗ (u1 ∧ · · · ∧ uk ) .

(34)

Here ev0 ◦π: P × Ck → L is the composition of P × Ck → P, f : P → L(L) and ev0 : L(L) → L. (ev0 ◦π)! is the integration along fiber. PROPOSITION 7.2. Ich∗ is a chain homomorphism The proof is straightforward. Next we recall Gerstenhaber bracket, which was introduced by Gerstenhaber (1963) to study deformation theory of associative algebra. We restrict ourselves to the case of Ω(L). A structure of diﬀerential graded Lie algebra can be defined on Hom(B(Ω(L)[1]), Ω(L)[1]) as following. Let ϕi ∈ Hom(Bki (Ω(L)[1]), Ω(L)[1]) and define (ϕ1 ◦ ϕ2 )(u1 ⊗ · · · ⊗ uk1 +k2 −1 ) = (−1)∗i ϕ1 (u1 ⊗ · · · ϕ2 (ui ⊗ · · · ⊗ ui+k2 −1 ) · · · ⊗ uk1 +k2 −1 ). (35) i

where ∗i = (deg ϕ2 )(deg u1 + · · · + deg ui−1 + i − 1). Then the Gerstenhaber bracket can be defined by: {ϕ1 , ϕ2 } = ϕ1 ◦ ϕ2 − (−1)deg ϕ1 deg ϕ2 ϕ2 ◦ ϕ1 .

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THEOREM 7.3 (Gerstenhaber, 1963). (Hom(B(Ω(L)[1]), Ω(L)[1]), δ, {·, ·}) is a diﬀerential graded Lie algebra. Now we have:

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PROPOSITION 7.4. δ Ich∗ (P, f, ω) = Ich∗ ∂(P, f, ω) and

Ich∗ ({(P1 , f1 , ω1 ), (P2 , f2 , ω2 )}) = Ich∗ (P1 , f1 , ω1 ) , Ich∗ (P2 , f2 , ω2 ) where {·, ·} in the left-hand side is loop bracket and {·, ·} in the right-hand side is Gerstenhaber bracket. The proof is straightforward and is omitted. The proposition implies that Ich∗ is a homomorphism of diﬀerential graded Lie algebra. REMARK 7.5. Chas and Sullivan (1999) already mentioned that their construction is an analogue of Gerstenhaber bracket. REMARK 7.6. Precisely speaking the loop bracket defines an L∞ structure since the Jacobi identity only holds modulo homotopy. However Jacobi iden tity on S D L(L) fails only because of parameterization. The diﬀerence of parameterization is killed by Ich∗ . 8.

A∞ deformation of de Rham complex

We now use the result of Sections 6 and 7 to define an A∞ deformation of the de Rham complex. We first recall the definition of filtered A∞ algebra. Let CR be a graded R vector space. We recall λi λ i Λ0 = ai T ∈ Λ λi ≥ 0 , Λ+ = ai T ∈ Λ λi > 0 . Λ0 has a filtration F λ Λ0 = T λ Λ0 which defines a (non-Archimedian) norm on it. k ⊗ ˆ Λ0 and C[1]k = C k+1 . Here and hereafter ⊗ˆ means that we are Let C k CR taking an appropriate completion with respect to the (non-Archimedian) norm on Λ0 . We omit the details and refer to Fukaya et al. (2000). Let Bk (C[1]) Bk (C[1]) = C[1] ⊗ˆ Λ0 · · · ⊗ˆ Λ0 C[1], B(C[1]) = 3456 k

k

and consider a series of Λ0 module homomorphisms mk : Bk (C[1]) → C[1] of odd degree. We assume that it is written as mk = T λi mk,i i

where λi ≥ 0, λi → +∞ and mk,i is induced by R linear map Bk (CR [1]) → CR , of degree di + 1. (Here di are even numbers and we assume that λi and di are independent of k.) We assume also λ0 = 0 and λi > 0 for i > 0.

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DEFINITION 8.1. We say that (C, mk ), k = 0, 1, . . ., is a filtered A∞ algebra if (−1)∗ mk (x1 ⊗ · · · m (xi ⊗ · · · ⊗ xi+−1 ) · · · ⊗ xn ) = 0 k+=n+1 i

where ∗ = deg x1 + · · · + deg xi−1 + i − 1 (deg is a degree before shift) and if m0 ≡ 0 mod Λ+ . k = Ωk (L) to be the de Rham complex and we define m in the Now take CR k following. Set mk,0 = mk . Recall that m1 = ±d, m2 = ±∧ and other mk are zero. We write mk,i ∈ Hom(Bk C[1], C[1]), ϕ = (ϕ0 , ϕ1 , . . .). (37) ϕk = i>0

The next proposition is in principle due to Gerstenhaber. PROPOSITION 8.2. mk is an filtered A∞ algebra if and only if δϕ + 12 {ϕ, ϕ} = 0. Here δ is as in (33). We remark that Gerstenhaber bracket {·, ·} can obviously be generalized to k Hom(Bk C[1], C[1]). Now let L be a Lagrangian submanifold of M. We assume that it is relatively spin and compact.

.

DEFINITION 8.3. α(L) =

# βi ). T E(β) M(L;

i

α(L) is a Λ0 valued approximate de Rham chain of L(L). Here βi ∈ π2 (M, L) such that 0 = E(β0 ) < E(β1 ) ≤ · · ·. We define ϕ = (ϕ0 , ϕ1 , . . .) = Ich∗ (α),

(38)

where Ich∗ is as in (34). We use ϕk to define mk by (37). THEOREM 8.4. The operator mk above defines a structure of filtered A∞ algebra on Ω(L) ⊗ˆ Λ0 . Proof. Theorem 8.4 follows immediately from Theorem 6.7, Propositions 7.4 and 8.2. To define a structure of filtered A∞ algebra on the cohomology group H ∗ (L; R) we use the following theorem. Let (C, mk ) be a filtered A∞ algebra. Note that

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K. FUKAYA

k → C k+1 . Using m m1,0 is induced from an R linear map m1 : CR 0,0 = 0, (which R follows from m0 ≡ 0 mod Λ+ ), we can show that m1 ◦ m1 = 0. Let H ∗ (CR ) be the cohomology group H ∗ (C; m1 ), which for our main example is nothing but the de Rham cohomology group. Let H ∗ (C; Λ0 ) = H ∗ (CR ) ⊗R Λ0 .

THEOREM 8.5 (Fukaya et al., 2000). There exists a structure of filtered A∞ algebra on H ∗ (C; Λ0 ) such that it is homotopy equivalent to (C, m). Theorem 8.5 is a filtered version of a classical result of homotopical algebra (see Kadeishvili, 1983, etc.) and is proved in the revised version of Fukaya et al. (2000) (see also Fukaya, 2003). We also refer to Fukaya et al. (2000) (and Fukaya, 2003) for the definition of homotopy equivalence of filtered A∞ algebras. Theorems 8.4 and 8.5 imply that we have a structure of filtered A∞ algebra on H ∗ (L; Λ0 ) for a compact relatively spin Lagrangian submanifold L. In the follows we show how the structures of Theorem 5.1 is deduced from it. First we consider m0 : B0 H(L; Λ0 ) = Λ0 → H(L; Λ0 ) and define oβi = m0,i (1) ∈ H 2+di (L; R). From Definition 8.3 it follows that di = −ηL (βi ). Hence oβi ∈ H 2−ηL (βi ) (L; R) as required. Suppose that oβ are all zero. Then (m1 ◦ m1 )(x) = ±m2 (m0 (1), x) ± m2 (x, m0 (1)) = 0 and we can define DEFINITION 8.6. HF(L, L; Λ0 ) =

Ker m1 . Im m1

The filtration F λ Λ0 = T λ Λ0 (λ ∈ R≥0 ) induces a filtration on H ∗ (L; Λ0 ). From it we construct the spectral sequence in Theorem 5.1. More explicitly, let F (k) H(L; Λ0 ) = H(L; Q) ⊗ T kλ Λ0 (k = 0, 1, 2, . . .) which is a filtration and gives rise to the spectral sequence. In the case when E(π2 (M, L)) = λZ, we use this particular λ to define F (k) H(L). Then the formula T λi m1,i m1 = i≥0

implies that the diﬀerential of the spectral sequence is induced by m1,i . Let dβi = m1,i .

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REMARK 8.7. In the case when L is not rational (that is E(π2 (M; L)) ⊂ R≥0 is not finitely generated), the natural filtration we have is only the one F λ H(L; Λ0 ) = H(L; Q) ⊗ T λ Λ0 which is parameterized by λ ∈ R≥0 . It is rather unusual to use such a filtration to construct a spectral sequence. Usually spectral sequence is induced by a filtration parameterized by integer or by natural number. Moreover the ring Λ0 is not Noetherian. This causes serious trouble in constructing spectral sequence and prove its convergence. This problem is resolved in Fukaya et al. (2000). REMARK 8.8. In the above argument, we use Λ0 in place of Λ. Of course Floer cohomology over Λ0 induces Floer homology over Λ. The reason we need to work over Λ is that 6 of Theorem 5.1 (or more generally the invariance of Floer cohomology under Hamiltonian deformation of Lagrangian submanifold) is not true over Λ0 and can be proved only with Λ coeﬃcient. See Fukaya et al. (2000). More arguments are needed to establish the properties asserted in Theorem 5.1 (Especially 4, 5, 6 of it.) We will not explain them here and refer to Fukaya et al. (2000). We mention one application of the construction described in this article. THEOREM 8.9. Let L be a compact relatively spin Lagrangian submanifold of M. We assume that L admits a metric of negative sectional curvature and dim L is even. Then oβ = 0. Moreover HF(L; L) H(L; Λ). # β) = n − 2 + ηL (β) is even, since Proof (Sketch). We remark that dim M(L; n is even and ηL (β) is even. Let us decompose L(L) into connected components L[γ] (L), where γ ∈ π1 (L) and [γ] is its conjugacy class. Since L has negative # β) is nonempty curvature Hi (L[γ] (L)) Hi (S 1 ) for γ 1. We remark that if M(L; # β) is nonempty then M(L; β) then it is at least one-dimensional. (In fact if M(L; # β) is homologous to zero if ∂β 1. Hence is nonempty.) This implies that M(L; mk,i is nonzero only for βi with ∂βi = 1. However since L is aspherical L[1] (L) is homotopy equivalent to L. We can use it to show that Ich∗ is trivial on L[1] (L). Thus the A∞ algebra (Ω(L) ⊗ˆ Λ0 , mk ) is homotopy equivalent to the de Rham complex. Theorem 8.9 follows. REMARK 8.10. Actually there is one point where the argument above is not # β) is not a cycle in general (see Theorem 6.7), it is suﬃcient. Namely since M(L; # β) to not clear how to use vanishing of cohomology of L[∂β] (L) to modify M(L; zero, without changing the homotopy type of filtered A∞ algebra induced by it on Ω(L) ⊗ˆ Λ0 . We can overcome this point in the following way. Firstly, there is a notion of gauge equivalence between elements satisfying Maurer-Cartan equation (that is the conclusion of Theorem 6.7), such that gauge

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# β) induces a homotopy equivalent A∞ structure on Ω(L) ⊗ˆ Λ0 . equivalent M(L; See Fukaya (2003). Secondly, we find that the set of homotopy equivalence class of elements satisfying Maurer – Cartan equation, is independent of the homotopy type of diﬀerential graded Lie algebra (or more generally of L∞ algebra). Thirdly, we can show that any diﬀerential graded Lie algebra C is homotopy equivalent to an L∞ algebra defined on cohomology group of C. This fact is an L∞ analogue of Theorem 8.5 and is proved by various people including Kadeishvili (1983). Since our solution α(L) of Maurer – Cartan equation has degree where the cohomology groups vanish, it follows that it is gauge equivalent to zero. Summing up we find that the A∞ structure induced by α(L) is homotopy equivalent to one induced by zero. Theorem 8.9 follows.

REMARK 8.11. Theorem 8.9 implies that negatively curved spin manifold of even dimension can not be embedded in C × M (for any symplectic manifold M which is convex at infinity) as a Lagrangian submanifold. This fact was established in stronger form by Viterbo and Eliashberg et al. (2000, Theorem 1.7.5). Namely negatively curved manifold can not be embedded as a Lagrangian submanifold in C × M or CPn (or more generally uniruled manifold of dimension > 2). They do not need to assume either that L is spin or that L is of even dimension. We will discuss related problems in Section 14.

REMARK 8.12. The idea of using Chan-Sullivan’s String topology to study open string theory is also applied by Cattaneo et al. (2003).3 Their interest is in its application to Physics. Here we emphasise its application to symplectic topology. The application of our approach to mirror symmetry will be discussed elsewhere. The idea of using loop space homology in Floer theory is independently proposed in Barraud and Cornea (2004), where Barraud and Cornea applied it in the case when there exist no pseudo-holomorphic disks and Floer homology is isomorphic to the usual homology.4 F. Lalonde informed the author that together with the authors of Barraud and Cornea (2004) he is now trying to apply it in a more general situation.

3 The author would like to thank A. Schwarz who pointed out the relation of Cattaneo et al. (2003) to the authors idea to use loop space homology and Chas – Sullivan bracket to study Floer homology. 4 Our main application in Sections 11, 12, 13, and 14 is in the case when Floer homology may not be well defined in the sense of Fukaya et al. (2000).

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9. S1 equivariant homology of loop space and cyclic A∞ algebra The loop space L(L) has a canonical S 1 action defined by (s · )(t) = (t + s). Chas and Sullivan (1999) also defined a bracket (which they call string bracket) on the 1 S 1 equivariant homology H∗S L(L) . Namely they define 1 1 S1 L(L) , (39) {·, ·}: HkS L(L) ⊗ HS L(L) → Hk+−n+2 by

I ∗ {x, y} = {I ∗ (x), I ∗ (y)} (40) ∗ where I : Hk L(L) → Hk+1 L(L) is the obvious map and {·, ·} in the right-hand side is the loop bracket. Similarly, we can construct it at the chain level using S 1 equivariant approximate de Rham chain of L(L). Here S 1 equivariant approximate de Rham chain of L(L) is an approximate de Rham chain (P, f, ω) where S 1 acts on P, f and ω are S 1 equivariant and that ιt ω = 0 where t is the generator of the Lie algebra of S 1 . We may regard M(L; β) as an S 1 equivariant approximate de Rham chain of L(L) of degree n − 3 + ηL (β). Then we have 1 {M(L; β1 ), M(L; β2 )} ∈ S D (L), (41) ∂M(L; β) + 2 β=β +β S1

1

2

where L is embedded in L(L) as the set of trivial loops. cyc Next we define a cyclic Bar complex Bk (C[1]) from the quotient of Bk (C[1]) by the equivalence relation generated by x1 ⊗ · · · ⊗ xk ∼ (−1)(deg xk +1)(deg x1 +··· deg xk−1 +k−1) xk ⊗ x1 ⊗ · · · ⊗ xk−1 . . cyc Then a Gerstenhaber bracket on k≥1 Hom(Bk (C[1]), R) can be defined similarly. There is a homomorphism cyc Ich: Bk (C[1]) → Ω∗S 1 L(L) k

where the right-hand side is the set of S 1 equivariant forms. (See Goodwillie, 1985.) Its dual Ich∗ is a chain homomorphism and sends string bracket to Gerstenhaber bracket. Now let α(L) = T E(β) M(L; βi ), i

equivariant approximate de Rham chain of L(L). Hence pulling which is an . cyc ∗ it back by Ich , we get an element Ich∗ α(L) of k≥1 Hom(Bk (Ω(L)[1]), Λ+ ) Then, by the fact that S D (L) is killed by iterated integral, we find that S1

δ(Ich∗ (α(L))) + 12 {Ich∗ (α(L)), Ich∗ (α(L))} = 0.

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It defines a family of operations m+k : Bk (Ω(L)[1]) → Λ0 , cyc

which are related to the operations mk in the last section by the formula mk (u1 , . . . , uk ), uk+1 = m+k+1 (u1 , . . . , uk , uk+1 )

(42)

if we take appropriate perturbations. Here ·, ·: Ωk (L) ⊗ Ωn−k (L) → R is defined by u ∧ v.

u, v = (−1)k(n−k+1) L

(42) implies that mk satisfies the following cyclic symmetry. mk (u1 , . . . , uk ), uk+1 = (−1)∗ mk (uk+1 , u1 . . . , uk−1 ), uk

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where ∗ = (deg xk+1 + 1)(deg x1 + · · · + deg xk + k). We will discuss the contents of this section in more detail in Fukaya (2005). REMARK 9.1. We constructed a filtered cyclic A∞ algebra in this section using de Rham cohomology. Hence it is defined over R. The author does not know how to do it keeping the cyclic symmetry over Z even in semi-positive case. (Actually he does not know how to do it for classical cohomology either.) Compare with Remark 6.1. REMARK 9.2. The author would like to thank P. Seidel discussions with whom let the author realize the importance of cyclic symmetry of Floer cohomology.

10.

L∞ structure on H(S1 × S n; Q)

In this section we consider the example of S 1 × S n in Cn+1 and study the leading term of the symmetrization of its A∞ structure. First we briefly discuss symmetrization of (filtered) A∞ algebra. Let Bk (C[1]) be as in Section 8. We divide it by the equivalence relation generated by x1 ⊗ · · · ⊗ xi ⊗ xi+1 ⊗ · · · ⊗ xk ∼ (−1)(deg xi +1)(deg xi+1 +1) x1 ⊗ · · · ⊗ xi+1 ⊗ xi ⊗ · · · ⊗ xk ,

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and denote the quotient by Ek (C[1]). Then mk induces lk : Ek (C[1]) → C[1] via lk ([x1 ⊗ · · · ⊗ xk ]) =

1 (−1)∗σ mk (xσ(1) ⊗ · · · ⊗ xσ(k) ) k! σ∈S k

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261

) here ∗σ = i< j;σ(i)>σ( j) (deg xi + 1)(deg x j + 1). In case mk = mk is induced by the structure of (graded commutative) diﬀerential graded algebra from (26), we can check by an easy calculation that the induced operations lk becomes zero. Hence the part of lk on Ek (Ω(L)[1]) induced by wedge product of diﬀerential forms vanishes. lk defines an L∞ structure and we define: DEFINITION 10.1. (C, lk ) is said to be a filtered L∞ algebra if the following holds: (−1)∗ l+1 (lk (xi1 ⊗ · · · xik ) ⊗ x j1 ⊗ · · · ⊗ x j ) = 0 k+=n I,J

where the second sum is taken over all I = {i1 , . . . , ik }, J = { j1 , . . . , j } with i1 < · · · < ik , j1 < · · · < j , I ∩ J = ∅, I ∪ J = {1, . . . , n} and ∗= (deg xia + 1)(deg x jb + 1). a,b;ia > jb

We thus obtained a filtered L∞ algebra (Ω(L), lk ) for a relatively spin compact Lagrangian submanifold L ⊂ M. We consider the case L = S 1 × S n ⊂ Cn+1 and calculate the leading term of lk . We put lk (u1 , . . . , uk ), uk+1 = l+k+1 (u1 , . . . , uk , uk+1 ). (46) Let us choose a generator γ ∈ π1 (S 1 × S n ) such that E(γ) = λ1 > 0. (Such γ exists by Theorem 2.3.) We consider the case ηL (γ) = 2, n + 1, which are the only ones that we have examples. Let a, b, c, e ∈ H ∗ (S 1 × S n ; Z) be generators of degree 1, n, n + 1, 0 respectively. THEOREM 10.2. If ηL (γ) = 2 then there exists an integer independent of k such that 1 l+k+1 (a, . . . , a, c) ≡ T λ1 mod T 2λ1 . k! + All other operations l among generators vanish. THEOREM 10.3. If ηL (γ) = n + 1 (n is odd) then l+k+2 (a, . . . , a, b, c) ≡ ±T λ1

1 mod T 2λ1 . k!

All other operations l+ among generators vanish. Proofs of Theorems 10.2 and 10.3 will be discussed later in this section. Details will appear in the revised version of Fukaya et al. (2000).

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REMARK 10.4. We remark that the filtered A∞ algebra and the filtered L∞ algebra associated to a Langrangian submanifold L (which is defined on the cohomology group of L) are well-defined only up to filtered A∞ or filtered L∞ isomorphism. (Here filtered L∞ isomorphism is an L∞ homomorphism which has an inverse.) So to be precise, we need to state Theorems 10.2 and 10.3 as such: “We may take filtered L∞ structure so that the following holds . . . .” To prove Theorems 10.2 and 10.3, we first remark that Theorem 6.7 implies # 1 × S n ; γ) is a cycle in S D L(S 1 × S n ). (This is because there is no that M(S γ ∈ π1 (S 1 × S n ) with 0 < E(γ ) < E(γ).) Actually we can prove more. Namely, since our Lagrangian submanifold is monotone we can use a result of Fukaya et al. (2000) to find an appropriate (single # 1 ×S n ; γ) is a cycle over valued) perturbation so that the fundamental chain of M(S Z. # 1 × S n ; γ)] ∈ Hn+1 (L(S 1 × Hence in case of Theorem 10.2, we have [M(S # 1 × S n ; γ)] ∈ H2n (L(S 1 × S n ); Z), and in case of Theorem 10.3 we have [M(S n S ); Z). We are going to study them below. We need the following: # 1 × S n ; γ)] is in the image of I ∗ : H S 1 (L(S 1 × S n ); Q) → LEMMA 10.5. [M(S k Hk+1 (L(S 1 × S n ); Q). This is a consequence of Section 9. Note that it is easy to show H∗ L(X × Y) = H∗ L(X) ⊗ H∗ L(Y) .

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Let γ0 ∈ π1 (S 1 ) is a generator and Lγ0 (S 1 ) be the component containing γ0 . It is 1 1 easy to see that H∗ Lγ0 (S 1 ) H∗ (S 1 ), H∗S Lγ0 (S 1 ) H∗S (S 1 ) Z. Now we consider the component of Lγ (S 1 ×S n ). By the commutative diagram L(S n ) ⊂

Lγ (S 1 × S n ) π

Lγ0 (S 1 )

/S 1

/ Lγ (S 1 × S n )/S 1 π

/S 1

/ one point

where the left vertical maps are fibrations, we find that 1

HkS (Lγ (S 1 × S n )) Hk (L(S n ))

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and the map I ∗ : HkS (Lγ (S 1 × S n )) → Hk+1 (Lγ (S 1 × S n )) is identified with x → [S 1 ] ⊗ x where we use the identification (47) and H∗ Lγ0 (S 1 ) H∗ (S 1 ). We now recall the calculation of homology group of loop space of S n . (See Vigu´e-Poirrier and Sullivan, 1976, for details.) Let E(a, b, . . . , c) be the free graded commutative graded algebra generated by a, b, . . . , c. (Namely if all of a, b, . . . , c are of even degree then E(a, b, . . . , c) are polynomial algebra and if all of them are of odd degree then E(a, b, . . . , c) is an exterior algebra.) We recall the following classical result (due to Serre). Let L0 (X) be the based loop space. 1

LEMMA 10.6. If n is odd then H ∗ (L0 (S n ); Q) E(x) with deg x = n − 1. If n is even then H ∗ (L0 (S n ); Q) E(x, y) with deg x = n − 1, deg y = 2n − 2. The cohomology of free loop space L(S n ) can be calculated with the Leray – Serre spectral sequence associated to the fiberation L0 (S n ) → L(S n ) → S n . Let [S n ] be the fundamental cohomology class of S n . Then the E 2 term of the spectral sequence is H ∗ (L0 (S n ); Q) ⊗ (Q[S n ] ⊕ Q[pt]). LEMMA 10.7. The boundary operator of the spectral sequence H ∗ (L0 (S n ); Q)⊗ (Q[S n ] ⊕ Q[pt]) is zero if n is odd and is given by d(x ⊗ [pt]) = 0,

d(y ⊗ [pt]) = 2x ⊗ [S n ].

if n is even. # 1 × S n ; γ)] ∈ Hn+1 (L(S 1 × S n ); Z) First consider ηL (γ) = 2. Then [M(S corresponds to an element of Hn (L(S n ); Q). Lemmas 10.6 and 10.7 imply that it corresponds to · 1 ⊗ [S n ] for some ∈ Z. We remark that the operations lk # 1 × S n ; γ). It is suﬃcient to consider depend only on the homology class of M(S # 1 × S n ; γ) = S 1 × S n and elements the case = 1. Hence we may assume that M(S 1 n (s, x) ∈ S × S corresponds to the curve t → (s + t, x) in S 1 × S n . Let dt be the one form on S 1 and Ω be the volume form on S n . Then a, b, c are the de Rham cohomology classes represented by dt, Ω and dt ∧ Ω respectively. Let us write ) iλ1 mk = ∞ i=0 T mk,iγ . Then, by definition, we have mk,γ (a, . . . , a), c 1 =± x∈S n

= ±1/k!,

t1 =0

t1

t2 =0

···

tk−1

tk =0

d(s + t1 ) ∧ · · · ∧ d(s + tk ) ∧ ds ∧ Ω

which implies Theorem 10.2. # 1 [M(S

S n ; γ)]

× ∈ Next, we consider the situation of Theorem 10.3. Then 1 n n H2n (L(S × S ); Z) corresponds to an element of H2n−1 (L(S ); Z). Now Lemmas 10.6 and 10.7 imply that it corresponds to x ⊗ [S n ] for some ∈ Z.

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LEMMA 10.8. = ±1. Before proving Lemma 10.8, let us complete the proof of Theorem 10.3. Let P → L(S n ) be the cycle representing x. For z ∈ P, let µz : S 1 → S n be the curve represented by it. We consider the map ev: P × S 1 → S n × S n defined by ev(z, t) = µz (0), µz (t) . By the definition of the class x, we find easily that ev: P × S 1 → S n × S n is of degree one. # 1 × S n ; γ)] is homologous to the cycle represented Now by Lemma 10.8, [M(S by f : P × S 1 → L(S 1 × S n ) given by f (z, s)(t) = s + t, µz (t) . Then, by definition, we have k

i

k−i

5634 5634 mk+1,γ (a, . . . , a, b, a, . . . , a), c

i=0

=±

(x,s)∈P×S 1

1 t=0

1

t1 =0

t1 t2 =0

···

tk−1 tk =0

d(s + t1 ) ∧ · · · ∧ d(s + tk ) ∧dt ∧ ev∗ (Ω ∧ Ω)

= ±1/k!. Theorem 10.3 follows. Proof. Let us prove Lemma 10.8. We can show (without using Lemma 10.8) that mk,γ (a, . . . , a, b), c = ±/k!. In particular m1,γ (b) = ±e. Since e should be in the image of m1 by Theorem 5.1, it follows that = ±1 as required. (We remark that since our Lagrangian submanifold is monotone we can work over integers. (See Fukaya et al., 2000.) Hence we can prove not only 0 but also = ±1.) REMARK 10.9. In Lemma 10.8 we proved = ±1. However the author does not know whether = ±1 if ηL (γ) = 2. The diﬀerence is the following. Let ηL (γ) = 2, then Theorem 10.2 implies that m1 (a) = ±e mod T 2λ1 . Note that e must be killed (over integer) by diﬀerentials of the spectral sequence. However this is not enough to show that = ±1. If ±1 then e is not yet killed over integer by the diﬀerential H n−1 (L) → H n (L). But there may be other diﬀerentials which kills e. If ηL (γ) = n + 1, for degree reason, except for the first one, no diﬀerential can kill e. See revised version of Fukaya et al. (2000) and Fukaya (2005) for more detail. REMARK 10.10. We remind the reader that we still have not yet presented the arguments that eliminate the possibility ηL (γ) = 2 − n when n is even. (We will do it in Section 13.) Let us look at the case of L ⊂ M with ηL (γ) = 2 − n and L S 1 × S n . (It seems very likely that this actually happens when M is a Calabi – Yau 3-fold and n = 2.)

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# 1 × S n ; γ)] ∈ H1 (Lγ (S 1 × S n )) Z, where γ ∈ In this case we have: [M(S # 1 × S n ; γ ) = ∅ for γ with 0 < E(γ ) < π2 (M, L) and we assume that M(S # 1 × S n ; γ)] is a generator. It follows E(γ). Let us suppose moreover that [M(S # 1 × S n ; γ)] S 1 such that s ∈ S 1 that, to calculate lk , we may assume that [M(S corresponds to the loop t → (s + t, x0 ) for some fixed x0 . Then we can calculate in a similar way as above and obtain l+k+1 (a, . . . , a, a) ≡ ±T λ1

1 mod T 2λ1 . k!

and all other operations are zero modulo T 2λ1 . Let us suppose l+k+1 (a, · · · , a, a) = T λ1 /k! for simplicity. Then lk (a, · · · , a) = T λ1 b/k!. We remark that lk (k = 0, 1, . . .) induces a coderivation dˆ on E(H(S 1 × S m )[1]) = E(a, b, c, e) in the same way as mk induces a coderivation on B(H(L)[1]). We then have, in our case, 1 ∂k dˆ = T λ1 b k , (49) k! ∂a ˆ f (a, b, c, e)) = T λ1 b f (a+1, b, c, e). (Here we identify element of E(a, b, c, e) i.e., d( as a (Λ coeﬃcient) polynomial of a, c tensored with an element of Q[1, b] ⊗ Q[1, e] E[b, e]. Note that a, c have even degree when shifted and b, e have odd degree when shifted.) From this calculation, we find easily that the cohomology ˆ vanishes. of (E(H(S 1 × S m )[1]), d) The calculation above means that, for L ⊂ Cn+1 , we can not use the calculation ˆ of d cohomology of E(H(S 1 × S m )[1]) to eliminate the possibility ηL (γ) = 2 − n. (Note that when L ⊂ Cn+1 , the dˆ cohomology of Lagrangian submanifold of Cn should vanish.) Actually we are going to use H(L(S 1 × S m )) (which is closely related to B(H(S 1 × S m )[1])) to eliminate the possibility ηL (γ) = 2 − n. 11. Lagrangian submanifolds of C3 We first state our main result on Lagrangian submanifolds of C3 . We recall that a 3-dimensional manifold L is said to be prime if L L1 L2 implies L1 S 3 or L2 S 3 . Here stands for connected sum and means diﬀeomorphism. Two Lagrangian immersions i0 : L → M and i1 : L → M are said to be exactly regular homotopic to each other if there exists a smooth family of Lagrangian immersions it : L → M connecting them such that E: π2 (M, it (L)) → R is independent of t. Recall that π2 (M, it (L)) π1 (L) is the set of homotopy classes of pairs of maps ( f, g), f : S 1 → L, g: D2 → M such that it ◦ f = g|∂D2 . THEOREM 11.1 (Fukaya, 2005). An oriented and connected prime 3-dimensional manifold L can be embedded to C3 as a Lagrangian submanifold if and only if L is diﬀeomorphic to S 1 × Σg where Σg is an oriented 2-dimensional manifold.

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Moreover a Lagrangian immersion i: L = S 1 × Σg → C3 is exactly regular homotopic to a Lagrangian embedding if and only if there exists γ ∈ π1 (L) such that E(γ) > 0, η(γ) = 2. REMARK 11.2. For any oriented 3-manifold L, we have T L ⊗ C C3 . Theorem 2.12 then implies that L has a Lagrangian immersion i: L → C3 . We may assume that the image of L is transversal to itself. Applying Lagrangian surgery (Lalonde and Sikorav, 1991; Polterovich, 1991b), we can show that there exists k such that L k(S 1 × S 2 ) is embedded as a Lagrangian submanifold of C3 , where k(S 1 × S 2 ) is a connected sum of k copies of S 1 × S 2 . The following seems to remain open. PROBLEM 11.3. Let Li be prime oriented 3-manifolds which are not diﬀeomorphic to Q homology sphere or S 1 × S 2 . Let L = L1 · · · Lk , k ≥ 2. Can any such L be embedded to C3 as a Lagrangian submanifold? The fact that S 1 × Σg can be embedded in C3 as a Lagrangian submanifold follows from Theorem 2.13. Carefully examining the proof of Theorem 2.13, we can prove that if E(S 1 ) > 0 and η(S 1 ) = 2 for an Lagrangian immersion i: S 1 × Σg → C3 then it is exactly regular homotopic to an embedding. So the main new part of the proof of Theorem 11.1 is the “only if” part. Let L ⊂ C3 be an embedded Lagrangian submanifold, which we assume to be oriented and prime. By Theorem 2.3, H1 (L; Q) 0. Then a well known result of 3-manifold topology (see for example (Hempel, 1976)) implies that L is diﬀeomorphic either to S 1 × S 2 or an aspherical manifold. A manifold L is said to be aspherical if πk (L) = 0 for k ≥ 2. We can generalize Theorem 11.1 in both cases to higher dimensions. We will discuss it in the next two sections and sketch the proofs. 12. Aspherical Lagrangian submanifolds In this section we consider an aspherical Lagrangian submanifold L of a symplectic manifold M of arbitrary dimension. We assume M is convex at infinity when M is noncompact. To prove Theorem 11.1 and its generalization in the case of aspherical Lagrangian submanifold L we are going to use the moduli space N(R, β) introduced in Section 3. To study this moduli space we need an assumption, which is stated below, similar to Assumption 3.6 in Section 3. Let H: M × [0, 1] → R be a compactly supported smooth function, then Ht = H(x, t): M → R is also compactly supported and smooth. We denote by XHt the Hamiltonian vector field generated by Ht , i.e., dHt = iXHt (ω). Let exptXH : M → M be a one-parameter

FLOER HOMOLOGY OF LANGRANGIAN SUBMANIFOLDS

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family of transformations associated to XH , satisfying ∂ XH = XHt0 exptX0H (x) . expt (x) ∂t t=t0 exptXH is a symplectic diﬀeomorphism for each t. ASSUMPTION 12.1. exp1XH (L) ∩ L = ∅. THEOREM 12.2. Let L ⊂ M be a Lagrangian submanifold. We assume that L is relatively spin and aspherical and Assumption 12.1. Then there exists β ∈ π2 (M, L) with the following properties: 1. E(β) > 0. 2. ηL (β) = 2. 3. ∂β ∈ π1 (L) is nonzero. Its centralizer Z∂β = {γ ∈ π1 (L) | γ(∂β) = (∂β)γ} is of finite index in π1 (L). REMARK 12.3. In the case of L = T n ⊂ CPn , the existence of β ∈ π2 (CPn , L) with ηL (β) = 2 was conjectured by M. Audin and is independently proved by Eliashberg (based on Eliashberg et al., 2000) as well as K. Cielieback. REMARK 12.4. (A) Let us assume that c1 ∩: π1 (M) → Z is zero in Theorem 12.2. Then ηL induces a homomorphism ηL : π1 (L) → Z. We now have an exact sequence ηL /2

0 → (Ker ηL ) ∩ Z∂β → Z∂β −−−→ Z → 0.

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Note that the image of ηL is even since L is orientable. Therefore Z∂β Z × ((Ker ηL ) ∩ Z∂β ) by 2. It follows that the finite covering space Lˆ of L with ˆ = Z∂β is homotopy equivalent to S 1 × L for a K(Z∂β /Z, 1) space L . π1 (L) (B) Under the assumption of Theorem 12.2, the finite covering space Lˆ of L ˆ = Z∂β is homotopy equivalent to an S 1 bundle over L for a with π1 (L) K(Z∂β /Z, 1) space L . If the image of c1 ∩: π1 (M) → Z is mZ, we can similarly show that the Euler class of this S 1 bundle is divisible by m. Let us consider the situation of Theorem 11.1. As we remarked before either L S 1 × S 2 or L is aspherical. We will discuss the first case in the next section. Here we may assume that L satisfies the assumption of Theorem 12.2. Recall that any oriented 3 manifold is spin. Since c1 (C3 ) = 0, we see that Z∂β Z× (Ker ηL )∩ Z∂β as in Remark 12.4(A). Using a standard result of 3-dimensional topology (see Hempel, 1976) and note that L is suﬃciently large in the sense of Waldhausen

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because H1 (L; Q) 0, we can prove that Lˆ is diﬀeomorphic to S 1 × Σg . Let G = $ ˆ It acts freely on Lˆ and L/G ˆ π1 (L)/π1 (L). = L. Since G = Ker ηL (Ker ηL ) ∩ Z∂β it follows that G acts freely on Σg and trivially on the S 1 factor. Hence Σg /G is again a Riemann surface. We can use this fact and Aut π1 (Σg ) ⊂ PSL(2; R) for g ≥ 2 to show that actually Z∂β = π1 (L). Namely L is diﬀeomorphic to S 1 × Σg . This proves Theorem 11.1 except for the case L = S 1 × S 2 . We now sketch the proof of Theorem 12.2. Consider a map ϕ = ϕ(τ, t): R × [0, 1] → M with the following properties. Here χR is as in Definition 3.2. ∂ϕ ∂ϕ (τ, t) = J (τ, t) − χR (τ)XHt ϕ(τ, t) , (51a) ∂τ ∂t ϕ(τ, 0), ϕ(τ, 1) ∈ L, (51b) ϕ∗ ω < ∞. (51c) R×[0,1]

(Compare with Definition 3.3.) Let N(R) be the set of all such ϕ. As in Section 3, we can extend ϕ to a map ϕ: (D2 , ∂D2 ) → (M, L). Let N(R; β) be the set of all ϕ ∈ N(R) such that the homotopy class of ϕ is β. Elements of N(R; β) may be regarded as maps (D2 , ∂D2 ) → (M, L). We then define the map ev: N(R; β) → L(L) by ev(ϕ) = ϕ|∂D2 . Let N(R; β) × {R} N(β) = R∈[0,∞)

and define ev: N(β) → L(L) in the obvious way. We note that dim N(β) = n + 1 + ηL (β).

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DEFINITION 12.5. B(L, H) is defined by B(L, H) = T β∩ω ev∗ [N(β)]. β

Here we may regard ev∗ [N(β)] as an approximate de Rham chain of L(L) as in Section 6. Then an analogue of Lemma 4.6 can be shown. Together with Gromov compactness it implies the following. LEMMA 12.6. There exists C such that N(β) is empty when β ∩ ω < −C. Moreover, for any C, there exists only a finite number of β such that β ∩ ω < C and N(β) ∅. Lemma 12.6 implies that B(L, H) ∈ S D (L) ⊗ˆ Λ, where ⊗ˆ means the completion of the algebraic tensor product. Now the main point of the proof of Theorem 12.2 is the following equality.

FLOER HOMOLOGY OF LANGRANGIAN SUBMANIFOLDS

269

THEOREM 12.7. Let L ⊂ M be a Lagrangian submanifold. We assume that L is relatively spin and Assumption 12.1. Then we have ∂B(L, H) + {α(L), B(L, H)} = [L].

(53)

Here L → L(L) is embedded as the set of constant maps. Hence [L] ∈ S nD L(L) . α(L) is defined in Definition 8.3. REMARK 12.8. We do not assume that L is aspherical in Theorem 12.7. Proof (Sketch). We are going to study the boundary of B(L, H). Let ϕi ∈ N(Ri ; β) be a divergent sequence. In a way similar to the proof of Proposition 3.8, we can show that Ri is bounded. (This is the place we use Assumption 12.1.) There exists pi = (τi , ti ) ∈ R × [0, 1] such that |d pi ϕi | = sup{|d x ϕi | | x ∈ R × [0, 1]}.

(54)

We then consider the following three cases separately. (Compare with the proof of Lemma 3.12.) Case 1 |d pi ϕi | = Di diverges. Di dist pi , ∂(R × [0, 1]) = Ci → ∞. Case 2 |d pi ϕi | = Di diverges. Di dist pi , ∂(R × [0, 1]) is bounded. Case 3 |d pi ϕi | = Di is bounded. |τi | diverges. Case 4 Ri → 0. Case 1 produces a sphere bubble, which happens in codimension two and does not contribute to ∂B(L, H). (We can make this argument rigorous in the same way as Fukaya and Ono, 1999.) Cases 2 and 3 correspond to bubbling at the boundary ∂D2 . We can show that they give the term {α(L), B(L, H)}. Let us consider Case 4. The equation (51a) then becomes equation for pseudo-holomorphicity when R = 0. Therefore, the limit limi→0 ϕi gives a pseudoholomorphic map ϕ: (D2 , ∂D2 ) → (M, L). The moduli space of such maps has an extra symmetry {g ∈ PSL(2; R) | g(1) = 1} = Aut(D1 , 1). The action of this group is nontrivial if β 0 and the contribution from Case 4 is zero as a chain, while when β = 0 we get [L]. We thus obtain the formula (53). The details will be in Fukaya (2005). To apply Theorem 12.7 to the proof of Theorem 12.2, we use a series of lemmas of purely topological nature. For γ ∈ π1 (L), let L[γ] (L) ⊂ L(L) be the set of all loops in the free homotopy class of γ, where [γ] stands for the free homotopy class of γ. Let Zγ ⊂ π1 (L) be the centralizer and Lˆ γ be the covering space of L corresponding to Zγ .

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LEMMA 12.9. If γ ∈ π1 (L) is nonzero, then the natural projection induces a homeomorphism, π∗ : L[γ] (Lˆ γ ) → L[γ] (L). Proof. Let : S 1 → L be a loop in L[γ] (L). Let p0 ∈ L be the base point. We can choose a path m: [0, 1] → L joining p0 to (1) such that m−1 ◦ ◦ m is homotopic ˜ = . −1 ◦ ˜ ◦ m in L[γ] (Lˆ γ ). It is easy to see that π∗ () to γ. It then lifts to a loop m Hence π∗ is surjective. Next, we assume that π∗ (˜1 ) = π∗ (˜2 ) = and ˜1 , ˜2 ∈ L[γ] (Lˆ γ ), then there exists g ∈ G = π1 (L)/Zγ such that g · ˜1 = ˜2 . Note that both ˜1 and ˜2 are in the free homotopy class of γ. Since γ is in the center of π1 (L[γ] ), it follows that g−1 γg = γ. Then g = 1 because g ∈ G = π1 (L)/Zγ . Thus π∗ is injective. LEMMA 12.10. Suppose L is a K(π, 1) space and γ ∈ π1 (L) = π, then the map ev: L[γ] (Lˆ γ ) → Lˆ γ is a homotopy equivalence. Proof. If γ = 1, then Lˆ γ = Lˆ 1 = L. Let L˜ be the universal covering space ˜ ˜ is of L. Then L[1] (L) = L[1] (L)/π. Since L˜ is contractible it follows that L[1] (L) contractible. Hence ev: L[1] (L) → L is a homotopy equivalence. ˜ we let X = {: R → L˜ | (t + 1) = γ(t)}. Zγ acts For γ 1, since γ acts on L, on X freely and X/Zγ = L[γ] (Lˆ γ ). It suﬃces to show that X is contractible. ˜ → (0), then X is homotopy equivalent to Consider the fibration ev0 : X → L: the fiber of ev0 . The fiber can be identified with the space of paths : [0, 1] → L˜ joining p˜ 0 to γ p˜ 0 , which is contractible. LEMMA 12.11. Let L is an n-dimensional aspherical manifold and γ ∈ π1 (L). Then Hk (L[γ] (L); Z) = 0 for k {0, . . . , n}. Moreover, if Hn (L[γ] (L); Z) 0 then Zγ is of finite index in π1 (L). Proof. Hk (L[γ] (L); Z) Hk (Lˆ γ ; Z) by Lemmas 12.9 and 12.10. Since Lˆ γ is a covering space of n-dimensional manifold L we see that Hk (Lˆ γ ; Z) = 0 for k {0, . . . , n}. If Hn (Lˆ γ ; Z) 0, then Lˆ γ is compact and hence Zγ is of finite index in π1 (L). LEMMA 12.12. Let L be an n-dimensional compact oriented aspherical manifold and [L] ∈ Hn (L0 (L); Q). Then we have {[L], [P]} = 0 for any [P] ∈ H∗ (L(L); Q). Proof. For p ∈ L let p be the constant loop at p. For x ∈ P let x be the corresponding loop. Now [P] ∗ [L] has support P ∗ L = {(p, x, t) ∈ L × P × S 1 | p (t) = x (0)}. If (p, x, t) ∈ P ∗ L then (p, x, t ) ∈ P ∗ L for any t . Moreover the images of (p, x, t) and (p, x , t) under ∗: P ∗ L → L(L) are the same. This implies that [P] ∗ [L] = 0. On the other hand, [L] ∗ [P] is supported at L ∗ P = {(x, p, t) ∈ L × P × S 1 | x (t) = p (0)}.

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If (p, x, t) ∈ L ∗ P then (p, x, t ) ∈ L ∗ P for any t . Moreover ∗(x, p, t) is diﬀerent from ∗(x, p, t ) only by parameterization. This implies that [L] ∗ [P] = 0. Now we go back to the (sketch of the) proof of Theorem 12.2. By (53) we have β ∈ π2 (M) such that {M(L; β), N(−β)} 0. Note that dim M(L; β) = n − 2 + ηL (β) and dim N(−β) = n + 1 − ηL (β) and by Lemma 12.11, we have n − 2 + ηL (β), n + 1 − ηL (β) ∈ {0, . . . , n}. Since ηL (β) is even, this implies ηL (β) = 2. Then because dim M(L; β) = n and is nonzero, it follows from Lemma 12.11 that the centralizer Zγ of γ = ∂β ∈ π1 (L) is of finite index in π1 (L). Theorem 12.2 follows. REMARK 12.13. Actually there is one point in the above proof we need to clarify. Namely the chain N(−β) is not necessary a cycle. So we need to work on the chain level and the way to apply Lemma 12.11 is not clear. We can overcome this in a way similar to Remark 8.10 by using a theorem that any L∞ algebra is homotopy equivalent to an L∞ algebra defined on its homology group. (We use also Lemma 12.12.) The details will appear in Fukaya (2005). REMARK 12.14. Loop space homology was used in Viterbo (1997) in a related context. To find relation of Floer homology to Viterbo (1997) was one of the motivations of the author to modify the construction of Fukaya et al. (2000) to the ones described in Sections 6, 7, 8, and 9. Viterbo (1990; 1997) used closed geodesic. Closed geodesic appears also in the study of Lagrangian submanifold using contact homology and in the approach by Eliashberg et al. (2000) and by Cielieback as mentioned in Remark 12.3. Closed geodesic is closely related to the homology of loop space. In a sense, our approach is a topological version of the one using closed geodesic. It seems possible to describe relation of Floer homology of Lagrangian submanifold to Viterbo (1997) and to Eliashberg et al. (2000), using the ideas developed in Sections 6, 7, 8, and 9. We will discuss it elsewhere.

13. Lagrangian submanifolds homotopy equivalent to S1 × S2m THEOREM 13.1. Let L ⊂ M be a Lagrangian submanifold. We assume that L is homotopy equivalent to S 1 × S 2m as well as Assumption 12.1. Then there exists β ∈ π2 (M, L) such that E(β) > 0, ηL (β) = 2 and ∂β ∈ π1 (L) is a generator.5 5 The author thanks to Prof. A. Kono who provides information on the homology of free loop space useful to prove this Theorem.

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Proof (Sketch). (We need to apply the same remark as Remark 12.13, to make the argument below precise.) Let n = 2m. Lemma 5.7 (and its proof) implies that, if the theorem is false, then there exists β ∈ π2 (M, L) with M(L; β) 0, ηL (β) = 2 − 2m. Moreover by Theorem 12.7, we may assume {M(L; β), N(−β)} = [L].

(55) Then dim N(−β) = 2n = 4m. Let γ = ∂β. Note that [N(−β)] ∈ Hn Lγ (S 1 × S 2m ) . Hence [N(−β)] is either of the form [pt] ⊗ a or [S 1 ] × a where H(Lγ (S 1 × S 2m )) H(Lγ (S 1 )) ⊗ H L(S 2m ) H(S 1 ) ⊗ H L(S 2m ) . By Lemma 10.7, x ⊗ [S 2m ] in the E2 term of the spectral sequence (converging to H(L(S 2m ))) does not survive. (Note deg(x⊗[S 2m ]) = deg a .) Using this fact, we can show that [N(−β)] lies in the image of H(S 1 ) ⊗ H L0 (S 4m ) , where L0 (S 4m ) denotes the based loop space. Define ev0 : Lγ (S 1 × S 2m ) → S 1 × S 2m by ev0 () = (0). Then ev0 ({P, Q}) ⊆ ev0 (P)∪ev0 (Q). Moreover the image of the elements of H(S 1 )⊗H L0 (S 2m ) under ev0 lie on S 1 × {p0 } where p0 ∈ S n is the base point. Note also that dim M(L; β) = 0. It follows that the support of ev0 ({M(L; β), N(−β)}) is contained in an onedimensional space. On the other hand, ev0∗ [L] = [L]. This contradicts (55).

14. Lagrangian submanifolds of CP n As mentioned in Remark 8.11, Viterbo and Eliashberg – Givental – Hofer proved that if L admits a metric of negative curvature it can not be embedded in CPn as a Lagrangian submanifold. Theorem 12.2 does not imply this result even in the spin case since Assumption 12.1 may not be satisfied. We however have an alternative argument which implies their result in the spin case. THEOREM 14.1. Let L be a Lagrangian submanifold of CPn . We assume that L is aspherical and spin. Then there exists β ∈ π2 (CPn , L) with the following properties: 1. E(β) > 0. 2. ηL (β) = 2. 3. ∂β ∈ π1 (L) is nonzero. Its centralizer Z∂β = {γ ∈ π1 (L) | γ(∂β) = (∂β)γ} is of finite index in π1 (L). REMARK 14.2. There are also results by Seidel (2000), Biran (2004), etc. on Lagrangian submanifold of CPn etc. The Lagrangian submanifolds discussed in

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those works seems to be on the opposite extreme to the ones in Theorem 14.1. Namely, in those works, Lagrangian submanifolds with small fundamental groups or homology groups are mainly studied, while main targets of Theorem 14.1 are the ones with huge fundamental groups. Proof (Sketch). We are going to construct an S 1 equivariant chain B(L) ∈ S D L(L) which satisfies an equation similar to (53). Fix p0 ∈ CPn \ L. For each β ∈ π2 (CPn , L), consider the moduli space of maps ϕ: D2 → CPn with the following properties: 1. ϕ is holomorphic. 2. ϕ(∂D2 ) ⊂ L. 3. The homotopy type of ϕ is β. 4. ϕ(0) = p0 . (Compare with Definition 3.1.) Let N (L; β) be the space of all such ϕ’s. Let Aut(D2 , 0) = {g ∈ PSL(2; R) | g(0) = 0} S 1 . It acts on N (L; β) in an obvious way. ϕ → ϕ|S 1 defines an S 1 equivariant map ev: N (L; β) → L(L). Hereafter we denote by S SD1 L(L) the set of all S 1 equivariant de Rham chains in L(L). We can use an argument similar to the one in Section 6 and may regard ev∗ [N (L; β)] ∈ S SD1 L(L) . ˆ is defined by DEFINITION 14.3. B(L) ∈ S SD1 L(L) ⊗Λ B(L) = T β∩ω ev∗ [N (β)]. β

In Section 9 we regarded M(L; β) as an S 1 equivariant approximate de Rham ) chain of L(L) of degree n−3+ηL (β) and used it to define α(L) = i T E(β) M(L; βi ) ∈ S SD1 L(L) . THEOREM 14.4. We normalize our symplectic form ω so that ω ∩ S 2 = 1 for the generator [S 2 ] ∈ H2 (CPn ; Z). We then have ∂B(L) + {α(L), B(L)} ≡ [L] mod T 2 . Proof (Sketch). Consider a divergent series of elements ϕi of N (L; β). Then, in the limit, one of the following occurs. 1. A bubble occurs at the boundary. 2. A bubble occurs at interior.

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1 gives the term {α(L), B(L)}. In general 2 is a phenomenon of codimension 2 and do not contribute to our formula. However there is an exception. Namely ϕi may converge to a union of trivial disk ϕ0 : D2 → C2 that is a constant map to p ∈ L, and a pseudoholomorphic sphere ϕ: S 2 → CPn such that ϕ(∞) = p0 and ϕ(0) = p = ϕ0 (D2 ). The reason that this is codimension one phenomenon is that the map ϕ0 ∨ ϕ: D2 ∨ S 2 → CPn is not stable, since its group of automorphisms is S 1 . We can analyse the neighborhoods of such maps in the moduli space and can show that such maps correspond to boundary points of the moduli space N (L; β). (We remark that in order such phenomenon to occur, ∂β must be zero.) We are interested in the case when ϕ ∩ ω = 1. It is easy to see that, for each p, there exists exactly one such ϕ and it gives [L]. (This is because there exists exactly one rational curve of degree one containing p and ∞.) Theorem 14.4 follows. (More details of the proof will be in the revised version of Fukaya et al., 2000 and in Fukaya, 2005.) Now we can use Theorem 14.4 in place of Theorem 12.7 and prove Theorem 14.1 in the same way as Theorem 12.2. We also replace Lemma 12.11 by the following: LEMMA 14.5. Let L be an n-dimensional aspherical manifold and γ ∈ π1 (L). If 1 x ∈ HkS (L[γ] (L); Z) for k {0, . . . , n − 1}, then {x, y} = 0 for any y. Moreover, if S 1 (L (L); Z) and {x, y} 0 for some y, then Z is of finite index in π (L). x ∈ Hn−1 [γ] γ 1 Lemma 14.5 follows immediately from Lemma 12.11 and (40).

REMARK 14.6. We remark that L is not assumed to be aspherical in Theorem 14.4. Thus the theorem can be used to study Lagrangian submanifold L of CPn for more general L. For example the case L = S 1 × S n can be studied similarly as in Section 13. The case where L is a rational homology sphere is also of interest since Gromov’s theorem 2.3 does not generalize directly to a Lagrangian submanifold of CPn . One may also study Lagrangian submanifolds of more general symplectic manifold M than CPn . For example the case when M is uniruled can be handled in a similar fashion. The author is planning to explore these points elsewhere.

References Arnol d, V. I. and Givental , A. V. (1985) Symplectic geometry, In V. I. Arnold and S. P. Novikov (eds.), Dynamical Systems IV, pp. 1–136, Berlin, Springer.

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Audin, M., F.Lalonde, and Polterovich, L. (1994) Symplectic rigidity: Lagrangian submanifolds, In Holomorphic Curves in Symplectic Geometry, Vol. 117 of Progr. Math, pp. 271–321, Basel, Birkh¨auser. Barraud, J.-F. and Cornea, O. (2004) Lagrangian intersections and the Serre spectral sequence, arXiv:math.DG/0401094. Biran, P. (2004) Lagrangian nonintersection, arXiv:math.SG/0412110. Cattaneo, A. S., Fr¨ohlich, J., and J¨urg, P. B. (2003) Topological field theory interpretation of string topology, Comm. Math. Phys. 240, 397 – 421. Chas, M. and Sullivan, D. (1999) String topology, arXiv:math.GT/9911159. Chen, K.-T. (1973) Iterated integrals of diﬀerential forms and loop space homology, Ann. of Math. (2) 97, 217 – 246. Eliashberg, Y., Givental, A., and Hofer, H. (2000) Introduction to symplectic field theory, Geom. Funct. Anal. Special Volume, Part II, 560–673. Eliashberg, Y. and Gromov, M. (1991) Convex symplectic manifolds, In Several Complex Variables and Complex Geometry II, Vol. 52 of Proc. Sympos. Pure Math., pp. 135–162, Providence, RI, Amer. Math. Soc. Eliashberg, Y. and Mishachev, N. (2002) Introduction to the h-Principle, Vol. 48 of Grad. Stud. Math., Providence, RI, Amer. Math. Soc. Floer, A. (1988) Morse theory for Lagrangian intersections, J. Diﬀerential Equations 28, 513 – 547. Fukaya, K. (2001) Floer homology and mirror symmetry I, In Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds, Vol. 23 of AMS/IP Stud. Adv. Math., Cambridge MA, 1999, pp. 15–43, Providence, RI, Amer. Math. Soc. Fukaya, K. (2003) Deformation theory, homological algebra and mirror symmetry, In Geometry and Physics of Branes, Como, 2001, Bristol, IOP. Fukaya, K. (2005), in preparation. Fukaya, K., Oh, Y. G., Ohta, H., and K.Ono (2000) Langrangian intersection Floer theory — anomaly and obstruction, http://www.kusm.kyoto-u.ac.jp/˜fukaya/fukaya.html. Fukaya, K. and Ono, K. (1999) Arnold conjecture and Gromov – Witten invariants, Topology 38, 933 – 1048. Fukaya, K. and Ono, K. (2001) Floer homology and Gromov – Witten invariant over integer of general symplectic manifolds — summary, In Adv. Stud. Pure Math., Vol. 31 of Adv. Stud. Pure Math., pp. 75–91, Tokyo, Math. Soc. Japan. Gerstenhaber, M. (1963) The cohomology structure of an associative ring, Ann. of Math. (2) 78, 267 – 288. Goodwillie, T. G. (1985) Cyclic homology, derivations, and the free loopspace, Topology p. 187 – 215. Gromov, M. (1985) Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82, 307 – 347. Gromov, M. (1986) Partial Diﬀerential Relations, Berlin, Springer. Hempel, J. (1976) 3-Manifolds, Vol. 86, Princeton, NJ, Princeton Univ. Press. Kadeishvili, T. V. (1983) The algebraic structure in the homology of an A(∞)-algebra, Soobshch. Akad. Nauk Gruzin. SSR 108, 249 – 252, Russian. Lalonde, F. and Sikorav, J.-C. (1991) Sous-vari´et´es lagrangiennes et lagrangiennes exactes des fibr´es cotangents, Comment. Math. Helv. 66, 18 – 33. Lees, J. A. (1976) On the classification of Lagrange immersions, Duke Math. J. 43, 217 – 224. Oh, Y.-G. (1992) Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions, Comm. Pure Appl. Math. 45, 121 – 139. Oh, Y.-G. (1993) Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks I, Comm. Pure Appl. Math. 46, 949 – 993.

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Oh, Y.-G. (1996) Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices 7, 305 – 346. Ohta, H. (2001) Obstruction to and deformation of Lagrangian intersection Floer cohomology, In Symplectic Geometry and Mirror Symmetry, Seoul, 2000, pp. 281–309, Singapore, World Sci. Publishing. Polterovich, L. (1991)a Monotone Lagrange submanifolds of linear spaces and the Maslov class in cotangent bundles, Math. Z. 207, 217 – 222. Polterovich, L. (1991)b The surgery of Lagrange submanifolds, Geom. Funct. Anal. 1, 198 – 210. Ruan, Y. (1999) Virtual neighborhoods and pseudo-holomorphic curves, Turkish J. Math. 23, 161 – 231. Seidel, P. (2000) Graded Lagrangian submanifolds, Bull. Soc. Math. France 128, 103 – 149. Silva, V. D. (1997) Products on symplectic Floer homology, Ph.D. thesis, Oxford University. Stasheﬀ, J. (1963) Homotopy associativity of H-spaces I, Trans. Amer. Math. Soc. 108, 275 – 292; II, 293 – 312. Sugawara, M. (1957) A condition that a space is group-like, Math. J. Okayama Univ. 7, 123 – 149. Vigu´e-Poirrier, M. and Sullivan, D. (1976) The homology theory of the closed geodesic problem, J. Diﬀerential Geom. 11, 633 – 644. Viterbo, C. (1990) A new obstruction to embedding Lagrangian tori, Invent. Math. 100, 301 – 320. Viterbo, C. (1997) Exact Lagrange submanifolds, periodic orbits and the cohomology of free loop spaces, J. Diﬀerential Geom. 47, 420 – 468.

THE LS-INDEX: A SURVEY MAREK IZYDOREK Gda´nsk University of Technology

Abstract. Let H be a Hilbert space. Consider a vector field f defined on H of the form f (x) = L(x) + K(x), where L is a strongly indefinite bounded linear operator and K is a completely continuous perturbation. In this paper we present an extension of the classical Conley index to flows generated by f as above. This topological invariant is called the LSindex. Furthermore, we describe an equivariant version of this index which generalizes the equivariant Conley index introduced by Floer. We also present cohomological LS-indices in both nonequivariant and equivariant cases. We discuss applications of these indices to existence and multiplicity results for periodic solutions of Hamiltonian systems. This survey is based on the author’s lectures delivered at the NATO-ASI – SMS 2004 at the Universit´e of Montr´eal. Key words: Hilbert space, strongly indefinite functional critical points, Hamiltonian system, periodic solution, Lie group, equivariant map, Morse decomposition, Conley index 2000 Mathematical Subject Classification: 37J45 (58E05 34C37 70H05)

1. Introduction Many dynamical processes in economy, social sciences, biology, physics, chemistry etc., can be described by diﬀerential equations depending on parameters which cannot be determined with a satisfactory precision. For the study of such systems it is important to determine those structural properties which remain invariant under “small” perturbations. Describing such properties in terms of appropriately chosen topological invariants seems to be a natural procedure. During the last three decades considerable progress took place in investigating dynamical systems from this point of view (see Benci, 1991; Conley and Zehnder, 1984; Degiovanni and Mrozek, 1993; Floer, 1990; Franzosa, 1989; Ge ba, 1997; Kurland, 1982, 1983; McCord, 1988; McCord and Mischaikow, 1992; Mischaikow, 1994; Montgomery, 1973; Reineck, 1988; Robbin and Salamon, 1988; Rybakowski, 1987, 1982, 1983; Rybakowski and Zehnder, 1985; Salamon, 1985, etc.). The Conley index theory, introduced by Charles Conley in late 60s, became one of the most powerful topological tools in this topic of research. The theory is based on the notion of the homotopy index (the Conley index), a topological invariant which is a far-reaching extension of the classical Morse index. Conley’s mono277 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 277–320. © 2006 Springer. Printed in the Netherlands.

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graph Isolated Invariant Sets and the Morse Theory (Conley, 1978) is the standard reference. Morse theory can only be used for flows generated by gradient vector fields and the Morse index is well defined solely for hyperbolic rest points. By contrast, the Conley theory is applicable to an arbitrary flow and the homotopy index is defined for a large class of compact sets, so called isolated invariant sets. Indeed, a hyperbolic stationary point provides the simplest example of an isolated invariant set. In Conley’s theory, the objects of primary interest are isolating neighbourhoods together with their associated isolated invariant sets. They are defined for flows on locally compact spaces. To be more precise, let us recall that a flow η: R × X → X on a locally compact metric space X is a continuous map satisfying: η(0, x) = x and η t, η(s, x) = η(t + s, x). A set S is an invariant set for a flow η if η(R, S ) := η(t, S ) = S . t∈R

An isolating neighbourhood is a compact set N such that its maximal invariant set lies in its interior, i.e., Inv(N, η) := {x ∈ N; η(R, x) ⊂ N} ⊂ int(N). An invariant set S is an isolated invariant set for a flow η if S = Inv(N, η) for some isolating neighbourhood N. Roughly speaking, Conley’s index theory associates to any compact isolated invariant set S of a flow η an index h(S ) which is the homotopy type of a compact, metrizable, pointed space. This homotopy type is invariant to continuation and is used basically to detect bounded orbits of (local) flows in locally compact spaces. Besides Conley (1978), other important references that should be mentioned are two papers containing new ideas and crucial improvements of the theory due to Mischaikow (1994) and Salamon (1985). As it was pointed out in Mischaikow (1994), instead of isolated invariant sets one can equivalently consider isolating neighbourhoods for flows. These are “stable” under perturbations, i.e., if N is an isolating neighbourhood for a particular flow η, then it is an isolating neighbourhood for all nearby flows ν. In contrast, isolated invariant sets can disappear, change their topological type or change their stability after a slight change of the flow. Moreover, working with isolating neighbourhoods one explicitly sees the analogy between the homotopy index and another topological invariant, the Brouwer degree. This observation aids considerably to finding a natural infinite-dimensional extension of Conley’s theory. For the classical theory, the local compactness of the phase space is a crucial condition. On the other hand, many problems that admit a variational structure appear in the setting of functional spaces. Examples of diﬀerential equations whose solutions correspond to critical points of nonlinear functionals defined in appropriate functional spaces are abundant in mathematical physics and mechanics: first

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and second order Hamiltonian systems, elliptic systems and wave equations are typical examples. Several methods have been worked out till now in order to study associated functionals (see, for instance, Blanchard and Br¨uning, 1992; Chang, 1993; Izydorek, 2000; Mawhin and Willem, 1989; Rabinowitz, 1986; Struwe, 1996). The basic questions concern the existence, mutiplicity and bifurcation of critical points. It also follows from this discussion that it is natural to search for new homotopy invariants which can detect critical points of functionals defined on infinite-dimensional, and thus, not locally compact spaces. Considering the existence and multiplicity problems for periodic solutions of Hamiltonian systems or solutions of certain elliptic systems one has to deal with so called strongly indefinite functionals, i.e., the second derivative of a functional at a critical point is a strongly indefinite operator (see Izydorek, 1999). This kind of functionals have been thoroughly studied by many mathematicians and the literature of this subject is vast. Let me only mention the following articles: Amann and Zehnder; (1980a; 1980b); Ambrosetti and Rabinowitz (1973); Benci (1982); Chang (1981); Conley and Zehnder (1984); Fei (1995a; 1995b); Li and Liu (1989); Li and Szulkin (1993); Long (1993; 1990); Rabinowitz (1986); Salamon (1990); Szulkin (1992; 1997) and books by Bartsch (1993), Mawhin and Willem (1989) and Chang (1993). In particular, rest points of flows generated by gradients (or pseudogradients) of energy functionals have been investigated. This immediately suggests to consider flows in functional spaces from a more general point of view. Namely, why not look at isolated invariant sets instead of looking at stationary points only. By dropping the compactness condition, we can give naive definitions of isolated invariant sets, isolating neighbourhoods, attractor-repeller pairs, Morse decompositions etc., for flows in infinite-dimensional spaces which are analogous to those from the classical Conley index theory. However, this naive approach does not work well for two main reasons: first, since a sphere in an infinite-dimensional Banach space is contractible we can not expect much information using standard methods of algebraic topology; second, in the classical theory, most proofs of the technical lemmas and theorems use local compactness in an essential way and therefore their naive extension in this way might not be possible or might be trivial. The first infinite-dimensional extension of Conley’s theory was made by Rybakowski (1982; 1983) in order to study nonlinear elliptic and parabolic equations (see Rybakowski, 1987). Later on the Conley index theory was further generalized by Benci (1991). However, both theories can give nontrivial results only for isolated invariant sets having finite-dimensional unstable (or stable) manifold. Hence, they are not applicable to flows generated by strongly indefinite functionals. Having in mind the analogy between the Conley index and the Brouwer topological degree it is possible to construct as in Ge ba et al. (1999) an infinite-

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dimensional version of the Conley index which extends the classical Conley index in the same way the Leray – Schauder degree extends the Brouwer degree. Consider a vector field f : H → H, f (x) = L(x) + K(x), where L: H → H is a linear bounded operator whose spectrum is separated from the imaginary axis and K is a completely continuous perturbation. Let Pl : H → H, l ∈ N be a sequence of H-orthogonal projectors such that, for every l ∈ N, the subspace Pl (H) is finitedimensional and L-invariant and such that Pl z → z in H, for every z ∈ H. For every l ∈ N define the map gl : Pl (H) → Pl (H) by gl (z) = Lz + Pl K(z),

z ∈ Pl (H).

Let N be a closed and bounded subset of H. If N is a isolating neighborhood for a flow generated by the ordinary diﬀerential equation z˙ = Lz + K(z)

(1)

on H, then there is a smallest l0 ∈ N such that, for every l ≥ l0 , the set N ∩ Pl (H) is an isolating neighborhood for the flow generated by the ordinary diﬀerential equation z˙ = gl (z) (2) on Pl (H). Thus, for every l ≥ l0 , the (finite-dimensional) Conley index h gl , N ∩ Pl (H) is defined. Now the LS-index h(L + K, N) is defined as the sequence . h(L + K, N) := h gl , N ∩ Pl (H) l≥l0

In other words, the LS-index of an isolating neighborhood N of a flow generated by the infinite-dimensional ODE (1) is the sequence of the Conley indices of N with respect to the finite-dimensional Galerkin approximations (2) of (1). One also proves that if the homotopy type a pointed space A is equal to h gl , N ∩ Pl (H) then there is a nonnegative integer ν such that the homotopy type of the ν-fold suspension of A is equal to h gl+1 , N ∩ Pl+1 (H) for every l ≥ l0 . This index, defined in Ge ba et al. (1999), enjoys the usual properties of the classical Conley index, like the nontriviality and the homotopy invariance property. Moreover, it generalizes infinite-dimensional Morse theory in the same way as the usual index extends classical Morse theory in the finite-dimensional case. It is also an extension of various Morse theories for indefinite and strongly indefinite functionals. The theory is easily applicable to Hamiltonian systems and also to strongly indefinite elliptic systems, giving nontrivial results. In Ge ba et al. (1999) the authors were motivated by Szulkin’s articles (Szulkin, 1992, 1997) devoted to various generalizations of Morse theory with applications to strongly indefinite problems. In particular, Szulkin (1992) contains description of a cohomology theory in a Hilbert space which is adapted to study that type of problem. In its turn,

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this approach was based on an infinite-dimensional cohomology and cohomotopy theory developed by Ge ba and Granas (1965a; 1965b; 1967a; 1967b; 1969; 1973). Later on, in Izydorek (2001), a cohomological LS-index in Hilbert spaces was defined and were proved generalized Morse inequalities for Morse decompositions of an isolated invariant sets. They were next applied to detect periodic solutions of Hamiltonian systems. It is worth pointing out that the gradient nature of the problem has been used in an essential way. This means that in cases considered in Izydorek (2001) Morse inequalities give more information about the existence of stationary points of flows than the topological degree of Leray – Schauder type. Considering systems of diﬀerential equations that admit symmetries of certain Lie group G one is lead to search critical points of associated functionals defined on a Hilbert representation of the group G which are G-invariant. This additional structure is often used to obtain multiplicity results for solutions of “symmetric” problems. Autonomous Hamiltonian systems provide typical example of diﬀerential equations with natural symmetries of the group S 1 = {z ∈ C; z = 1}. Till now, several approaches have been worked out to search for critical points of G-invariant functionals. The vast literature on the subject can be found in Bartsch (1993) and Chang (1993). One can therefore ask if there is a topological invariant having properties similar to those of the Conley index which respects symmetries of a group G. A G-equivariant Conley index in locally compact spaces was defined by Floer (1987). Relationships between equivariant Conley index and the degree for G-equivariant gradient maps have been established by Ge ba (1997). An infinite-dimensional extension of this theory was made by the author in Izydorek (2002). Izydorek and Rybakowski (2002b) presented a version of the LS-index theory for certain classes of infinite-dimensional ODEs without the uniqueness property of the Cauchy problem. Next, in Izydorek and Rybakowski (2002a) this new invariant has been applied to strongly indefinite elliptic systems of the form −∆u = ∂v Q(u, v, x) −∆v = ∂u Q(u, v, x) u = 0, v = 0

in Ω, in Ω, in ∂Ω

on a smooth bounded domain Ω in RN for Hamiltonians Q of class C 2 satisfying subcritical growth assumptions on their first order derivatives. The authors obtained an elementary proof of the existence of nontrivial solutions of the above problem, a result previously established by Angenent and van der Vorst (1999) via Morse – Floer homology. Finally, in Izydorek and Rybakowski (2003) a Z2 -equivariant LS-index has been applied to prove a multiplicity result for solutions of the above elliptic system with Z2 -symmetries which was conjectured in Angenent and van der Vorst (1999) .

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The survey is organized as follows. In Section 2, we give the description of the LS-index and we present its basic properties. Also we recall some definitions and facts concerning flows in functional spaces. In Section 3 we describe a cohomology theory for spectra. Attractor-repeller pairs, Morse decompositions and the Morse inequalities are discussed in Section 4. Equivariant flows and the G-LSindex for arbitrary compact Lie groups G are considered in Section 5. Finally, in Section 6, we discuss applications and examples of the LS-index. 2. The LS-index 2.1. BASIC DEFINITIONS AND FACTS

The homotopy Conley index for a class of flows generated by LS-vector fields in Hilbert spaces has been recently defined in Ge ba et al. (1999). These flows appear, for instance, when one applies variational methods to prove the existence or multiplicity results for periodic solutions of certain types of Hamiltonian systems (see Amann and Zehnder, 1980b; Chang, 1993; Conley and Zehnder, 1984; Szulkin, 1992), second order ODE’s (see Benci, 1991; Mawhin and Willem, 1989 and references therein) as well as some elliptic and hyperbolic problems (see Chang et al., 1982; Izydorek and Rybakowski, 2002a, 2003; Rabinowitz, 1986, 1978; Szulkin, 1997). In what follows, we recall basic definitions and facts which will be used in our constructions. Let H = (H, ·, ·) be a real Hilbert space and L: H → H be a linear bounded operator with spectrum σ(L) such that: 7∞ H.1 H = k=0 Hk with all subspaces Hk being mutually orthogonal and of finite dimension; H.2 L(H0 ) ⊂ H0 , H0 is the invariant subspace of L corresponding to the part of spectrum σ0 (L) := iR ∩ σ(L) lying on the imaginary axis and L(Hk ) = Hk for all k > 0; H.3 σ0 (L) is isolated in σ(L), i.e., σ0 (L) ∩ cl σ(L) \ σ0 (L) = ∅. Throughout the paper Λ denotes a compact metric space. A continuous map η: D(η) → H is a local flow on H if − D(η) is an open subset of R × H, such that {0} × H ⊂ D(η),

− for every x ∈ H there exist α(x), ω(x) ∈ R ∪ {−∞, ∞} such that α(x), ω(x) = {t ∈ R; (t, x) ∈ D(η)}; − η t, η(s, x) = η(t + s, x) for all x ∈ H and t, s ∈ α(x), ω(x) such that t + s ∈ α(x), ω(x) .

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A local flow η is a flow on H if D(η) = R × H. If η is a local flow on H and X ⊂ H then

Inv(X) = Inv(X, η) := x ∈ X; η(t, x) ∈ X for all t ∈ α(x), ω(x) is the maximal η-invariant subset of X. We say that S ⊂ H is an invariant set (with respect to a flow η) if S = Inv(S , η). A continuous map η: D(η) → H is a family of local flows on H (indexed by Λ) if D(η) is an open subset of R × H × Λ and ηλ (where ηλ (t, x) = η(t, x, λ)) is a local flow for every λ ∈ Λ. If D(η) = R × H × Λ then η is called a family of flows on H (indexed by Λ). If η is a family of local flows on X ⊂ H then we let Inv(X, η) := {(x, λ) ∈ X × Λ; x ∈ Inv(X, ηλ )}. Let W be one of the spaces H, H × Λ or R × H × Λ. A map F: W → H is said to be completely continuous if F is continuous and for any bounded subset A ⊂ W the closure of F(A) is a compact subset of H. DEFINITION 2.1. A flow η: R × H → H is called an LS-flow if η(t, x) = etL x + U(t, x) where U: R × H → H is completely continuous. DEFINITION 2.2. LS-flows if

A family of flows η: R × H × Λ → H is called a family of η(t, x, λ) = etL x + U(t, x, λ)

where U: R × H × Λ → H is completely continuous. The LS-flows have a crucial compactness property which we now formulate. THEOREM 2.3 (Ge ba et al., 1999, Proposition 2.3). Let Λ be a compact metric space and let η: R × H × Λ → H be a family of LS-flows. If X ⊂ H is closed and bounded then S := Inv(X × Λ, η) is a compact subset of X × Λ. As we will see later, the above result plays an essential role in the construction of the LS-homotopy index. DEFINITION 2.4. A map f : H → H is called an LS-vector field if there exists a completely continuous and locally Lipschitzian map K: H → H such that f (x) = Lx + K(x) for all x ∈ H.

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DEFINITION 2.5. A map f : H ×Λ → H is called a family of LS-vector fields if there exists a completely continuous and locally Lipschitzian map K: H × Λ → H such that f (x, λ) = Lx + K(x, λ) for all (x, λ) ∈ H × Λ. If f : H → H is an LS-vector field and x ∈ H then it is well known (e.g., see Deimling, 1977) that there exists the maximal C 1 -curve α(x), ω(x) t → η(t, x) ∈ H satisfying

  dη    = f ◦η dt    η(0, x) = x

Moreover, if we set D(η) := {(t, x) ∈ R × H: α(x) < t < ω(x)}, then D(η) ⊂ R × H is open and η: D(η) (t, x) → η(t, x) ∈ H is a local flow. In what follows we call η the local flow generated by f . Let f , f (x) = Lx+ K(x) be an LS-vector field. Then the local flow η generated by f can be written in the form η(t, x) = etL x + U(t, x) where U: D(η) → H is completely continuous (see Rabinowitz, 1986). DEFINITION 2.6. Let η: D(η) → H be a local flow on H. A bounded and closed subset X ⊂ H is an isolating neighbourhood for η if Inv(X) ⊂ Int(X). We say that S ⊂ H is an isolated invariant set if there exists an isolating neighbourhood X for η such that S = Inv(X, η). In contrast to invariant sets, isolating neighbourhoods are robust in the sense given by the following. THEOREM 2.7 (Ge ba et al., 1999, Theorem 2.8). Let Λ be a compact metric space and let η: R × H × Λ → H be a family of LS-flows. Assume that X ⊂ H is an isolating neighbourhood for a flow ηλ0 for some λ0 ∈ Λ. Then there is an open neighbourhood V ⊂ Λ of λ0 such that X is an isolating neighbourhood for any flow ηλ whenever λ ∈ V.

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2.2. SPECTRA

Because of the stability property given by Theorem 2.7, it is more convenient to associate the homotopy LS-index to isolating neighbourhoods of associating it to isolated invariant sets. In our definition of the homotopy LS-index we will use the notion of spectrum. Let M0 be the category of compact metrizable spaces with a base point. If (X, x0 ), (Y, y0 ) are objects in M0 then the set of morphisms Mor(X, Y) consists of all continuous maps f : X → Y preserving base points. A closed inclusion ι ∈ Mor(A, X) is a cofibration if for any topological pointed space (Z, z0 ) and any continuous map G: X × {0} ∪ A × [0, 1] → Z satisfying G(x0 , t) = z0 for all t ∈ [0, 1], there is an extension of G to the space X × [0, 1]. A pair of spaces (X, A) in M0 is a pair of objects from the category M0 such that A is a closed subset of X and base points of A and X coincide. If in addition the inclusion map ι: (A, x0 ) → (X, x0 ) is a cofibration then (X, A) is called a cofibration pair in M0 . Clearly, if A is a base point in X then (X, A) is a cofibration pair in M0 . A map of pairs f : (X, A) → (Y, B) is any continuous map from X into Y preserving base points and such that f (A) ⊂ B. Let (X, A) be a pair in M0 . Then the quotient space X/A is obtained from X by collapsing A to a point, the base point of X/A; X/A is an object of M0 . If X, Y are objects in M0 (with base points x0 , y0 resp.) then the cartesian product X × Y is also an object in M0 (with base point (x0 , y0 )). Moreover, their wedge X ∨ Y = X × {y0 } ∪ {x0 } × Y is a closed subspace of X × Y and (X × Y, X ∨ Y) is a pair in M0 . Hence, the smash product X ∧ Y defined as a quotient space (X × Y)/(X ∨ Y) is an object in M0 . Additionally to that, if f : X → Y and g: X → Y are morphisms in the category M0 then the induced map f ∧ g: X ∧ X → Y ∧ Y is a morphism in M0 as well. Recall that the smash product is commutative, associative and distributive over the wedge up to the natural homeomorphism. Denote by I the unit interval with base point {0}, ∂I = S 0 the subspace {0, 1} of I, S = S 1 = I/∂I. In fact, the smash product is a functor in M0 and therefore it induces the suspension functor defined by S X := S 1 ∧ X. For any m ∈ N we define S m X = S (S m−1 X) (note that S 0 ∧ X is naturally homeomorphic with X). It follows immediately from the basic properties of the smash product mentioned above that S X ∧ Y and X ∧ S Y are naturally homeomorphic. Finally, let us recall that f ∈ Mor(X, Y) is a homotopy equivalence if there is g ∈ Mor(Y, X) such that g ◦ f is homotopic with idX and f ◦ g is homotopic with idY , both homotopies are relative base points. If f : X → Y is a homotopy equivalence then we say that spaces X and Y are homotopy equivalent or they have the same homotopy type. Standard references for this section is Whitehead (1970).

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Let ν: N ∪ {0} → N ∪ {0} be a fixed “suspension” map. Suppose that (En )∞ n=n(E) is a sequence of objects in M0 . Let (εn : S ν(n) En → En+1 )∞ be a sequence of n=n(E) morphisms. ∞ DEFINITION 2.8. We say that a pair E := (En )∞ n=n(E) , (εn )n=n(E) is a spectrum if there exists n0 ≥ n(E) such that εn : S ν(n) En → En+1 is a homotopy equivalence for all n ≥ n0 . DEFINITION 2.9. A map of spectra f : E → E is a sequence of maps ( fn )∞ n=n0 , fn ∈ Mor(En , En ) n0 ≥ max{n(E), n(E )} such that diagrams S ν(n) En

S ν(n) fn

n

εn

En+1

/ S ν(n) E

fn+1

ε n

/ E . n+1

are homotopy commutative for all n ≥ n0 . The category of spectra with a function ν is denoted E(ν). DEFINITION 2.10. Two maps of spectra f, f : E → E are homotopic if there is n1 ∈ N ∪ {0} such that fn fn whenever n ≥ n1 . DEFINITION 2.11. We say that f : E → E is a homotopy equivalence of spectra E and E if there exists g: E → E such that g ◦ f : E → E is homotopic with the identity map idE and f ◦ g: E → E is homotopic with idE . DEFINITION 2.12. Two spectra E, E are said to be homotopy equivalent or they have the same homotopy type if there is a homotopy equivalence f : E → E . A homotopy type of spectrum is denoted by [E]. Given two spectra E and E their wedge E w = E ∨ E is defined as follows. For any n ≥ n(E w ) = max{n(E), n(E )} we put Enw = En ∨ En . A map εwn : S ν(n) Enw → w is defined as the composition En+1 χn

εn ∨ε n

w = En+1 , S ν(n) Enw = S ν(n) (En ∨ En ) −→ S ν(n) En ∨ S ν(n) En −−−−→ En+1 ∨ En+1

where

χn : S ν(n) (En ∨ En ) → S ν(n) En ∨ S ν(n) En

is the natural homeomorphism. Clearly, εwn is a homotopy equivalence for n suﬃciently large. Thus w ∞ E w = (Enw )∞ n=n(E w ) , (εn )n=n(E w )

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is an object in E(ν). The category E(ν) is not closed with respect to the smash product defined below. Let E be an object of E(µ) and E be an object of E(ν). Define E s = E ∧ E as follows. For each n ≥ n(E s ) = max{n(E), n(E )} we put Ens = En ∧ En and s define a map εns : S (µ+ν)(n) Ens → En+1 as the composition ψn

εn ∧ε n

s = En+1 , S (µ+ν)(n) Ens = S µ(n)+ν(n) (En ∧ En ) −→ S µ(n) En ∧ S ν(n) En −−−−→ En+1 ∧ En+1

where

ψn : S µ(n)+ν(n) (En ∧ En ) → S µ(n) En ∧ S ν(n) En

is the natural homeomorphism. Clearly, εns is a homotopy equivalence if n is s ∞ suﬃciently large and therefore E s = (Ens )∞ n=n(E s ) , (εn )n=n(E s ) is an object in the category E(µ + ν). Let S be a spectrum such that for each n = 0, 1, 2, . . ., En = S 1 , the unit sphere, εn = idS 1 and µ(n) = 0. Then S E := S ∧ E is the suspension functor in the category of spectra. Note that if E is an object in E(ν) then S E is an object of E(ν) as well. For any m ∈ N we define S m E = S m−1 (S E). It is clear that both operations “∨” and “∧” preserve homotopy type of spectra. This leads us to a conclusion that one can define “wedge” and “smash” of homotopy types of spectra putting [E] ∨ [E ] := [E ∨ E ] and [E] ∧ [E ] := [E ∧ E ], respectively. In particular, the suspension functor is defined, we put S [E] := [S E]. ∞ REMARK 2.13. For a given spectrum E = (En )∞ , (ε ) n n=n(E) n=n(E) its homotopy type is uniquely determined by the homotopy type of a pointed space En with n suﬃciently large. In particular, if in the spectrum E the sequence (εn )∞ n=n(E) is replaced by another sequence of homotopy equivalences (ε n )∞ then the n=n(E) resulting spectrum has the same homotopy type as the original one. Therefore, in order to define the homotopy type [E] one only needs a sequence of spaces ν(n) E is homotopy equivalent to E E = (En )∞ n n+1 for n suﬃciently n=n(E) such that S large. Denote by 0 a spectrum such that for each n ≥ 0, the space En consists only of a base point, εn maps the point in En into the point in En+1 . The suspension functor acts trivially on such spaces i.e. S En = En and therefore 0 is an object of E(ν) for arbitrary ν. DEFINITION 2.14. We say that the homotopy type of spectrum E is trivial if E is homotopy equivalent with 0. DEFINITION 2.15. Let A and E be objects of the category E(ν). We say that A is a subspectrum of E if there is n0 ∈ N such that for all n ≥ n0 one has: (a) (En , An ) is a cofibration pair in M0 ;

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(b) the inclusion map ι = (ιn : An → En )∞ n=n0 is a map of spectra . We say that (E, A) is a pair of spectra if A is a subspectrum of E. 2.3. THE LS-INDEX

In this subsection we present the construction of the LS-homotopy index and recall its basic properties. DEFINITION 2.16. that

We say that f is subquadratic if there exist a, b > 0 such |(K(x), x)| ≤ ax2 + b

∀x ∈ H

(3)

The following result is a direct consequence of Troestler and Willem (1996, Theorem 2.1). PROPOSITION 2.17 (Ge ba et al., 1999, Proposition 2.6). Let f , f (x) = Lx+K(x), be an LS-vector field and let η be the local flow generated by f . If f satisfies (3) then η is an LS-flow, i.e, D(η) = R × H. 7n − Let Pn : H → H be the orthogonal projection onto H n := i=0 Hi . Let Hn + (resp. Hn ), n ≥ 1, denote the L-invariant subspace of Hn corresponding to the part of spectrum of L with the negative (resp. positive) real part. Define ν: N ∪ {0} → − . N ∪ {0} by ν(n) := dim Hn+1 We begin with a special case. Thus we assume f : H → H, f (x) = Lx + K(x) is an LS-vector field satisfying condition (3), i.e., f is subquadratic. Let η: R × H → H be the LS-flow generated by f and let X ⊂ H be an isolating neighbourhood for the flow η. Define fn : H n → H n and Fn : H n+1 × [0, 1] → H n+1 by fn (x) := Lx + Pn K(x) and Fn (x, t) := Lx + (1 − t)Pn K(x) + tPn+1 K(x) . Let ηn : R × H n → H n denote the flow induced by fn and ξn : R × H n+1 × [0, 1] → H n+1 denote the family of flows induced by Fn . As a consequence of Theorem 2.3 we obtain that Xn := X ∩ H n is an isolating neighbourhood for the flow ηn and for the family of flows ξn for n suﬃciently large, say n ≥ n0 . Choose n ≥ n0 and set S n := Inv(Xn , ηn ). Thus S n admits an index pair (Yn , Zn ) and the Conley index of S n is the homotopy type of the pointed space Yn /Zn . Let D+n := {x ∈ Hn+ ; x ≤ 1}, D−n := {x ∈ Hn− ; x ≤ 1}, ∂D−n := {x ∈ Hn− ; x = 1}. Let S n+1,n := Inv(Xn+1 × [0, 1], ξn ), S n+1,n (t) := {x ∈ Xn+1 ; (x, t) ∈ S n+1,n }.

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Consider a family of flows θn : R × Hn+1 × [0, 1] → H n+1 generated by hn : H n+1 × [0, 1] → H n+1 , hn (x, s) = Lx + Pn K Pn x + s(x − Pn x) . Clearly (Yn × D+n+1 × D−n+1 , Zn × D+n+1 × D−n+1 ∪ Yn × D+n+1 × ∂D−n+1 ) is an index pair for the isolated invariant set Inv Xn+1 , θn (·, ·, 0) = S n . Thus, the Conley index of S n = S n+1,n (0) with respect to θn (·, ·, 0) equals the homotopy type of (Yn × D+n+1 × D−n+1 )/(Zn × D+n+1 × D−n+1 ∪ Yn × D+n+1 × ∂D−n+1 ) which in turn is equal to the homotopy type of S ν(n) (Yn /Zn ). Moreover, Xn+1 is an isolating neighbourhood for both families θn (·, ·, s), and ξn (·, ·, s), s ∈ [0, 1] and θn (·, ·, 1) = ξn (·, ·, 0). Therefore, by the continuation property of the Conley index (see Conley, 1978; Salamon, 1985), S ν(n) (Yn /Zn ) is homotopy equivalent with Yn+1 /Zn+1 . Thus, in view of Remark 2.13, the sequence ∞ (En )∞ n=n0 := (Yn /Zn )n=n0

determines uniquely the homotopy type [E]. DEFINITION 2.18. Let η be an LS-flow generated by a subquadratic LS-vector field and let X be an isolating neighbourhood for η. Define hLS (X, η) := [E]. We call hLS (X, η) the LS-homotopy index of X with respect to η or simply the LS-index. Turning to the general case assume that f : H → H, f (x) = Lx + K(x) is an LS-vector field and X ⊂ H is an isolating neighbourhood for the local flow η generated by f . Choose s ∈ R such that X is contained in a ball B(0, s) centered at the origin with radius s and define maps µ: R → R,   1 if t ≤ s,    1 + s − t if s
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neighbourhood for η1 . Note also that if η2 is another LS-flow generated by a subquadratic LS-vector field f2 such that f2 (x) = f (x) for all x ∈ X then hLS (X, η1 ) = hLS (X, η2 ). DEFINITION 2.19. Let f be an LS-vector field, η the local flow generated by f , and let X be an isolating neighbourhood for η. Define hLS (X, η) := hLS (X, η1 ) We call hLS (X, η) the LS-homotopy index of X with respect to η or the LS-index. The following propositions give the basic properties of the LS-index. PROPOSITION 2.20 (Nontriviality, Ge ba et al., 1999, Proposition 4.4). Let η: D(η) → H be a local flow generated by an LS-vector field and let X ⊂ H be an isolating neighbourhood for η. If the LS-homotopy index hLS (X, η) 0 then Inv(X, η) ∅. PROPOSITION 2.21 (Continuation, Ge ba et al., 1999, Proposition 4.5). Let Λ be a compact, connected metric space. Assume that η: D(η) → H is a family of local flows generated by a family of LS-vector fields f : H × Λ → H. Let X be an isolating neighbourhood for a flow ηλ for some λ ∈ Λ. Then there is a compact neighbourhood C ⊂ Λ of λ (λ ∈ Int(C)) such that hLS (X, ηµ ) = hLS (X, ην ) for all µ, ν ∈ C.

3. Cohomology of spectra ˇ In what follows, Hˇ denotes the reduced Cech cohomology theory with coeﬃcients in s ome fixed ring Z. This particular cohomology is chosen because it is defined for compact spaces and has the continuity property i.e., Hˇ ∗ (X) = lim Hˇ ∗ (Xn ) ←−

1 if X = Xn . Let A = (An , αn ) be a subspectrum of E = (En , εn ). We define its quotient C = E/A (which is not necessarily a spectrum) as follows. Choose n0 as in Definition 2.15. For each n ≥ n0 we put Cn := En /An . Since ιn : An → En is a cofibration and ι: A → E is a map of spectra there is a map of pairs cn : (S ν(n) En , S ν(n) An ) → (En+1 , An+1 )

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such that cn : S ν(n) En → En+1 is homotopic with εn and the restriction of cn to S ν(n) An considered as a map into An+1 is equal to αn . Thus, the induced homomorphisms c∗n : Hˇ ∗ (En+1 ) → Hˇ ∗ (S ν(n) En )

and

c∗n : Hˇ ∗ (An+1 ) → Hˇ ∗ (S ν(n) An )

are isomorphisms. This implies that c∗n : Hˇ ∗ (En+1 , An+1 ) → Hˇ ∗ (S ν(n) En , S ν(n) An ) is an isomorphism as well. On the other hand the map cn induces a map γn : S ν(n)Cn → Cn+1 . If (X, A) is a pair in M0 then Hˇ ∗ (X, A) ≈ Hˇ ∗ (X/A) and therefore γn∗ : Hˇ ∗ (Cn+1 ) → Hˇ ∗ (S ν(n)Cn ) is an isomorphism. Now, by the quotient E/A of a pair (E, A) we understand the ∞ pair C = (Cn )∞ n=n0 , (γn )n=n0 . It may happen that the maps γn are not homotopy equivalences and therefore C may not be a spectrum. However, as far as the LSindex is concerned we will always be able to arrange things so that without any extra assumptions the quotient of a pair of spectra is a spectrum itself. Define a map ρ: N ∪ {0} → N ∪ {0}, ρ(0) = 0 and ρ(n) =

n−1

ν(i) n ≥ 1.

i=0

∞ Let E = (En )∞ n=n(E) , (εn )n=n(E) be a spectrum. For a fixed q ∈ Z consider a sequence of cohomology groups Hˇ q+ρ(n) (En ),

n ≥ n(E)

and define a sequence of homomorphisms ε

q+ρ(n+1)

(S ∗ )−ν(n)

n hn+1 : Hˇ q+ρ(n+1) (En+1 ) −−−−−−→ Hˇ q+ρ(n+1) (S ν(n) En ) −−−−−−→ Hˇ q+ρ(n) (En )

where S ∗ denotes the suspension isomorphism. DEFINITION 3.1. limit group

The qth cohomology group of a spectrum E is the inverse H q (E) := lim{Hˇ q+ρ(n) (En ), hn }. ←−

Since E is a spectrum there is n0 ∈ N such that εn is a homotopy equivalence whenever n ≥ n0 . Thus hn+1 : Hˇ q+ρ(n+1) (En+1 ) → Hˇ q+ρ(n) (En ) is an isomorphism if n ≥ n0 and the sequence of groups Hˇ q+ρ(n) (En ) stabilizes. Consequently we have the following

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REMARK 3.2. For a fixed spectrum E there is n0 ∈ N ∪ {0} such that H q (E) Hˇ q+ρ(n) (En ) for all n ≥ n0 . Here are some consequences of this observation. 1. the graded group H ∗ (E) is finitely generated if Hˇ ∗ (En0 ) is finitely generated; 2. the spectrum E is of finite type (i.e., H ∗ (E) is finitely generated and almost all groups are zero) if the space En0 is of finite type. REMARK 3.3. Cohomology groups of E can be nontrivial both for positive and negative q ∈ Z. Indeed, put ν(n) = 2 for n ∈ N ∪ {0} and let En := S 2n−1 ∨ S 2n+1 , the wedge product of two spheres, n ≥ 1. Then ρ(n) = 2n and Z for q = −1 or 1 q q+ρ(n) ˇ (En ) = H (E) H 0 else Let f : E → E be a map of spectra and let {Hˇ q+ρ(n) (En ), hn }, {Hˇ q+ρ(n) (En ), h n } be inverse systems of groups constructed for E and E , respectively. By functoriality of the suspension isomorphism the sequence of group homomorphisms q+ρ(n) ˇ q+ρ(n) :H (En ) → Hˇ q+ρ(n) (En ) induced by f satisfies fn q+ρ(n+1)

hn ◦ fn+1

q+ρ(n)

= fn

◦ h n

for all n ≥ max{n(E), n(E )} and thus it defines a group homomorphism f q : H q (E ) → H q (E) on inverse limits. Clearly, two homotopic maps of spectra f and g induce the same homomorphism on cohomology groups. Let (E, A) be a pair of spectra. For each q ∈ Z the quotient ∞ E/A = (Cn )∞ n=n0 , (γn )n=n0 defines an inverse system of cohomology groups q+ρ(n) }. {Hˇ q+ρ(n) (Cn ), (S ∗ )−ν(n−1) ◦ γn

We let

}. H q (E/A) := lim{Hˇ q+ρ(n) (Cn ), (S ∗ )−ν(n−1) ◦ γn ←− q+ρ(n) ˇ q+ρ(n) Since (S ∗ )−ν(n−1) ◦ γn :H (Cn ) → Hˇ q+ρ(n−1) (Cn−1 ) is an isomorphism for n > n0 the sequence of groups Hˇ q+ρ(n) (Cn ) stabilizes and therefore H q (E/A) Hˇ q+ρ(n) (Cn ) for n suﬃciently large. Define the relative cohomology groups as follows. q+ρ(n)

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DEFINITION 3.4. For a pair of spectra (E, A) and q ∈ Z we define the qth cohomology group of (E, A) H q (E, A) := H q (E/A). Let (E, A) be a pair of spectra. For every pair (En , An ) there is a long exact sequence ∗−1

j∗

∗−1

∗

∗

ι δ ι δ · · · −−→ Hˇ ∗−1 (An ) −−−→ Hˇ ∗ (En , An ) −→ Hˇ ∗ (En ) − → Hˇ ∗ (An ) −→ · · ·

with n suﬃciently large. Passing to the inverse limits we obtain the long exact sequence ι∗−1

j∗

δ∗−1

ι∗

δ∗

· · · −−→ H ∗−1 (A) −−−→ H ∗ (E, A) −→ H ∗ (E) − → H ∗ (A) −→ · · ·

(4)

which is functorial. Denote by rq (E) the rank (dimension if the coeﬃcient ring is the field) of the qth cohomology group H q (E) and by dq (E, A) the rank of the image of δq : H q (A) → H q+1 (E, A). Assuming all groups in the sequence (4) are of finite rank we define the following generalized formal power series: P(t, E) = rq (E) · tq , q∈Z

Q(t, E, A) =

dq (E, A) · tq .

q∈Z

If and are 0 for all q less than some fixed q0 ∈ Z then P(t, E) and Q(t, E, A) are called the generalized Poincar´e series. rq (E)

dq (E, A)

THEOREM 3.5 (Izydorek, 2001, Lemma 3.5). Let (E, A) be a pair of spectra such that P(t, E), P(t, A), P(t, E/A) and Q(t, E, A) are generalized Poincar´e series. Then P(t, E/A) + P(t, A) = P(t, E) + (1 + t) · Q(t, E, A). REMARK 3.6. In fact, the formal power series P and Q are well-defined for the homotopy types of spectra. This observation will be used in the next section. 4. Attractors, repellers and Morse decompositions In classical Conley index theory it is well known that a Morse decomposition of an isolated invariant set is “robust” with regards to perturbations (cf. Mischaikow,

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1994). This is because of the local compactness of the phase space. As we show below the same result holds true for LS-flows. Recall that for a flow η: R × H → H and every x ∈ H one can define its ω and α limits: ω(x) := cl η([t, ∞), x) , α(x) := cl η((−∞, t], x) . t

t

Let η: R × H × Λ → H be a family of LS-flows and assume that X ⊂ H is an isolating neighbourhood for the flow ηλ , where λ ∈ Λ is fixed. Suppose that (A, A∗ ) is an attractor-repeller pair in the invariant set S = Inv(X, ηλ ). Both sets A and A∗ are isolated invariant sets themselves and therefore there are isolating neighbourhoods XA , XA∗ ⊂ H for ηλ such that A = Inv(XA , ηλ ) and A∗ = Inv(XA∗ , ηλ ), respectively. Obviously, we may suppose that XA ∪ XA∗ ⊂ X and XA ∩ XA∗ = ∅. THEOREM 4.1 (Izydorek, 2001, Theorem 4.1). There is an open neighbourhood V of λ in Λ such that for each µ ∈ V one has: 1. X, XA , XA∗ are isolating neighbourhoods for ηµ , 2. if x ∈ Inv(X, ηµ ) \ Inv(XA , ηµ ) ∪ Inv(XA∗ , ηµ ) then ω(x) ⊂ Inv(XA , ηµ ) and α(x) ⊂ Inv(XA∗ , ηµ ). Let η, X ⊂ H be as above and let S = Inv(X, ηλ ). Then the finite collection {M(π); π ∈ D} of compact invariant sets in S is said to be a Morse decomposition of S if there exists an ordering π1 , . . . , πn of D such that for every x ∈ S \ π∈D M(π) there exist indices i, j ∈ {1, 2, . . . , n} such that i < j and ω(x) ⊂ M(πi ), α(x) ⊂ M(π j ). Every ordering of D with this property is said to be admissible. The sets M(π) are called Morse sets. THEOREM 4.2 (Izydorek, 2001, Theorem 4.2). Let {M(π); π ∈ D} be a Morse decomposition of S = Inv(X, ηλ ) with D having n elements. There are closed ∗ subsets X1 , . . . , Xn = X = X0∗ , . . . , Xn−1 of X and an open neighbourhood V of λ ∈ Λ such that for each µ ∈ V the following conditions are satisfied: 3. Xi , X ∗j i ∈ {1, . . . , n}, j ∈ {0, . . . , n − 1} are isolating neighbourhoods for a flow ηµ ; ∗ , η ) = M(π ), i = 1, . . . , n, 4. Inv(Xi ∩ Xi−1 λ i ∗ , η ); i = 1, . . . , n} is a Morse decomposition of Inv(X, η ). 5. {Inv(Xi ∩ Xi−1 µ µ

Let η: R × H → H be an LS-flow generated by an LS-vector field f : H → H, f (x) = Lx + K(x). Assume that X ⊂ H is an isolating neighbourhood for η. Let (A, A∗ ) be an attractor-repeller pair in the invariant set S = Inv(X, η). Denote by XA and XA∗ isolating neighbourhoods for η such that A = Inv(XA , η) and A∗ = Inv(XA∗ , η).

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THEOREM 4.3 (Izydorek, 2001, Theorem 4.3). Under the above assumptions there exist spectra ES , E A and E A∗ representing LS-homotopy indices of X, XA , and XA∗ , respectively, such that the sequence ι∗−1

δ∗−1

j∗

ι∗

δ∗

→ H ∗ (E A ) −→ · · · · · · −−→ H ∗−1 (E A ) −−−→ H ∗ (E A∗ ) −→ H ∗ (ES ) −

(5)

is exact. Since ([ES ], [E A ]) is a pair of homotopy type of spectra, [E A∗ ] = [ES /E A ] is the homotopy type of a quotient spectrum and all those spectra are of finite type we have the following PROPOSITION 4.4 (Izydorek, 2001, Proposition 4.4). Under the assumptions of Theorem 4.3 one has P(t, [E A∗ ]) + P(t, [E A ]) = P(t, [ES ]) + (1 + t) · Q(t) where all coeﬃcients of the generalized power series Q(t) are nonnegative integers. The following result is a direct consequence of Ge ba et al. (1999, Proposition 2.3) and the classical version of the above proposition, see, e.g., Salamon (1985, Theorem 5.8). PROPOSITION 4.5 (Izydorek, 2001, Proposition 4.5). Let (A, A∗ ) be an attractor-repeller pair for the isolated invariant set S and suppose that S = A ∪ A∗ . Then [ES ] = [E A ] ∨ [E A∗ ] = [E A ∨ E A∗ ]. An important consequence of the above is PROPOSITION 4.6 (Izydorek, 2001, Corollary 4.6). Assume that for some q ∈ Z the boundary homomorphism δq from the long exact sequence (5) is nonzero. Then there is a connecting orbit in S joining A and A∗ . In the sequel we use notation as in Theorem 4.2. Let {M(π); D} be a Morse decomposition of S = Inv(X, η) with D having n elements. Denote by [E M(πi ) ] the ∗ and by [E ] the LS-index of X. LS-homotopy index of Xi ∩ Xi−1 S THEOREM 4.7 (Morse inequalities). Under the above assumptions one has n

P(t, [E M(πi ) ]) = P(t, [ES ]) + (1 + t) · Q(t)

(6)

i=1

where the coeﬃcients of the generalized power series Q(t) are nonnegative integers.

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M. IZYDOREK

5. Equivariant LS-flows and the G-LS-index Throughout this section we will assume that G is a compact Lie group. 5.1. SYMMETRIES

An action of G on a topological space X is a continuous map ρ: G × X → X such that (a) ρ g, ρ(h, x) = ρ(gh, x) for g, h ∈ G, x ∈ X; (b) ρ(e, x) = x for x ∈ X and the unit e ∈ G. A G-space is a pair (X, ρ) consisting of a space X with an action ρ of G on X. Usually the G-space (X, ρ) is denoted by its underlying topological space X and ρ(g, x) is denoted by gx. A subset A of a G-space X is an invariant subset (or G-subset) of X, if for all x ∈ A and g ∈ G one has gx ∈ A. Given G-spaces X, Y, a map f : X → Y is called a G-equivariant map (shortly G-map), if it is continuous and f (gx) = g f (x) for all g ∈ G and x ∈ X. A map h: X × [0, 1] → Y is called a G-equivariant homotopy (shortly G-homotopy), if it is continuous and h(gx, t) = gh(x, t) for all g ∈ G, x ∈ X and t ∈ [0, 1]. Clearly, two G-maps f0 , f1 : X → Y are G-homotopic, if there is a G-homotopy h: X × [0, 1] → Y such that h(·, 0) = f0 and h(·, 1) = f1 . For each x ∈ X the set Gx := {gx ∈ X; g ∈ G} is called the orbit through x. The set G x := {g ∈ G; gx = x} is a closed subgroup of G and called the isotropy group of x. The normalizer of a closed subgroup H ⊂ G is denoted by NH. Set X H := {x ∈ X; H ⊂ G x }. The space X H is closed in X and invariant with respect to the action of NH, i.e. gx ∈ X H for every g ∈ NH and x ∈ X H . It is then an NH-space. A G-map f : X → Y defines an NH-map f H : X H → Y H , f H (x) := f (x), x ∈ X H . A pointed G-space is a pair (X, x0 ) consisting of a G-space X and a point x0 ∈ X (called the base point) such that gx0 = x0 for all g ∈ G. Let V be a linear normed space over real numbers. Let ρ: G × V → V be an action of G on V. We say that V is a real representation of G (or G-representation), if ρ(g, ·): V → V is an automorphism for every g ∈ G. Throughout the paper, all representations are assumed to be real and orthogonal, i.e. they are equipped with an inner product ·, ·: V × V → V and gx, gy = x, y for every x, y ∈ V and g ∈ G. Since the material is rather standard for all details we refer the reader to Bredon (1972) and tom Dieck (1987). 5.2. ISOLATING NEIGHBOURHOODS AND THE EQUIVARIANT LS-INDEX

In order to fix the notation we give equivariant versions of the definitions and results from the previous sections. Let H = (H, ·, ·) be a real orthogonal Hilbert representation of a compact Lie group G. Let L: H → H be a linear bounded

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297

and G-equivariant operator with spectrum σ(L) such that (H.1)–(H.3) hold and all spaces Hn are G-representations. A flow η: R × H → H is a G-flow if η(t, gx) = gη(t, x), for every t ∈ R, x ∈ H and g ∈ G. Given a G-flow η and a G-subset X of H we define Inv(X, η) := {x ∈ X; η(t, x) ∈ X for all t ∈ R} which is the maximal η-invariant subset of X. Clearly, it is also a G-subset of X. A continuous map η: R × H × Λ → H is a family of G-flows indexed by Λ if ηλ (where ηλ (t, x) := η(t, x, λ)) is a G-flow on H for all λ ∈ Λ. If η: R × H × Λ → H is a family of G-flows and X is a G-subset of H then we let Inv(X × Λ, η) := {(x, λ) ∈ X × Λ; η(t, x, λ) ∈ X for all t ∈ R}. A family of G-flows η: R × H × Λ → H is said to be a family of G-LS-flows if η(t, x, λ) = etL x + U(t, x, λ) where U: +R × H × Λ → H is completely continuous and G-equivariant with respect to the variable x. Define an action of G on X × Λ by g(x, λ) := (gx, λ), g ∈ G, x ∈ X, λ ∈ Λ. A direct consequence of the nonequivariant result (see in Ge ba et al., 1999, Proposition 2.3) is the following PROPOSITION 5.1 (Izydorek, 2002, Proposition 2.2). Let Λ be a compact metric space and let η: R × H × Λ → H be a family of G-LS-flows. If X is a closed and bounded G-subset of H, then S := Inv(X × Λ, η) is a compact G-subset of X × Λ. We say that f : H × Λ → H is a family of G-LS-vector fields if there exists a completely continuous and locally Lipschitz continuous G-map K: H × Λ → H such that f (x, λ) = Lx + K(x, λ) for all (x, λ) ∈ H × Λ. In the case when Λ is a one pointed space we will call f a G-LS-vector field. It follows directly from the nonequivariant case that every G-LS-vector field f : H → H generates a local G-LS-flow on H. A bounded and closed G-subset X of H is an isolating G-neighbourhood for a G-flow η if and only if Inv(X) ⊂ int(X), the maximal η-invariant G-subset of X is included in the interior of X The following theorem is a direct consequence of Proposition 5.1. THEOREM 5.2 (Izydorek, 2002, Theorem 2.5). Let Λ be a compact metric space and let η: R × H × Λ → H be a family of G-LS-flows. Assume that X ⊂ H is an isolating G-neighbourhood for a flow ηλ0 for some λ0 ∈ Λ. Then there is an open neighbourhood of λ0 , V ⊂ Λ such that X is an isolating G-neighbourhood for any flow ηλ whenever λ ∈ V.

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M. IZYDOREK

0 of compact metrizable G-spaces with a base point. Consider the category MG 0 , then the set of morphisms Mor (X, Y) consists If (X, x0 ), (Y, y0 ) are objects in MG G of all continuous G-maps f : X → Y preserving base points. A G-cofibration and cofibration pair of G-spaces are defined in the obvious way. For a finite-dimensional real representation V of G we let S V := V ∪ {∞} to be a one-point-compactification with {∞} as base point. The smash product 0 defined by S V X := S V ∧ X. It is well induces the suspension functor on MG known that for any two representations V, W of G one has a canonical pointed G-homeomorphism S V⊕W S V ∧ S W (7)

The book tom Dieck (1987) is the standard reference. Let ξ = (Vn )∞ 0 be a fixed sequence of real G-representations with dim Vn < ∞. 0 Vn ∞ Suppose that (En )∞ n(E) is a sequence of objects in MG . Let (εn : S E n → E n+1 )n(E) be a sequence of morphisms. ∞ DEFINITION 5.3. We say that a pair E(ξ) := (En )∞ , (ε ) n n(E) n(E) is a G-spectrum of type ξ, if there exists n0 ≥ n(E) such that εn : S Vn En → En+1 is a G-homotopy equivalence for all n ≥ n0 . DEFINITION 5.4. A G-map of spectra f : E(ξ) → E (ξ) is a sequence of maps ( f n )∞ n0 , fn ∈ MorG (E n , E n ), n0 ≥ max{n(E), n(E )} such that diagrams S Vn En

S Vn fn

n

ε n

εn

En+1

/ S Vn E

fn+1

/ E n+1

are G-homotopy commutative for all n ≥ n0 . Clearly, two G-maps of spectra f, f : E(ξ) → E (ξ) are G-homotopic if there is n1 ≥ 0 such that fn is G-homotopic with fn whenever n ≥ n1 . A map f : E(ξ) → E (ξ) is a G-homotopy equivalence of G-spectra E(ξ) and E (ξ) if there exists g: E (ξ) → E(ξ) such that g ◦ f : E(ξ) → E(ξ) is G-homotopic with the identity map idE(ξ) and f ◦ g: E (ξ) → E (ξ) is G-homotopic with idE (ξ) . Spectra E(ξ), E (ξ) are said to be G-homotopy equivalent, or they have the same G-homotopy type if there is a G-homotopy equivalence f : E(ξ) → E (ξ). The G-homotopy type of a G-spectrum E(ξ) will be denoted by [E(ξ)]. DEFINITION 5.5. Let A and E be G-spectra. We say that A is a G-subspectrum of E, if there is n0 ∈ N such that for all n ≥ n0 one has: 0; (a) (En , An ) is a cofibration pair in MG

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(b) the inclusion map ι = (ιn : An → En )∞ n0 is a G-map of spectra. We say that (E, A) is a pair of G-spectra if A is a G-subspectrum of E. The construction of the G-equivariant LS-homotopy Conley index is basically the same as in the case without a group action. However, instead of the classical Conley’s theorem we used Floer’s result saying that in a locally compact G-space an isolated η− and G-invariant set admits a G-index pair (see Floer, 1987). Thus, the G-LS-index, hG (X, η), is by definition the G-homotopy type of a certain GLS spectrum obtained with help of Galerkin approximations of an initial G-LS-flow. Clearly, the G-LS-index has the following basic properties PROPOSITION 5.6 (Nontriviality, Izydorek, 2002, Proposition 2.15). Let η be a local flow on H generated by a G-LS-vector field and let X ⊂ H be an isolating G-neighbourhood for η. If the G-LS-index hG (X, η) 0 then Inv(X, η) ∅. LS PROPOSITION 5.7 (Continuation, Izydorek, 2002, Proposition 2.16). Let Λ be a compact, connected metric space. Assume that η is a family of local flows on H generated by a family of G-LS-vector fields f : H × Λ → H. Let X be an isolating G-neighbourhood for a flow ηλ for every λ ∈ Λ. Then for arbitrary µ, ν ∈ Λ one has G hG LS (X, ηµ ) = hLS (X, ην ). The cohomological version of the G-LS-index diﬀers much from the nonsymmetric case. The main problem is to choose an appropriate finite-dimensional cohomology theory which allows to define cohomology of G-spectra. For this purpose we used the Borel cohomology. Let p: EG → BG denote the universal principal G-bundle. To every G-space X one associates fibre bundle with fibre X, pX : XG = EG ×G X → BG. We apply the Alexander – Spanier cohomology to the bundle pX and for a pointed G-space (X, x0 ) we set: HG∗ (X) := H ∗ (XG , {x0 }G ) = H ∗ (EG ×G X, EG ×G {x0 }). If A is a G-subspace of X, then AG is a subspace of XG . We set HG∗ (X, A) := H ∗ (XG , AG ). The fibre map pX defines H ∗ (BG)-module structure in HG∗ (X, A); for every a ∈ H ∗ (BG) and u ∈ HG∗ (X, A) we let a ∗ u := p∗X (a) ∪ u, where · ∪ ·: H ∗ (EG ×G X) ⊗ HG∗ (X, A) → HG∗ (X, A) is the cup-product (see Spanier, 1966). In particular, HG∗ (X) = HG∗ (X, {x0 }) is a H ∗ (BG)-module. A G-map

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M. IZYDOREK

f : (X, A) → (Y, B) defines fG := id ×G f : (XG , AG ) → (YG , BG ) whose induced homomorphism fG∗ : HG∗ (Y, B) → HG∗ (X, A) preserves H ∗ (BG)-module structure. The functor HG∗ (·) is called the G-equivariant Borel cohomology (see tom Dieck, 1987). Let V be an orthogonal representation of G of dimension n, with unit ball (resp. unit sphere) DV (resp. S V). Assume the action on V is orientation preserving, i.e., for each g ∈ G the linear map corresponding to g is an element of GL+ (V). Then, the vector bundle ζ: EG ×G (V × X) → EG ×G X is orientable over integers. Using Thom’s isomorphism for ζ T ∗ : H ∗ (EG ×G X) → H ∗+n EG ×G (DV × X), EG ×G (S V × X) one defines the suspension isomorphism of H ∗ (BG)-modules S∗ : HG∗ (X) → HG∗+n (S V X).

(8)

REMARK 5.8. If V is a unitary representation of G and r(V) is the underlying real G-representation, then the action of G on r(V) is orientation preserving. On the other hand, each vector bundle is orientable over Z2 . Now, similarly to the nonequivariant case, we are able to define appropriate inverse systems of groups and consequently obtain the Borel cohomology of a Gq q spectrum HG (E), or of a pair of G-spectra HG (E, A). For more details see Izydorek (2002). Here is an important analogue of the theorem about the existence of a long exact sequence for a pair of spectra. THEOREM 5.9 (Izydorek, 2002, Proposition 3.4). Let (E, A) be a pair of G-spectra. There is a long exact sequence ι∗−1

j∗

δ∗−1

ι∗

δ∗

· · · −−→ HG∗−1 (A) −−−→ HG∗ (E, A) −→ HG∗ (E) − → HG∗ (A) −→ · · ·

(9)

which is functorial. Furthermore, there is an exact triangle of H ∗ (BG)-modules HG∗ (E, A)

9 ss ss s ss ss ∗ H (A) o

KK ∗ KK j KK KK K%

δ∗

G

ι∗

(10)

HG∗ (E)

which is functorial. Let η: R × H → H be a flow generated by a G-LS-vector field f : H → H, f (x) = Lx + K(x). Assume that X ⊂ H is an isolating G-neighbourhood for η. Let (A, A∗ ) be a G-attractor-repeller pair in the set S = Inv(X, η). Denote by XA and XA∗ isolating G-neighbourhoods for η such that A = Inv(XA , η) and A∗ = Inv(XA∗ , η).

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THEOREM 5.10 (Izydorek, 2002, Theorem 4.4). Let the above assumptions be satisfied and let ES , E A and E A∗ be G-spectra representing G-LS-indices of X, XA , and XA∗ , respectively. Then HG∗ (E A∗ )

KKK ∗ KKKj KKK K%

s9

δ∗ ssss

ss sss H ∗ (E A ) o G

ι∗

(11)

HG∗ (ES )

is an exact triangle of H ∗ (BG)-modules. As we have already mentioned, the equivariant cohomology admits a nontrivial H ∗ (BG)-module structure. It turns out that there are strong relations between this structure and the Fadell–Rabinowitz index of a G-space. This observation is used in estimations of the number of periodic solutions of certain autonomous Hamiltonian systems. Let X be a paracompact G-space. If P is a one-pointed G-space, then the constant G-map h: X → P defines a map hG : EG ×G X → EG ×G P ≈ BG. ∗ : H ∗ (BG, Z) → H ∗ (EG × X, Z) be the induced homomorphism. Here Z Let hG G stands for a commutative ring with unity. The following definition has been introduced by Fadell and Rabinowitz in Fadell and Rabinowitz (1978): ∗ in H ∗ (BG) is called the G-index DEFINITION 5.11. The ideal IG (X) := ker hG of X.

The following property of G-index is a straightforward consequence of its definition: PROPERTY (Monotonicity). If X and Y are paracompact G-spaces and f : X → Y is a G-map then IG (Y) ⊂ IG (X). We say that M is a trivial H ∗ (BG)-module if a ∗ u = 0 for every a ∈ H p (BG) with p > 0 and u ∈ M. Using simple but crucial observation which says that 7 ∗ ∗ p H (EG ×G X) is a trivial H (BG)-module if and only if p>0 H (BG) ⊂ IG (X), one proves the following. PROPOSITION 5.12 (Izydorek, 2002, Proposition 3.11). Assume that a G-orbit O is an isolated invariant set for a local flow η generated by a G-LS-vector field f : H → H and that H ∗ (EG×G O) is a trivial H ∗ (BG)-module. Let X be an isolating G-neighbourhood for η, with Inv(X, η) = O, and let E X be a corresponding Gspectrum. Then HG∗ (E X ) is a trivial H ∗ (BG)-module.

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Let G = S 1 := {z ∈ C; |z| = 1}. Choose real numbers R as a coeﬃcients ring in the S 1 -equivariant Borel cohomology theory HS∗ 1 (·, R). It is well known that H ∗ (BS 1 , R) ≈ R[ω] the polynomial ring with one generator ω ∈ H 2 (BS 1 , R). Here is a direct consequence of (5.12). THEOREM 5.13 (Izydorek, 2002, Corollary 3.12). Let for some x ∈ H the orbit through x, S 1 x, be an isolated invariant set for a local S 1 -LS-flow η. Assume that the isotropy group G x S 1 . If X is an isolating S 1 -neighbourhood for S 1 x and E X is a corresponding S 1 -spectrum then HS∗ 1 (E X , R) is a trivial R[ω]-module. Let f : H → H be a G-LS-vector field. Let X be an isolating G-neighbourhood for the local flow η generated by f , and let E X stands for a corresponding spectrum. DEFINITION 5.14. We call u ∈ HG∗ (E X ) a nontorsion element if the map H ∗ (BG) → HG∗ (E X ): a → a ∗ u is a monomorphism. DEFINITION 5.15. We say that S = Inv(X, η) is homologically pt-hyperbolic p of index p ∈ Z if there exists u ∈ HG (E X ), so that the map H ∗ (BG) → HG∗ (E X ): a → a ∗ u is an isomorphism. Let η be a local flow generated by an S 1 -LS-vector field F: H → H. Let M = {Mr ; r = 1, . . . , k} be an admissible S 1 -Morse decomposition of some bounded isolated S 1 - and η-invariant set T . Put T r = {x ∈ T ; ω(x) and α(x) ⊂ M0 ∪ M1 ∪ · · · ∪ Mr }

r = 0, 1, . . . , k

and notice, that (T r−1 , Mr ) is an S 1 -attractor-repeller pair in T r . Let Xr , Yr stand for isolating G-neighbourhoods for η with Inv(Xr , η) = Mr and Inv(Yr , η) = T r . Denote by E r and F r corresponding spectra for Xr and Yr , respectively. Let Ml be a distinguished set in M for some 0 ≤ l ≤ k. THEOREM 5.16 (Izydorek, 2002, Theorem 5.1). Suppose that HG∗ (E l ) and HG∗ (F k ) contain nontorsion elements of minimal even dimensions 2p and 2q, respectively. Let HG∗ (E r ) r = 0, . . . , k, r l be trivial R[ω]-modules. Then, k ≥ |p − q|.

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303

The above result is an infinite-dimensional version of theorem proved by Floer and Zehnder in Floer and Zehnder (1988). 6. Applications 6.1. A GENERAL SETTING

The LS-index theory turned out to be a powerful tool in studying certain strongly indefinite variational problems. In this section we present several results and discuss examples for which our theory works eﬃciently. We will be concerned with the existence and multiplicity results for periodic solutions of autonomous and nonautonomous Hamiltonian systems. Given a Hamiltonian function Q ∈ C 1 (R2m × R, R) which is 2π-periodic with respect to t and satisfies certain growth conditions (for instance, the following generalized Lipschitz condition: there are constants C, s > 0 such that for all z, w ∈ R2m ∇Q(z)−∇Q(w) ≤ Cz−w(1+z s +w s )), consider the Hamiltonian system of diﬀerential equations z˙ = J∇Q(z, t) where J =

8

9

0 −I I 0

(12)

is the standard symplectic matrix and ∇ denotes the gradient

with respect to z ∈ R2m . Let us denote by H = H 1/2 (S 1 , R2m ) the Hilbert space of 2π-periodic, R2m -valued functions z(t) = a0 +

∞ an cos(nt) + bn sin(nt) ,

where a0 , an , bn ∈ R2m

n=1

with the inner product given by

z, z H =

2πa0 , a 0

+π

∞

n(an , a n + bn , b n ).

(13)

n=1

where a, b denotes the standard inner product in R2m . If Q is independent of t then (12) becomes an autonomous system. Such a system admits natural symmetries of the group S 1 . Identifying S 1 with the quotient group R/2πZ we define an action of S 1 on H by the time shift, i.e.,: (gz)(t) := z(t + g)

z ∈ H, t, g ∈ S 1 .

Clearly, that action is linear and invariant with respect to the inner product (13) and therefore H is an orthogonal representation of S 1 .

304

M. IZYDOREK

The associated functional Φ: H → R defined by the formula 1 Φ(z) = − Lz, zH − φ(z), 2

(14)

where Lz, zH =

2π

J˙z(t), z(t) dt

and

2π

φ(z) =

0

Q z(t) dt.

(15)

0

is continuously diﬀerentiable. Each map z(t) is a 2π-periodic solution of (12) if and only if it is a critical point of Φ. Moreover, −∇Φ(z) = Lz + ∇φ(z) is an LSvector field in H. If the action of the group S 1 is involved one has ∇Φ(gz) = g∇Φ(z) for all g ∈ S 1 and z ∈ H. Choose e1 , . . . , e2m the standard basis in R2m and denote H0 = span{e1 , . . . , e2m }

Hn+ = span (cos(nt))e j + sin(nt) Je j : j = 1, . . . , 2m , Hn− = span cos(nt) e j − sin(nt) Je j : j = 1, . . . , 2m},

n ∈ N, n ∈ N.

(16)

In the equivariant case all spaces are representations 7 of S 1 . ∞ + − Put Hn = Hn ⊕ Hn , n = 1, 2, . . .. Obviously, H = n=0 Hn , spaces Hn are mutually orthogonal and H0 = ker L. Conditions (H.1) – (H.3) are satisfied. In the nonsymmetric case the suspension map ν: N ∪ {0} → N ∪ {0} is defined by − )∞ if S 1 -symmetries are considered. ν(n) = 2m. We put ξ = (Hn+1 n=0 Finally, if A is a symmetric 2m × 2m-matrix and Q(z) = 12 Az, z, then (12) becomes a linear Hamiltonian system z˙ = JAz. The vector field ∇Φ: H → H corresponding to that system preserves all spaces Hn and the restriction of ∇Φ to Hn , n ≥ 1, may be identified with the linear map on R4m whose matrix is : 1 ; − n A −J T n (A) = (17) J − 1n A and with −A on R2m if n = 0. The following numbers has been defined by Amann and Zehnder (1980b) i− (A) := M − (−A) +

∞

M − T n (A) − 2m

n=1

i0 (A) := M 0 (−A) +

∞ n=1

M 0 T n (A) ,

(18)

THE LS-INDEX: A SURVEY

305

where M − (U) is the number (with multiplicity) of negative eigenvalues of a symmetric matrix U and M 0 (U) is the dimension of its kernel. 6.2. APPLICATIONS OF THE LS-INDEX

The examples discussed in this section show that the LS-index theory can sometimes give better results than other Morse type theories recently developed. If i0 (A) = 0 then ∇Φ is a linear isomorphism. In such a case S := {0} is an isolated invariant set for the flow η induced by −∇Φ. Moreover, if r > 0 then D(r) := {z ∈ H; z ≤ r} is an isolating neighbourhood and hLS (D(r), η) = [E] where E is a spectrum such that En = S p(n) with p(n) = i− (A) + n · 2m for suﬃciently large n. Thus, using elementary properties of hLS we obtain the following two observations: Remark. [Ge ba et al., 1999, Remark 5.1]Assume that Q(z, t) = 12 A0 z, z + q(z, t), where A0 is linear symmetric and ∇q(z, t) = o(|z|) uniformly in t as z → 0. If i0 (A0 ) = 0 then for r suﬃciently small and positive D(r) := {z ∈ H; z ≤ r} is an isolating neighbourhood of S = {0} and hLS (D(r), η) = [E] where En = S p(n) for n suﬃciently large. Remark. [Ge ba et al., 1999, Remark 5.2]Assume that Q(z, t) = 12 A∞ z, z + q(z, t), where A∞ is linear symmetric and ∇q(z, t) is bounded. If i0 (A∞ ) = 0 then for r suﬃciently large D(r) is an isolating neighbourhood and hLS (D(r), η) = [E] where En = S p(n) for n suﬃciently large. We begin our discussion with the following two propositions. PROPOSITION 6.1 (Ge ba et al., 1999, Lemma 5.3). Let B be a Banach space and F: B × Rn (x, y) → F(x, y) ∈ B be a C p -map. Suppose that F(0, 0) = 0, DB F(0, 0): B → B is an isomorphism and partial derivatives DiRn F(0, 0): Rn × · · · × Rn → B are zero for i = 1, 2, . . . , p. Let φ: Rn ⊃ V y → φ(y) ∈ B be the implicit function defined by F(φ(y), y) = 0, V is an open neighbourhood of 0 ∈ Rn . Then the derivatives Di φ(0): Rn × · · · × Rn → B are zero maps for i = 1, 2, . . . , p.

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PROPOSITION 6.2 (Ge ba et al., 1999, Lemma 5.4). Let F: B × Rn → B × Rn be a C p+1 -map, F(x, y) = F1 (x, y), F2 (x, y) . Assume that F1 : B × Rn → B satisfies assumptions of Proposition 6.1 and let φ: Rn ⊃ V y → φ(y) ∈ B be the implicit function defined by F1 (φ(y), y) = 0, y ∈ V. Additionally, suppose that DB F2 (0, 0): B → Rn is zero. Define ψ: Rn ⊃ V y → ψ(y) ∈ Rn , ψ(y) = F2 (φ(y), y). Then, Di ψ(0) = DiRn F2 (0, 0): Rn × · · · × Rn → Rn for i = 1, 2, . . . , p + 1. As an immediate consequence of the above results we obtain FACT 6.3 (Ge ba et al., 1999, Corollary 5.5). Assume that the first nonvanishing derivative DiRn F2 (0, 0): Rn × · · · × Rn → Rn is nondegenerate, i.e., DiRn F2 (0, 0)(y, . . . , y) = 0

if and only if

y = 0.

Then the origin (0, 0) ∈ B × Rn is isolated in the set of zeroes F −1 (0) of F: B × Rn → B × Rn . Now, we turn to our examples. EXAMPLE 6.4. Let m = 1 and define Q: R2 × R → R by: Q(x, y, t) = 12 (x2 + y2 ) + (x3 − 3xy2 ) cos(3t), i.e., Q(z, t) = 12 A0 z, z + q(z, t), where z = (x, y) ∈ R2 , A0 = Id and q(z, t) = (x3 − 3xy2 ) cos(3t). Let A0 denote the derivative of −∇Φ at 0. Evidently it is a selfadjoint operator and ker A0 = H(1) ⊂ H1 is a subspace of dimension 2 spanned by u1 , u2 , where u1 (t) = cos(t)e1 + sin(t)e2

and

u2 (t) = − sin(t)e1 + cos(t)e2 .

Subspaces V1 = im A0 and V2 = ker A0 are orthogonal in H and H = V1 ⊕ V2 . Let π1 , π2 : H → H denote the orthogonal projections onto V1 , V2 , respectively. Define F1 : V1 ⊕ V2 → V1 , F1 (v1 , v2 ) = −π1 ◦ ∇Φ(v1 , v2 ), and F2 : V1 ⊕ V2 → V2 , F2 (v1 , v2 ) = −π2 ◦ ∇Φ(v1 , v2 ). Clearly, DV1 F1 (0, 0) = π1 ◦ A0 |V1 : V1 → V1 is an isomorphism and DV2 F1 (0, 0) = π1 ◦ A0 |V2 : V2 → V1 is zero. By Proposition 6.1 the implicit function φ: U → V1 defined by F1 (φ(v2 ), v2 ) = 0 satisfies Dφ(0) = 0, where U is an open neighbourhood of the origin in V2 . Consequently, by Proposition 6.2 Di ψ(0) = DiV2 F2 (0, 0) for i = 1, 2, where ψ(v2 ) = F2 (φ(v2 ), v2 ). Choosing the basis {u1 (t), u2 (t)} we introduce a coordinate system in V2 . For each v2 ∈ V2 one has v2 (t) = au1 (t) + bu2 (t) = a cos(t) − b sin(t) e1 + a sin(t) + b cos(t) e2 ,

a, b ∈ R.

THE LS-INDEX: A SURVEY

307

We replace x by a cos(t) − b sin(t) and y by a sin(t) + b cos(t) in ∇Q. The resulting map u(t) is an element of H 1/2 (S 1 , R2 ) and π2 u(t) = u(t), u1 (t)H u1 (t) + (t), u2 (t)H u2 (t). Using the above procedure one shows that up to a positive multiplier F2 0, (a, b) = (a2 − b2 , −2ab) in some neighbourhood of 0 ∈ V2 . Thus DV2 F2 (0, 0) = 0 and D2V2 F2 (0, 0) is nondegenerate at 0. Therefore, 0 ∈ H is isolated in ∇Φ−1 (0) by Fact 6.3. Since we deal with a gradient vector field 0 ∈ H is an isolated invariant set for the semiflow generated by F = −∇Φ. Now, if the local flow on V2 is generated by F2 (0, ·): U → V2 then we easily compute the Conley index of {0} which is equal to [S 1 ∨ S 1 , ∗], the homotopy type of the join of 2 copies of 1-dimensional pointed spheres. Using (17) we then find that the LS-index of {0} in H is equal to the homotopy type of spectrum E for which one has En = S 2n+3 ∨ S 2n+3 , the wedge of two pointed spheres of dimension 2n + 3, n = 1, 2, . . .. EXAMPLE 6.5. Let m = 1 and define Q: R2 × R → R by: Q(x, y, t) = 12 (x2 + y2 ) + (x2 + y2 )2 + h(x, y, t). The same arguments as in Example 6.4 show us that 0 ∈ H is an isolated invariant set for the semiflow η generated by F = −∇Φ, where Φ: H → R is the functional corresponding to the system (12) with Q as above. This time we find that up to a positive multiplier F2 0, (a, b) = (a3 + ab2 , a2 b + b3 ) in a neighbourhood of 0 ∈ V2 . Now, the LS-index of {0} is the homotopy type of spectrum E for which one has En = S 2n+2 , a pointed sphere of dimension 2n + 2, n = 1, 2, . . .. Throughout the rest of this subsection we will be concerned with Hamiltonian Q ∈ C 2 (R2 × R, R), 2π-periodic in t and satisfying the following two conditions: (Q.0) there exists c0 > 0 such that |z| < c0 implies Q(z, t) = 12 A0 z, z + q0 (z, t), where A0 is a linear symmetric and ∇q0 (z, t) = o(|z|) uniformly in t as z → 0; (Q.∞) there exists c∞ > 0 such that |z| > c∞ implies Q(z, t) = 12 A∞ z, z + q∞ (z, t), and ∇q∞ is bounded.

308

M. IZYDOREK

If Q satisfies the above assumptions then 0 is a critical point of ∇Φ. Generalizing some earlier results, Szulkin has proved in Szulkin (1992, Theorem 7.2) that if i− (A∞ ) [i− (A0 ), i0 (A0 ) + i− (A0 )] (19) then (12) has a nontrivial solution (in addition to the trivial one z = 0). The following examples show how the LS-index can be used to obtain similar results even if (19) is not satisfied. EXAMPLE 6.6. Let 1. Q(x, y, t) = 12 (x2 + y2 ) + (x3 − 3xy2 ) · cos(3t) if x2 + y2 < c1 , for some c1 > 0; 2. Q(x, y, t) = 12 d(x2 + y2 ) + q(x, y, t) if x2 + y2 > c2 for some c2 > c1 with q having a bounded derivative and d ∈ (0, 1) ∪ (1, 2). From Example 6.4 we know that S = {0} is an isolated invariant set for the LSflow generated by −∇Φ whose LS-index is the homotopy type of the spectrum E with En = S 2n+3 ∨ S 2n+3 . It is also evident, that if we take U to be an open ball in H of a suﬃciently large radius, then U is an isolating neighbourhood. Let S ∞ denote the maximal invariant subset contained in U. The LS-index of S ∞ is the homotopy type of spectrum E such that: (a) En = S 2n+2 a pointed sphere of dimension 2n + 2 if d ∈ (0, 1), (b) En = S 2n+4 a pointed sphere of dimension 2n + 4 if d ∈ (1, 2) if n is suﬃciently large. Since [E] [E ] we conclude that S ∞ {0} and therefore there must be another zero of ∇Φ which gives a nontrivial solution of (12). Finally one easily verify that i− (A0 ) = 2, i0 (A0 ) = 2, i− (A∞ ) = 2 if d ∈ (0, 1) and i− (A∞ ) = 4 if d ∈ (1, 2). This shows that (19) is not satisfied. However our example has also a disadvantage. Consider a linear isomorphism ˜ H→H L: 7∞ k=1 Hk . ˜L = Lz if z ∈ z if z ∈ H0 . Let U ⊂ H be an open and bounded set and let f : U → H be a map of the ˜ + K(z), where K: U → H is completely continuous and f has no form f (z) = L(z) zeroes on the boundary of U. Then one defines the Leray – Schauder degree with respect to L˜ deg( f, U, 0) := degLS (Id + L˜ −1 ◦ K, U, 0). Now, an easy computation shows that in our example the local degree of ˜ + K(z) ˜ −∇Φ(z) = Lz + K(z) = Lz

THE LS-INDEX: A SURVEY

309

at 0 is equal to −2 and if U is a disc centered at 0 with radius suﬃciently large then deg(−∇Φ, U, 0) is equal to 1. Consequently, there is at least one point z 0 such that ∇Φ(z) = 0. In the next example that kind of argument can not be used. EXAMPLE 6.7. Let 1. Q(x, y, t) = 12 (x2 + y2 ) + (x2 + y2 )2 + h(x, y, t) if x2 + y2 ≤ c1 for some c1 > 0, with h being a smooth perturbation of order higher than 4 (with respect to x and y variables); 2. Q(x, y, t) = 12 d(x2 + y2 ) + q(x, y, t), if x2 + y2 ≥ c2 for some c2 > c1 with q having a bounded derivative and d ∈ (1, 2). From Example 6.5 we know that S = {0} is an isolated invariant subset for the LS-flow generated by −∇Φ whose LS-index is the homotopy type of the spectrum E with En = S 2n+2 . Again it is clear that an open ball U ⊂ H of a suﬃciently large radius is an isolating neighbourhood. The LS-index of S ∞ = Inv(U) is the homotopy type of spectrum E such that En = S 2n+4 , a pointed sphere of dimension 2n + 4, with n being suﬃciently large. We have [E] [E ] and therefore S ∞ {0}. Since we work with a gradient vector field S ∞ has to have at least two stationary points. Note, that in this example we have i− (A0 ) = 2, i0 (A0 ) = 2, i− (A∞ ) = 4 which gives i− (A∞ ) ∈ [i− (A0 ), i− (A0 ) + i0 (A0 )] and thus (19) is not satisfied. The Leray – Schauder degree gives us no extra information as well. The local degree at 0 and ˜ are equal to 1. the degree on a suﬃciently large disc of −∇Φ (with respect to L) 6.3. APPLICATIONS OF THE COHOMOLOGICAL LS-INDEX

In what follows we consider a Hamiltonian function, 2π-periodic in t, such that the corresponding Hamiltonian system is asymptotically linear at trivial (constant) solutions and at the infinity. In both examples discussed in this subsection we have resonance at some trivial solutions. Using Morse inequalities (6) we are able to prove the existence of at least one periodic solution of a given problem in addition to trivial ones. EXAMPLE 6.8. Let n = 1 and Q: R2 × R → R be such that: 1. Q(x, y, t) = 12 (x2 + y2 ) + (x2 + y2 )2 + h(x, y, t) if x2 + y2 ≤ α1 , α1 > 0, with h(x, y, t) being a smooth perturbation of order higher than 4 (with respect to x and y variables); 2. Q(x, y, t) = 12 (x − x0 )2 + 12 (y − y0 )2 + (x − x0 )3 − 3(x − x0 )(y − y0 )2 cos(3t) if (x − x0 )2 + (y − y0 )2 ≤ α2 for some (x0 , y0 ) (0, 0), α2 > 0; 3. Q(x, y, t) = 12 d(x2 + y2 ) + q(x, y, t), if x2 + y2 ≥ α3 , α3 > 0, d > 0 is not an integer and the derivative of q(x, y, t) is bounded.

310

M. IZYDOREK

Clearly z(t) = 0 and z(t) = (x0 , y0 ) are trivial solutions of (12). It follows from our assumptions that an LS-vector field −∇Φ: H → H has derivatives A0 at 0, Az0 at z0 = (x0 , y0 ) and A∞ at the infinity which are selfadjoint operators. Additionally, A∞ is an isomorphism and ker A0 = ker Az0 is the space of solutions of the linearizations of system (12) z˙ = J∇2 Q(0, t)z = J∇2G(z0 , t)z = Jz z = (x, y) ∈ R2

(20)

at trivial solutions. This means in particular that 0 and z0 are degenerate critical points for Φ: H → R and thus Morse indices are not defined in these points. Fortunately, we are still able to show that there are isolating neighbourhoods X0 and Xz0 for the LS-flow η generated by −∇Φ such that Inv(X0 , η) = {0} and Inv(Xz0 , η) = {z0 }. The LS-index of Xz0 with respect to η, hLS (Xz0 , η) is equal to the homotopy type of spectrum E for which one has Ek = S 2k+3 ∨ S 2k+3 , the wedge of two pointed spheres of dimension 2k + 3, k = 1, 2, . . .. (compare Example 6.4) and the index of X0 , hLS (X0 , η) is equal to the homotopy type of spectrum E such that Ek = S 2k+2 a pointed sphere of dimension 2k + 2, k = 1, 2, . . .. (compare Example 6.5). Since the derivative of −∇Φ at the infinity is an isomorphism there is an isolating neighbourhood X∞ for η such that S = Inv(X∞ , η) is the maximal bounded isolated invariant set for η in H. The LS-index hLS (X∞ , η) is the homotopy type of spectrum E such that Ek = S 2k+2a , a pointed sphere of dimension 2k + 2a if d ∈ (a − 1, a), a ∈ Z and k is suﬃciently large. From the inequality [E∨E ] [E ] we obtain S {0, z0 }. It might happen that S consists of two stationary points and a set of connecting orbits. We show that except {0} and {z0 } there is another rest point of η in S . Suppose, on the contrary, that {0} and {z0 } are the only stationary points in S . Since η is generated by the gradient vector field both points are Morse sets in S and the collection of sets

M1 = {0}, M2 = {z0 } defines a Morse decomposition of S . We easily compute the cohomology groups of spectra E, E , and E : Z ⊕ Z for q = 3, Z for q = 4, q q H (E) = H (E ) = 0 for q 3, 0 for q 4, Z for q = 2a, H q (E ) = 0 for q 2a. Applying Theorem 4.7 we obtain the equality 2t3 + t4 = t2a + (1 + t)Q(t)

(21)

which can not be true due to the fact that all coeﬃcients of Q are nonnegative. This proves that except two trivial solutions z0 (t) = 0 and z1 (t) = (x0 , y0 ) the Hamiltonian system (12) satisfying 1, 2, and 3 has a third periodic solution.

THE LS-INDEX: A SURVEY

311

However, if in (21) t is replaced by −1 we obtain −1 = 1 which obviously says that (21) does not hold even if we know nothing about the polynomial Q(t). This may suggest that we are able to prove the existence of a third critical point of Φ without using the gradient structure of the vector field ∇Φ and consequently without Morse theory methods. Indeed, a third solution can be deduced with help of the Leray – Schauder degree theory as it is shown in Izydorek (2001). In the next example this kind of argument can not be used. EXAMPLE 6.9. Assume that additionally to 1, 2, and 3 a map Q: R2 × R → R satisfies: 4. there are points z1 , z2 ∈ R2 , z1 = (x1 , y1 ), z2 = (x2 , y2 ), z1 z2 such that: Q(x, y, t) = 12 di (x − xi )2 + (y − yi )2 + hi (x, y, t) if (x − xi )2 + (y − yi )2 ≤ α4 , α4 > 0, i = 1, 2. where: − hi (x, y, t) is a smooth perturbation of order higher than 2 at (xi , yi ) (with respect to x and y variables), i = 1, 2; − d1 , d2 > 0 are not integers. This time the LS-vector field −∇Φ: H → H has derivatives at four points: 0, z0 , z1 , and z2 which are critical points for Φ and has the derivative at the infinity. One easily checks that the derivatives Az1 , Az2 of −∇Φ at z1 and z2 are isomorphisms. As z1 and z2 are nondegenerate critical points for Φ there are isolating neighbourhoods Xi for η such that Inv(Xi , η) = {zi } for i = 1, 2. Now, the LSindex hLS (Xi , η) = [E i ] with Eki = S 2k+2ai , a pointed sphere of dimension 2k + 2ai if di ∈ (ai − 1, ai ), i = 1, 2, and k = 1, 2, . . .. Arguing as in the previous example since [E ∨ E ∨ E 1 ∨ E 2 ] [E ] we conclude that S {0, z0 , z1 , z2 }. It turns out that one can give suﬃcient conditions for the existence of at least 5 rest points of η in S in terms of a1 , a2 , and a. Indeed, assume that η has exactly four rest points in S . Thus a Morse decomposition of S is defined by the collection of sets {M1 = {0}, Mi = {zi }; i = 0, 1, 2}. We compute the cohomology groups of spectra E i : Z for q = 2ai , H q (E i ) = i = 1, 2. 0 for q 2ai , Applying Theorem 4.7 we obtain the equality 2t3 + t4 + t2a1 + t2a2 = t2a + (1 + t)Q(t). Now, the equation (22) does not hold in the following cases: T1 a = 1 and (a1 , a2 ) ∈ N × N \ {(1, 1), (2, 1), (1, 2)};

(22)

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T2 a > 1 and (a1 , a2 ) ∈ N × N \ {(1, a), (2, a), (a, 1), (a, 2)} which is a consequence of the fact that Q(t) has nonnegative coeﬃcients. Thus, if either T1 or T2 is satisfied then η has at least 5 rest points in S . Notice, that both sides of (22) coincide at t = −1 so the polynomial Q(t) has been used in an essential way.

6.4. APPLICATIONS OF THE EQUIVARIANT LS-INDEX

An important consequence of Theorem 5.16 is the following THEOREM 6.10 (Izydorek, 2002, Corollary 5.3). Let η be a flow generated by a gradient S 1 -LS-vector field F: H → H, i.e., F = ∇ f for some S 1 -invariant functional f : H → R. Assume that: 1. {0} ⊂ H is homologically pt-hyperbolic isolated invariant set for η of index 2p ∈ Z; 2. there is a homologically pt-hyperbolic isolated (G- and η-)invariant set T for η of index 2q ∈ Z and 0 ∈ T ; 3. if x ∈ T ∩ F −1 (0) and x 0 then G x S 1 . Then, except 0 there are at least |p − q| orbits in T ∩ F −1 (0). The above result plays a crucial role in problems concerning estimations of the number of periodic solutions of (12) in the autonomous case. Here is one of its consequences. THEOREM 6.11 (Izydorek, 2002, Theorem 5.4). Let Q ∈ C 1 (R2m , R) satisfy the following conditions: 1.1. Q(z) = 12 az2 + q0 (z) in a neighbourhood of 0 ∈ R2m , and ∇q0 (z) = o(z) as z → 0, a > 0 is not an integer; 1.2. Q(z) = 12 bz2 + q∞ (z) in a neighbourhood of ∞, and ∇q∞ is bounded, b > 0 is not an integer; 1.3. 0 ∈ R2m is the unique critical point of Q. Then, system (12) possesses at least m|[a] − [b]| nonconstant, 2π-periodic solutions ([c] denotes the integer part of a number c).

THE LS-INDEX: A SURVEY

313

Proof. The constant map z(t) = 0 is a trivial solution of (12). It follows from 1.1 and 1.2 that an S 1 -LS-vector field −∇Φ: H → H has derivatives A0 at 0 ∈ H and A∞ at the infinity defined by (17) for A equal to aId and bId, respectively. Since a and b are not integers, A0 and A∞ are isomorphisms. Thus, the origin in H is an isolated invariant set for the flow η generated by −∇Φ and there is a maximal bounded isolated invariant set T for η in H. Clearly, T is S 1 -invariant and 0 ∈ T . By 1.3, the isotropy group of each point in T \ {0} is diﬀerent from S 1 . Let X0 , X∞ stand for isolating G-neighbourhoods for η with Inv(X0 , η) = {0} and Inv(X∞ , η) = T . By (17), the S 1 -LS-index of X0 is equal to S 1 -homotopy type of spectrum E such that Ei = S H0 ⊕···⊕Hi r − − Ei = S H ⊕Hr+1 ⊕···⊕Hi

i = 0, 1, . . . , r, i > r, .

(23) (24)

where r = [a]. Hence, ∗+ρ(r)

HS∗ 1 (E, R) ≈ HS 1

(Er , R) = HS∗+2rm (S H , R). 1 r

The suspension isomorphism (for V = H r , with dim H r = 4rm + 2m) S∗ : HS∗−2rm−2m (S 0 , R) → HS∗+2rm (S H , R), 1 1 r

where S 0 stands for zero-dimensional sphere with a base point, gives us (S 0 , R) ≈ ωm(r+1) ∗ R[ω], HS∗ 1 (E, R) ≈ HS∗−2m(r+1) 1 thus {0} is homologically pt-hyperbolic of index 2m(r + 1). Similarly, if E is a spectrum corresponding to X∞ , then HS∗ 1 (E , R) ≈ ωm(s+1) ∗ R[ω], where s = [b]. Thus, the set T is homologically pt-hyperbolic of index 2m(s + 1). Consequently, by Corollary 6.10 there is at least |m(r + 1) − m(s + 1)| = m|[a] − [b]| nontrivial S 1 -orbits of zeroes of −∇Φ in T . That proves our theorem.

The above theorem is a special case of a result obtained by Degiovanni and Olian Fannio (see Degiovanni and Olian Fannio, 1996, Theorem 1.2). It can be easily generalized as follows. Let A and B be symmetric 2m × 2m-matrices. Consider system (12) with Q(z) = 12 Az, z. If i0 (A) = 0, then ∇Φ is a linear isomorphism and {0} ⊂ H is an isolated invariant set for the flow η induced by −∇Φ. Using the same arguments as in the proof of Theorem 6.11 one shows, that {0} is homologically pt-hyperbolic of index i− (A). Similarly, if Q(z) = 12 Bz, z

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for a symmetric 2m × 2m-matrix B in a neighbourhood of the infinity, then there exists a maximal bounded isolated invariant set T for the corresponding flow η. The set T is homologically pt-hyperbolic of index i− (B). Consequently, we obtain the following. THEOREM 6.12 (Izydorek, 2002, Theorem 5.5). Let A, B be symmetric, 2m × 2m-matrices. Let Q ∈ C 1 (R2m , R) satisfy the following conditions: − Q(z) = 12 Az, z + q0 (z) in a neighbourhood of 0 ∈ R2m , and ∇q0 (z) = o(z) as z → 0; − Q(z) = 12 Bz, z + q∞ (z) in a neighbourhood of ∞, and ∇q∞ is bounded; − 0 ∈ R2m is the unique critical point of Q. If i0 (A) = i0 (B) = 0, then system (12) possesses at least 1 − 2 |i (A)

− i− (B)|

nonconstant 2π-periodic solutions. The above theorem generalizes Costa and Willem result from Costa and Willem (1986) obtained for positive definite matrices A, B and for strictly convex Hamiltonian functions (see also Mawhin and Willem, 1989). Theorems 6.10 and 6.12 have been applied to Hamiltonian systems with nonconvex Hamiltonian function in Izydorek (2001). In what follows U stands for an open halfline (α, ∞), where α is negative or −∞. Consider a map F: U → R of class C 1 such that F.1 F (x) 0 for each x ∈ U; F.2 there exists lim x→∞ F (x) (finite or infinite); F.3 there exists F (0). Let B: R2m → R2m be a linear isomorphism. We will be concerned with Hamiltonian functions Q: R2m → R, Q(z) = F(Bz2 ).

(25)

Since the case lim x→∞ F (x) = a 0 is rather standard we discuss the most interesting extreme cases only, i.e., lim F (x) = 0

x→∞

Here are the results.

and

lim F (x) = ±∞.

x→∞

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THEOREM 6.13 (Izydorek, 2001, Theorem 3.5). Let a Hamiltonian function Q be of the form (25) with F satisfying lim x→∞ F (x) = 0. Then, system (12) possesses at least (a) 12 |i− (2F (0)BT B) − 2m| if F (0) > 0, (b) 12 |i− (2F (0)BT B)| if F (0) < 0, nonconstant 2π-periodic solutions. Proof. By our assumptions Q is continuously diﬀerentiable in R2m and ∇Q(z) = 2F (Bz2 )BT Bz, where BT denotes the conjugated map. Since for each x ∈ U F (x) 0, one has ∇Q(z) = 0 iﬀ z = 0. Furthermore, the vector field ∇Q: R2m → R2m is asymptotically linear, i.e., admits derivatives at the origin and at the infinity. The derivative at 0 ∈ R2m A0 = 2F (0)BT B is selfadjoint and positively or negatively definite, depending on a sign of F (0). We claim that the derivative at the infinity A∞ = 0 so that we have a resonance. It is enough to show that ∇Q(z) lim = 0. z→∞ z Indeed, we have 2F (Bz2 )BT Bz |F (Bz2 )|z ≤ 2BT B lim = 0. z→∞ z→∞ z z

0 ≤ lim

Given β ∈ R such that F (x) = β for some x > 0. Set y = max{x ∈ U; F (x) = β} and define Fβ : U → R, F(x) if x ≤ y, Fβ (x) = F(y) + β(x − y) if x > y. Then, Fβ is a C 1 -map and Fβ (x)

=

F (x) if x ≤ y, β if x > y.

The gradient of the modified Hamiltonian function Qβ (z) = Fβ (Bz2 ) is given by the formula ∇Q(z) if Bz2 ≤ y, ∇Qβ (z) = (26) 2βBT B if Bz2 > y.

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Obviously, ∇Qβ (z) is asymptotically linear and its derivatives at 0 ∈ R2m and at β β the infinity are equal to A0 = A0 = 2F (0)BT B and A∞ = 2βBT B, respectively. Since lim x→∞ F (x) = 0 and BT B is positively definite one can choose β so close to 0 that β β i0 (A∞ ) = 0 and i− (A∞ ) = M − (−2βBT B). Moreover, as βF (0) > 0 one has M − (−2βBT B) = M − (−2F (0)BT B). If i0 (A0 ) = 0 then by Theorem 6.12 the Hamiltonian system z˙ = J∇Qβ (z)

(27)

possesses at least 1 − T 2 |i (2F (0)B B)

− M − (−2F (0)BT B)| β

nonconstant 2π-periodic solutions. As i0 (A∞ ) = 0 the system z˙ = 2βJBT Bz does not have nonconstant 2π-periodic solutions and therefore each 2π-periodic solution of (27) is a solution of (12). Finally, let us notice that M − (−2F (0)BT B) is equal to 0 if F (0) < 0 and is equal to 2m if F (0) > 0. It is worth pointing out that the above asymptotically linear problem is degenerate at the infinity. EXAMPLE 6.14. Choose F(x) = 12 e−x and B = aId, where Id is the identity map in R2m and k < a2 < k + 1 for some k ∈ N ∪ {0}. Then Q(z) = 12 e−az

2

and system (12) possesses at least k · 2m nonconstant 2π-periodic solutions. THEOREM 6.15 (Izydorek, 2001, Theorem 3.6). Let a Hamiltonian function Q be of the form (25) with F satisfying lim x→∞ F (x) = ∞. Then system (12) possesses infinitely many nonconstant 2π-periodic solutions. Proof. We use similar methods as in the proof of Theorem 6.13. This time however, ∇Q is not asymptotically linear. By our assumptions F (x) > 0 for every x ∈ U. Since the Hessian of Q at 0 ∈ R2m , A0 = 2F (0)BT B is positively definite it follows from (21) that there is a increasing sequence of real numbers (βn )∞ 1 such that: − limn→∞ βn = ∞; − i0 (βn BT B) = 0 for every n ∈ N;

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− limn→∞ i− (βn BT B) = ∞. By Theorem 6.12, if i0 (A0 ) = 0 then for each n ∈ N the Hamiltonian system z˙ = J∇Qβn (z) possesses at least

1 − 2 |i (A0 )

(28)

− i− (βn BT B)|

nonconstant 2π-periodic solutions, where Qβ is defined by (26). The condition i0 (βn BT B) = 0 implies that all 2π-periodic solutions of (28) are also solutions of (12). Since limn→∞ |i− (βn BT B)| = ∞ our assertion is concluded. This time the problem is not asymptotically linear itself but it can be “approximated” by a sequence of asymptotically linear problems. As an example one can choose f (x) = (1 + x) s with s > 1, f (x) = e x , etc. Clearly, there are also functions for which the corresponding Hamiltonian function is nonconvex, e.g., f (x) = 2x − x3 /3! + x5 /5!. References Amann, H. and Zehnder, E. (1980)a Nontrivial solutions for a class of nonresonance problems and applications to nonlinear diﬀerential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7, 539 – 603. Amann, H. and Zehnder, E. (1980)b Periodic solutions of asymptotically linear Hamiltonian systems, Manuscripta Math. 32, 149 – 189. Ambrosetti, A. and Rabinowitz, P. H. (1973) Dual variational methods in critical point theory and applications, J. Functional Analysis 14, 349 – 381. Angenent, S. and van der Vorst, R. (1999) A superquadratic indefinite elliptic system and its Morse – Conley – Floer homology, Math. Z. 231, 203 – 248. Bartsch, T. (1993) Topological Methods for Variational Problems with Symmetries, Vol. 1560 of Lecture Notes in Math., Berlin, Springer. Benci, V. (1982) On critical point theory for indefinite functionals in the presence of symmetries, Trans. Amer. Math. Soc. 274, 533 – 572. Benci, V. (1991) A new approach to the Morse – Conley theory and some applications, Ann. Mat. Pura Appl. (4) 158, 231 – 305. Blanchard, P. and Br¨uning, E. (1992) Variational Methods in Mathematical Physics, Texts Monogr. Phys., Berlin, Springer. Bredon, G. E. (1972) Introduction to compact transformation groups, Vol. 46 of Pure Appl. Math., New York, Academic Press. Chang, K. C. (1981) Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34, 693 – 712. Chang, K. C. (1993) Infinite-Dimensional Morse Theory and Multiple Solution Problem, Vol. 6 of Progr. Nonlinear Diﬀerential Equations Appl., Boston, MA, Birkh¨auser. Chang, K. C., Wu, S. P., and Li, S. J. (1982) Multiple periodic solutions for an asymptotically linear wave equation, Indiana Univ. Math. J. 31, 721 – 731. Conley, C. C. (1978) Isolated Invariant Sets and the Morse Index, Vol. 38 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc.

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Conley, C. C. and Zehnder, E. (1984) Morse-type index theory for flows and periodic solutions for Hamiltonian equations, Comm. Pure Appl. Math. 37, 207 – 253. Costa, D. G. and Willem, M. (1986) Lusternik – Schnirelman theory and asymptotically linear Hamiltonian systems, In Diﬀerential Equations: Qualitative Theory, Vol. 47 of Colloq. Math. Soc. J´anos Bolyai, Szeged, 1984, pp. 179–191, Amsterdam, North Holland. Degiovanni, M. and Mrozek, M. (1993) The Conley index for maps in the absence of compactness, Proc. Roy. Soc. Edinburgh Sect. A 123, 75 – 94. Degiovanni, M. and Olian Fannio, L. (1996) Multiple periodic solutions of asymptotically linear Hamiltonian systems, Nonlinear Anal. 26, 1437 – 1446. Deimling, K. (1977) Ordinary Diﬀerential Equations in Banach Spaces, Vol. 596 of Lecture Notes in Math., Berlin, Springer. Fadell, E. and Rabinowitz, P. (1978) Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems, Invent. Math. 45, 139 – 174. Fei, G. H. (1995)a Maslov-type index and periodic solution of asymptotically linear Hamiltonian systems which are resonant at infinity, J. Diﬀerential Equations 121, 121 – 133. Fei, G. H. (1995)b Relative Morse index and its application to Hamiltonian systems in the presence of symmetries, J. Diﬀerential Equations 122, 302 – 315. Floer, A. (1987) A refinement of the Conley index and an application to the stability of hyperbolic invariant sets, Ergodic Theory Dynam. Systems 7, 93 – 103. Floer, A. (1990) A topological persistence theorem for normally hyperbolic manifolds via the Conley index, Trans. Amer. Math. Soc. 321, 645 – 657. Floer, A. and Zehnder, E. (1988) The equivariant Conley index and bifurcations of periodic solutions of Hamiltonian systems, Ergodic Theory Dynam. Systems 8*, 87 – 97. Franzosa, R. D. (1989) The connection matrix theory for Morse decompositions, Trans. Amer. Math. Soc. 311, 561 – 592. Ge ba, K. (1997) Degree for gradient equivariant maps and equivariant Conley index, In M. Matzeu and A. Vignoli (eds.), Topological Nonlinear Analysis II, Vol. 27 of Progr. Nonlinear Diﬀerential Equations Appl., Frascati, 1995, pp. 147–272, Boston, MA, Birkh¨auser. Ge ba, K. and Granas, A. (1965)a Algebraic topology in linear normed spaces I, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 13, 287 – 290. Ge ba, K. and Granas, A. (1965)b Algebraic topology in linear normed spaces II, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 13, 341 – 345. Ge ba, K. and Granas, A. (1967)a Algebraic topology in linear normed spaces III, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 15, 137 – 143. Ge ba, K. and Granas, A. (1967)b Algebraic topology in linear normed spaces IV, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 15, 145 – 152. Ge ba, K. and Granas, A. (1969) Algebraic topology in linear normed spaces V, Bull. Acad. Polon. Sci. S´er. Sci. Math. Astronom. Phys. 17, 123 – 130. Ge ba, K. and Granas, A. (1973) Infinite-dimensional cohomology theories, J. Math. Pures Appl. (9) 52, 145 – 270. Ge ba, K., Izydorek, M., and Pruszko, A. (1999) The Conley index in Hilbert spaces and its applications, Studia Math. 134, 217 – 233. Izydorek, M. (1999) Bourgin – Yang type theorem and its applications to Z2 -equivariant Hamiltonian systems, Trans. Amer. Math. Soc. 351, 2807 – 2831. Izydorek, M. (2000) Multiple solutions for an asymptotically Linear wave equation, Diﬀerential Integral Equations 13, 289 – 310. Izydorek, M. (2001) A cohomological Conley index in Hilbert spaces and applications to strongly infinite problems, J. Diﬀerential Equations 170, 22 – 50.

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Izydorek, M. (2002) Equivariant Conley index in Hilbert spaces and applications to strongly indefinite problems, Nonlinear Anal. 51, 33 – 66. Izydorek, M. and Rybakowski, K. P. (2002)a The Conley index in Hilbert spaces and a problem of Angenent and van der Vorst, Fund. Math. 173, 77 – 100. Izydorek, M. and Rybakowski, K. P. (2002)b On the Conley index in Hilbert spaces in the absence of uniqueness, Fund. Math. 171, 31 – 52. Izydorek, M. and Rybakowski, K. P. (2003) Multiple solutions of indefinite elliptic systems via a Galerkin-type Conley index theory, Fund. Math. 176, 233 – 249. Kurland, H. L. (1982) Homotopy invariants of repeller-attractor pairs I: The Puppe sequence of an R-A pair, J. Diﬀerential Equations 46, 1 – 31. Kurland, H. L. (1983) Homotopy invariants of repeller-attractor pairs II: Continuation of R-A pairs, J. Diﬀerential Equations 49, 281 – 329. Li, S. J. and Liu, J. Q. (1989) Morse theory and asymptotic linear Hamiltonian systems, J. Diﬀerential Equations 78, 53 – 73. Li, S. J. and Szulkin, A. (1993) Periodic solutions of an asymptotically linear wave equation, Topol. Methods Nonlinear Anal. 1, 211–230. Long, Y. M. (1990) Maslov-type index, degenerate critical points, and asymptotically linear Hamiltonian systems, Sci. China Ser. A 33, 1409 – 1419. Long, Y. M. (1993) The Index Theory of Hamiltonian Systems with Applications, Beijing, Science Press. Mawhin, J. and Willem, M. (1989) Critical Point Theory and Hamiltonian Systems, Vol. 74 of Appl. Math. Sci., New York, Springer. McCord, C. (1988) Mappings and homological properties in the homology Conley index, Ergodic Theory Dynam. Systems 8*, 175 – 198. McCord, C. and Mischaikow, K. (1992) Connected simple systems, transition matrices and heteroclinic bifurcations, Trans. Amer. Math. Soc. 333, 379–422. Mischaikow, K. (1994) Conley Index Theory, In R.Johnson (ed.), Dynamical Systems, Vol. 1609 of Lecture Notes in Math., Montecatini Terme, 1994, pp. 119–207, Berlin, Springer. Montgomery, J. (1973) Cohomology of isolated invariant sets under perturbation, J. Diﬀerential Equations 13, 257–299. Rabinowitz, P. H. (1978) Free vibrations for a semilinear wave equation, Comm. Pure Appl. Math. 31, 31 – 68. Rabinowitz, P. H. (1986) Minimax Methods in Critical Point Theory with Applications to Differential Equations, Vol. 35 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc. Reineck, J. (1988) Connecting orbits in one-parameter families of flows, Ergodic Theory Dynam. Systems 8*, 359 – 374. Robbin, J. W. and Salamon, D. (1988) Dynamical systems, shape theory and the Conley index, Ergodic Theory Dynam. Systems 8*, 375–394. Rybakowski, K. P. (1982) On the homotopy index for infinite-dimensional semiflows, Trans. Amer. Math. Soc. 269. Rybakowski, K. P. (1983) The Morse index, releller-attractor pairs and the connection index for semiflows on noncompact spaces, J. Diﬀerential Equations 47, 66 – 98. Rybakowski, K. P. (1987) The Homotopy Index and Partial Diﬀerential Equations, Universitext, Berlin, Springer. Rybakowski, K. P. and Zehnder, E. (1985) A Morse equation in Conley’s index theory for semiflows on metric spaces, Ergodic Theory Dynam. Systems 5, 123 – 143. Salamon, D. (1985) Connected simple systems and the Conley index of isolated invariant sets, Trans. Amer. Math. Soc. 291, 1 – 41.

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Salamon, D. (1990) Morse theory, the Conley index and Floer homology, Bull. London Math. Soc. 22, 113–140. Spanier, E. H. (1966) Algebraic Topology, New York, McGraw-Hill. Struwe, M. (1996) Variational Methods, Vol. 34 of Ergeb. Math. Grenzgeb. (3), Berlin, Springer. Szulkin, A. (1992) Cohomology and Morse theory for strongly indefinite functionals, Math. Z. 209, 375 – 418. Szulkin, A. (1997) Index theories for indefinite functionals and applications, In Topological and variational methods for nonlinear boundary value problems, Vol. 365 of Pitman Res. Notes Math. Ser., Chol´ın, 1995, pp. 89–121, Harlow, Longman. tom Dieck, T. (1987) Transformation Groups, Vol. 8 of de Gruyter Stud. Math., Berlin, de Gruyter. Troestler, C. and Willem, M. (1996) Nontrivial solution of a semilinear Schr¨odinger equation, Comm. Partial Diﬀerential Equations 21, 1431 – 1449. Whitehead, G. W. (1970) Recent Advances in Homotopy Theory, Vol. 5 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc.

LECTURES ON FLOER THEORY AND SPECTRAL INVARIANTS OF HAMILTONIAN FLOWS YONG-GEUN OH∗ University of Wisconsin

Abstract. The main purpose of this lecture is to provide a coherent explanation of the chain level Floer theory and its applications to the study of geometry of the Hamiltonian diﬀeomorphism group of closed symplectic manifolds. In particular, we explain the author’s recent construction of spectral invariants of Hamiltonian paths and an invariant norm of the Hamiltonian diﬀeomorphism group on nonexact symplectic manifolds. Key words: action functional, Hamiltonian paths, Floer homology, Novikov – Floer cycles, energy estimates, Hofer’s norm, mini-max theory, spectral invariants

1. Introduction The main purpose of this lecture note is to provide a coherent explanation on the chain level Floer theory and its applications to the study of geometry of the Hamiltonian diﬀeomorphism group of closed symplectic manifolds (M, ω), which has been systematically developed in a series of papers (Oh 2002, 2005b, 2005d, 2005a, 2005c, 2004b, 2004a). This study is based on a construction of certain invariants, which we call spectral invariants, of one-periodic Hamiltonian functions H: S 1 × M → R satisfying the normalization condition Ht dµ = 0 M

where dµ is the Liouville measure of (M, ω). We denote the set of such functions by ∞ 1 (S × M, R) Hm := Cm where “m” stands for “mean zero.” The construction of these invariants is through a Floer theoretic version of the mini-max theory of the associated perturbed action functional AH 1 ∗ AH (γ, w) = − w ω − H t, γ(t) dt 0 ∗

Partially supported by the NSF grant #DMS 0203593, a grant of the 2000 Korean Young Scientist Prize, and the Vilas Research Award of University of Wisconsin

321 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 321–416. © 2006 Springer. Printed in the Netherlands.

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defined for the pairs (γ, w) of smooth maps γ: S 1 = R/Z × M and w: D2 → M satisfying w|∂D2 = γ. These invariants form a function ρ: Hm × QH∗ (M) → R whose value ρ(H; a) is the mini-max value of the action functional over the Novikov – Floer cycles representing the Floer homology class a which is ‘dual’ to the quantum cohomology class a. In the classical mini-max theory for the indefinite functionals as in Rabinowitz (1978) and Benci and Rabinowitz (1979), there was implicitly used the notion of ‘semi-infinite cycles’ to carry out the mini-max procedure. There are two essential ingredients needed to prove existence of actual critical values out of the mini-max values: one is the finiteness of the mini-max value, or the linking property of the (semi-infinite) cycles associated to the class a and the other is to prove that the corresponding mini-max value is indeed a critical value of the action functional. When the global gradient flow of the action functional exists as in the classical critical point theory (Benci and Rabinowitz, 1979) this point is closely related to the well-known Palais – Smale condition and the deformation lemma which are essential ingredients needed to prove the criticality of the mini-max value. Partly because we do not have the global flow, we need to geometrize all these classical mini-max procedures. It turns out that the Floer homology theory in the chain level is the right framework for this purpose. The idea of construction of spectral invariants is originated from the author’s Floer theoretic construction (Oh, 1997) of Viterbo’s invariants (Viterbo, 1992) of Lagrangian submanifolds in the cotangent bundle, and is also based on the framework of the mini-max theory over natural semi-infinite cycles on the covering 0 (M). We call the corresponding semi-infinite cycles the Novikov – Floer space L cycles. In this construction, the ‘finiteness’ condition in the definitions of the Novikov ring and the Novikov – Floer cycles is fully exploited in the proofs of various existence results of pseudo-holomorphic curves. Now the organization of the content of the paper is in order. In Section 2, we view the free loop space of the symplectic manifold as an infinite-dimensional (weakly) symplectic manifold with the natural symplectic action of S 1 induced by the domain rotation. Its lifted action to the universal covering space then has the associated moment map which is nothing but the unperturbed action functional. After then, we compute the first variation and the gradient equation of the action functional with respect the L2 -metric induced by a one-periodic family J = {Jt }0≤t≤1 of compatible almost complex structures. We also look at the nonautonomous version of the gradient equation associated to each two parameter family j: [0, 1] × S 1 → Jω

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and a cut-oﬀ function ρ: R → [0, 1]. In Section 3, we review construction of the Floer complex and of the various basic operators in the chain level Floer theory. While these constructions are standard by now (see Floer, 1989; Salamon and Zehnder, 1992), we add some novelty in our exposition which is needed in our construction of the spectral invariants and their applications. In Section 4, we carefully study the energy estimates and the change of action levels under the Floer trajectories, and explain its relation to the L(1,∞) norm of Hamiltonian functions which arise naturally in this study of energy estimates. In Section 5, we give the definition of ρ(H; a) and prove their basic properties, especially the well-definedness and the finiteness of its value. In Section 6, we discuss the so called, spectrality, i.e., whether the mini-max value ρ(H; a) is indeed a critical value of AH . We give the proof, coming from Oh (2005a), of the spectrality for an arbitrary smooth H on rational symplectic manifolds. For the nonrational (M, ω), we just state the theorem from Oh (2004a) that the spectrality holds for nondegenerate Hamiltonian function H, whose proof is referred to Oh (2004a). In Section 7, we follow Oh (1999), Schwarz (2000), and Entov (2000) and explain the pants product in Floer homology and prove the triangle inequality ρ(H#K; a · b) ≤ ρ(H; a) + ρ(K; b). In Section 8, we explain our construction of the spectral norm, denoted by γ: Ham(M, ω) → R+ , which was carried out in Oh (2005c). As illustrated by Ostrover (2003), this norm is not the same as but smaller than the Hofer norm. Along the way, we also introduce certain geometric invariants of the pair (H, J) and also their family versions. These geometric invariants play crucial roles in our proof of nondegeneracy of the spectral norm γ. We call these invariants the -regularity type invariants in general because their nontriviality strongly relies on the so called the -regularity theorem, which was first introduced by Sacks and Uhlenbeck (1981) in the context of harmonic maps. In Section 9, we explain a simple criterion for the length minimizing property of the Hamiltonian paths in terms of the spectral invariant ρ(H; 1) stated in Oh (2005d). An analogous criterion had been used by Hofer (1993) and by Bialy and Polterovich (1994) in Cn . We illustrate its application to the study of length minimizing property of some autonomous Hamiltonians. Besides this criterion, this application is based on a construction of an optimal Floer cycle as done in Polterovich (2001), Oh (2002), and Kerman and Lalonde (2003), especially the one used by Kerman and Lalonde (2003). We refer readers to Section 9 for more detailed explanations. In Appendix, starting from the definition of the Conley – Zehnder index µH ([z, w]) given in Salamon and Zehnder (1992) and Hofer and Salamon (1995),

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but using a diﬀerent convention of the canonical symplectic form on Cn R2n from Salamon and Zehnder (1992) and Hofer and Salamon (1995), we provide complete details of the proof of the following index formula in our convention: µH ([z, w ]) = µH ([z, w]) − 2c1 ([w # w])

(1)

µH ([z, A # w]) = µH ([z, w]) − 2c1 (A).

(2)

or There are many diﬀerent conventions used in the literature of symplectic geometry concerning the definitions of Hamiltonian vector fields, the canonical symplectic form on the cotangent bundle, the action functional and others. And partly because there is no literature which provides detailed explanations of the index formula in any fixed convention, this formula has been a source of confusion at least for the present author, especially concerning the sign in front of the first Chern number term in the formula. We set the record straight here once and for all by announcing that the sign is ‘-’ in our convention which has been used by the author here and Oh (2002; 2005b; 2005d; 2005a; 2005c; 2004b; 2004a). And we also emphasize that the form of this index formula has nothing to do with whether one use the homological or the cohomological version of the Floer homology as long as they fix the definition of the Conley – Zehnder index of the symplectic path in Sp(n) as in Salamon and Zehnder (1992). We like to refer readers to Entov (2004) and Entov and Polterovich (2003) for other interesting applications of the spectral invariants in which a construction of quasi-morphisms on Ham(M, ω) is given for some class of symplectic manifolds. We refer to Polterovich (2005) for a survey of these works. To get the main stream of ideas transparent without getting bogged down with technicalities related with transversality question of various moduli spaces, we assume that (M, ω) is strongly semi-positive in the sense of Seidel (1997) and Entov (2000): A closed symplectic manifold is called strongly semi-positive if there is no spherical homology class A ∈ π2 (M) such that ω(A) > 0,

2 − n ≤ c1 (A) < 0.

Under this condition, the transversality problem concerning various moduli spaces of pseudo-holomorphic curves is standard. We will not mention this generic transversality question at all in the main body of the paper unless it is absolutely necessary. In Section 10, we will briefly explain how this general framework could be incorporated in our proofs in the context of Kuranishi structure (Fukaya and Ono, 1999). In this lecture, we will be very brief in explaining the Fredholm theory and compactness properties of the Floer moduli space the details of which are by now well-known and standard in the literature, at least for the semi-positive cases. We refer readers to Hofer and Salamon (1995) and Salamon and Zehnder (1992) for

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such details in the semi-positive case. Instead, we will put more emphasis on the calculations involved in the analysis of the filtration changes under the chain map, and on explaining the chain level arguments used in our Floer mini-max theory to overcome the diﬃculties arising from the nonexactness and the nonrationality of general symplectic manifolds. These chain level arguments also require one to closely examine all the basic constructions in Floer theory, especially in the choice of compatible almost complex structures and its relation to the given Hamiltonian functions. These materials have recently appeared in the series of our papers (Oh, 2002, 2005b, 2005d, 2005a, 2005c, 2004b, 2004a) and are less known in the standard Floer theory. We believe that these details deserve more attention and scrutiny in the future. Another exposition of spectral invariants, based on the approach using the so called “PSS-isomorphism,” has been given by McDuﬀ and Salamon (2004) for the rational case. However, to make this approach well-founded, it remains to fill some nonstandard analytic details in the proof of isomorphism property of the PSS-map which is used in the various construction carried out in Piunikhin et al. (1996) and McDuﬀ and Salamon (2004). We would like to thank the organizers of the summer school in CRM for running a successful school, and also thank the speakers in the school for delivering stimulating lectures and discussions. CONVENTION AND NOTATIONS

− The Hamiltonian vector field X f associated to a function f on (M, ω) is defined by d f = ω(X f , ·). − The multiplication F # G and the inverse G on the set of time periodic Hamiltonians C ∞ (M × S 1 ) are defined by F # G(t, x) = F(t, x) + G t, (φtF )−1 (x) t G(t, x) = −G t, φG (x) . − L(M) = Map(S 1 , M) − L0 (M) = the connected component of L0 (M) consisting of contractible loops. 0 (M) = the universal covering space or the Γ-covering space of L0 (M) − L depending on the circumstances. − Hm = Hm (M) = C ∞ (S 1 × M, R) − Jω = Jω (M) = the set of compatible almost complex structures −

jω = C ∞ (S 1 , Jω )

− P(Hm ) = C ∞ ([0, 1], Hm )

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− P( jω ) = C ∞ ([0, 1], jω ) 2. The free loop space and the action functional 2.1. THE FREE LOOP SPACE AND THE S 1 -ACTION IN GENERAL

Let M be a general smooth manifold, not necessarily symplectic. We denote by L(M) := Map(S 1 , M) be the free loop space, i.e., the set of smooth maps γ: S 1 = R/Z → M. We emphasize the loops have a marked point 0 ∈ R/Z and often parameterize them by the unit interval [0, 1] with the periodic boundary condition γ(0) = γ(1). L(M) has the distinguished connected component of contractible loops, which we 0 (M), can denote by L0 (M). The universal covering space of L0 (M), denoted by L be expressed by {[γ, w] | γ ∈ L0 (M) and w: D2 → M satisfying ∂w =: w|∂D2 = γ} where [γ, w] is the set of homotopy classes of w relative to ∂w = γ. Here we identify ∂D2 with S 1 . We call such w a bounding disc of γ. The deck transformation 0 (M) → L0 (M) is realized by the operation of of the universal covering space L “gluing a sphere” (γ, w) → (γ, w # u) (3) for a (and so any) sphere u: S 2 → M representing the given class A ∈ π2 (M) π1 L0 (M) . There is a natural circle action on L(M) induced by the time translation γ → γ ◦ Rϕ = γ(· + ϕ)

(4)

where Rϕ : S 1 → S 1 is the map given by Rϕ (t) = t + ϕ,

ϕ ∈ S 1.

The infinitesimal generator of this action is the vector field X on L(M) provided by X(γ) = γ˙ . The fixed point set of this S 1 action is the set of constant loops M → L0 (M). This action lifts to an action on the set of pairs (γ, w) → (γ ◦ Rϕ , w ◦ Rϕ )

(5)

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induced by the complex multiplication, which we again denote by Rϕ : z ∈ D2 ⊂ C → e2πiϕ z. 0 (M) forms a principal bundle of The fixed point set of the induced S 1 action on L π2 (M) over M with π2 (M)-action induced by (5): Obviously S 1 acts trivially on the constant loops γ ≡ x ∈ M. On the other hand, it acts trivially on the homotopy class of the pairs (x, w) because the pair (x, w) and (x ◦ Rϕ , w ◦ Rϕ ) are homotopic as Rϕ is homotopic to the identity. 2.2. THE FREE LOOP SPACE OF SYMPLECTIC MANIFOLDS

- Now we specialize our discussion on the loop space to the case of a symplectic manifold (M, ω). In this case, L(M) carries a canonical (weak) symplectic form defined by 1 Ω(ξ1 , ξ2 ) := ω ξ1 (t), ξ2 (t) dt : (6) 0

the closedness of Ω is a consequence of the closedness of ω together with the fact that S 1 has no boundary, and the (weak) nondegeneracy follows from the nondegeneracy of ω. The S 1 action (5) is symplectic, i.e., preserves Ω LX Ω = 0. 0 (M) by the pullObviously Ω induces a symplectic form on the covering space L back under the projection L0 (M) → L0 (M), which we denote by Ω. LEMMA 2.1. The form X+Ω is a closed one form on L(M). Proof. Since the closedness is local, it is enough to construct a function A = A0 defined in a neighborhood of any given loop γ0 that satisfies dA = X+Ω.

(7)

Note that for any path γ suﬃciently C ∞ close to a given γ0 , we have a distinguished path to γ defined by uγ0 γ : s ∈ [0, 1] → expγ0 sE(γ0 , γ) , E(γ0 , γ) := (expγ0 )−1 (γ) which defines a distinguished homotopy class of paths [uγ0 γ ] with fixed ends, u(0) = γ0 ,

u(1) = γ.

We ambiguously denote the associated map uγ0 γ : [0, 1] × S 1 → M also by uγ0 γ . We then define the locally defined function A by the formula A(γ; γ0 ) = 0 − u∗γ0 γ ω

(8)

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where ‘0’ should be regarded as the value A(γ0 ; γ0 ), which can be chosen arbitrarily. Now we verify (7). We first note that for any loop γ nearby γ0 and for a tangent vector ξ = du/ds| s=0 ∈ T γ L(M), we have d (9) dA(γ)(ξ) = − u∗γγs ω ds s=0 But we derive, after a change of variables, s 1 ∂u ∂u d d , u∗γγs ω = − ω dt ds − ds s=0 du u=0 0 0 ∂s ∂t 1 −ω ξ(t), γ˙ (t) dt = X(γ)+Ω. = 0

This combined with (9) finishes the proof of (7) and hence the lemma.

REMARK 2.2. In the point of view of de Rham theory of the loop space (Chen, 1973; Getzler et al., 1991), a symplectic form ω on M induces a canonical cohomology class of degree one induced by the closed one form X+Ω, which is obtained by the iterated integrals. This one form is not exact in general. Exactness of this one form is precisely the so called the weakly exactness of the symplectic form ω. The Floer homology can be considered as a version of the Novikov – Morse homology of this closed one form, or the Morse homology of the circle valued functions on L(M). If we restrict this closed one-form to L0 (M) and consider its lifting to the 0 (M), the formula (8) has a global lifting induced by universal covering space L the function of the pairs (γ, w), again denoted by A = A0 A0 (γ, w) = − w∗ ω considering w as a path from a constant path w(0) to γ = ∂w. We call this the unperturbed action functional. It satisfies dA0 = X+Ω

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0 (M) is Hamiltonian and its asso0 (M). In other words, the S 1 -action on L on L 0 (M) → R (see Weinstein, ciated moment map is nothing but the function A0 : L 1978, for more detailed discussions). 2.3. THE NOVIKOV COVERING

Following Floer (1989) and Hofer and Salamon (1995), we now introduce a notion of the Novikov covering space of L0 (M).

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DEFINITION 2.3. Let (γ, w) be a pair of γ ∈ L0 (M) and w be a disc bounding γ. We say that (γ, w) is Γ-equivalent to (γ, w ) if and only if ω([w # w]) = 0 and

c1 ([w # w]) = 0

where w is the map with opposite orientation on the domain and w # w is the obvious glued sphere. Here Γ stands for the group Γ= We denote

π2 (M) . ker ω|π2 (M) ∩ ker c1 |π2 (M)

Γω := ω(Γ) = ω π2 (M) ⊂ R

and call it the (spherical) period group of (M, ω). DEFINITION 2.4. We call (M, ω) rational if Γω ⊂ R is a discrete subgroup, and irrational otherwise. EXAMPLE 2.5. The product S 2 (r1 ) × S 2 (r2 ) with the product symplectic form ω1 ⊕ ω2 is rational if and only if the ratio r22 /r12 is rational. REMARK 2.6. We note that for an irrational (M, ω), the period group is a countable dense subset of R. In general, the dynamical behavior of the Hamiltonian flow on an irrational symplectic manifold is expected to become much more complex than on a rational symplectic manifold. The period group Γω is the simplest indicator of this distinct dynamical behavior. From now on, we exclusively denote by [γ, w] the Γ-equivalence class of (γ, w) 0 (M) the set of Γ-equivalence classes. We denote by π : L 0 (M) → and by L L0 (M) the canonical projection. We call L0 (M) the Γ-covering space of L0 (M). We denote by A or qA the image of A ∈ π2 (M) under the projection π2 (M) → Γ. There are two natural invariants associated to A: the valuation v(A) v: Γ → R;

v(A) = ω(A)

d: Γ → Z;

d(A) = c1 (A).

(11)

and the degree d(A) (12) qA

In general these two invariants are independent and so is a formal parameter depending on two variables. In that sense, we may also denote qA = T ω(A) ec1 (A) with two diﬀerent formal parameters T and e.

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The (unperturbed) action functional A0 defined above obviously projects down to the Γ-covering space by the same formula A0 ([γ, w]) = − w∗ ω as in Section 2.2. This functional provides a natural increasing filtration on the 0 (M): for each λ ∈ R, we define space L λ (M) := {[z, w] ∈ L 0 (M) | A0 ([z, w]) ≤ λ}. L 0 We note that

λ (M) ⊂ L λ (M) L 0 0

if λ ≤ λ .

0 (M) → R It follows from (10) that the critical set, denoted by Crit A0 , of A0 : L is the disjoint union of copies of M g·M Crit A0 (M) = g∈Γ

where M → L0 (M); x → [x, xˆ] is the canonical inclusion, where xˆ is the constant disc xˆ ≡ x. The following is well-known and straightforward to check. LEMMA 2.7. At each [x, xˆ # A] ∈ Crit A0 , the Hessian d2 A0 defines a bilinear form on 0 (M) T x L0 (M) T [x, xˆ#A] L which is (weakly) nondegenerate in the normal direction to Crit A0 . In particular, A0 is a Bott – Morse function. For the convenience of notations, we also denote [x, xˆ] = xˆ,

[x, xˆ # A] = xˆ ⊗ qA .

2.4. PERTURBED ACTION FUNCTIONALS AND THEIR ACTION SPECTRA

When a one-periodic Hamiltonian H : (R/Z) × M → R is given, we consider the 0 (M) → R defined by perturbed functional AH : L ∗ AH ([γ, w]) = A0 − H t, γ(t) dt = − w ω − H t, γ(t) dt. (13) Unless otherwise stated, we will always consider one-periodic normalized Hamiltonian functions H: [0, 1] × M → R.

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LEMMA 2.8. The set of critical points of AH is given by Crit(AH ) = {[z, w] | z ∈ Per(H), ∂w = z} 0 (M) canonically restricts. to which the Γ action on L DEFINITION 2.9. We define the action spectrum of H by 0 (M), z ∈ Per(H)}, Spec(H) := {AH (z, w) ∈ R | [z, w] ∈ Ω → R. For each given z ∈ Per(H), we i.e., the set of critical values of AH : L(M) denote Spec(H; z) = {AH (z, w) ∈ R | (z, w) ∈ π−1 (z)}. Note that Spec(H; z) is a principal homogeneous space modelled by the period group Γω . We then have Spec(H; z). Spec(H) = z∈Per(H)

Recall that Γω is either a discrete or a countable dense subgroup of R. The following was proven in Oh (2002). PROPOSITION 2.10. Let H be any periodic Hamiltonian. Spec(H) is a measure zero subset of R for any H. We note that when H = 0, we have Spec(H) = Γω . The following definition is standard. DEFINITION 2.11. We say that two Hamiltonians H and F are homotopic if φ1H = φ1F and their associated Hamiltonian paths φH , φK ∈ P(Ham(M, ω), id) are path-homotopic relative to the boundary. In this case we denote H ∼ F and denote * the set of equivalence classes by Ham(M, ω). The following lemma was proven in the aspherical case in Schwarz (2000) and Polterovich (2001). We refer the reader to Oh (2005b) for complete details of its proof in the general case. PROPOSITION 2.12. Suppose that F, G are normalized. If F ∼ G, we have Spec(F) = Spec(G) as a subset of R.

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This enables us to make this definition DEFINITION 2.13. * Ham(M, ω)

∞ (S 1 × M), we define the spectrum of h ∈ For any H ∈ Cm

Spec(h) := Spec(F) for a (and so any) normalized Hamiltonian F with [φ, F] = h. 2.5. THE L2 -GRADIENT FLOW AND PERTURBED CAUCHY – RIEMANN EQUATIONS

The Floer homology theory (Floer, 1989; Hofer and Salamon, 1995) is the semi-infinite version of the Novikov’s circle valued Morse theory (Novikov, 1981, 1982) of AH on the space L0 (M) of contractible free loops. To do the Morse 0 (M). We do this by first defining theory of AH , we need to provide a metric on L 0 (M). Note that any S 1 -family a metric on L0 (M) and then pulling it back to L {gt }t∈S 1 of Riemannian metrics on M induces an L2 -type metric on L(M) by the formula 1 ξ1 , ξ2 = gt ξ1 (t), ξ2 (t) dt (14) 0

for ξ1 , ξ2 ∈ T γ L(M). On the symplectic manifold (M, ω), we will particularly consider the family of almost K¨ahler metrics induced by the almost complex structures compatible to the symplectic form ω. Following Gromov (1985), we give the following definition. DEFINITION 2.14. to ω, if J satisfies

An almost complex structure J on M is called compatible

1. ω(v, Jv) ≥ 0 and equality holds only when v = 0

(Tameness)

2. ω(v1 , Jv2 ) = ω(v2 , Jv1 ).

(Symmetry)

We denote by Jω = Jω (M) the set of compatible almost complex structures. Gromov’s lemma (Gromov, 1985) says that Jω is a contractible infinite-dimensional (Fr´echet) manifold. We denote the associated family of metrics on M by g J = ω(·, J·) and its associated norm by | · | J . When we are given a one-periodic family J = {Jt }t∈S 1 , it induces the associated L2 -metric on L(M) by ·, · which can be written as 1 ξ1 , ξ2 J = ω ξ1 (t), Jt ξ2 (t) dt. (15) 0

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From now on, we will always denote by J an S 1 -family of compatible almost complex structures unless otherwise stated, and denote jω := C ∞ (S 1 , Jω ). If we denote by grad J AH the associated L2 -gradient vector field, (10) and (15) imply that grad J AH has the form (16) grad J AH ([γ, w])(t) = Jt γ˙ (t) − XH t, γ(t) which we will simply write J γ˙ − XH (γ) . It follows from this formula that the gradient is projectable to L0 (M). Therefore when we project the negative gradient 0 (M) to L0 (M), it has the form flow equation of a path u: R → L ∂u ∂u +J − XH (u) = 0 (17) ∂τ ∂t if we regard u as a map u: R × S 1 → M. We call this equation Floer’s perturbed Cauchy – Riemann equation or simply as the perturbed Cauchy-Riemann equation associated to the pair (H, J). The Floer theory largely relies on the study of the moduli spaces of finite energy solutions u: R × S 1 → M of the kind (17) of perturbed Cauchy – Riemann equations. The relevant energy function is given by DEFINITION 2.15 (Energy). For a given smooth map u: R × S 1 → M, we define the energy, denoted by E(H,J) (u), of u by 2 1 ∂u 2 ∂u + − XH (u) dt dτ. E(H,J) (u) = 2 ∂τ Jt ∂t Jt The following lemma exemplifies significance of the finite energy condition. Although the proof is standard, we provide details of the proof for the reader’s convenience to illustrate the kind of analytic arguments used in the study of perturbed Cauchy – Riemann equations. PROPOSITION 2.16. Let H: S 1 × M → R be any Hamiltonian. Suppose that u: R × S 1 → M is a finite energy solution of (17). Then there exists a sequence τk → ∞ (respectively τk → −∞) such that the loop zk := u(τk ) = u(τk , ·) C ∞ converges to a one-periodic solution z: S 1 → M of the Hamilton equation x˙ = XH (x). Proof. Since u satisfies (17), the energy of u can be re-written as 2 ∂u − XH (u) dt dτ. E(H,J) (u) = ∂t Jt

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Therefore the finite energy condition, in particular, implies existence of τk , ∞ such that 1 2 ∂u (τ , ·) − X u(τ , ·) dt → 0 (18) k H k J 0 ∂t t as k → ∞. Since M is compact, XH is bounded and so (18) implies 1 |˙zk |2Jt dt → 0

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0

for some C > 0 independent of k. Equation (19) implies the equicontinuity of zk and so there exists a subsequence, which we still denote by τk , such that zk → z∞ in C 0 -topology. Furthermore Fatou’s lemma implies 1 1 2 |˙z − XH (z)| Jt dt ≤ lim inf |˙zk − XH (zk )|2Jt dt → 0. k

0

0

Therefore z is a weak solution of x˙ = XH (x), which lies in W 1,2 . In particular, z˙ lies in W 2,2 (S 1 ), which follows from diﬀerentiating z˙ = XH (z). Then the Sobolev embedding W 2,2 (S 1 ) → C 1 (S 1 ) implies that z is C 1 and satisfies x˙ = XH (x). Once we know z is C 1 , the boot-strap argument by diﬀerentiating z˙ = XH (z) implies z is smooth. Finally since zk → z in C 0 , so does XH (zk ) → XH (z), which in turn implies zk → z in C 1 . Similar boot-strap argument then implies the C ∞ -convergence of zk → z. This finishes the proof. We denote by M(H, J) = M(H, J; ω) the set of finite energy solutions of (17) for general H not necessarily nondegenerate. Similar discussion can be carried out for the nonautonomous version of (17), which we now describe. We first denote Hm = Hm (M) := {H : S 1 × M → R | H is normalized}. Consider the R-family HR : R → Hm ; jR : R → Jω ;

τ → H(τ) τ → J(τ)

that are asymptotically constant, i.e., H(τ) = H ±∞ ,

J(τ) = J ±∞

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for some H ±∞ ∈ Hm and J ±∞ ∈ jω if |τ| > R for a suﬃciently large constant R. To any such pair is associated the following nonautonomous version of (17) ∂u ∂u + J(τ) − XH(τ) (u) = 0. (20) ∂τ ∂t The associated energy function is given by 2 ∂u 1 ∞ 1 ∂u 2 + − XH(τ) (u) dt dτ. E(HR , jR ) (u) = 2 −∞ 0 ∂τ J(τ) ∂t J(τ) We denote by M(HR , jR ) = M(HR , jR ; ω) the set of finite energy solutions of (20). Here is the analog to Proposition 2.16, whose proof is essentially the same as Proposition 2.16 due to the asymptotically constant condition on (H, j). PROPOSITION 2.17. Let HR and jR be as above. Suppose that u: R×S 1 → M is a finite energy solution of (20). Then there exists a sequence τk → ∞ (respectively τk → −∞) such that the loop zk := u(τk ) = u(τk , ·) C ∞ converges to a one-periodic solution z: S 1 → M of the Hamilton equation x˙ = XH ±∞ (x) respectively. A typical way how such an asymptotically constant family appears is through an elongation of a given smooth one-parameter family over [0, 1]. DEFINITION 2.18. A continuous map f : [0, 1] → T for any topological space T is said to be boundary flat if the map is constant near the boundary ∂[0, 1] = {0, 1}. Let H: [0, 1] → Hm be a homotopy connecting two Hamiltonians Hα , Hβ ∈ Hm , and j: [0, 1] → Jω connecting Jα , Jβ ∈ Jω . We denote P( jω ) := C ∞ ([0, 1], jω ) P(Hm ) := C ∞ ([0, 1], Hm ). We define a function ρ: R → [0, 1] of the type 0 for τ ≤ −R ρ(τ) = 1 for τ ≥ R

(21)

for some R > 0. We call ρ a (positively) monotone cut-oﬀ function if it satisfies ρ (τ) ≥ 0 for all τ’s in addition. Each such pair (H, j), combined with a cut-oﬀ function ρ, defines a pair ρ (H , jρ ) of asymptotically constant R-families HR = H ρ ,

jR = jρ

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where H ρ is the reparameterized homotopy H ρ = {H ρ }τ∈R defined by τ → H ρ (τ, t, x) = H(ρ(τ), t, x). We call H ρ the ρ-elongation of H or the ρ-elongated homotopy of H. The same definition applies to j. Therefore such a triple (H, j; ρ) gives rise to the nonautonomous equation ∂u ∂u (22) + J ρ(τ) − XH ρ(τ) (u) = 0. ∂τ ∂t We denote by M(H, j; ρ) the set of finite energy solutions of (22). 2.6. COMPARISON OF TWO CAUCHY – RIEMANN EQUATIONS

In this subsection, we explain the relation between Floer’s standard perturbed Cauchy – Riemann equation (17) for u: R × S 1 → M and its mapping cylinder version v: R × R → M   ∂v ∂v    + Jt = 0    ∂τ ∂t 2 (23)    ∂v < ∞   φ v(τ, t + 1) = v(τ, t),  ∂τ J  t

where φ = φ1H . We often restrict v to R × [0, 1] and consider it as a map from R × [0, 1] that satisfies φ v(τ, 1) = v(τ, 0). A similar correspondence had been exploited in Oh (1997; 1999) in the ‘open string’ context of Lagrangian submanifolds for the same purpose, and call the former version of Floer homology the dynamical and the latter geometric. We do the same here. For any given solution u = u(τ, t): R×S 1 → M, we ‘open up’ u along t = 0 ≡ 1 and define the map v: R × [0, 1] → M by

v(τ, t) = (φtH )−1 u(τ, t) (24) and then extend to R so that φ v(τ, t + 1) = v(τ, t). A simple computation shows that when u satisfies (17) the map v satisfies (23), provided the family J = {Jt }0≤t≤1 is defined by Jt = (φtH )∗ Jt for the given periodic family J used for the equation (17), and vice versa. By definition, this family J of almost complex structure satisfies J (t + 1) = φ∗ J (t).

(25)

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One can even fix J(0) = J0 for any given almost complex structure J0 which leads to the following definition (Oh, 2005a) DEFINITION 2.19. Let J0 ∈ Jω and φ ∈ Ham(M, ω). We define j(φ,J0 ) by j(φ,J0 ) := {J : [0, 1] → Jω | J (t + 1) = φ∗ J (t), J (0) = J0 }. The condition

φ v(τ, t + 1) = v(τ, t)

(26) (27)

enables us to consider the map v: R × R → M as a pseudo-holomorphic section of the ‘mapping cylinder’ Eφ := R × Mφ = R × R × M/ τ, t + 1, φ(x) ∼ (τ, t, x) where Mφ is the mapping circle defined by Mφ := R × M/ t + 1, φ(x) ∼ (t, x). Note that the product symplectic form dτ ∧ dt + ω on R × R × M naturally projects to Eφ since φ is symplectic, and so Eφ has the structure of a Hamiltonian fibration. In this setting, v: R × R → M can be regarded as the section s: R × S 1 → Eφ defined by s(τ, t) = [τ, t, v(τ, t)] which becomes a pseudo-holomorphic section of Eφ for a suitably defined almost complex structure. One advantage of the mapping cylinder version over the more standard dynamical version (17) is that its dependence on the Hamiltonian H is much weaker than in the latter. Indeed, this mapping cylinder version can be put into the general framework of Hamiltonian fibrations with given fixed monodromy of the fibration at infinity as in Entov (2000). This framework turns out to be essential to prove the triangle inequality of the spectral invariants. (See Schwarz, 2000; Oh, 2005a; or Section 7 later in this paper). Another important ingredient is the comparison of two diﬀerent energies E(H,J) (u) and E J (v): for the given J = {Jt }0≤t≤1 ∈ j(J0 ,φ) , we define the energy of the map v: R × [0, 1] → M by ∂v 2 ∂v 2 1 + dt dτ. E J (v) = 2 R×[0,1] ∂τ Jt ∂t Jt This energy is the vertical part of the energy of the section s: R×S 1 → Eφ defined above with respect to a suitably chosen almost complex structure J˜ on Eφ . (See

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Oh, 2005c, Section 3, for more explanation.) Note that because of (25) – (27), one can replace the domain of integration R × [0, 1] by any fundamental domain of the covering projection R × R → R × (R/Z) without changing the integral. The choice of R × [0, 1] is one such choice. The following identity plays an important role in the proof of the nondegeneracy of the invariant norm we construct later. The proof is a straightforward computation left to the readers. LEMMA 2.20. Let J = {Jt }0≤t≤1 be a periodic family and define J = {Jt }0≤t≤1 by Jt = (φtH )∗ Jt . Let u: R × S 1 → M be any smooth map and v: R × [0, 1] → M be the map defined by v(τ, t) = (φtH )−1 u(τ, t) . Then we have E(H,J) (u) = E J (v).

3. Floer complex and the Novikov ring In this section we provide the details of construction of the Floer complex and its basic operators. The details of construction are given in Floer (1989) and Salamon and Zehnder (1992), for example. But we closely follow the exposition given in Oh (2005a). 3.1. NOVIKOV – FLOER CHAINS AND THE NOVIKOV RING

Suppose that φ ∈ Ham(M, ω) is nondegenerate. For each nondegenerate H: S 1 × M → R, we know that the cardinality of Per(H) is finite. We consider the free Q vector space generated by the critical set of AH 0 (M) | z ∈ Per(H)}. Crit AH = {[z, w] ∈ Ω To be able to define the Floer boundary operator correctly, we need to complete this vector space downward with respect to the real filtration provided by the action AH ([z, w]) of the element [z, w] as in Floer (1989) and Hofer and Salamon (1995). More precisely, we give the following definitions slightly streamlining those from Oh (2002).

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DEFINITION 3.1. Consider the formal sum β= a[z,w] [z, w],

a[z,w] ∈ Q

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[z,w]∈Crit AH

1. We call those [z, w] with a[z,w] 0 generators of the sum β and write [z, w] ∈ β. We also say that [z, w] contributes to β in that case. 2. We define the support of β by supp(β) := {[z, w] ∈ Crit AH | a[z,w] 0 in the sum (28)}. 3. We call the formal sum β a Novikov – Floer chain (or simply a Floer chain) if #(supp(β) ∩ {[z, w] | AH ([z, w]) ≥ λ}) < ∞

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for any λ ∈ R. We denote by CF∗ (H) the set of Floer chains. Note that CF∗ (H) is a Q-vector space which is always infinite-dimensional in general, unless (M, ω) is symplectically aspherical. Since the aspherical case was studied in Oh (1999) and Schwarz (2000) before, we will focus on the general case where the quantum contributions could be present. There is a natural grading on CF∗ (H): we associate the Conley – Zehnder index, denote by µH ([z, w]) to each generator [z, w] ∈ Crit AH . We refer to Conley and Zehnder (1984), Salamon and Zehnder (1992), and Hofer and Salamon (1995) for the definition of µH ([z, w]). For readers’ convenience, we recall the definition in Appendix in the course of proving the index formula in our convention. Now consider a Floer chain β= a[z,w] [z, w], a[z,w] ∈ Q. Following Oh (2002), we introduce the following notion which is a crucial concept for the mini-max argument that we carry out in this paper. DEFINITION 3.2. Let β 0 be a Floer chain in CF∗ (H). We define the level of the chain β and denote it by λH (β) = max{AH ([z, w]) | [z, w] ∈ supp(β)}, [z,w]

and set λH (0) = −∞. We call a generator [z, w] ∈ β satisfying AH ([z, w]) = λH (β) a peak of β, and denote it by peak(β). We emphasize that it is the Novikov condition 3 of Definition 3.1 that guarantees that λH (β) is well-defined. The following lemma illustrates optimality of the definition of the Novikov covering space.

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LEMMA 3.3. Let β 0 be a homogeneous Floer chain. Then the peak of β over a fixed periodic orbit is unique. Proof. By the assumption of homogeneity, the generators of β have the same Conley – Zehnder indices. Let [z, w] and [z, w ] be two such peaks of β. Then we have AH ([z, w]) = λH (β) = AH ([z, w ]) which in turn implies ω([w]) = ω([w ]). By the homogeneity assumption, we also have µH ([z, w]) = µH ([z, w ]). It follows from the definition of Γ-equivalence classes that [z, w] = [z, w ], which finishes the proof. So far we have defined CF∗ (H) as a Z-graded Q-vector space with Crit AH as its generating set which has infinitely many elements, unless (M, ω) is symplectically aspherical. We now explain the description of CF(H) as a module over the Novikov ring as in Floer (1989) and Hofer and Salamon (1995). We consider the group ring Q[Γ] consisting of the finite sum R=

k

ri qAi ∈ Q([Γ])

i=1

and define its support by supp R = {A ∈ Γ | A = Ai in this sum}. We recall the valuation v: Γ → R and the degree map d: Γ → R. We now define a valuation v: Q[Γ] → R by ↓

v(R) = v (R) = v

↓

k

ri q

Ai

:= max{ω(Ai ) | Ai ∈ supp R}.

i=1

This satisfies the following non-Archimedean triangle inequality v(R1 + R2 ) ≤ max{v(R1 ), v(R2 )}

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and so induces a natural metric topology on Q[Γ]. DEFINITION 3.4. The (downward) Novikov ring is the downward completion Q[[Γ] of Q[Γ] with respect to the valuation v: Q[Γ] → R. We denote it by Λ↓ω . More concretely we have the description ↓ A Λω = rA q ∀λ ∈ R, #{A ∈ Γ | rA 0, ω(A) > λ} < ∞ . A∈Γ

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Similarly we define the upward Novikov ring, denoted by Λ↑ω , by ↑ −A Λω = rA q ∀λ ∈ R, #{A ∈ Γ | rA 0, ω(−A) < λ} < ∞ . A∈Γ

Since we will mostly use the downward Novikov ring in this lecture, we will just denote Λω = Λ↓ω dropping the arrow. Then we have the valuation on Λω given by v(R) = max{ω(A) | A ∈ supp R}.

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We recall that Γ induces a natural action on Crit AH by ‘gluing a sphere’ [z, w] → [z, w # A] which in turn induces the multiplication of Λω on CF(H) by the convolution product. This enables one to regard CF(H) as a Λω -module. We will try to consistently denote by CF(H) as a Λω -module, and by CF∗ (H) as a graded Q vector space. The action functional provides a natural filtration on CF∗ (H): for any given λ ∈ R \ Spec(H), we define

CFλ∗ (H) = α ∈ CF∗ (H) | AH peak(α) ≤ λ and denote the natural inclusion homomorphism by iλ : CFλ∗ (H) → CF∗ (H). 3.2. DEFINITION OF THE FLOER BOUNDARY MAP

Suppose H is a nondegenerate one-periodic Hamiltonian function and J a oneperiodic family of compatible almost complex structures. We first recall Floer’s construction of the Floer boundary map, and the transversality conditions needed to define the Floer homology HF∗ (H, J) of the pair. The following definition is useful for the later discussion. DEFINITION 3.5. Let z, z ∈ Per(H). We denote by π2 (z, z ) the set of homotopy classes of smooth maps u: [0, 1] × S 1 := T → M relative to the boundary u(0, t) = z(t),

u(1, t) = z (t).

We denote by [u] ∈ π2 (z, z ) its homotopy class and by C a general element in π2 (z, z ).

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We define by π2 (z) the set of relative homotopy classes of the maps w: D2 → M;

w|∂D2 = z.

We note that there is a natural action of π2 (M) on π2 (z) and π2 (z, z ) by the obvious operation of a ‘gluing a sphere.’ Furthermore there is a natural map of C ∈ π2 (z, z ) (·) # C: π2 (z) → π2 (z ) induced by the gluing map w → w # u. More specifically we will define the map w # u: D2 → M in the polar coordinates (r, θ) of D2 by the formula w(2r, θ) for 0 ≤ r ≤ 12 (32) w # u: (r, θ) = w(2r − 1, θ) for 12 ≤ r ≤ 1 once and for all. There is also the natural gluing map π2 (z0 , z1 ) × π2 (z1 , z2 ) → π2 (z0 , z2 ) (u1 , u2 ) → u1 # u2 . We also explicitly represent the map u1 # u2 : T → M in the standard way once and for all similarly to (32). DEFINITION 3.6. We define the relative Conley – Zehnder index of C ∈ π2 (z, z ) by µH (z, z ; C) = µH ([z, w]) − µH ([z , w # C]) for a (and so any) representative u: [0, 1] × S 1 × M of the class C. We will also write µH (C), when there is no danger of confusion on the boundary condition. It is easy to see that this definition does not depend on the choice of bounding disc w of z, and so the function µH : π2 (z, z ) → Z is well-defined. REMARK 3.7. In fact, the function µH : π2 (z, z ) → Z can be defined without assuming z0 , z1 being contractible, as long as z0 and z1 lie in the same component of Ω(M): For any given map u: T → M, choose a marked symplectic trivialization Φ: u∗ T M → T × R2n

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that satisfies Φ ◦ Φ−1 |[0,1]×{1} = id. We know that z0 (t) = φtH (p0 ) and z1 (t) = φtH (p1 ) for p0 , p1 ∈ Fix(φ1H ). Then we have two maps αΦ,i : [0, 1] → Sp(2n), i = 0, 1, such that Φ ◦ dφtH (pi ) ◦ Φ−1 (i, t, v) = (i, t, αΦ,i (t)v) for v ∈ R2n and t ∈ [0, 1]. By the nondegeneracy of H, the maps αΦ,i define elements in SP∗ (1). Then we define µH (z, z ; C) := µCZ (αΦ,0 ) − µCZ (αΦ,1 ). It is easy to check that this definition does not depend on the choice of marked symplectic trivializations. We now denote by M(H, J; z, z ; C) the set of finite energy solutions of (17) with the asymptotic condition and the homotopy condition u(−∞) = z,

u(∞) = z ;

[u] = C.

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Here we remark that although u is a priori defined on R × S 1 , it can be compactified into a continuous map u¯ : [0, 1] × S 1 → M with the corresponding boundary condition u¯ (0) = z, u¯ (1) = z due to the exponential decay property of solutions u of (42), recalling we assume H is nondegenerate. We will call u¯ the compactified map of u. By some abuse of notation, we will also denote by [u] the class [¯u] ∈ π2 (z, z ) of the compactified map u¯ . The Floer boundary map ∂(H,J) : CFk+1 (H) → CFk (H) is defined under the following conditions. (See Floer, 1989; Hofer and Salamon, 1995.) DEFINITION 3.8 (The boundary map). Let H be nondegenerate. Suppose that J satisfies the following conditions:

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1. For any pair (z0 , z1 ) ⊂ Per(H) satisfying µH (z0 , z1 ; C) = µH ([z0 , w0 ]) − µH ([z1 , w0 # C]) = 0, M(H, J; z0 , z1 ; C) = ∅ unless z0 = z1 and C = 0. When z0 = z1 and C = 0, the only solutions are the stationary solution, i.e., u(τ) ≡ z0 = z1 for all τ ∈ R. 2. For any pair (z0 , z1 ) ⊂ Per(H) and a homotopy class C ∈ π2 (z0 , z1 ) satisfying µH (z0 , z1 ; C) = 1, M(H, J; z0 , z1 ; C)/R is transverse and compact and so a finite set. We denote n(H, J; z0 , z1 ; C) = #(M(H, J; z0 , z1 ; C)/R) the algebraic count of the elements of the space M(H, J; z0 , z1 ; C)/R. We set n(H, J; z0 , z1 : C) = 0 otherwise. 3. For any pair (z0 , z2 ) ⊂ Per(H) and C ∈ π2 (z0 , z2 ) satisfying µH (z0 , z2 ; C) = 2, M(H, J; z0 , z2 ; C)/R can be compactified into a smooth one-manifold with boundary comprising the collection of the broken trajectories [u1 ] #∞ [u2 ] where u1 ∈ M(H, J; z0 , y; C1 ) and u2 ∈ M(H, J; y, z2 ; C2 ) for all possible y ∈ Per(H) and C1 ∈ π2 (z0 , y), C2 ∈ π2 (y, z2 ) satisfying C1 # C2 = C;

[u1 ] ∈ M(H, J; z0 , y; C1 )/R,

[u2 ] ∈ M(H, J; y, z2 ; C2 )/R

and µH (z0 , y; C1 ) = µH (y, z2 ; C2 ) = 1. Here we denote by [u] the equivalence class represented by u. We call any such J H-regular and call such a pair (H, J) Floer-regular. The upshot is that for a Floer-regular pair (H, J) the Floer boundary map ∂ = ∂(H,J) : CF∗ (H) → CF∗ (H) is defined and satisfies ∂∂ = 0, which enables us to take its homology. We now explain this construction in detail. For each given [z− , w− ] and + [z , w+ ], we collect the elements C ∈ π2 (z− , z+ ) satisfying 0 (M) [z+ , w+ ] = [z+ , w− #C] in L

(34)

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and define the moduli space M(H, J; [z− , w− ], [z+ , w+ ]) :=

{M(H, J; z− , z+ ; C) | C satisfies (34)}. C

We like to note that there could be more than one C ∈ π2 (z− , z+ ) that satisfies (34) 0 (M). The following lemma according to the definition of the Γ-covering space L is an easy consequence of a standard compactness argument. LEMMA 3.9. This union is a finite union. In other words, for any given pair ([z− , w− ], [z+ , w+ ]), there are only a finite number of C ∈ π2 (z− , z+ ) that satisfies (34) and M(H, J; z− , z+ ; C) ∅. 0 (M), we write the Now considering u as a path in the covering space L − − + + asymptotic condition of u ∈ M(H, J; [z , w ], [z , w ]) as u(−∞) = [z− , w− ],

u(∞) = [z+ , w+ ].

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The Floer boundary map ∂ = ∂(H,J) : CF∗ (H) → CF∗ (H) is defined by its matrix coeﬃcient n(H,J) (z− , z+ ; C), ∂([z− , w− ]), [z+ , w+ ] := C

where C is as in (34) and the Conley – Zehnder indices of [z− , w− ] and [z+ , w+ ] satisfy µH ([z− , w− ]) − µH ([z+ , w+ ]) = µ(z− , z+ ; C) = 1, We set the matrix coeﬃcient to be zero otherwise. ∂ = ∂(H,J) has degree −1 and satisfies ∂ ◦ ∂ = 0. DEFINITION 3.10. We say that a Floer chain β ∈ CF(H) is Floer cycle of (H, J) if ∂β = 0, i.e., if β ∈ ker ∂(H,J) , and a Floer boundary if β ∈ im ∂(H,J) . Two Floer chains β, β are said to be homologous if β − β is a boundary. We denote ZF∗ (H, J) = ker ∂,

BF∗ (H, J) = im ∂

and then the Floer homology is defined by HF∗ (H, J) := ZF∗ (H, J)/ BF∗ (H, J). One may regard this either as a graded Q-vector space or as a Λω -module. We will mostly consider it as a graded Q-vector space in this lecture, because it well 0 (M). suits the mini-max theory of the action functional on L

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3.3. DEFINITION OF THE FLOER CHAIN MAP

When we are given a family (H, j) with H = {H s }0≤s≤1 and j = {J s }0≤s≤1 and a cut-oﬀ function ρ: R → [0, 1], the chain homomorphism hH = h(H, j) : CF∗ (Hα ) → CF∗ (Hβ ) is defined by considering the nonautonomous equation (20). It may be instructive to mention that (14) is not a gradient-like flow unlike (17). We now provide the details. Consider the pair (HR , jR ) that are asymptotically constant, i.e., there exists R > 0 such that J(τ) ≡ J(∞), H(τ) ≡ H(∞) for all τ with |τ| ≥ R. DEFINITION 3.11 (The chain map). We say that (HR , jR ) is Floer-regular if the following holds: 1. For any pair z0 ∈ Per(H0 ) and z1 ⊂ Per(H1 ) satisfying µHR (z0 , z1 ; C) = 0, M(HR , jR ; z0 , z1 ; C) is transverse and compact, and so a finite set. We denote n(HR , jR ; z0 , z1 ; C) := # M(HR , jR ; z0 , z1 ; C) the algebraic count of the elements in M(HR , jR ; z0 , z1 ; C). Otherwise, we set n(HR , jR ; z0 , z1 : C) = 0. 2. For any pair z0 ∈ Per(H0 ) and z1 ∈ Per(H1 ) satisfying µHR (z0 , z2 ; C) = 1, M(H, J; z0 , z2 ; C) is transverse and can be compactified into a smooth onemanifold with boundary comprising the collection of the broken trajectories u1 #∞ u2 where (u1 , u2 ) ∈ M(HR , jR ; z0 , y : C1 ) × M(H(∞), J(∞); y, z2 : C 2 ); µHR (z0 , y; C1 ) = 0, µH (y, z2 ; C2 ) = 1 or (u1 , u2 ) ∈ M(H(−∞), J(−∞); z0 , y : C 1 ) × M(HR , jR ; y, z2 : C1 ); µHR (z0 , y; C1 ) = 1, µH (y, z2 ; C2 ) = 0 and C1 # C2 = C for all possible such y ∈ Per(H) and C1 ∈ π2 (z0 , y), C2 ∈ π2 (y, z2 ).

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We say that (HR , jR ) are Floer-regular if it satisfies these conditions. Now suppose that (H, j) is a homotopy connecting two Floer-regular pairs (Hα , Jα ) and (Hβ , Jβ ). Choose a cut-oﬀ function ρ: R → [0, 1]. For each such pair (H, j) and a cut-oﬀ function ρ, we consider the ρ-elongations H ρ and jρ respectively. Therefore to such a triple (H, j; ρ) is associated the nonautonomous equation (20) with the boundary condition u(−∞) = z0 ,

u(∞) = z1

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and the homotopy condition [u] = C ∈ π2 (z0 , z1 ) for a fixed C. Now for each given pair of [zα , wα ] ∈ Crit AHα and [zβ , wβ ] ∈ Crit AHβ , we define M (H, j; ρ); [zα , wα ], [zβ , wβ ] := M (H, j; ρ); zα , zβ ; C C

where C ∈ π2 (zα , zβ ) are the elements satisfying [zβ , wβ ] = [zβ , wα # C]

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similarly as in (34). We say that (H, j; ρ) is Floer-regular if the ρ-elongation (H ρ , jρ ) is Floer-regular in the sense of Definition 3.11. Under the condition in Definition 3.11, we can define a map of degree zero h(H, j;ρ) : CF(Hα ) → CF(Hβ ) by the matrix element n(H, j;ρ) ([zα , wα ], [zβ , wβ ]) similarly as for the boundary map. The conditions in Definition 3.11 then also imply that h(H, j) has degree 0 and satisfies the identity h(H, j;ρ) ◦ ∂(Hα ,Jα ) = ∂(Hβ ,Jβ ) ◦ h(H, j;ρ) . Two such chain maps h( j1 ,H 1 ) , h( j2 ,H 2 ) are also chain homotopic (Floer, 1989). 3.4. SEMI-POSITIVITY AND TRANSVERSALITY

In this subsection, we briefly discuss the hypotheses imposed in Definitions 3.8 and 3.11. For the case of the boundary map ∂(H,J) , Hofer and Salamon (1995) prove the following THEOREM 3.12. Suppose (M, ω) satisfies that there is no A ∈ π2 (M) such that ω(A) > 0

and

4 − n ≤ c1 (A) < 0.

Then the hypotheses stated in Definition 3.8 hold.

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For the case of the chain map h(H, j;ρ) , they prove THEOREM 3.13. Suppose (M, ω) satisfies that there is no A ∈ π2 (M) such that ω(A) > 0

and

3 − n ≤ c1 (A) < 0.

Then the hypotheses stated in Definition 3.11 hold. This leads one to introduce the following definition. DEFINITION 3.14. A symplectic manifold (M, ω) is called semi-positive if it satisfies that there is no A ∈ π2 (M) such that ω(A) > 0

and

3 − n ≤ c1 (A) < 0.

For the later purpose of studying the pants product on the Floer complex, following Seidel (1997) and Entov (2000), we introduce DEFINITION 3.15. A symplectic manifold (M, ω) is called strongly semi-positive if it satisfies that there is no A ∈ π2 (M) such that ω(A) > 0

and

2 − n ≤ c1 (A) < 0.

For the general symplectic manifolds, one needs to use the concept of virtual moduli cycle and abstract multivalued perturbations in the context of the Kuranishi structure (Fukaya and Ono, 1999). We will make further remarks in Section 10 in relation to the numerical estimates concerning the energy of solutions and the levels of the Novikov – Floer cycles. 3.5. COMPOSITION LAW OF FLOER’S CHAIN MAPS

In this section, we examine the composition raw hαγ = hβγ ◦ hαβ of the Floer’s canonical isomorphism hαβ : HF∗ (Hα ) → HF∗ (Hβ ).

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Although the above isomorphism in homology depends only on the end Hamiltonians Hα and Hβ , the corresponding chain map depends on the homotopy H = {H(η)}0≤η≤1 between Hα and Hβ , and also on the homotopy j = {J(η)}0≤η≤1 . Let us fix nondegenerate Hamiltonians Hα , Hβ and a homotopy H between them. We then fix a homotopy j = {J(η)}0≤η≤1 of compatible almost complex structures and a cut-oﬀ function ρ: R → [0, 1].

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We recall that we have imposed the homotopy condition [w+ ] = [w− # u];

[u] = C in π2 (z− , z+ ) (39) in the definitions of M(H, J; [z−, w− ], [z+ , w+ ]) and M (H, j; ρ); [zα , wα ], [zβ , wβ ] . One consequence of (39) is [z+ , w+ ] = [z+ , w− # u] in Γ but the latter is a weaker condition than the former. In other words, there could be more than one distinct elements C1 , C2 ∈ π2 (z− , z+ ) such that µ(z− , z+ ; C1 ) = µ(z− , z+ ; C2 ),

ω(C1 ) = ω(C2 ).

When we are given a homotopy ( ¯, H) of homotopies with ¯ = { jκ }, H = {Hκ }, we also define the elongations H ρ¯ of Hκ by a homotopy of cut-oﬀ functions ρ¯ = {ρκ }: we have ρ H ρ¯ = {Hκ κ }0≤κ≤1 . Consideration of the parameterized version of (22) for 0 ≤ κ ≤ 1 defines the chain homotopy map HH : CF∗ (Hα ) → CF∗ (Hβ ) which has degree +1 and satisfies h( j1 ,H1 ;ρ1 ) − h( j0 ,H0 :ρ0 ) = ∂(J 1 ,H 1 ) ◦ HH + HH ◦ ∂(J 0 ,H 0 ) .

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Again the map HH depends on the choice of a homotopy ¯ and ρ¯ = {ρκ }0≤κ≤1 connecting the two functions ρ0 , ρ1 . Therefore we will denote HH = H(H, ¯;ρ) ¯ as well. Equation (40) in particular proves that two chain maps for diﬀerent homotopies ( j0 , H0 ; ρ0 ) and ( j1 , H1 ; ρ1 ) connecting the same end points are chain homotopic (Floer, 1989) and so proves that the isomorphism (38) in homology is independent of the homotopies (H, ¯) or of ρ. ¯ Now we re-examine the equation (20). One key analytic fact in the study of the Floer moduli spaces is an a priori upper bound of the energy, which we will explain in the next section. Next, we consider the triple (Hα , Hβ , Hγ ) of Hamiltonians and homotopies H1 , H2 connecting from Hα to Hβ and Hβ to Hγ respectively. We define their concatenation H1 # H2 = {H3 (s)}1≤s≤1 by H1 (2s) 0 ≤ s ≤ 12 H3 (s) = H2 (2s − 1) 12 ≤ s ≤ 1.

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We note that due to the choice of the cut-oﬀ function ρ, the continuity equation (20) is autonomous for the region |τ| > R i.e., is invariant under the translation by τ. When we are given a triple (Hα , Hβ , Hγ ), this fact enables us to glue solutions of two such equations corresponding to the pairs (Hα , Hβ ) and (Hβ , Hγ ) respectively. Now a more precise explanation is in order. For a given pair of cut-oﬀ functions ρ = (ρ1 , ρ2 ) and a positive number R > 0, we define an elongated homotopy of H1 # H2 H1 #(ρ;R) H2 = {H(ρ;R) (τ)}−∞<τ<∞ by

H(ρ;R) (τ, t, x) =

Note that H(ρ;R)

H1 (ρ1 (τ + 2R), t, x) τ ≤ 0 H2 (ρ2 (τ − 2R), t, x) τ ≥ 0.

  H    α Hβ ≡   H γ

for τ ≤ −(R1 + 2R) for −R ≤ τ ≤ R for τ ≥ R2 + 2R

for some suﬃciently large R1 , R2 > 0 depending on the cut-oﬀ functions ρ1 , ρ2 and the homotopies H1 , H2 respectively. In particular this elongated homotopy is always smooth, even when the usual glued homotopy H1 # H2 may not be so. We define the elongated homotopy j1 #(ρ;R) j2 of j1 # j2 in a similar way. For an elongated homotopy ( j1 #(ρ;R) j2 , H1 #(ρ,R) H2 ), we consider the associated perturbed Cauchy – Riemann equation  ∂u  ρ(τ) ∂u   (u) =0 + J − X ρ(τ)   ∂τ 3 H3 ∂t     lim u(τ) = z− , lim u(τ) = z+  τ→−∞

τ→∞

with the condition (39). Now let u1 and u2 be given solutions of (22) associated to ρ1 and ρ2 respectively. If we define the pre-gluing map u1 #R u2 by the formula u (τ + 2R, t) for τ ≤ −R u1 #R u2 (τ, t) = 1 u2 (τ − 2R, t) for τ ≥ R and a suitable interpolation between them by a partition of unity on the region −R ≤ τ ≤ R, the assignment defines a diﬀeomorphism (u1 , u2 , R) → u1 #R u2 from M( j1 , H1 ; [z1 , w1 ], [z2 , w2 ]) × M( j2 , H2 ; [z2 , w2 ], [z3 , w3 ]) × (R0 , ∞)

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onto its image, provided R0 is suﬃciently large. Denote by ∂¯ (H, j;ρ) the corresponding perturbed Cauchy-Riemann operator ∂u ρ(τ) ∂u + J3 − XH ρ(τ) (u) u → 3 ∂τ ∂t acting on the maps u satisfying the asymptotic u(± < in f ty) = z± and fixed homotopy condition [u] = C ∈ π2 (z− , z+ ). By perturbing u1 #R u2 by the amount that is smaller than the error for u1 #R u2 to be a genuine solution, i.e., less than a weighted L p -norm, for p > 2, ∂¯ (H, j;ρ) (u1 #(ρ;R) u2 ) p in a suitable W 1,p space of u’s (see Floer, 1988, 1989), one can construct a unique genuine solution near u1 #R u2 . By an abuse of notation, we will denote this genuine solution also by u1 #R u2 . Then the corresponding map defines an embedding M( j1 , H1 ; [z1 , w1 ], [z2 , w2 ]) × M( j2 , H2 ; [z2 , w2 ], [z3 , w3 ]) × (R0 , ∞) → M( j1 #(ρ;R) j2 , H1 #(ρ;R) H2 ; [z1 , w1 ], [z3 , w3 ]). Especially when we have µHβ ([z2 , w2 ]) − µHα ([z1 , w1 ]) = µHγ ([z3 , w3 ]) − µHβ ([z2 , w2 ]) = 0 both M( j1 , H1 ; [z1 , w1 ], [z2 , w2 ]) and M( j2 , H2 ; [z2 , w2 ], [z3 , w3 ]) are compact, and so consist of a finite number of points. Furthermore the image of the above-mentioned embedding exhausts the ‘end’ of the moduli space M( j1 #(ρ;R) j2 , H1 #(ρ;R) H2 ; [z1 , w1 ], [z3 , w3 ]) and the boundary of its compactification consists of the broken trajectories u1 #(ρ;∞) u2 = u1 #∞ u2 . This then proves the following gluing identity PROPOSITION 3.16. There exists R0 > 0 such that for any R ≥ R0 we have h(H1 , j1 )#(ρ;R) (H2 , j2 ) = h(H1 , j1 ;ρ1 ) ◦ h(H2 , j2 ;ρ2 ) as a chain map from CF∗ (Hα ) to CF∗ (Hγ ). Here we remind the readers that the homotopy H1 #(ρ;R) H2 itself is an elongated homotopy of the glued homotopy H1 # H2 . This proposition then gives rise to the composition law hαγ = hβγ ◦ hαβ in homology.

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4. Energy estimates and Hofer’s geometry 4.1. ENERGY ESTIMATES AND THE ACTION LEVEL CHANGES

Let us fix the Hamiltonians Hα , Hβ and a homotopy H between them. We emphasize that Hα and Hβ are not necessarily nondegenerate for the discussion of this section. We choose a homotopy j = {J(η)}0≤η≤1 of compatible almost complex structures and a cut-oﬀ function ρ: R → [0, 1]. We would like to mention that the homotopies can be constant when Hα = Hβ . Now we re-examine the equation (20) (also (17) as a special case where H ≡ H and j ≡ J). One key analytic fact on the study of moduli spaces of the equations is an a priori upper bound of the energy 2 ∂u 1 ∞ 1 ∂u 2 E(H, j;ρ) (u) := + − XH ρ(τ) (u) dt dτ 2 −∞ 0 ∂τ J ρ(τ) ∂t J ρ(τ) for the solutions u of (20) with (39). In this respect, the following identity is crucial. LEMMA 4.1. Let (H, j) be any pair and ρ be any cut-oﬀ function as above. Suppose that u satisfies (20) with (39), has finite energy and satisfies lim u(τ−j ) = z− ,

j→∞

lim u(τ+j ) = z+

j→∞

for some sequences τ±j with τ−j → −∞ and τ+j → ∞. Then we have AF ([z+ , w+ ]) − AH ([z− , w− ]) ∞ 1 s 2 ∂H ∂u − ρ (τ) t, u(τ, t) dt dτ = − ∂τ J ρ(τ) ∂s s=ρ(τ) −∞ 0

(41)

COROLLARY 4.2. Let (H, j; ρ) and u be as in Lemma 4.1. 1. Suppose that ρ is monotone. Then we have 2 1 ∂H s ∂u t + + − − + − min dt (42) AF ([z , w ]) − AH ([z , w ]) ≤ − ρ (τ) x,s ∂τ J 1 ∂s 0 1 ∂H s t − min dt. (43) ≤ x,s ∂s 0 And (42) can be rewritten as the upper bound for the energy 1 2 ∂H s ∂u t + + − − ≤ A ([z , w ]) − A ([z , w ]) + − min dt. H F ∂τ J ρ1 (τ) x,s ∂s 0

(44)

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2. For a general ρ, we instead have

2 1 s ∂H ∂u + max t dt (45) AF ([z , w ]) − AH ([z , w ]) ≤ − ρ (τ) x,s ∂τ J 1 ∂s 0 1 s ∂H (46) ≤ max t dt. x,s ∂s 0 +

+

−

−

And (46) can be rewritten as the upper bound for the energy 2 1 s ∂u ∂Ht dt. + + − − ≤ A ([z , w ]) − A ([z , w ]) + max H F ∂τ J ρ1 (τ) 0 x, s ∂s

(47)

The proof is an immediate consequence of (41) and omitted. Here we would like to emphasize that the above various energy upper bounds do not depend on u or on the choice of j or ρ, but depend only on the homotopy H itself and the asymptotic condition of u. Motivated by the upper estimate (43), we introduce the following definition DEFINITION 4.3. Let H = {H(s)}0≤s≤1 be a homotopy of Hamiltonians. We define the negative part of the variation and the positive part of the variation of H by 1 1 ∂H s ∂H s t t − + E (H) := − min max dt, E (H) := dt. x,s x,s ∂s ∂s 0 0 And we define the total variation E(H) of H by E(H) = E − (H) + E + (H). If we denote by H −1 the time reversal of H, i.e., the homotopy given by H −1 : s ∈ [0, 1] → H 1−s then we have the identity E ± (H −1 ) = E ± (H)

and

E(H −1 ) = E(H).

With these definitions, applied to a pair (H, j) such that their ends H(0) and H(1) are nondegenerate, the a priori energy estimate (43) can be written as 2 ∂u − ∂τ J ρ(τ) ≤ −AF u(∞) + AH u(−∞) + E (H) for a monotone ρ, and (46) as 2 ∂u ∂τ J ρ(τ) ≤ −AF u(∞) + AH u(−∞) + E(H) for a general ρ. Here we recall that, when the Hamiltonian is nondegenerate, any finite energy solution has well-defined asymptotic limits as τ → ± (Floer, 1988).

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COROLLARY 4.4. Let (H, J) and be given. Then for any finite energy solution u of (17) with (39), we have 2 ∂u + + − − AH ([z , w ]) − AH ([z , w ]) ≤ − ≤ 0. (48) ∂τ J In particular, when (H, J) is Floer-regular, then the associated boundary map ∂(H,J) satisfies ∂(H,J) CFλ∗ (H) ⊂ CFλ∗ (H) and hence canonically restricts to a boundary map ∂(H,J) : CFλ∗ (H), ∂(H,J) → CFλ∗ (H), ∂(H,J) for any real number λ ∈ R. We denote by HFλ∗ (H, J) the associated filtered homology and call it the filtered Floer homology group. COROLLARY 4.5. Suppose (H 0 , J 0 ) and (H 1 , J 1 ) are Floer-regular, (H, j) is a Floer-regular path between them, and ρ is as before. Then the chain map h(H, j;ρ) satisfies − (H) (H 1 ) h(H, j;ρ) CFλ∗ (H 0 ) ⊂ CFλ+E ∗ and so canonically restricts to a chain map − (H) h(H, j;ρ) : CFλ∗ (H 0 ), ∂(H 0 ,J 0 ) → CFλ+E (H 1 ), ∂(H 1 ,J 1 ) . ∗ One particular case of Corollary 4.2 and Corollary 4.5 is worthwhile to mention separately which will be used in the construction of the spectral invariants ρ(H; a) later. The same result was used in Oh (1997) for the spectral invariants of Lagrangian submanifolds on the cotangent bundle. COROLLARY 4.6. Let H be given. Consider two J 0 and J 1 , a cut-oﬀ function ρ and the homotopy (H, j) between (H, J 0 ) and (H, J 1 ) satisfying H ≡ H. Then for any finite energy solution u of (20) with (39), we have 2 ∂u AH ([z+ , w+ ]) − AH ([z− , w− ]) ≤ − ≤ 0. (49) ∂τ J ρ(τ) In particular, when H is nondegenerate and J 0 , J 1 are H-regular and (H, j) is generic, then the associate chain map h(H, j);ρ satisfies h(H, j);ρ CFλ∗ (H) ⊂ CFλ∗ (H) and hence canonically restricts to a chain map hλ(H, j);ρ : CFλ∗ (H), ∂(H,J 0 ) → CFλ∗ (H), ∂(H,J 1 ) and induces an isomorphism in homology for any real number λ ∈ R.

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Proof. It remains to prove that hλ(H, j);ρ induces an isomorphism in homology. For this, we choose any homotopy j connecting from J 1 to J 0 such that (H, j) is Floer-regular, and a cut-oﬀ function ρ. Then we consider the j # j which connects from J 0 to J 0 . Now we deform j # j to the constant homotopy jconst ≡ J 0 . We denote the homotopy of homotopy by ¯ connecting from jconst to j # j . Then by (49), (H, ¯) provides a chain homotopy from h(H, jconst ;ρ) and h(H, j)#(ρ;R) (H, j ) . We ρ note that since jconst ≡ J 0 , the elongated homotopy of (H ρ , jconst ) becomes the constant homotopy (H, J 0 ). Therefore by the Floer-regularity hypothesis of (H, J 0 ) as a family, we derive h(H, jconst ;ρ) = id. On the other hand, by choosing R > 0 suﬃciently large, we have the gluing identity h(H, j)#(ρ;R) (H, j ) = h(H, j;ρ) ◦ h(H, j ;ρ) Therefore we have proved h(H, j;ρ) ◦ h(H, j ;ρ) = id on HFλ∗ (H, J 0 ). By the same argument, we also have h(H, j ;ρ) ◦ h(H, j;ρ) = id on HFλ∗ (H, J 1 ).

4.2. ENERGY ESTIMATES AND HOFER’S NORM

We first recall some basic definitions and facts used in Hofer’s geometry of the Hamiltonian diﬀeomorphism group. In this section, we consider general time dependent Hamiltonian functions which are not necessarily one-periodic. We call a smooth map λ: [0, 1] → Ham(M, ω) a Hamiltonian path. According to Banyaga’s theorem [Ba], to any such path issued at the identity of Ham(M, ω) is associated a unique normalized smooth Hamiltonian H: [0, 1] × M → R satisfying λ(t) = φtH . We will denote the Hamiltonian path generated by H by φH . Therefore there is a one-one correspondence ∞ ([0, 1] × M, R) ↔ P(Ham(M, ω), id). Cm

(50)

For a given Hamiltonian diﬀeomorphism φ ∈ Ham(M, ω), we denote H → φ if φ = φ1H . REMARK 4.7. We remind the readers that the one-one correspondence (50) holds only in the smooth category. It is a fundamental task to understand what is happening when we go down to the Hamiltonians with low regularity, especially in the continuous category. We refer to Oh (2004b) for a detailed study of this issue.

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We recall the standard definitions 1 − E (H) = − min Ht dt,

+

1

E (H) = max Ht dt x x 0 0 1 (max Ht − min Ht ) dt H = E(H) = E + (H) + E − (H) = 0

x

x

used in Hofer’s geometry. (See Polterovich, 2001, for example.) Note that when H is the linear homotopy H lin : s → (1 − s)H1 + sH2 between H1 and H2 , E ± (H lin ) and E(H lin ) just become E ± (H2 − H1 ), and H2 − H1 , respectively. In fact, E ± (H) or E(H) correspond to the variations of the linear path s ∈ [0, 1] → sH in the sense of Definition 4.3. On the other hand, when H is nonautonomous, this linear path does not seem to have much intrinsic meaning in terms of the geometry of Ham(M, ω) itself. It would be desirable to discover more intrinsic invariants attached to a Hamiltonian path λ ∈ P(Ham(M, ω), id). We first state the following basic facts in the algebra of Hamiltonian functions. (See Hofer, 1990). PROPOSITION 4.8. Let H and F be arbitrary Hamiltonians, not necessarily one-periodic. 1. If H → φ, H → φ−1 where H is defined by H(t, x) := −H t, φtH (x) . 2. If H → φ, F → ψ, then we have H # F → φ ◦ ψ where H # F is the Hamiltonian defined by (H # F)(t, x) := H(t, x) + F t, (φtH )−1 (x) . COROLLARY 4.9. Ham(M, ω) ⊂ Symp(M, ω) forms a Lie subgroup and its ∞ (M), {·, ·}) where {·, ·} is the associated Lie algebra is (anti)-isomorphic to (Cm Poisson bracket associated to ω. REMARK 4.10. We would like to mention that, even when H ∼ F, H F.

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Therefore the map H → H does not push down to the universal (´etale) covering * space π : Ham(M, ω) → Ham(M, ω). One standard way of defining an invariant * for the elements h ∈ Ham(M, ω) is by taking the infimum h := inf H = inf leng(φH ). [H]=h

This function

(51)

[H]=h

* h ∈ Ham(M, ω) → h ∈ R+

* is not a priori continuous with respect to the natural topology on Ham(M, ω). However we will see later that our construction in Oh (2005c; 2004a) naturally * provides a continuous invariant of h ∈ Ham(M, ω). DEFINITION 4.11 (The Hofer norm). For φ ∈ Ham(M, ω), the Hofer norm, denoted by φ, is defined by φ := inf H H→φ

(= inf h). π(h)=φ

Then except the proof of nondegeneracy, the proof of the following theorem is straightforward. Nondegeneracy was proven by Hofer (1990) for Cn , by Polterovich (1993) for tame rational symplectic manifolds, and by Lalonde and McDuﬀ (1995a) in complete generality. THEOREM 4.12. Let φ, ψ ∈ Ham(M, ω). Then we have 1. φ−1 = φ

(Symmetry)

2. φψ ≤ φ + ψ

(Triangle inequality)

3. ηφη−1 = φ for any symplectic diﬀeomorphism η. (Symplectic invariance) 4. φ = 0 if and only if φ = id.

(Nondegeneracy)

We now note that for the linear path H lin : s → (1 − s)Hα + sHβ we have

E ± (H lin ) = E ± (Hβ − Hα )

and in particular, the pseudo-norms E ± (H) and H correspond to the variation of the linear path s → sH connecting the zero Hamiltonian to H. Taking the infimum of E(H) over all H with fixed end points H(0) = H 0 and H(1) = H 1 , we have the inequality inf {E(H) | H(0) = H 0 , H(1) = H 1 } ≤ H 1 − H 0 H

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which is a strict inequality in general. It seems to be an interesting problem to investigate the geometric meaning of the quantity in the left-hand side. Next, we consider the triple (Hα , Hβ , Hγ ) of Hamiltonians and homotopies H1 , H2 connecting from Hα to Hβ and Hβ to Hγ respectively. We define their concatenation H1 # H2 as defined in Section 3.5. From the definitions of E ± and E for the homotopy H above, we immediately have the following lemma LEMMA 4.13. All E ± and E are additive under the concatenation of homotopies. In other words, for any triple (Hα , Hβ , Hγ ) and homotopies H1 , H2 as above, we have E ± (H1 # H2 ) = E ± (H1 ) + E ± (H2 ). The same additivity holds for E. 4.3. LEVEL CHANGES OF FLOER CHAINS UNDER THE HOMOTOPY

In this subsection, we consider nondegenerate Hamiltonians H and the Floer regular pairs (H, J). Similarly we will only consider the Floer regular homotopy (H, j) connecting those Floer regular pairs. We also consider homotopy of homotopies, (H, ¯) with H = {Hκ }0≤κ≤1 a nd ¯ = { jκ }0≤κ≤1 and the induced chain homotopy map HH = H(H, ¯;ρ) . The following proposition shows how the level of α changes under the various Floer operators. PROPOSITION 4.14. Suppose that ρ is a (positively) monotone cut-oﬀ function. 1. λH ∂(H,J) (α) < λH (α) for an arbitrary Floer chain α. 2. λH 1 h(H, j;ρ) (α) ≤ λH 0 (α) + E − (H) for an arbitrary choice of ρ 3. λH 1 HH (α) ≤ λH 0 (α) + maxκ∈[0,1] E − (Hκ ). Proof. 1 and 2 are immediate consequences of Corollary 4.2. For the proof of 3, let [z , w ] ∈ Crit AH 1 be the peak of the chain. By the definition of the chain map HH (α), there exists a generator [z, w] ∈ Crit AH 0 and a parameter κ ∈ (0, 1) such that the equation ∂u ∂u (52) + J κ,ρ − XH κ,ρ (u) = 0 ∂τ ∂t with the asymptotic condition u(−∞) = [z, w],

u(∞) = [z , w ]

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has a solution for some generator [z, w] of α. Then by (45), we derive AH 1 ([z , w ]) − AH 0 ([z, w]) ≤ E − (Hκ ) i.e.,

AH 1 ([z , w ]) ≤ AH 0 ([z, w]) + E − (Hκ ).

(53)

Since we have chosen [z , w ] to be the peak of HH (α), applying (45) for the pair (Hκ , jκ ) using the arguments similar to the above, we prove AH 1 HH (α) ≤ λH 0 (α) + E − (Hκ ). By taking the supremum of the right hand side of this inequality over κ ∈ (0, 1), we have proved (3). REMARK 4.15. We would like to emphasize that the (Hκ , jκ ) is not Floerregular, but a minimally degenerate pair, and that 3 is not a special case of 2. We denote

E − (H) := max E − (Hκ ). κ∈[0,1]

Then we have the following corollary of Proposition 4.14 3. COROLLARY 4.16. Let (H 0 , J 0 ) and (H 1 , J 1 ) be two Floer regular pairs. Consider a generic homotopy of homotopies, (H, ¯) with H = {Hκ }0≤κ≤1 ,

¯ = { jκ }0≤κ≤1

where each Hκ is a homotopy connecting (H 0 , J 0 ) and (H 1 , J 1 ). Then the induced chain homotopy map HH = H(H, ¯;ρ) satisfies − HH CFλ (H 0 ) ⊂ CFλ+E (H) (H 1 ). 4.4. THE -REGULARITY TYPE INVARIANTS

We recall a well-known invariant of the almost K¨ahler structure (M, ω, J0 ) defined by A(ω, J0 ) := inf{ω(v) | v: S 2 → M is nonconstant and ∂¯ J0 v = 0}. This is known to be positive. We call A(ω, J0 ) the -regularity invariant because its positivity is a consequence of the so called the -regularity theorem in the geometric analysis (Sacks and Uhlenbeck, 1981). We refer to Oh (2005c) for the details of such a proof.

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We now generalize this invariant for any compact family K ⊂ Jω of compatible almost complex structures. Let K: [0, 1]n → Jω be a continuous n-parameter family in the C 1 -topology, and define A(ω; K) be the constant

A(ω; K) = inf n A ω, J(κ) . κ∈[0,1]

This is always positive (see Proposition 8.6), and enjoys the following lower semicontinuity property. We refer to Oh (2004a) for its proof. PROPOSITION 4.17. A(ω; K) is lower semi-continuous in K. In other words, for any given K and 0 < < A(ω; K), there exists some δ = δ(K, ) > 0 such that for any K with K − KC 1 ≤ δ we have A(ω; K ) ≥ A(ω; K) − . We now introduce two other invariants of the -regularity type associated to the perturbed Cauchy – Riemann equations. We first remark that our family J = {Jt }0≤t≤1 is a special case of the compact family K above with n = 1. Let H be a given nondegenerate Hamiltonian function and consider the perturbed Cauchy – Riemann equation ∂u ∂u +J − XH (u) = 0 ∂τ ∂t for each H-regular J. We call a solution u stationary if it is τ-independent. We define ∂u 2 A(H,J) := inf ∂τ J u satisfies (17) and is not stationary and µ A(H,J)

∂u 2 u satisfies (17) and µ (u) = 1 . := inf H ∂τ J

The positivity of A(H,J) is an easy consequence of the Gromov compactness type µ theorem (see Oh, 2005b, for details of such a proof). Obviously we have A(H,J) ≥ A(H,J) . Then we can strengthen the statement 1 of Lemma 4.14 to the following inequality µ λH ∂(H,J) (α) ≤ λH (α) − A(H,J) (54) for an arbitrary Floer chain α.

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5. Definition of spectral invariants and their axioms 5.1. FLOER COMPLEX OF A SMALL MORSE FUNCTION

We start this section with the study of the Floer complex CF(H), ∂(H,J) , as a complex with the Novikov ring as its coeﬃcients, for the special case H = f,

J ≡ J0

when > 0 is suﬃciently small. Here f is any Morse function. The following theorem was essentially proven by Floer (1989). Also see Hofer and Salamon (1995), Fukaya and Ono (1999), and Liu and Tian (1998). THEOREM 5.1. Let f be any small Morse function on M and J0 be a compatible almost complex structure such that f is Morse – Smale with respect to the metric g J0 . Then there exists 0 > 0 such that for all 0 < ≤ 0 we have the chain isomorphism ↓ CF∗ ( f ), ∂( f,J0 ) CM∗ (− f ), ∂Morse (− f,g J ) ⊗ Λω . 0

Once we have this theorem, applying the Poincar´e duality Morse CM∗ (− f ), δMorse (− f,g J ) CM2n−∗ (− f ), ∂(− f,g J ) , 0

0

we have the natural canonical isomorphism H ∗ (M) ⊗ Λ↑ HF∗ ( f, J0 ) =: HF∗ ( f ) as a Q-vector space. Here the grading ∗ in HF∗ stands for the degree of the Floer cycle α which is provided by the Conley – Zehnder index of its generators. We refer to Oh (2005a) (and also to Section 7.2) for a detailed discussion on this grading problem. We also recall that H ∗ (M) ⊗ Λ↑ω is isomorphic to the quantum cohomology ∗ QH (M) as a Λ↑ω -module, by definition. In this sense, the complex ↑ CM∗ (− f ), δMorse (− f,g J ) ⊗ Λω 0

provides the chain complex of the quantum cohomology. Composing the isomorphism QH∗ (M) H ∗ (M) ⊗ Λ↑ HF∗ ( f ) HF∗ (H) after incorporating a grading consideration, we obtain a natural isomorphism QHn−k (M) HFk (H)

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as a graded Q-vector space. We would like to emphasize that for nonexact (M, ω), there will be no isomorphism between them, unless we reverse the direction of the Novikov rings. We recall that QH∗ (M) is a module over Λ↑ω , while HF∗ (H) is one over Λ↓ω . 5.2. DEFINITION OF SPECTRAL INVARIANTS

For each given (homogeneous) quantum cohomology class a ∈ QH∗ (M), we denote by a = aH ∈ HF∗ (H, J) the image under the above isomorphism. We denote by iλ : HFλ∗ (H, J) → HF∗ (H, J) the canonical inclusion induced homomorphism. DEFINITION 5.2. Let H be a nondegenerate Hamiltonian and J be H-regular. For any given 0 a ∈ QH∗ (M), we consider Floer cycles α ∈ ZF∗ (H, J) ⊂ CF∗ (H) of the pair (H, J) representing a . Then we define ρ (H, J); a := inf λH (α), α;[α]=a

or equivalently ρ (H, J); a := inf{λ ∈ R | a ∈ im −iλ ⊂ HF∗ (H, J)}. We will mostly use the first definition in our exposition, which is more intuitive and flexible to use in practice. The following theorem was proved in Oh (2005a). Because its proof illustrates the typical argument in our chain level mini-max theory, we provide more details of the proof than we did in Oh (2005a). THEOREM 5.3. Suppose that H is nondegenerate and let 0 a ∈ QH∗ (M). 1. We have ρ (H, J); a > −∞ for any H-regular J. 2. The definition of ρ (H, J); a does not depend on the choice of H-regular J’s. We denote by ρ(H; a) the common value. Proof. We will give the proof in several steps. We write the quantum cohomology class a as a= aA q−A . A∈Γ

Let Γ(a) ⊂ Γ be the support of a, i.e., the set of A ∈ Γ with aA 0 in this sum. By the definitions of the quantum cohomology and of the Novikov ring, we can enumerate Γ(a) = {A j } j∈Z+ so that ω(−A1 ) < ω(−A2 ) < · · · < ω(−A j ) < · · · → ∞

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or equivalently ω(A1 ) > ω(A2 ) > · · · > ω(A j ) → −∞ In that case, we denote A1 =: Aa . DEFINITION 5.4. For each homogeneous element a = ΣA∈Γ aA q−A ∈ QHk (M),

aA ∈ H ∗ (M, Q)

(55)

of degree k, we define v(a) by v(a) = ω(−A1 ) and call it the level of a. And we define the leading order term of a by σ(a) := aA1 q−A1 . We also call aA1 the leading order coeﬃcient of a. Note that the leading order term σ(a) of a homogeneous element a is unique among the summands in the sum by the definition of Γ. Step 1. We first prove the following proposition. PROPOSITION 5.5. Suppose that J is H-regular and ρ (H, J); a is finite, i.e., ρ (H, J); a > −∞. Then ρ (H, J ); a is also finite for any other H-regular J and satisfies ρ( H, J); a = ρ (H, J ); a . Proof. Let α ∈ CF(H) be a Floer cycle of (H, J ) with [α ] = a . We choose a generic homotopy j = {J (s)}0≤s≤1 satisfying J (0) = J and J (1) = J the constant homotopy H = H, and pick a cut-oﬀ function ρ . We then consider the corresponding chain map h(H, j );ρ : CF(H) → CF(H) and the image cycle h(H, j );ρ (α ) of (H, J). Since [h(H, j);ρ (α )] = a in HF∗ (H, J), we have λH h(H, j );ρ (α ) ≥ ρ (H, J); a . (56) On the other hand, Corollary 4.6 implies λH h(H, j );ρ (α ) ≤ λH (α )

(57)

Combining (56) and (57), we have obtained λH (α ) ≥ ρ (H, J); a . By taking the infimum over all Floer cycles α of (H, J ), we obtain ρ (H, J ); a ≥ ρ (H, J); a .

(58)

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In particular, we have also proven that ρ (H, J ); a is finite. Once we have proved finiteness of ρ (H, J ); a , we can change the role of J and J , we have also obtain the opposite inequality ρ (H, J); a ≥ ρ (H, J ); a and hence ρ (H, J ); a = ρ (H, J); a . This finishes the proof. Step 2. Let f be any Morse function and J0 be a compatible almost complex structure such that the pair (− f, g J0 ) is Morse – Smale. We fix a suﬃciently small > 0 so that Theorem 5.1 holds. We will prove the finiteness of ρ(( f, J0 ); a), which corresponds to the linking property of the classical critical point theory (see Benci and Rabinowitz, 1979 for example). Let α ∈ ZF( f, J0 ) ⊂ CF( f ) be a Floer cycle representing a . It follows from Theorem 5.1 that α has the form α= αA qA A∈Γ

where αA ∈ CM∗ (− f ) is a Morse cycle of (− f, g J0 ), i.e., ∂Morse (− f,g J ) αA = 0 0

and its corresponding homology class satisfies [αA ] = PD(aA ), the Poincar´e dual to aA . Since [α] = a 0, there is at least one αA whose Morse homology class of − f is not zero. By removing the boundary terms from α, which only possibly decreases the level of chains, we obtain the following lemma, whose proof we refer to Oh (2005a). LEMMA 5.6. There exists another Floer cycle α ∈ ZF∗ ( f, J0 ) such that α and α are homologous, and α has the form α A qA α = A∈Γ(a)

such that [α A ] = PD(aA ) and λ f (α ) ≤ λ f (α). The upshot of this lemma is that, as far as the mini-max process is concerned, we can safely fix the support of α to be the set Γ(a) ⊂ Γ when we choose the minimaxing cycle α in the class a , which does not depend on α but depends only on the class a. The following is a standard fact in the finite dimensional critical point theory. LEMMA 5.7. For a given singular homology class B ∈ H∗ (M), we have λMorse − f (γ) ≥ min(− f ) ≥ − max f for any Morse cycle γ with [γ] = B.

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Lemma 5.7 and 5.6 then imply λ f (α) ≥ λ f (α ) ≥ − max f − ω(Aa ) > −∞. Then by taking the infimum over all α with [α] = a, we have obtained ρ(( f, J0 ); a) = inf λ f (α) ≥ − max f − ω(Aa ) > −∞. [α]=a

Once have proven the finiteness of this infimum for the pair ( f, J0 ), Proposition 5.5 implies that ρ(( f, J); a) does not depend on the choice f -regular J. Step 3. Let (H, J) be any Floer-regular pair. We consider any generic path H satisfying H(0) = H and H(1) = f , j with J(0) = J and J(1) = J0 and a cut-oﬀ function ρ, such that (H, j; ρ) is Floer-regular. Let h(H, j);ρ) : CF(H) → CF( f ) be the associated chain map. By the similar argument used in Step 1 using (46) applied to the homotopy H, we have obtain ρ(( f, J0 ); a) ≤ ρ (H, J); a + E − (H) and so

(59) ρ (H, J); a ≥ ρ(( f, J0 ); a) − E − (H) > −∞. This finishes the proof of finiteness of ρ (H, J); a . Then Proposition 5.5 proves that ρ (H, J); a does not depend on the choice of H-regular pair. Hence the proof of the theorem. The following proposition can be proven by the similar arguments used in the proof of (59) by considering the homotopy connecting H and K that is arbitrarily close to the linear homotopy s → (1 − s)H + sK. We omit its proof referring to Oh (2005a) for the details of the proof. PROPOSITION 5.8. For any nondegenerate H, K, we have −E + (H − K) ≤ ρ(H; a) − ρ(K; a) ≤ E − (H − K). In particular, ρa : H → ρ(H; a) is continuous in the C 0 -topology or (in the L(1,∞) 0 ([0, 1] × M; R). topology) and hence can be continuously extended to Cm 5.3. AXIOMS OF SPECTRAL INVARIANTS

In this subsection, we state basic properties of the function ρ in a list of axioms.

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THEOREM 5.9. Let (M, ω) be arbitrary closed symplectic manifold. For any given quantum cohomology class 0 a ∈ QH∗ (M), we have a continuous function denoted by ρ : Hm × QH∗ (M) → R such that they satisfy the following axioms: Let H, F ∈ Hm be smooth Hamiltonian functions and a 0 ∈ QH∗ (M). Then ρ satisfies the following axioms: 1. (Projective invariance) ρ(H; λa) = ρ(H; a) for any 0 λ ∈ Q. ) 2. (Normalization) For a = A∈Γ aA q−A , we have ρ(0; a) = v(a) where 0 is the zero function and v(a) := min{ω(−A) | aA 0} = − max{ω(A) | aA 0}. is the (upward) valuation of a. 3. (Symplectic invariance) ρ(η∗ H; a) = ρ(H; a) for any symplectic diﬀeomorphism η. 4. (Triangle inequality) ρ(H # F; a · b) ≤ ρ(H; a) + ρ(F; b). 5. (C 0 -continuity) |ρ(H; a) − ρ(F; a)| ≤ H # F = H − F where · is the Hofer’s pseudo-norm on Hm . In particular, the function ρa : H → ρ(H; a) is C 0 -continuous. Proof. The projective invariance is obvious from the construction. The C 0 continuity is an immediate consequence of Proposition 5.8. We postpone the proof of triangle inequality to Section 7. For the proof of symplectic invariance, we consider the symplectic conjugation φ → ηφη−1 ;

Ham(M, ω) → Ham(M, ω)

for any symplectic diﬀeomorphism η: (M, ω) → (M, ω). Recall that the pull-back function η∗ H given by η∗ H(t, x) = H t, η−1 (x) generates the conjugation ηφη−1 when H → φ. We summarize some basic facts on this conjugation relevant to the filtered Floer homology here: 1. when H → φ, η∗ H → ηφη−1 ; 2. if H is nondegenerate, η∗ H is also nondegenerate; 3. if (H, J) is Floer-regular, then so is (η∗ J, η∗ H); 4. there exists natural bijection η∗ : Ω0 (M) → Ω0 (M) defined by η∗ ([z, w]) = ([η ◦ z, η ◦ w]) under which we have the identity AH ([z, w]) = Aη∗ H (η∗ [z, w]);

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5. the L2 -gradients of the corresponding action functionals satisfy η∗ (grad J AH )([z, w]) = gradη∗ J Aη∗ H )(η∗ ([z, w]) ; 6. if u: R × S 1 → M is a solution of perturbed Cauchy – Riemann equation for (H, J), then η∗ u = η ◦ u is a solution for the pair (η∗ J, η∗ H). In addition, all the Fredholm properties of (J, H, u) and (η∗ J, η∗ H, η∗ u) are the same. These facts imply that the conjugation by η induces a canonical filtration preserving chain isomorphism η∗ : CFλ∗ (H), ∂(H,J) → CFλ∗ (η∗ H), ∂(η∗ H,η∗ J) for any λ ∈ R \ Spec(H) = R \ Spec(η∗ H). In particular it induces a filtration preserving isomorphism η∗ : HFλ∗ (H, J) → HFλ∗ (η∗ H, η∗ J). in homology. The symplectic invariance is then an immediate consequence of our mini-max procedure used in the construction of ρ(H; a). By the one-one correspondence between (normalized) H and its associated Hamiltonian path φH : t → φtH , one can regard the spectral function ∞ ([0, 1] × M) → R ρa : C m

as a function defined on P(Ham(M, ω); id), i.e., ρa : P(Ham(M, ω), id) → R as described in Oh (2004b). Here we denote by P(Ham(M, ω), id) the set of * smooth Hamiltonian paths in Ham(M, ω) and by Ham(M, ω) the set of path homotopy classes on P(Ham(M, ω), id), i.e., the (´etale) universal covering space of * Ham(M, ω) in the sense of Oh (2004b, Appendix 2). We equip Ham(M, ω) with the quotient topology. An important question to ask is then whether we have the equality ρ(H; a) = ρ(K; a) for any smooth functions H ∼ K satisfying [H] = [K] so that ρa pushes down to * Ham(M, ω). We will discuss this in the next section. 6. The spectrality axiom One of the most nontrivial properties of the spectral invariants ρ(H; a) is the following property.

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AXIOM (Spectrality). For any H and a ∈ QH∗ (M), we have ρ(H; a) ∈ Spec(H).

(60)

We proved this spectrality axiom on any rational symplectic manifold in Oh (2005a). On the other hand, we have proven only the following weaker version on irrational symplectic manifolds (Oh, 2004a). We suspect that the spectrality could fail if the Hamiltonian is highly degenerate on nonrational symplectic manifolds. AXIOM (Nondegenerate Spectrality). QH∗ (M), (60) holds.

For any nondegenerate H and a ∈

Before studying these axioms in general, let us state one important consequence thereof. 6.1. A CONSEQUENCE OF THE NONDEGENERATE SPECTRALITY AXIOM

* The following proposition shows that the function ρa pushes down to Ham(M, ω) as a continuous function. THEOREM 6.1 (Homotopy invariance). Let (M, ω) be an arbitrary closed symplectic manifold. Suppose that nondegenerate spectrality axiom holds for (M, ω). Then we have ρ(H; a) = ρ(K; a) (61) for any smooth functions H ∼ K satisfying [H] = [K]. Proof. We first consider nondegenerate Hamiltonians H, K with H ∼ K. We now recall the following basic facts: 1. Nondegeneracy of a Hamiltonian function H depends only on its time one map φ = φ1H . 2. The set Spec(H) ⊂ R, which is the set of critical values of the action functional AH is a set of measure zero (see Oh, 2002, Lemma 2.2). 3. For any two Hamiltonian functions H, H → φ such that [φ, H] = [φ, H ], we have Spec(H) = Spec(H ) as a subset of R provided H, H satisfy the normalization condition (2) (see Oh, 2005b, for the proof). 4. The function H → ρ(H; a) is continuous with respect to the smooth topology ∞ (S 1 × M) (see Oh, 2002, for its proof). on Cm 5. The only continuous functions on a connected space (e.g., the interval [0, 1]) to R, whose values lie in a measure zero subset, are constant functions.

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Since H ∼ K, we have a smooth family H = {H(s)}0≤s≤1 with H(0) = H and H(1) = K. We define a function λ: [0, 1] → R by λ(s) = ρ(H(s); a). Note that H(s) is nondegenerate since their time one map is φ1H(s) = φ1H for all s ∈ [0, 1], and that its image is contained in the fixed subset Spec(h) ⊂ R independent of s, where h is the path homotopy class [H] = [K]. This subset has measure zero by (1) above and so totally disconnected. Therefore since the function λ is continuous by the C 0 -continuity axiom, λ must be constant and so ρ(H; a) = λ(0) = λ(1) = ρ(K; a), which finishes the above proof for the nondegenerate Hamiltonians. We like to emphasize that at this moment, because we do not know validity of the spectrality axiom for degenerate Hamiltonians, the scheme of the above proof used for the nondegenerate case cannot be applied to degenerate Hamiltonians. Suppose H ∼ K which are not-necessarily nondegenerate. We approximate H and K by sequences of nondegenerate Hamiltonians Hi and Ki in the C ∞ topology respectively. We note that the Hamiltonian K # Hi # K generates the flow φtK ◦ φtHi ◦ (φtK )−1 , which is conjugate to the flow φtHi and is nondegenerate. Therefore we have ρ(Hi ; a) = ρ(K # Hi # K; a)

(62)

by the symplectic invariance of ρ. On the other hand, since H ∼ K, we have K # Hi # K ∼ K # Hi # H. Since both are nondegenerate, the above proof of (61) for the nondegenerate Hamiltonians implies ρ(K # Hi # K; a) = ρ(K # Hi # H; a).

(63)

By taking the limits of (62) and (63) and using the continuity of ρ(·; a), we get ρ(H; a) = ρ(K # H # K; a) = ρ(K # H # H; a) = ρ(K; a) where the last equality comes since H # H = 0. Hence the proof.

* Therefore we can define the function ρa : Ham(M, ω) → R by setting ρ(h; a) := ρ(H; a)

(64)

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for a (and so any) H satisfying [H] = h, whether h is nondegenerate or not. This defines a well-defined function * ω) → R. ρa : Ham(M, * THEOREM 6.2. The function ρa defined by (64) is continuous to Ham(M, ω) in the quotient topology of H am(M, ω) induced from P(Ham(M, ω), id). Proof. Recall the definition of the quotient topology under the projection * π: P(Ham(M, ω), id) → Ham(M, ω). We have proved that the assignment H → ρ(H; a)

(65)

is continuous on C ∞ ([0, 1] × M). By the definition of the quotient topology, the function * ω) → R ρa : Ham(M, is continuous, because the composition ρa ◦ π: P(Ham(M, ω), id) → R, which is nothing but (65), is continuous.

6.2. SPECTRALITY AXIOM FOR THE RATIONAL CASE

In this subsection, we will prove the full Spectrality Axiom for the rational symplectic manifolds (Oh, 2005a). We first recall a useful notion of canonical thin cylinder between two nearby loops. For the reader’s convenience, we provide its precise description following Oh (2005d). We denote by Jref a fixed compatible almost complex structure and by exp the exponential map of the metric g := ω(·, Jref ·). Let ι(g) be the injectivity radius of the metric g. As long as d(x, y) < ι(g) for the given two points of M, we can write y = exp x (ξ) for a unique vector ξ ∈ T x M. As usual, we write the unique vector ξ as ξ = (exp x )−1 (y).

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Therefore if the C 0 distance dC 0 (z, z ) between the two loops z, z : S 1 → M is smaller than ι(g), we can define the canonical map 1 ucan zz : [0, 1] × S → M

by

−1 or ξzz (t) = expz(t) z (t) .

ucan zz (s, t) = expz(t) ξzz (t) ,

(66)

It is important to note that the image of ucan zz is contained in a small neighborhood of z (or z ), and uniformly converges to z∞ when z and z converge to a loop z∞ in the C 1 topology. Therefore ucan zz also picks out a canonical homotopy class, can denoted by [uzz ], among the set of homotopy classes of the maps u: [0, 1] × S 1 → M satisfying the given boundary condition u(0, t) = z(t),

u(1, t) = z (t).

The following lemma is an important ingredient in our proof. LEMMA 6.3. Let z, z : S 1 → M be two smooth loops and ucan be the above canonical cylinder. Then as dC 1 (z, z ) → 0, then the map ucan zz converges in the 1 can C -topology, and its geometric area Area(u ) converges to zero. In particular, we have the followings: 1. For any bounding disc w of z, the bounding disc w := w # ucan zz of w is pre-compact in the C 1 -topology of the maps from the unit disc. 2.

ucan zz

ω→0

(67)

as dC 1 (z, z ) → 0 as z → z. Proof. 1 is an immediate consequence of the explicit form of ucan zz above and from the standard property of the exponential map. On the other hand, from the explicit expression of the canonical thin cylinder and from the property of the exponential map, it follows that the geometric area Area(ucan i∞ ) converges to zero as dC 1 (z, z ) → 0 by an easy area estimate. Since z, z 1 are assumed to be C 1 , it follows ucan zz is C and hence the inequality . Area(ucan ) ≥ ω i∞ ucan i∞

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This implies

lim

j→∞

ucan i∞

ω = 0,

which finishes the proof.

The following theorem was proved by Oh (2005a) from which we borrow its proof verbatim. THEOREM 6.4. Suppose that (M, ω) is rational. Then for any smooth one-periodic Hamiltonian function H: S 1 × M → R, we have ρ(H; a) ∈ Spec(H) for each given quantum cohomology class 0 a ∈ QH∗ (M). Proof. We need to show that the mini-max value ρ(H; a) is a critical value, or 0 (M) such that that there exists [z, w] ∈ Ω AH ([z, w]) = ρ(H; a) dAH ([z, w]) = 0,

i.e., z˙ = XH (z).

The finiteness of the value ρ(H; a) was already proved in Section 5.2. If H is nondegenerate, we just use the fixed Hamiltonian H. If H is degenerate, we approximate H by a sequence of nondegenerate Hamiltonians Hi in the C 2 topology. Let peak(αi ) = [zi , wi ] ∈ Crit AHi be the peak of the Floer cycle αi ∈ CF∗ (Hi ), such that lim AHi ([zi , wi ]) = ρ(H; a). (68) j→∞

Such a sequence can be chosen by the definition of ρ(·; a) and its finiteness property. Since M is compact and Hi → H in the C 2 topology, and z˙i = XHi (zi ) for all i, it follows from the standard boot-strap argument that zi has a subsequence, which we still denote by zi , converging to some loop z∞ : S 1 → M satisfying z˙ = XH (z). 0 (M). Since we fix Now we show that the sequence [zi , wi ] are pre-compact on Ω ∗ the quantum cohomology class 0 a ∈ QH (M) (or more specifically since we fix its degree) and since the Floer cycle is assumed to satisfy [αi ] = a , we have µHi ([zi , wi ]) = µH j ([z j , w j ]). 0 (M) is a closed subset of R LEMMA 6.5. When (M, ω) is rational, Crit AK ⊂ Ω for any smooth Hamiltonian K, and is locally compact in the subspace topology of the covering space 0 (M) → Ω0 (M). π: Ω

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Proof. First note that when (M, ω) is rational, the covering group Γ of π above is discrete. Together with the fact that the set of solutions of z˙ = XK (z) is compact (on compact M), it follows that 0 (M) | z˙ = XK (z)} Crit(AK ) = {[z, w] ∈ Ω

is a closed subset which is also locally compact. Now consider the bounding discs of z∞ given by w i = wi # ucan i∞

can for all suﬃciently large i, where ucan i∞ = uzi z∞ is the canonical thin cylinder between zi and z∞ . We note that as i → ∞ the geometric area of ucan i∞ converges to 0. We compute the action of the critical points [z∞ , w i ] ∈ Crit AH ,

AH ([z∞ , w i ])

= −

w i

ω−

= −

1

ω− 0

ucan i∞

wi

= − ω− wi

H(t, z∞ (t)) dt

1

(69)

1

ω− 0

H(t, z∞ (t)) dt

Hi (t, zi (t)) dt −

0

1

H(t, z∞ (t)) dt − 0 = AHi ([zi , wi ]) − ω −

−

0

ucan i∞

1

ω ucan i∞

1 0

H(t, z∞ (t)) dt −

1

Hi (t, zi (t)) dt .

0

Therefore combining (68), (69) and (71), we derive lim AH ([z∞ , w i ]) = ρ(H; a).

i→∞

In particular AH ([z∞ , w i ]) is a Cauchy sequence, which implies ω− ω = AH ([z∞ , wi ]) − AH ([z∞ , w j ]) → 0 w i

w j

(70)

0

Since zi converges to z∞ uniformly and Hi → H, we have 1 1 H t, z∞ (t) dt − H t, zi (t) dt → 0. − 0

Hi (t, zi (t)) dt

(71)

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i.e.,

Since Γ is discrete and

w i #w j

, w i #w j

ω → 0.

ω ∈ Γ, this indeed implies that w i #w j

ω=0

(72)

,

for all suﬃciently large i, j ∈ Z+ . Since the set w ω i∈Z+ is bounded, we coni , clude that the sequence w ω eventually stabilize, by choosing a subsequence if i necessary. Going back to (69), we derive that the actions AH ([z∞ , w i ]) themselves stabilize and so we have AH ([z∞ , w N ]) = lim AH ([z∞ , w i ]) = ρ(H; a) i→∞

for a fixed suﬃciently large N ∈ Z+ . This proves that ρ(H; a) is indeed the value of AH at the critical point [z∞ , w N ]. This finishes the proof.

6.3. SPECTRALITY FOR THE IRRATIONAL CASE

In fact, an examination of the proof of Theorem 6.4 proves a stronger fact which we now explain. We recall that if H, H are nondegenerate and suﬃciently C 2 -close, there exists a canonical one-one correspondence between the sets of associated Hamiltonian periodic orbits. We call an associated pair any such pair (z, z ) of Hamiltonian periodic orbits of H, H mapped to each other under this correspondence. The following is proved in the appendix of Oh (2004a) whose proof we omit. PROPOSITION 6.6. Suppose that H, H are nondegenerate and suﬃciently C 2 close. Let (z, z ) be an associated pair of H, H . Then we have µH ([z, w]) = µH ([z , w # ucan zz ]).

(73)

We derive can 2c1 ([w i # w j ]) = 2c1 ([wi # ucan i∞ # w j # u j∞ ])

¯ can = 2c1 ([wi # ucan i∞ # u j∞ # w j ]) ¯ can = µHi ([zi , wi ]) − µHi ([zi , w j # ucan j∞ # u i∞ ]).

(74)

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The third equality comes from the index formula µH ([z, w # A]) = µH ([z, w]) − 2c1 (A). On the other hand, we derive can can µHi ([zi , w j # ucan j∞ # ui∞ ]) = µHi ([zi , w j # uz j z ]) = µH j ([z j , w j ]) i

(75)

when i, j are suﬃciently large. Here the first equality follows since ucan ¯ can i∞ # u i∞ is can homotopic to the canonical thin cylinder uz j z , and the second comes from (73). i On the other hand, [zi , wi ] and [z j , w j ] satisfy µHi ([zi , wi ]) = µH j ([z j , w j ])

(76)

because they are generators of Floer cycles αi and α j both representing the same Floer homology class a and so having the same degree. Hence combining (73) – (76), we obtain (77) c1 ([w i # w j ]) = 0 for all suﬃciently large i, j. Combining (72) and (77), we have proved 0 (M). [z∞ , w i ] = [z∞ , w j ] in Ω 0 (M), we have proven that the If we denote by [z∞ , w∞ ] this common element of Ω sequence [zi , wi ] converges to a critical point [z∞ , w∞ ] of AH in the topology of 0 (M) → Ω0 (M). This finishes the proof. the covering space π : Ω For the irrational case, the sequence [z∞ , w i ] used in the above proof will not stabilize, and more seriously the action values AH ([z∞ , w i ]) may accumulate at a value in R \ Spec(H). Recall that in the irrational case, Spec(H) is a dense subset of R. Therefore in the irrational case, one needs to directly prove that the sequence 0 (M). It turns out that has a convergent subsequence in the natural topology of Ω the above limiting arguments used for the rational case cannot be carried out due to the possibility that the discs wi could behave wildly in the limiting process. We 0 (M) → Ω0 (M) defines a emphasize that in the irrational case, the projection π: Ω covering only in the e´ tale sense (see Oh, 2004b, Appendix, for the precise meaning of this), but not in the ordinary sense. As a result, proving such a convergence is not possible in general even for the nondegenerate case for a given mini-max sequence of critical points [zi , wi ] satisfying (69). One needs to use a mini-max sequence of cycles instead. This scheme is exactly what we have carried out in Oh (2004a), which however turns out to be a highly nontrivial matter to carry out. THEOREM 6.7. Let (M, ω) be an arbitrary closed symplectic manifold. Then the nondegenerate spectrality axiom holds.

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We refer to Oh (2004a) for the complete details of the proof and many other basic ingredients in the chain level Floer theory To go to the case of degenerate Hamiltonians from this theorem, it is unavoidable to use the approximation arguments above as in the rational case. Therefore one has to work with the action functional AH in the spirit of the general critical point theory as in Benci and Rabinowitz (1979). One important point of the chain level theory we develop in Oh (2002; 2005b; 2005d; 2005a; 2005c; 2004b; 2004a) is that it has certain continuity property when Hamiltonian functions become degenerate, even in the irrational case where Spec(H) is a dense subset of R. Our chain level Floer theory developed in Oh (2002; 2005b; 2005d; 2005a; 2005c; 2004b; 2004a) should be regarded as the mini-max theory of the action functional, while the usual Floer homology theory is the Morse theory of the action functional. For this reason, we call our chain level theory the Floer mini-max theory. However this mini-max theory still meets the same kind of diﬃculty mentioned above, and cannot prove the spectrality axiom for general degenerate Hamiltonians on irrational symplectic manifolds. (See Remark 2.6 for some related comments.) It would be very interesting to see if this diﬃculty is something intrinsic for this case. * We summarize the basic axioms of the invariant ρ: Ham(M, ω)×QH∗ (M) → R in the following theorem, whose proofs immediately follow from Theorems 5.9 and 6.7 THEOREM 6.8. Let (M, ω) be any closed symplectic manifold. Let h, k ∈ * Ham(M, ω) and 0 a ∈ QH∗ (M). Then for each 0 a ∈ QH∗ (M), the function * ω) → R ρa : Ham(M, is continuous, and the function * ρ : Ham(M, ω) × QH∗ (M) → R satisfies the following axioms: 1. (Nondegenerate spectrality) For any nondegenerate h, ρ(h; a) ∈ Spec(h) for all 0 a ∈ QH∗ (M). ˜ λa) = ρ(φ; ˜ a) for any 0 λ ∈ Q. 2. (Projective invariance) ρ(φ; ) 3. (Normalization) For a = A∈Γ aA q−A , we have ρ(0; a) = v(a) where 0 is the * identity in Ham(M, ω) and v(a) := min{ω(−A) | aA 0} = − max{ω(A) | aA 0}. A

is the (upward) valuation of a. 4. (Symplectic invariance) ρ(ηhη−1 ; a) = ρ(h; a) for any symplectic diﬀeomorphism η

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5. (Triangle inequality) ρ(h · k; a · b) ≤ ρ(h; a) + ρ(k; b) 6. (C 0 -continuity) |ρ(h; a) − ρ(k; a)| ≤ h ◦ k−1 where · is the Hofer’s * pseudo-norm on Ham(M, ω). In particular, the function ρa : h → ρ(h; a) is 0 C -continuous.

7. Pants product and the triangle inequality 7.1. QUANTUM COHOMOLOGY IN THE CHAIN LEVEL

We first recall the definition of the quantum cohomology ring QH∗ (M). As a module, it is defined as QH∗ (M) = H ∗ (M, Q) ⊗ Λ↑ω where Λ↑ω is the (upward) Novikov ring ↑ −A aA q aA ∈ Q, # A | ai 0, Λω =

−A

A∈Γ

ω < λ < ∞, ∀λ ∈ R .

Due to the finiteness assumption on the Novikov ring, we have the natural (upward) valuation v : QH∗ (M) → R defined by v aA q−A = min{ω(−A) : aA 0} (78) A∈Γω

which satisfies that for any a, b ∈ QH∗ (M) v(a + b) ≥ min{v(a), v(b)}. The product on QH∗ (M) is defined by the usual quantum cup product, which we denote by “·” and which preserves the grading, i.e, satisfies QHk (M) × QH (M) → QHk+ (M). Often the homological version of the quantum cohomology is also useful, sometimes called the quantum homology, which is defined by QH∗ (M) = H∗ (M) ⊗ Λ↓ω . We define the corresponding (downward) valuation by v aB qB = max{ω(B) : aB 0} B∈Γ

(79)

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which satisfies that for f, g ∈ QH∗ (M) v( f + g) ≤ max{v( f ), v(g)}. We like to point out that the summand in Λ↓ω is written as bB qB while the one in Λ↑ω as aA q−A with the minus sign. This is because we want to clearly show which one we use. Obviously v satisfies the axiom of non-Archimedean norm which induce a topology on QH∗ (M) and QH∗ (M) respectively. The finiteness assumption in the definition of the Novikov ring allows us to enumerate supp(a) so that λ1 > λ2 > · · · > λ j > · · · → −∞ with λ j = ω(B j ) for B j ∈ supp(a) when a ∈ QH∗ (M) We have a canonical isomorphism : QH∗ (M) → QH∗ (M); ai q−Ai → PD(ai )qAi and its inverse : QH∗ (M) → QH∗ (M);

b j qB j →

PD(b j )q−B j .

We denote by a and b# the images under these maps. There exists the canonical nondegenerate pairing ·, ·: QH∗ (M) ⊗ QH∗ (M) → Q defined by

<

−Ai

ai q

,

b jq

Bj

=

=

(ai , b j )δAi B j

(80)

where δAi B j is the delta-function and (ai , b j ) is the canonical pairing between H ∗ (M, Q) and H∗ (M, Q). Note that this sum is always finite by the finiteness condition in the definitions of QH∗ (M) and QH∗ (M) and so is well-defined. This is equivalent to the Frobenius pairing in the quantum cohomology ring. However we would like to emphasize that the dual vector space (QH∗ (M))∗ of QH∗ (M) is not isomorphic to QH∗ (M) even as a Q-vector space. Rather the above pairing induces an injection QH∗ (M) → (QH∗ (M))∗ whose images lie in the set of continuous linear functionals on QH∗ (M) with respect to the topology induced by the valuation v (79) on QH∗ (M). We refer to the appendix of Oh (2005a) for further discussions on this matter. Let (C∗ , ∂) be any chain complex on M whose homology is the singular homology H∗ (M). One may take for C∗ the usual singular chain complex or a Morse chain complex. However since we need to take a nondegenerate pairing in the

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chain level, we should use a model which is finitely generated. We will always prefer to use the Morse homology complex CM∗ (− f ), ∂Morse (− f,g J ) 0

of the pair (− f, g J0 ) for a suﬃciently small > 0, because it is finitely generated and avoids some technical issue related to singular degeneration problem of the type studied in (Fukaya and Oh, 1997). The negative sign in CM∗ (− f ), ∂Morse (− f,g J0 ) is put to make the correspondence between the Morse homology and the Floer homology consistent with our conventions of the Hamiltonian vector field and the action functional. In our conventions, solutions of negative gradient of − f correspond to ones for the negative gradient flow of the action functional A f . We denote by CM∗ (− f ), δMorse (− f,g J ) 0

the corresponding cochain complex, i.e, CMk := Hom(CMk , Q), δ− f = ∂∗(− f,gJ ) . 0 Morse Now we extend the complex CM∗ (− f ), ∂(− f,gJ ) to the quantum chain 0 complex, denoted by (CQ∗ (− f ), ∂Q ) CQ∗ (− f ) := CM∗ (− f ) ⊗ Λω , ∂Q := ∂Morse (− f,g J ) ⊗ Λω . 0

This coincides with the Floer complex (CF∗ ( f ), ∂) as a chain complex if is sufficiently small (Theorem 5.1). Similarly we define the quantum cochain complex (CQ∗ (− f ), δQ ) by changing the downward Novikov ring to the upward one. In other words, we define CQ∗ (− f ) := CM2n−∗ ( f ) ⊗ Λ↑ ,

δQ := ∂( f,gJ0 ) ⊗ Λ↑ω .

Again we would like to emphasize that CQ∗ (− f ) is not isomorphic to the dual space of CQ∗ (− f ) as a Q-vector space. To emphasize the role of the Morse function in the level of complex, we denote the corresponding homology by HQ∗ (− f ) QH∗ (M). With these definitions, we have the obvious nondegenerate pairing CQ∗ (− f ) ⊗ CQ∗ (− f ) → Q induced by the duality pairing (not the Poincar´e pairing!) CM2n−∗ ( f ) ⊗ CM∗ (− f ) → Q which also induces the pairing above in homology.

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We now choose a generic Morse function f and an almost complex structure J0 as before. Then for any given homotopy (H, j) with H = {H s } s∈[0,1] with H 0 = f and H 1 = H, we denote by h(H, j) : CQ∗ (− f ) = CF∗−n ( f ) → CF∗−n (H)

(81)

the standard Floer chain map from f to H via the homotopy H. This induces a homomorphism h(H, j) : HQ∗ (− f ) HF∗−n ( f, J0 ) → HF∗−n (H, J).

(82)

Although (81) depends on the choice (H, j), (82) is canonical, i.e, does not depend on the homotopy (H, j). One confusing point in this isomorphism is the issue of grading. See the next section for a review of the construction of this chain map and the issue of grading of HF∗ (H, J). 7.2. GRADING CONVENTION

We set up our grading convention of the Floer homology. We denote by µH ([z, w]) the Conley-Zehnder index of [z, w] for the Hamiltonian H. The convention of the grading of CF∗ (H) from Oh (2005c) is deg([z, w]) = µH ([z, w])

(83)

for [z, w] ∈ Crit AH . This convention is the analog to the one we use in Oh (1999) in the context of Lagrangian submanifolds. We next compare this grading and the Morse grading of the Morse complex of the negative gradient flow equation of − f , (i.e., of the positive gradient flow of f χ˙ − grad f (χ) = 0. This corresponds to the negative gradient flow of the action functional A f ). This gives rise to the relation between the Morse indices µMorse − f (p) and the Conley – Zehnder indices µ f ([p, p]) ˆ in our convention (See Salamon and Zehnder, 1992, Lemma 7.2, but with some care about the diﬀerent convention of the Hamiltonian vector field. Their definition of XH is −XH in our convention.): µ f ([p, p]) ˆ = µMorse − f (p) − n or ˆ +n µMorse − f (p) = µ f ([p, p]) Recalling that we chose the Morse complex CM∗ (− f ) ⊗ Λ↓

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for the quantum chain complex CQ∗ (− f ), we have the following grading preserving isomorphism QHn−k (M) → QHn+k (M) HQn+k (− f ) → HFk ( f, J0 ) → HFk (H, J). We will also show in Section 7.3 that this grading convention makes the pants product, denoted by ∗, has the degree −n ∗: HFk (H) ⊗ HF (K) → HF(k+)−n (H # K)

(84)

which will be compatible with the degree-preserving quantum product ·: QHa (M) ⊗ QHb (M) → QHa+b (M) under the ring isomorphism between QH∗ and HF∗ (Piunikhin et al., 1996; Liu, 1999). Finally we state an important identity relating the Conley – Zehnder index and the first Chern number c1 (A) under the action by ‘gluing a sphere’ [z, w] → [z, A # w]. We like to emphasize that in our convention, the sign in front of the first Chern number term in the formula is ‘−.’ The diﬀerence of the sign from the formula in Hofer and Salamon (1995) is due to the diﬀerent convention of the canonical symplectic form on Cn : when we identify R2n T ∗ Rn and denote by (q1 , . . . , qn , p1 , . . . , pn ) the corresponding canonical coordinates, then the canonical symplectic form is given by ω0 = dqi ∧ d pi in our convention, while it is given by ω 0 = −ω0 =

d pi ∧ dqi .

according to the convention of (Hofer and Salamon, 1995), (Salamon and Zehnder, 1992), or (Polterovich, 2001). We will provide a complete self-contained proof starting from the definition of the Conley – Zehnder index from (Salamon and Zehnder, 1992). THEOREM 7.1. Let z: S 1 = R/Z → M be a given one-periodic solution of x˙ = XH (x) and w, w two given bounding discs. Then we have the identity µH ([z, w ]) = µH ([z, w]) − 2c1 ([w # w]).

(85)

In particular we have µH ([z, A # w]) = µH ([z, w]) − 2c1 (A).

(86)

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7.3. HAMILTONIAN FIBRATIONS AND THE PANTS PRODUCT

To start with the proof of the triangle inequality, we need to recall the definition of the “pants product” HF∗ (H, J 1 ) ⊗ HF∗ (F, J 2 ) → HF∗ (H # F, J 3 ). For the purpose of studying the eﬀect on the filtration under the product, we need to define this product in the chain level in an optimal way as in Oh (1999) and Schwarz (2000). For this purpose, we will mostly follow the description provided by Entov (2000) with few notational changes and diﬀerences in the grading. Except the grading convention, the conventions in Entov (2000; 2004) on the definition of Hamiltonian vector field and the action functional coincide with our conventions in Oh, (1997; 1999; 2002; 2005b; 2005d; 2005a; 2005c; 2004b; 2004a) and also here. Let Σ be the compact Riemann surface of genus 0 with three punctures. We fix a holomorphic identification of a neighborhood of each puncture with either [0, ∞)×S 1 or (−∞, 0]×S 1 with the standard complex structure on the cylinder. We call punctures of the first type negative and the second type positive. In terms of the “pair-of-pants” Σ\∪i Di , the positive puncture corresponds to the outgoing ends and the negative corresponds to the incoming ends. We denote the neighborhoods of the three punctures by Di , i = 1, 2, 3, and the identification by ϕ+i : Di → (−∞, 0] × S 1

for i = 1, 2

for positive punctures and ϕ−3 : D3 → [0, ∞) × S 1 for negative punctures. We denote by (τ, t) the standard cylindrical coordinates on the cylinders. We fix a cut-oﬀ function ρ+ : (−∞, 0] → [0, 1] defined by 1 τ ≤ −2 ρ= 0 τ ≥ −1 and ρ− : [0, ∞) → [0, 1] by ρ− (τ) = ρ+ (−τ). We will just denote by ρ these cut-oﬀ functions for both cases when there is no danger of confusion. We now consider the (topologically) trivial bundle P → Σ with fiber isomorphic to (M, ω) and fix a trivialization Φi : Pi := P|Di → Di × M on each Di . On each Pi , we consider the closed two form of the type ωPi := Φ∗i (ω + d(ρHt dt))

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for a time periodic Hamiltonian H: [0, 1] × M → R. The following is an important lemma whose proof we omit (see Entov, 2000). LEMMA 7.2. Consider three normalized Hamiltonians Hi , i = 1, 2, 3. Then there exists a closed 2-form ωP such that 1. ωP |Pi = ωPi 2. ωP restricts to ω in each fiber 3. ωn+1 P =0 Such ωP induces a canonical symplectic connection ∇ = ∇ωP (Guillemin and Lerman, 1996; Entov, 2000). In addition it also fixes a natural deformation class of symplectic forms on P obtained by those ΩP,λ := ωP + λωΣ where ωΣ is an area form , and λ > 0 is a suﬃciently large constant. We will always normalize ωΣ so that Σ ωΣ = 1. Next let J˜ be an almost complex structure on P such that 1. J˜ is ωP -compatible on each fiber and so preserves the vertical tangent space 2. the projection π: P → Σ is pseudo-holomorphic, i.e, dπ ◦ J˜ = j ◦ dπ. When we are given three t-periodic Hamiltonian H = (H1 , H2 ; H3 ), we say that J˜ is (H, J)-compatible, if J˜ additionally satisfies 3. For each i, (Φi )∗ J˜ = j ⊕ JHi where JHi (τ, t, x) = (φtHi )∗ J for some t-periodic family of almost complex structure J = {Jt }0≤t≤1 on M over a disc D i ⊂ Di in terms of the cylindrical coordinates. Here D i = 1 −1 1 ϕ−1 i ((−∞, −Ki ] × S ), i = 1, 2 and ϕ3 ([K3 , ∞) × S ) for some Ki > 0. See Oh ˜ (2005c) for a more detailed discussion on J. ˜ The condition 3 implies that the J-holomorphic sections v over D i are precisely the solutions of the equation ∂u ∂u + Jt − XHi (u) = 0 (87) ∂τ ∂t if we write v(τ, t) = τ, t, u(τ, t) in the trivialization with respect to the cylindrical coordinates (τ, t) on D i induced by φ±i above. Now we are ready to define the moduli space which will be relevant to the definition of the pants product that we need to use. To simplify the notations, we denote zˆ = [z, w] in general and zˆ = (ˆz1 , zˆ2 , zˆ3 ) where zˆi = [zi , wi ] ∈ Crit AHi for i = 1, 2, 3.

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DEFINITION 7.3. Consider the Hamiltonians H = {Hi }1≤i≤3 with H3 = H1 # H2 , ˜ zˆ) and let J˜ be a H-compatible almost complex structure. We denote by M(H, J; ˜ the space of all J-holomorphic sections u: Σ → P that satisfy 1 1. The maps ui := u ◦ (ϕ−1 i ): (−∞, Ki ] × S → M which are solutions of (87), satisfy lim ui (τ, ·) = zi , i = 1, 2 τ→−∞

and similarly for i = 3 changing −∞ to +∞. 2. The closed surface obtained by capping oﬀ pr M ◦u(Σ) with the discs wi taken with the same orientation for i = 1, 2 and the opposite one for i = 3 represents zero in π2 (M). ˜ zˆ) depends only on the equivalence class of zˆ’s: we say that Note that M(H, J; ∼ zˆ if they satisfy z i = zi , w i = wi # Ai ) for Ai ∈ π2 (M) and 3i=1 Ai represents zero (mod) Γ. The (virtual) dimension of ˜ zˆ) is given by M(H, J; zˆ

˜ zˆ) = 2n − (−µH1 (z1 ) + n) − (−µH2 (z2 ) + n) − (µH3 (z3 ) + n) dim M(H, J; = −n + µH1 (z1 ) + µH2 (z2 ) − µH3 (z3 ) . ˜ zˆ) = 0, we have Note that when dim M(H, J; n = −µH3 (ˆz3 ) + µH1 (ˆz1 ) + µH2 (ˆz2 ) which is equivalent to µH3 (ˆz3 ) = µH1 (ˆz1 ) + µH2 (ˆz2 ) − n which provides the degree of the pants product (84) in our convention of the grading of the Floer complex we adopt in the present paper. Now the pair-of-pants product ∗ for the chains is defined by ˜ zˆ)zˆ3 zˆ1 ∗ zˆ2 = # M(H, J; (88) zˆ3

for the generators zˆi and then by linearly extending over the chains in CF∗ (H1 ) ⊗ CF∗ (H2 ). Our grading convention makes this product is of degree −n. Now with this preparation, we are ready to prove the triangle inequality.

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Let α ∈ CF∗ (H) and β ∈ CF∗ (F) be Floer cycles with [α] = [β] = a and consider their pants product cycle α ∗ β: = γ ∈ CF∗ (H # F). Then we have [α ∗ β] = (a · b) and so ρ(H # F; a · b) ≤ λH#F (α ∗ β).

(89)

Let δ > 0 be any given number and choose α ∈ CF∗ (H) and β ∈ CF∗ (F) so that δ 2 δ λF (β) ≤ ρ(F; b) + . 2

λH (α) ≤ ρ(H; a) +

(90) (91)

Then we have the expressions α=

i

and β=

j

ai [zi , wi ] with AH ([zi , wi ]) ≤ ρ(H; a) +

δ 2

δ a j [z j , w j ] with AF ([z j , w j ]) ≤ ρ(F; b) + . 2

Now using the pants product (88), we would like to estimate the level of the chain α ∗ β ∈ CF∗ (H # F). The following is a crucial lemma whose proof we omit but refer to Schwarz (2000, Section 4.1) or Entov (2000, Section 5). ˜ zˆ) is nonempty. Then we have the identity LEMMA 7.4. Suppose that M(H, J; v∗ ωP = −AH1 #H2 ([z3 , w3 ]) + AH1 ([z1 , w1 ]) + AH2 ([z2 , w2 ]) ˜ zˆ) for any ∈ M(H, J; ˜ Now since J-holomorphic and J˜ is ΩP,λ -compatible, we have ∗ ∗ ∗ 0≤ v ΩP,λ = v ω P + λ v ωΣ = v∗ ωP + λ. LEMMA 7.5 (Entov, 2000, Theorems 3.6.1 & 3.7.4). Let Hi ’s be as in Definition 7.3. Then for any given δ > 0, we can choose a closed 2-form ωP so that ΩP,λ = ωP + λωΣ becomes a symplectic form for all λ ≥ δ.

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We recall that from the definition of ∗ that for any [z3 , w3 ] ∈ α ∗ β there exist ˜ H; zˆ) is nonempty with the asymptotic [z1 , w1 ] ∈ α and [z2 , w2 ] ∈ β such that M( J, condition zˆ = ([z1 , w1 ], [z2 , w2 ]; [z3 , w3 ]). Applying this and the above two lemmata to H and F for λ arbitrarily close to 0, and also applying (89) – (91), we immediately derive AH#F ([z3 , w3 ]) ≤ AH ([z1 , w1 ]) + AF ([z2 , w2 ]) + δ ≤ λH (α) + λF (β) + δ ≤ ρ(H; a) + ρ(F; b) + 2δ

(92)

for any [z3 , w3 ] ∈ α ∗ β. Combining (89), (90) – (92), we derive ρ(H # F; a · b) ≤ ρ(H; a) + ρ(F; b) + 2δ Since this holds for any δ, we have proven ρ(H # F; a · b) ≤ ρ(H; a) + ρ(F; b).

This finishes the proof. 8. Spectral norm of Hamiltonian diﬀeomorphisms

In this section, we will explain our construction of an invariant norm of Hamiltonian diﬀeomorphisms following Oh (2005c), which we call the spectral norm. This involves a careful usage of the spectral invariant ρ(H; 1) corresponding to the quantum cohomology class 1 ∈ QH∗ (M). 8.1. CONSTRUCTION OF THE SPECTRAL NORM

Using ρ(H; 1), we define a function ∞ ([0, 1] × M) → R γ: Cm

by γ(H) = ρ(H; 1) + ρ(H; 1),

(93)

∞ ([0, 1] Cm

on × M). Obviously we have γ(H) = γ(H) for any H. The general triangle inequality ρ(H; a) + ρ(F; b) ≥ ρ(H # F; a · b) for the spectral invariants restricted to a = b = 1, and the normalization axiom ρ(id; 1) = 0 imply γ(H) = ρ(H; 1) + ρ(H; 1) ≥ ρ(0; 1) = 0.

(94)

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Here a · b is the quantum product of the quantum cohomology classes a, b ∈ QH∗ (M) and 0 is the zero function. The following theorem generalizes the inequality (96) proven in Oh (1997; 1999) for the exact case to the general quantum cohomology classes on nonexact symplectic manifolds. THEOREM 8.1. For any H and 0 a ∈ QH∗ (M), we have −E + (H) + v(a) ≤ ρ(H; a) ≤ E − (H) + v(a).

(95)

In particular for any classical cohomology class b ∈ H ∗ (M) → QH∗ (M), we have −E + (H) ≤ ρ(H; b) ≤ E − (H)

(96)

for any Hamiltonian H. Proof. We first recall the following general inequality 1 1 − max(H − K) dt ≤ ρ(H, a) − ρ(K, a) ≤ − min(H − K) dt. 0

0

proven in Oh (2005a), which can be rewritten as 1 ρ(K; a) + − max(H − K) dt ≤ ρ(H; a) ≤ ρ(K; a) + 0

1

− min(H − K) dt.

0

Now let K → 0 which results in 1 − max(H) dt ≤ ρ(H; a) ≤ ρ(0; a) + ρ(0; a) + 0

1

− min(H) dt.

(97)

0

By the normalization axiom, we have ρ(0; a) = v(a) which turns (97) to v(a) − E + (H) ≤ ρ(H; a) ≤ v(a) + E − (H) for any H. Equation (96) immediately follow from the definitions and the identity v(b) = 0 for a classical cohomology class b. This finishes the proof. Applying the right-hand side of (96) to b = 1, we derive ρ(H; 1) ≤ E − (H) and ρ(H; 1) ≤ E − (H) for arbitrary H. On the other hand, we also have E − (H) = E + (H) for arbitrary H’s and hence γ(H) ≤ H. The nonnegativity (94) leads us to the following definition. DEFINITION 8.2. We define γ: Ham(M, ω) → R+ by γ(φ) := inf ρ(H; 1) + ρ(H; 1) . H→φ

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THEOREM 8.3. Let γ be as above. Then γ: Ham(M, ω) → R+ defines an invariant norm, i.e., it enjoys the following properties. 1. φ = id if and only if γ(φ) = 0. 2. γ(η−1 φη) = γ(φ) for any symplectic diﬀeomorphism η. 3. γ(φψ) ≤ γ(φ) + γ(ψ) . 4. γ(φ−1 ) = γ(φ). 5. γ(φ) ≤ φ. In the remaining subsection, we will give the proofs of these statements postponing the most nontrivial statement, nondegeneracy, to the next subsection modulo the Fundamental Existence Theorem whose proof we refer either to Oh (2005a) or Oh (2005c). We split the proof of this theorem item by item. Proof of 2. We recall the symplectic invariance of spectral invariants ρ(H; a) = ρ(η∗ H; a). Applying this to a = 1, we derive the identity γ(φ) = inf ρ(H; 1) + ρ(H; 1) H→φ

= inf ρ(η∗ H; 1) + ρ(η∗ H; 1) = γ(η−1 φη), H→φ

which finishes the proof. Proof of 3. We first recall the triangle inequality ρ(H # K; 1) ≤ ρ(H; 1) + ρ(K; 1)

(98)

ρ(K # H; 1) ≤ ρ(K; 1) + ρ(H; 1).

(99)

and Adding up (98) and (99), we have ρ(H # K; 1) + ρ(H # K; 1) = ρ(H # K; 1) + ρ(K # H; 1) ≤ ρ(H; 1) + ρ(H; 1) + ρ(K; 1) + ρ(K; 1) . (100) Now let H → φ and K → ψ. Because H # K generates φψ, we have γ(φψ) ≤ ρ(H # K; 1) + ρ(H # K; 1) and hence

γ(φψ) ≤ ρ(H; 1) + ρ(H; 1) + ρ(K; 1) + ρ(K; 1)

from (100). By taking the infimum of the right-hand side over all H → φ and K → ψ, 3 is proved.

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Proof of 4. The proof immediately follows from the observation that the definition of γ is symmetric over the map φ → φ−1 . Proof of 5. By taking the infimum of γ(H) ≤ H over H → φ, we have proved γ(φ) ≤ φ.

It now remains to prove nondegeneracy of γ, which we will do in the next two sections. We like to mention that our proof of nondegeneracy of γ provides another proof of nondegeneracy of the Hofer norm via the inequality γ(φ) ≤ φ. 8.2. THE -REGULARITY THEOREM AND ITS CONSEQUENCES

The entirety of this and the next subsections will be occupied with the proof of nondegeneracy of the semi-norm γ: Ham(M, ω) → R+ defined in Section 8. First we note that the null set (γ) := {φ ∈ Ham(M, ω) | γ(φ) = 0} is a normal subgroup of Ham(M, ω) by the symplectic invariance of γ. Therefore by Banyaga’s theorem (Banyaga, 1978), it is enough to exhibit one φ such that γ(φ) 0. We will prove that γ(φ) > 0 for any nondegenerate Hamiltonian diﬀeomorphism and so for all φ id. Suppose φ is a nondegenerate Hamiltonian diﬀeomorphism. Denote by J0 a compatible almost complex structure on (M, ω). For given such a pair (φ, J0 ), we consider the set of paths J j(φ,J0 ) = {J : [0, 1] → Jω | J (0) = J0 , We extend J to R so that

J (1) = φ∗ J0 }.

J (t + 1) = φ∗ J (t).

For each given J ∈ j(φ,J0 ) , we define the constant AS (φ, J0 ; J ) = inf{ω([u]) | u: S 2 → M nonconstant and satisfying ∂¯ Jt u = 0 for some t ∈ [0, 1]}. A priori it is not obvious whether AS (φ, J0 ; J ) is not zero. This is an easy consequence of the so called -regularity theorem, first introduced by Sacks and Uhlenbeck (1981) in the context of harmonic maps. We state a parameterized version of this theorem in the context of pseudo-holomorphic curves from Oh (1992).

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LEMMA 8.4 (-Regularity Theorem). Let g be any given background almost K¨ahler metric of (M, ω). We denote by D = D2 (1) the unit open disc . Let J0 be any almost complex structure and let u: D → M be , a J0 -holomorphic map. Then there exists some = (g, J0 ) > 0 such that if D |Du|2 < , then for any smaller disc D = D2 (r) with D ⊂ D, we have Du∞,D := max |Du(z)| ≤ C z∈D

where C > 0 depends only on g, , J0 , and D , not on u. Furthermore, the same C 1 bound holds for any compact family K of compatible almost complex structures with = (g, K) and C = C(g, , K, D ) depending on K. An immediate corollary of this -regularity theorem is the following uniform C 1 -estimate of pseudo-holomorphic curves whose derivation is the standard covering method in the geometric analysis. We refer to Oh (2005c) for its complete proof especially in the parametric form. COROLLARY 8.5. Let J ∈ j(φ,J0 ) . Then there exists an = (J ) > 0 such that if ω(u) < , then we have Du∞ := max |Du(z)| ≤ C z∈S 2

for any Jt -holomorphic sphere u: S 2 → M and for any t ∈ [0, 1] where C = C(, J ) does not depend on u. The following positivity is an important consequence of the above uniform C 1 estimate for a pseudo-holomorphic map with small energy. To illustrate the usage of this C 1 -estimate, we provide a complete proof borrowed from (Oh, 2005c). PROPOSITION 8.6. Let φ, J0 and J be as above. Then we have AS (φ, J0 ; J ) > 0. Proof. Suppose AS (φ, J0 = 0. Then there exists a sequence t j ∈ [0, 1] and a sequence of nonconstant maps u j : S 2 → M such that u j is Jt j -holomorphic and ; J )

ω(u j ) = E Jt j (u j ) → 0 as j → ∞. By choosing a subsequence of t j , again denoted by t j , we may assume that t j → t∞ ∈ [0, 1] and so Jt j converges to Jt∞ in the C ∞ -topology. We choose suﬃciently large N ∈ Z+ so that E Jt j (u j ) = ω(u j ) < (J )

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for all j ≥ N, where (J ) is the constant provided in Corollary 8.5. Then we have the uniform C 1 -bound 0 < Du j ∞ ≤ C(, J ). The Ascoli – Arzela theorem then implies that there exists a subsequence, again denoted by u j , such that u j converges uniformly to a continuous map u∞ : S 2 → M. Recalling that all the u j are Jt j -holomorphic and Jt j converges to Jt∞ in the C ∞ topology, the standard bootstrap argument implies that {u j } converges to u∞ in the C 1 topology (and so in the C ∞ -topology). However we have E Jt∞ (u∞ ) = lim E Jt j (u j ) = 0 j→∞

and hence u∞ must be a constant map, say u∞ ≡ x ∈ M. Therefore {u j } converges to the point x in the C ∞ -topology. In particular, if j is suﬃciently large, then the image of u j is contained in a (contractible) Darboux neighborhood of x. Therefore we must have ω([u j ]) = 0 and in turn E Jt j (u j ) = 0 for all suﬃciently large j, because E Jt j (u) = ω(u) holds for any Jt j -holomorphic curve u. This contradicts the assumption that u j is nonconstant. This finishes the proof. Next for each given J ∈ j(φ,J0 ) , we consider the equation of v : R × R → M  ∂v ∂v    + Jt = 0    ∂τ ∂t 2 (101)   ∂v < ∞.    φ v(τ, t + 1) = v(τ, t), R×[0,1] ∂τ Jt Now it is a crucial matter to produce a nonconstant solution of (101). For this purpose, using the fact that φ id, we choose a symplectic ball B(λ) such that φ B(λ) ∩ B(λ) = ∅ (102) where B(λ) is the image of a symplectic embedding g: B2n (r) → B(λ) ⊂ M of the standard Euclidean ball B2n (r) ⊂ Cn of radius r with λ = πr2 . We then study (101) together with v(0, 0) ∈ B(λ). (103) Because of (102) and (103), it follows v(±∞) ∈ Fix φ ⊂ M \ B(λ). Therefore any such solution cannot be constant.

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We now define the constant AD (φ, J0 ; J ) := inf v

v ω v nonconstant solution of (101) ∗

R×[0,1]

for each J ∈ j(φ,J0 ) as in Section 2.6. Obviously we have AD (φ, J0 ; J ) ≥ 0. We will prove AD (φ, J0 ; J ) 0. We first derive the following simple lemma. LEMMA 8.7. Let H be nondegenerate. Suppose that u: R × S 1 → M is any finite energy solution of  ∂u ∂u    (u) =0 + J − X  t H   ∂τ ∂t  (104)  ∂u 2     < ∞.   ∂τ Jt that satisfies Then

u(−∞, t) = u(∞, t).

, R×S 1

u∗ ω converges, and we have E J (u) =

R×S 1

u∗ ω.

(105)

(106)

Proof. First note that when H is nondegenerate, any finite energy solution has well-defined asymptotic limits z± = u(±∞). Then we pick any bounding discs w± of z± such that w+ ∼ w− # u. Now (106) is an immediate consequence of (41) applied to H ≡ H, since we assume (105), i.e., z+ = z− . With this proposition, we are ready to prove positivity of AD (φ, J0 ; J ). PROPOSITION 8.8. Suppose that φ is nondegenerate, and J0 and J ∈ j(φ,J0 ) as above. Then we have AD (φ, J0 ; J ) > 0. Proof. We prove this by contradiction. Suppose AD (φ, J0 ; J ) = 0 so that there exists a sequence of nonconstant maps v j : R × [0, 1] → M that satisfy (101) and E J (v j ) → 0 Therefore we have

as j → ∞.

E J (v j ) < (J )

for all suﬃciently large j’, where (J ) is the constant in Lemma 8.4 and Corollary 8.5. In particular, the sequence v j cannot bubble oﬀ. This implies that v j locally uniformly converge, and in turn that v j must (globally) uniformly converge to a constant map because E J (v j ) → 0. Since there are only finitely many

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fixed points of φ by the nondegeneracy hypothesis, by choosing a subsequence if necessary, we conclude v j (−∞) = v j (∞) = p (107) for all j’s for some p ∈ Fix φ. Now we fix any Hamiltonian H: [0, 1] × M → R that is zero near t = 0, 1 and with H → φ, and consider the following maps u j : R × S 1 → M,

u j (τ, t) := (φtH )(v j (τ, t)).

It follows from (107) that u j (−∞, t) = u j (∞, t). Furthermore for the family J = {Jt }0≤t≤1 with Jt := (φtH )∗ (Jt ), the u j ’s satisfy the perturbed Cauchy – Riemann equation (104). We note that (105) and the exponential convergence of u j (τ) to u j (±∞), as τ → ±∞ respectively, allows us to compactify the maps u j and , consider each of them as a cycle defined over a torus T 2 . Therefore the integral u∗j ω depends only on the homology class of the compactified cycles. Now, because v j : R × [0, 1] → M uniformly converges to the constant map p ∈ Fix φ, the image of u j will be contained in a tubular neighborhood of the p closed orbit zH : S 1 → M of x˙ = XH (x) given by p

zH (t) = φtH (p).

, In particular, u∗j ω = 0 because the cycle [u j ] is homologous to the one-dip mensional cycle [zH ]. Then Lemma 8.7 implies the energy E J (u j ) = 0. But by the choice of J above, Lemma 2.20 implies E J (v j ) = 0, a contradiction to the hypothesis that v j are nonconstant. This finishes the proof of Proposition 8.8. We then define A(φ, J0 ; J ) = min{AS (φ, J0 ; J ), AD (φ, J0 ; J )}. Propositions 8.6 and 8.8 imply A(φ, J0 ; J ) > 0. The finiteness

A(φ, J0 ; J ) < ∞

is a consequence of the Fundamental Existence Theorem, Theorem 8.15 in the next section. Finally we define A(φ, J0 ) := sup A(φ, J0 ; J ) J ∈ j(φ,J0 )

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and A(φ) = sup A(φ, J0 ). J0

By definition, we have A(φ, J0 ) > 0 and so we have A(φ) > 0. However a priori it is not obvious whether they are finite, which will be again a consequence of the Fundamental Existence Theorem. 8.3. PROOF OF NONDEGENERACY

With the definitions and preliminary studies of the invariants of A(φ, J0 ; J ), the following is the main theorem we will prove in this section, modulo the proof of Theorem 8.15 which we refer to Oh (2005c) and omit here. THEOREM 8.9. Suppose that φ is nondegenerate. Then for any J0 and J ∈ j(φ,J0 ) , we have γ(φ) ≥ A(φ, J0 ; J ) (108) and hence γ(φ) ≥ A(φ). In particular, A(φ) is finite. We have the following two immediate corollaries. The first one proves nondegeneracy of γ and the second provides a new lower bound for the Hofer norm itself. COROLLARY 8.10. The pseudo-norm is nondegenerate, i.e., γ(φ) = 0 if and only if φ = id. COROLLARY 8.11. Let φ be as in Theorem 8.9. Then we have φ ≥ A(φ). REMARK 8.12. The function φ → A(φ) is not C 0 -continuous. However there is another geometric invariant A(φ; 1) introduced in Oh (2005c) which enjoys better C 0 -continuity property than A(φ) and which we call the homological area of φ. This invariant A(φ; 1) satisfies A(φ; 1) ≥ A(φ) and is more computable than A(φ). Furthermore in Oh (2005c) we proved results stronger than those of Theorem 8.9 and Corollary 8.10 by replacing A(φ) by A(φ; 1). We expect that A(φ; 1) is C 0 continuous. We refer to Oh (2005c) for further discussions on A(φ; 1) in relation to the optimal energy capacity inequality.

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The rest of this section will be occupied by the proof of Theorem 8.9. Let φ be a nondegenerate Hamiltonian diﬀeomorphism with φ id. In particular, we can choose a small symplectic ball B(λ) with λ = πr2 such that B(λ) ∩ φ(B(λ)) = ∅. By the definition of γ, for any given δ > 0, we can find H → φ such that ρ(H; 1) + ρ(H; 1) ≤ γ(φ) + δ.

(109)

For any Hamiltonian H → φ, we know that H → φ−1 . However we will use another Hamiltonian x) := −H(1 − t, x) H(t, generating φ−1 , which is more useful than H, at least in the study of duality and pants product. We refer to Oh (2005c) for the proof of the following lemma. → φ−1 and H ∼ H, LEMMA 8.13. Let H be a Hamiltonian generating φ. Then H i.e., [φ−1 , H] = [φ−1 , H]. a). In particular, we have ρ(H; a) = ρ(H; over H is that the time reversal One advantage of using the representative H t → 1 − t acting on the loops z: S 1 → M induces a natural one-one correspondence between Furthermore the space-time reversal Crit(H) and Crit(H). (τ, t) → (−τ, 1 − t) acting on the maps u: R × S 1 → M induces a bijection between the moduli J) ˜ of the perturbed Cauchy – Riemann equations correspaces M(H, J) and M(H, ˜ respectively, where J˜t = J1−t . This correspondence sponding to (H, J) and (H, J) reverses the direction of the Cauchy – Riemann flow and the corresponding actions satisfy ]) = −AH ([z, w]). AH([˜z, w (110) ] is the class corresponding to z˜(t) := z(1 − t) and w = w ◦ c where Here [˜z, w c: D2 → D2 is the complex conjugation of D2 ⊂ C. The following estimate of the action diﬀerence is an important ingredient in our proof of nondegeneracy. The proof here is similar to the analogous nontriviality proof for the Lagrangian submanifolds studied in Oh (1999, Sections 6 – 7).

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PROPOSITION 8.14. Let J0 be any compatible almost complex structure, J ∈ j(φ,J0 ) and J be the one-periodic family Jt = (φtH )∗ Jt . Let H be any Hamiltonian with H → φ. Consider the equation  ∂u ∂u    + J − X (u) =0  t H   ∂τ ∂t (111)   u(−∞) = [z− , w− ], u(∞) = [z+ , w+ ]    w # u ∼ w , u(0, 0) = q ∈ B(λ) − + for a map u: R × S 1 → M. If (111) has a broken trajectory solution (without sphere bubbles attached) u1 # u2 # · · · · · · # uN which is a connected union of solutions of (111) for H that satisfies the asymptotic condition uN (∞) = [z , w ], u j (0, 0) = q

u1 (−∞) = [z, w] for some j.

(112)

] ∈ Crit AH, then we have For some [z, w] ∈ Crit AH and [˜z , w AH u(−∞) − AH u(∞) ≥ AD (φ, J0 ; J ). Proof. Suppose u is such a solution. Opening up u along t = 0, we define a map v: R × [0, 1] → M by v(τ, t) = (φtH )−1 u(τ, t) . It is straightforward to check that v satisfies (101). Moreover we have 2 2 ∂v = ∂u < ∞ ∂τ J ∂τ J t t

(113)

from Lemma 2.20. Since φ(B(λ)) ∩ B(λ) = ∅, we have v(±∞) ∈ Fix φ ⊂ M \ B(λ). On the other hand since v(0, 0) = u(0, 0) ∈ B(λ), v cannot be a constant map. In particular, we have 2 ∂v = v∗ ω ≥ AD (φ, J0 ; J ). ∂τ J t Combining this and (113), we have proven AH u(−∞) − AH u(∞) =

2 ∂u ≥ A (φ, J ; J ). D 0 ∂τ J t

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This finishes the proof.

This proposition demonstrates relevance of the existence result of the equation (111) to Theorem 8.9. However we still need to control the asymptotic condition (112) and to establish some relevance of the asymptotic condition to the inequality (108). For this, we will use (110) and impose the condition u(−∞) = [z, w], in (112) so that [z, w] ∈ αH ,

u(∞) = [z , w ] ] ∈ βH [ z , w

(114)

for the suitably chosen fundamental Floer cycles αH of H and βH of H. We recall from (109) that we have 1) ≤ γ(φ) + δ. ρ(H; 1) + ρ(H; We choose H → φ so that 1) ≤ γ(φ) + δ. ρ(H; 1) + ρ(H; By the definition of ρ and from (25), there exist αH ∈ CFn (H) and βH ∈ CFn (H) representing 1 = [M] such that δ 2 δ 1) ≤ λ (β ) ≤ ρ(H; 1) + . ρ(H; H H 2 ρ(H; 1) ≤ λH (αH ) ≤ ρ(H; 1) +

(115) (116)

Once we have these, by adding (115) and (116), we obtain 1) ≤ λH (αH ) + λ (β ) 0 ≤ ρ(H; 1) + ρ(H; H H 1) + δ. ≤ ρ(H; 1) + ρ(H; The fundamental cycles αH and βH that satisfy (115) and (116) respectively will be used as the asymptotic boundary condition required in (114). The following is the fundamental existence theorem of the Floer trajectory with its asymptotic limits lying near the ‘top’ of the given Floer fundamental cycles which will make the diﬀerence ]) AH ([z, w]) − AH ([z , w ]) = AH ([z, w]) + AH([˜z , w 1) as possible. We refer readers to Oh (2005c) for other as close to ρ(H; 1) + ρ(H; interesting consequences of this theorem besides the proof of nondegeneracy of γ.

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THEOREM 8.15 (Fundamental Existence Theorem). Let φ, H, J0 and J ∈ j(φ,J0 ) , be as in Proposition 8.14. and let q ∈ M \ Fix(φ) be given. Choose any δ such that 0 < δ < AD (φ, J0 ; J ). Then there exist some fundamental cycles αH of (H, J) for Jt = (φtH )∗ Jt , and βH J) ˜ such that of (H, δ 2 1) + δ λH(βH) ≤ ρ(H; 2

λH (αH ) ≤ ρ(H; 1) +

] ∈ βH that satisfy the and we can find some generators [z, w] ∈ αH and [˜z , w following alternative: 1. equation (111) has a broken-trajectory solution (without sphere bubbles attached) u1 # u2 # · · · # uN which is a connected union of Floer trajectories for H that satisfies the asymptotic condition uN (∞) = [z , w ],

u1 (−∞) = [z, w],

u j (0, 0) = q ∈ B(λ)

for some 1 ≤ j ≤ N, (and hence AH ([z, w]) − AH ([z , w ]) ≥ AD (φ, J0 ; J ) from Proposition 8.14) or, 2. there exists a Jt -holomorphic sphere v∞ : S 2 → M for some t ∈ [0, 1] passing through the point q ∈ B(λ), and hence AH ([z, w]) − AH ([z , w ]) ≥ AS (φ, J0 ; J ) − δ. This in particular implies 1) + δ < ∞ A(φ, J0 ; J ) < ρ(H; 1) + ρ(H;

(117)

for any φ and J0 . Finish-up of the proof of nondegeneracy. Let φ be a nondegenerate Hamiltonian diﬀeomorphism. From the definition of γ(φ), since δ and H are arbitrary as long as H → φ, we immediately derive, from (117), A(φ, J0 ; J ) ≤ γ(φ)

(118)

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for all J0 and J ∈ j(φ,J0 ) . Next by taking the supremum of A(φ, J0 ; J ) over all J0 and J ∈ j(φ,J0 ) in (118), we also derive A(φ) ≤ γ(φ). This finishes the proof of Theorem 8.9 and so the proof of nondegeneracy.

9. Applications to Hofer geometry of Ham(M, ω) 9.1. QUASI-AUTONOMOUS HAMILTONIANS AND THE MINIMALITY CONJECTURE

In this section, we drop the one-periodicity of the Hamiltonian function H, unless otherwise stated. The norm of H 1 (max Ht − min Ht ) dt H = 0

can be identified with the Finsler length 1 max H t, (φtH )(x) − min H(t, (φtH )(x) dt leng(φH ) = 0

x

x

of the path φH : t → φtH where the Banach norm on T id Ham(M, ω) C ∞ (M)/R defined by h = osc(h) = max h − min h for a normalized function h: M → R. DEFINITION 9.1 (The Hofer topology). Consider the metric d: P(Ham(M, ω), id) → R+ defined by

d(λ, µ) := leng(λ−1 ◦ µ)

where λ−1 ◦ µ is the Hamiltonian path t ∈ [0, 1] → λ(t)−1 µ(t). We call the induced topology on P(Ham(M, ω), id) the Hofer topology. The Hofer topology on Ham(M, ω) is the strongest topology for which the evaluation map λ → λ(1) is continuous. It is easy to see that this definition of the Hofer topology of Ham(M, ω) coincides with the usual one induced by the Hofer norm function given in Definition 4.11, which also shows that the Hofer topology is metrizable. Of course nontriviality of the topology is not a trivial matter which was proven by Hofer

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(1990) for Cn and by Lalonde and McDuﬀ (1995a) in its complete generality. It is also immediate to check that the Hofer topology of Ham(M, ω) is locally path-connected (see the proof of Oh (2004b, Theorem 3.15) for the relevant argument). Hofer (1993) also proved that the path of any compactly supported autonomous Hamiltonian on Cn is length minimizing as long as the corresponding Hamilton’s equation has no nonconstant periodic orbit of period less than or equal to one. This result has been generalized in Entov (2000), McDuﬀ and Slimowitz (2001), and Oh (2002; 2005b; 2005d) under the additional hypothesis that the linearized flow at each fixed point is not overtwisted i.e., has no closed trajectory of period less than one. Bialy and Polterovich (1994) and Lalonde and McDuﬀ (1995b) proved that any length minimizing (respectively, locally length minimizing) Hamiltonian path is generated by quasi-autonomous (respectively, locally quasi-autonomous) Hamiltonian paths. DEFINITION 9.2. A Hamiltonian H is called quasi-autonomous if there exists two points x− , x+ ∈ M such that H(t, x− ) = min H(t, x), x

H(t, x+ ) = max H(t, x) x

for all t ∈ [0, 1]. We now recall the Ustilovsky – Lalonde – McDuﬀ’s necessary condition on the stability of geodesics. Ustilovsky (1996) and Lalonde and McDuﬀ (1995b) proved that for a generic φ in the sense that all its fixed points are isolated, any stable geodesic φt , 0 ≤ t ≤ 1 from the identity to φ must have at least two fixed points which are undertwisted. DEFINITION 9.3. Let H: [0, 1] × M → R be a Hamiltonian which is not necessarily time-periodic and φtH be its Hamiltonian flow. 1. We call a point p ∈ M a time T periodic point if φTH (p) = p. We call t ∈ [0, T ] → φtH (p) a contractible time T -periodic orbit if it is contractible. 2. When H has a fixed critical point p over t ∈ [0, T ], we call p overtwisted as a time T -periodic orbit if its linearized flow dφtH (p); t ∈ [0, T ] on T p M has a closed trajectory of period less than or equal to T . Otherwise we call it undertwisted. If in addition the linearized flow has only the origin as the fixed point, then we call the fixed point generically undertwisted. Here we follow the terminology used by Kerman and Lalonde (2003) for the “generically undertwisted.” Note that under this definition of the undertwistedness, undertwistedness is C 2 -stable property of the Hamiltonian H. The following conjecture was raised by Polterovich (1998, Conjecture 12.6.D). (See also Polterovich, 2001; Lalonde and McDuﬀ, 1995b; McDuﬀ and Slimowitz, 2001.)

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CONJECTURE (Minimality Conjecture). Any autonomous Hamiltonian path that has no contractible periodic orbits of period less than equal to one is Hofer-length minimizing in its path-homotopy class relative to the boundary.

9.2. LENGTH MINIMIZING CRITERION VIA ρ(H; 1)

In this subsection, we describe a simple criterion of the length minimizing property of Hamiltonian paths in terms of the spectral invariant ρ(H; 1), which was given in Oh (2005d). The criterion is similar to the one used in Hofer (1993) and in Bialy and Polterovich (1994) for the case of Cn . In fact, Bialy and Polterovich (1994) predicted existence of such a criterion via the Floer homology on general symplectic manifolds, and this criterion indeed confirms their prediction. To describe this criterion, we recall H = E − (H) + E + (H) where

−

1

E (H) = +

− min H dt

0 1

E (H) =

max H dt. 0

These are called the negative Hofer-length and the positive Hofer-length of H respectively. We will consider them separately. First note E + (H) = E − (H). THEOREM 9.4. Let G: [0, 1] × M → R be any Hamiltonian that satisfies ρ(G; 1) = E − (G)

(119)

Then H is negative Hofer-length minimizing in its homotopy class with fixed ends. In particular, G must be quasi-autonomous. Proof. Let F be any Hamiltonian with F ∼ G. Then we have a string of equalities and inequality E − (G) = ρ(G; 1) = ρ(F; 1) ≤ E − (F) from (119), (61) for a = 1, (96) respectively. The last statement follows from Bialy – Polterovich, Ustilovsky, and Lalonde – McDuﬀ’s criterion for the minimality. This finishes the proof.

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On the other hand, if G is one-periodic, we can consider the associated action functional AG . Then AG has two obvious critical values of AG for a quasi-autonomous Hamiltonian G given by 1 − − AG ([x , xˆ ]) = −G(t, x− ) dt 0 1 −G(t, x+ ) dt AG ([x+ , xˆ+ ]) = 0

which coincide with −

E (G) =

1

− min Gt dt

0

E + (G) =

1

max Gt dt 0

respectively. We note that when G is one-periodic and quasi-autonomous having given by x− and x+ its uniform minimum and maximum points, then G x) = −G(1 − t, x) G(t, is also one-periodic and quasi-autonomous and has x+ and x− as a uniform minimum and a maximum point respectively. We also know (Lemma 8.13) that ∼ G. G Now we explain how to dispose the periodicity and extend the definition of ρ(H; a) for arbitrary time dependent Hamiltonians H: [0, 1] × M → R. Note that it is obvious that the semi-norms E ± (H) and H are defined without assuming the periodicity. For this purpose, the following lemma from Oh (2002) is important. We leave its proof to readers or to Oh (2002). LEMMA 9.5. Let H be a given Hamiltonian H: [0, 1] × M → R and φ = φ1H be its time-one map. Then we can re-parameterize φtH in time so that the re-parameterized Hamiltonian H satisfies the following properties: 1. φ1H = φ1H 2. H ≡ 0 near t = 0, 1 and in particular H is time periodic 3. Both E ± (H − H) can be made as small as we want 4. If H is quasi-autonomous, then so is H 5. For the Hamiltonians H , H generating any two such re-parameterizations of φtH , there is canonical one-one correspondences between Per(H ) and Per(H ), and Crit AH and Crit AH with their actions fixed. Furthermore this re-parametrization is canonical with the “smallness” in 3 can be chosen uniformly over H depending only on the C 0 -norm of H.

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Using this lemma, we can now define ρ(H; a) for arbitrary H by ρ(H; a) := ρ(H ; a) where H is the Hamiltonian generating the canonical re-parametrization of φtH in time provided in Lemma 9.5. It follows that this definition is well-defined because any such re-parameterizations are homotopic to each other with fixed ends. This being said, we will always assume that our Hamiltonians are time one-periodic without mentioning further in the rest of the paper. 9.3. CANONICAL FUNDAMENTAL FLOER CYCLES

Now we are ready to introduce the following concept of homological essentialness in the chain level theory, which is the heart of matter in the chain level Floer theory. DEFINITION 9.6. We call a Floer cycle α ∈ CF(H) tight if it satisfies the following nonpushing down property under the Cauchy – Riemann flow (31): for any Floer cycle α ∈ CF(H) homologous to α (in the sense of Definition 3.10 2), it satisfies λH (α ) ≥ λH (α). In terms of the length-minimizing criterion in Theorem 9.4, we would like to construct a tight fundamental Floer cycle of G whose level is precisely E − (G) for a quasi-autonomous Hamiltonian G. As often done in Oh (2002), one natural way of constructing a Floer fundamental cycle of general Hamiltonian H is to transfer a Morse cycle using Floer’s chain map. More precisely, we consider a Morse function f and the fundamental Morse cycle α of − f for a suﬃciently small > 0 such that Theorem 5.1 holds. Then α also becomes a Floer cycle of f . We then transfer α and define a fundamental Floer cycle of H as αH := hL (α) ∈ CF(H) where hL is the Floer chain map over the canonically given linear path L : s → (1 − s) f + sH. We call any of such transferred cycle a canonical fundamental Floer cycle of H as in Oh (2005c). We however note that this cycle depends on the choice of the Morse function f . In general, we do not expect this cycle will be tight even when H is quasi-autonomous. Now we apply this construction to a quasi-autonomous Hamiltonian G that has the unique nondegenerate global minimum x− that is undertwisted for all

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t ∈ [0, 1] which was studied by Kerman and Lalonde (2003). In this case, they made the following particular choice of the Morse function f in the above linear path L so that 1. f has a global minimum point at x− 2. f satisfies

f (x− ) = 0,

f (x− ) < f (x j )

for all other critical points x j . Having f adapted to the given G this way, Kerman and Lalonde (2003) proved the following basic result on the transferred cycle αG for the aspherical manifold. Their proof was then generalized by Oh (2005d) for general symplectic manifolds. We refer readers to Oh (2005d) for the details of the proof. PROPOSITION 9.7. Suppose that G is a generic one-periodic Hamiltonian such that Gt has the unique nondegenerate global minimum x− which is fixed and undertwisted for all t ∈ [0, 1]. Suppose that f : M → R is a Morse function such that f has the unique global minimum point x− and f (x− ) = 0. Then the canonical fundamental cycle has the expression αG = [x− , xˆ− ] + β ∈ CF(G) for some Floer – Novikov chain β ∈ CF(G) with the inequality 1 − − −G(t, x− ) dt. λG (β) < λG ([x , xˆ ]) =

(120)

(121)

0

In particular its level satisfies λG (αG ) = λG ([x− , xˆ− ]) 1 − −G(t, x ) dt = = 0

1

− min G dt.

(122)

0

9.4. THE CASE OF AUTONOMOUS HAMILTONIANS

In this section, we will restrict to the case of autonomous Hamiltonians G. The following result was proven in Oh (2005d). THEOREM 9.8. Let (M, ω) be an arbitrary closed symplectic manifold. Suppose that G is an autonomous Hamiltonian such that 1. it has no nonconstant contractible periodic orbits “of period one” 2. it has a maximum and a minimum that are generically undertwisted

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3. all of its critical points are nondegenerate in the Floer theoretic sense (i.e., the linearized flow of XG at each critical point has only the zero as a periodic orbit). t is length minimizing in its homotopy class with Then the one parameter group φG fixed ends for 0 ≤ t ≤ 1.

And the same result with the condition 1 is replaced by the one in which the phrase “of period one” replaced by “of period less than equal to one” was proven in Oh (2002) earlier. There are also similar results proven in McDuﬀ and Slimowitz (2001), and Entov (2000) (for the strongly semi-positive case) with slightly diﬀerent hypotheses. The improvement of the phrase “of period less than equal to one” being replaced by “of period one” is due to Kerman and Lalonde (2003) in the case of symplectically aspherical (M, ω). To prove Theorem 9.8, according to the criterion Theorem 9.4, it will be enough to prove that the value AG ([x− , xˆ− ]) = E − (G) coincides with the minimax value ρ(G; 1). This latter fact is an immediate consequence of the following theorem, which is a special case of the main theorem in Oh (2005d) restricted to the strongly semi-positive case. Here we provide details of the proof for the strongly semi-positive case. THEOREM 9.9. Suppose that G is an autonomous Hamiltonian satisfying the hypotheses in Theorem 9.8. Then the canonical fundamental cycle αG constructed in Proposition 9.7 is tight, i.e., ρ(G; 1) = λG (αG ) = −G(x− ) = E − (G) . Proof. Note that the conditions in Theorem 9.8 in particular imply that G is nondegenerate. We fix a time-independent J0 which is G-regular. Suppose that α is homologous to the canonical fundamental Floer cycle αG , i.e., α = αG + ∂G (γ) (123) for some Floer Novikov chain γ ∈ CF∗ (G). When G is autonomous and J ≡ J0 is t-independent, there is no nonstationary t-independent trajectory of AG landing at [x− , xˆ− ] because any such trajectory comes from the negative Morse gradient flow of G but x− is the minimum point of G. Therefore any nonstationary Floer trajectory u landing at [x− , xˆ− ] must be t-dependent. Because of the assumption that G has no nonconstant contractible periodic orbits of period one, any critical points of AG has the form [x, w] with x ∈ Crit G. Let u be a trajectory starting at [x, w], x ∈ Crit G with µ([x, w]) − µ([x− , xˆ− ]) = 1,

(124)

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and denote by M(G,J0 ) ([x, w], [x− , xˆ− ]) the corresponding Floer moduli space of connecting trajectories. The general index formula shows µ([x, w]) = µ([x, w x ]) − 2c1 ([w]).

(125)

We consider two cases separately: the cases of c1 ([w]) = 0 or c1 ([w]) 0. If c1 ([w]) 0, we derive from (5.4), (5.5) that x x− . This implies that any such trajectory must come with (locally) free S 1 -action, i.e., the moduli space #(G,J ) ([x, w], [x− , xˆ− ]) = M(G,J ) ([x, w], [x− , xˆ− ])/R M 0 0 and its stable map compactification have a locally free S 1 -action without fixed points. Then it follows from the S 1 -equivariant transversality theorem from (Floer #(G,J ) ([x, w], [x− , xˆ− ]) becomes empty for a suitable choice of et al., 1995) that M 0 an autonomous J0 . This is because the quotient has the virtual dimension -1 by the assumption (124). We refer to Floer et al. (1995) for more explanation on this S 1 -invariant regularization process. Now consider the case c1 ([w]) = 0. First note that (124) and (125) imply that x x− . On the other hand, if x x− , the same argument as above shows that the perturbed moduli space becomes empty. It now follows that there is no trajectory of index 1 that land at [x− , xˆ− ]. Therefore ∂G (γ) cannot kill the term [x− , xˆ− ] in (123) away from the cycle αG = [x− , xˆ− ] + β and hence we have

λG (α) ≥ λG ([x− , xˆ− ])

by the definition of the level λG . Combined with (122), this finishes the proof.

10. Remarks on the transversality for general (M, ω) Our construction of various maps in the Floer homology works as they are in the previous section for the strongly semi-positive case (Seidel, 1997; Entov, 2000) by the standard transversality argument. On the other hand in the general case where constructions of operations in the Floer homology theory requires the machinery of virtual fundamental chains through multi-valued abstract perturbation, we need to explain how this general machinery can be incorporated in our construction. The full details will be provided elsewhere. We will use the terminology ‘Kuranishi structure’ adopted by Fukaya and Ono (1999) for the rest of the discussion.

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One essential point in our proofs is that various numerical estimates concerning the critical values of the action functional and the levels of relevant Novikov cycles do not require transversality of the solutions of the relevant pseudo-holomorphic sections, but depends only on the nonemptiness of the moduli space ˜ zˆ) M(H, J; which can be studied for any, not necessarily generic, Hamiltonian H. Since we always have suitable a priori energy bound which requires some necessary homotopy assumption on the pseudo-holomorphic sections, we can compactify the corresponding moduli space into a compact Hausdorﬀ space, using a variation of the notion of stable maps in the case of nondegenerate Hamiltonians H. We denote this compactification again by ˜ zˆ). M(H, J; This space could be pathological in general. But because we assume that the Hamiltonians H are nondegenerate, i.e, all the periodic orbits are nondegenerate, the moduli space is not completely pathological but at least carries a Kuranishi structure in the sense of Fukaya-Ono (Fukaya and Ono, 1999) for any This enables us to apply the abstract multi-valued perturbation H-compatible J. theory and to perturb the compactified moduli space by a Kuranishi map Ξ so that the perturbed moduli space ˜ zˆ, Ξ) M(H, J; is transversal in that the linearized equation of the perturbed equation ∂¯ J˜(v) + Ξ(v) = 0 is surjective and so its solution set carries a smooth (orbifold) structure. Furthermore the perturbation Ξ can be chosen so that as Ξ → 0, the perturbed moduli ˜ zˆ, Ξ) converges to M(H, J; ˜ zˆ) in a suitable sense (see Fukaya and space M(H, J; Ono, 1999 for the precise description of this convergence). Now the crucial point is that nonemptiness of the perturbed moduli space will be guaranteed as long as certain topological conditions are met. For example, the followings are the prototypes that we have used in this paper: 1. hH : CF0 ( f ) → CF0 (H) is an isomorphism in homology and so [hH (1 )] 0. This is immediately translated as an existence result of solutions of the perturbed Cauchy-Riemann equation. 2. The definition of the pants product ∗ and the identity [α ∗ β] = (a · b) in homology guarantee nonemptiness of the relevant perturbed moduli space ˜ zˆ, Ξ) for α ∈ CF∗ (H1 ), β ∈ CF∗ (H2 ) with [α] = a and [β] = b M(H, J; respectively.

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˜ zˆ, Ξ) and an a priori energy bound Once we prove nonemptiness of M(H, J; for the nonempty perturbed moduli space and if the asymptotic conditions zˆ are ˜ zˆ, Ξ j ) as Ξ j → 0 fixed, we can study the convergence of a sequence v j ∈ M(H, J; by the Gromov – Floer compactness theorem. However a priori there are infinite possibility of asymptotic conditions for the pseudo-holomorphic sections that we are studying, because we typically impose that the asymptotic limit lie in certain Novikov cycles like zˆ1 ∈ α, zˆ2 ∈ β, zˆ3 ∈ α ∗ β Because the Novikov – Floer cycles are generated by an infinite number of critical points [z, w] in general, one needs to control the asymptotic behavior to carry out compactness argument. For this purpose, we need to establish a lower bound for the actions which will enable us to consider only a finite number of possibilities for the asymptotic conditions because of the finiteness condition in the definition of Novikov chains. We would like to emphasize that obtaining a lower bound is the heart of matter in the classical mini-max theory of the indefinite action functional which requires a linking property of semi-infinite cycles. On the other hand, obtaining an upper bound is usually an immediate consequence of the identity like (50). With such a lower bound for the actions, we may then assume, by taking a subsequence if necessary, that the asymptotic conditions are fixed when we take the limit and so we can safely apply the Gromov – Floer compactness theorem to ˜ zˆ). This produce a (cusp)-limit lying in the compactified moduli space M(H, J; would then justify all the statements and proofs in this paper for the complete generality, without assuming the strong semi-positivity assumption. Appendix A. Proof of the index formula In this appendix, we give the proof of the index formula (1), Theorem 7.1. The only thing that enters in the definition of the Conley – Zehnder index is a periodic solution of the Hamilton’s equation x˙ = XH (x) on a symplectic manifold (M, ω) for a one-periodic Hamiltonian function H : S 1 × M → R. We will give the proof of the index formula in several steps. 0. Other convention. There is another package of conventions that have been consistently used by Salamon and Zehnder (1992), Polterovich (2001), and others. In that convention, there are two things to watch out in relation to the index formula, when compared to our convention. The first thing is that their definition

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of the Hamiltonian vector field, also called as the symplectic gradient and denoted by sgrad H, is given by sgrad H+ω = −dH. (126) Therefore we have XH = − sgrad H. The second thing is that their definition of the canonical symplectic form on T ∗ Rn = R2n Cn in the coordinates z j = q j + ip j is given by d p j ∧ dq j = −ω0 (127) ω 0 = j=1

Cancelling out two negatives, the definition of the Hamiltonian vector field of a function H on R2n in this package becomes the same vector field as ours that is given by J0 ∇H where ∇H is the usual gradient vector field of H with respect to the standard Euclidean inner product on R2n . 1. Canonical symplectic form. Our convention of the canonical symplectic form of on T ∗ Rn = R2n Cn in the coordinates z j = q j + ip j is given by dq j ∧ d p j . (128) ω0 = j=1

This means that on R2n J0 is the standard complex structure on R2n Cn obtained by multiplication by the complex number i. 2. Canonical complex structure. In our convention of the canonical symplectic form ω0 Cn , the associated Hermitian structure ·, ·: Cn × Cn → C becomes complex linear in the first argument, but anti-linear in the second argument. In other words, the Hermitian inner product is given by u, v = g(u, v) − iω0 (u, v).

(129)

We like to note that this Hermitian structure on Cn is the conjugate to that of Hofer and Salamon (1995), Salamon and Zehnder (1992), and Polterovich (2001), which corresponds to u, v = g(u, v) + iω0 (u, v). (130) (See the remark right before Lemma 5.1 in Salamon and Zehnder, 1992.) Equivalently, the latter Hermitian structure is associated to the almost K¨ahler structure (g, ω 0 , J0 )

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where J0 is the almost complex structure conjugate to J0 . This change of complex structure on Cn aﬀects the sign of the first Chern number of general complex vector bundles E: we recall the following general formula for the Chern classes of the complex vector bundle E ck (E) = (−1)k ck (E). 3. The Conley – Zehnder index on SP∗ (1). We follow the definition from Salamon and Zehnder (1992) of the Conley – Zehnder index, denoted by indCZ (α) as in Floer and Hofer (1993), for a paths α lying in SP∗ (1) : we denote SP∗ (1) = {α: [0, 1] → Sp(2n, R) | α(0) = id, det(α(1) − id) 0}

(131)

following the notation from Salamon and Zehnder (1992). Note that the definition of Sp(2n, R) are the same in both of the above conventions. We will define the Conley – Zehnder index function indCZ : SP∗ (1) → Z to be the same as that of Salamon and Zehnder (1992). This index is then characterized by Floer and Hofer (1993, Proposition 5). 0 (M) determines a pre4. Symplectic trivialization. A given pair [γ, w] ∈ Ω ferred homotopy class of trivialization of the symplectic vector bundle γ∗ T M on S 1 = ∂D2 that extends to a trivialization Φw : w∗ T M → D2 × (R2n , ω0 ) over D2 of where D2 ⊂ C is the unit disc with the standard orientation. A symplectic trivialization Φw : w∗ T M → D2 × (R2n , ω 0 ) of w∗ T M in terms of (R2n , ω 0 ) is then obtained by the composition Φw = c ◦ Φw ;

Φw (z, v) := Φw (z, v), (z, v) ∈ w∗ T M

(132)

where c is the complex conjugation on R2n Cn in the obvious sense. 5. The Conley – Zehnder index, µH ([z, w]). Let z : R/Z × M be a one-periodic solution of x˙ = XH (x). Any such one-periodic solution has the form z(t) = φtH (p) for a fixed point p = z(0) ∈ Fix(φ1H ). When we are given a one-periodic solution z and its bounding disc w : D2 → M, we consider the one-parameter family of the symplectic maps dφtH z(0) : T z(0) M → T z(t) M and define a map α[z,w] : [0, 1] → Sp(2n, R) by α[z,w] (t) = Φw z(t) ◦ dφtH z(0) ◦ Φw z(0) −1 .

(133)

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Obviously we have α[z,w] (0) = id, and nondegeneracy of H implies that det(α[z,w] (1) − id) 0 and hence

α[z,w] ∈ SP∗ (1).

Then according to the definition of Salamon and Zehnder (1992) and Hofer and Salamon (1995) the Conley – Zehnder index of [z, w] is defined by µH ([z, w]) := indCZ (α[z,w] )

(134)

where α[z,w] = c ◦ α[z,w] . 6. When we are given two maps w, w : D2 → M with w|∂D2 = w |∂D2 , we define the glued map u = w#w : S 2 → M in the following way: w(z) z ∈ D+ u(z) = w (1/¯z) z ∈ D− . Here D+ is D2 with the same orientation, and D− with the opposite orientation. This is a priori only continuous but we can deform to a smooth one without changing its homotopy class by ‘flattening’ the maps near the boundary: In other words, we may assume w(z) = w(z/|z|) for |z| ≥ 1 − for suﬃciently small > 0. We will always assume that the bounding disc will be assumed to be flat in this sense. With this adjustment, u defines a smooth map from S 2 . 7. The marking condition. For the given [z, w], [z, w ] with a periodic solution z(t) = φtH (z(0)), we impose the additional marking condition Φw (z(0)) = Φw (z(0))

(135)

as a map from T z(0) M to R2n for the trivialization Φw , Φw : w∗ T M → D2 × (R2n , ω0 ) which is always possible. With this additional condition, we can write α[z,w ] (t) = S w w (t) · α[z,w] (t)

(136)

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where S w w : S 1 = R/Z → Sp(2n, R) is the loop defined by the relation (136). Note that this really defines a loop because we have α[z,w ] (0) = α[z,w] (0) (= id) α[z,w ] (1) = α[z,w] (1)

(137) (138)

where (138) follows from the marking condition (135). In fact, it follows from the definition of (136) and (135) that we have the identity S w w (t) = Φw (z(t)) ◦ dφtH z(0) ◦ Φw z(0) −1 ◦ Φw z(t) ◦ dφtH z(0) ◦ Φw z(0) −1 −1 = Φw (z(t)) ◦ dφtH z(0) ◦ Φw z(0) −1 ◦ Φw z(0) ◦ (dφtH )−1 z(0) −1 ◦ Φw z(t) . (139) Then the marking condition (135) implies the middle terms in (139) are cancelled away and hence we have proved S w w (t) = Φw (z(t)) ◦ Φw (z(t))−1

(140)

Then we derive the following formula, from the definition µCZ in Conley and Zehnder (1984) and from (140), indCZ (α[z,w ] ) = 2 wind S w w + indCZ (α[z,w] ) (141) where S w w : S 1→ U(n) is a loop in U(n) that is homotopic to S w w inside Sp(2n, R). (See Floer and Hofer, 1993, Proposition 5, for this formula.) Such a homotopy always exists and is unique upto homotopy because U(n) is a deformation retract to Sp(2n, R). And wind( S w w ) is the degree of the obvious determinant map S w w ): S 1 → S 1 . detC ( 8. Normailization of c1 . Finally, we recall the definition of the first Chern class c1 of the symplectic vector bundle E → S 2 . We normalize the Chern class so that the tangent bundle of S 2 CP1 has the first Chern number 2, which also coincides with the standard convention in the literature. We like to note that this normalization is compatible with the Hermitian structure on Cn given by (129) in our convention. (See Milnor and Stasheﬀ (1974), p. 167.) We decompose S 2 = D+ ∪ D− and consider the symplectic trivializations Φ+ : E|D+ → D2 × (R2n , ω0 ) and Φ− : E|D− → D2 × (R2n , ω0 ). Note that under the Hermitian structure on Cn in our convention, these are homotopic to a unitary trivialization, while in other convention they are homotopic to a conjugate unitary trivialization.

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Denote by the transition matrix loop φ+− : S 1 → Sp(2n, R) which is the loop determined by the equation Φ+ |S 1 ◦ (Φ− |S 1 )−1 (t, ξ) = (t, φ+− (t)ξ) for (t, ξ) ∈ E|S 1 , where S 1 = ∂D+ = ∂D− . Then, by definition, we have c1 (E) = wind(φˆ +− )

(142)

in our convention. Equivalently, we have c1 (E) = − wind(φˆ +− ).

(143)

Now we apply this to u∗ (T M) where u = w # w and Φw and Φw are the trivializations given in 4. It follows from (140) that S w w is the transition matrix loop between Φw and Φw . Then by definition, the first Chern number c1 (u∗ T M) is provided by the winding number wind( S w w ) of the loop of unitary matrices S w w (t); S w w : t →

S 1 → U(n)

in the Hermitian structure of Cn in our convention. One can easily check that this winding number is indeed 2 when applied to the tangent bundle of S 2 and so consistent with the convention of the Chern class that we are adopting. 9. Wrap-up of the proof. These steps, in particular, Step 2 and Steps 7 and 8 combined, (134), (141), and (143) turn into the index formula we want to prove. We restate this in the following theorem. THEOREM A.1. Let (M, ω) be a symplectic manifold and XH a Hamiltonian vector field defined by XH +ω = dH of any contractible one-periodic Hamiltonian function H: [0, 1] × M → R. For a given one-periodic solution z: S 1 = R/Z → M of x˙ = XH (x) and two given bounding discs w, w , we have the identity µH ([z, w ]) = µH ([z, w]) − 2c1 ([w # w]).

References Banyaga, A. (1978) Sur la structure du groupe des diﬀ´eomorphismes qui pr´eservent une forme symplectique, Comment. Math. Helv. 53, 174 – 227.

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Benci, V. and Rabinowitz, P. H. (1979) Critical point theorems for indefinite functionals, Invent. Math. 52, 241 – 273. Bialy, M. and Polterovich, L. (1994) Geodesics of Hofer’s metric on the group of Hamiltonian diﬀeomorphisms, Duke Math. J. 76, 273 – 227. Chen, K. T. (1973) Iterated integrals of diﬀerential forms and loop space homology, Ann. of Math. (2) 97, 217 – 246. Conley, C. C. and Zehnder, E. (1984) Morse-type index theory for flows and periodic solutions of Hamiltonian equations, Comm. Pure Appl. Math. 37, 207 – 253. Entov, M. (2000) K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math. 146, 93 – 141. Entov, M. (2004) Commutator length of symplectomorphisms, Comment. Math. Helv. 79, 58 – 104. Entov, M. and Polterovich, L. (2003) Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30, 1635 – 1676. Floer, A. (1988) The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41, 775 – 813. Floer, A. (1989) Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120, 575 – 611. Floer, A. and Hofer, H. (1993) Coherent orientations for periodic orbit problems in symplectic geometry, Math. Z. 212, 13 – 38. Floer, A., Hofer, H., and Salamon, D. (1995) Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80, 251 – 292. Fukaya, K. and Oh, Y.-G. (1997) Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1, 96 – 180. Fukaya, K. and Ono, K. (1999) Arnold conjecture and Gromov – Witten invariants, Topology 38, 933 – 1048. Getzler, E., Jones, J. D. S., and Petrack, S. (1991) Diﬀerential forms on loop spaces and the cyclic bar complex, Topology 30, 339 – 371. Gromov, M. (1985) Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82, 307 – 347. Guillemin, V. and Lerman, E.and Sternberg, S. (1996) Symplectic Fibrations and Multiplicity Diagrams, Cambridge, Cambridge Univ. Press. Hofer, H. (1990) On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115, 25 – 38. Hofer, H. (1993) Estimates for the energy of a symplectic map, Comment. Math. Helv. 68, 48 – 72. Hofer, H. and Salamon, D. (1995) Floer homology and Novikov rings, In H. Hofer, C. H. Tauber, A. Weinstein, and E. Zehnder (eds.), The Floer Memorial Volume, Vol. 133 of Progr. Math, pp. 483–524, Boston, MA, Birkh¨auser. Kerman, E. and Lalonde, F. (2003) Length minimizing Hamiltonian paths for symplectically aspherical manifolds, Ann. Inst. Fourier (Grenoble) 53, 1503 – 1526. Lalonde, F. and McDuﬀ, D. (1995)a The geometry of symplectic energy, Ann. of Math. (2) 141, 349 – 371. Lalonde, F. and McDuﬀ, D. (1995)b Hofer’s L∞ -geometry: energy and stability of Hamiltonian flows. I, Invent. Math. 122, 1 – 33; II, 35 – 69. Liu, G. and Tian, G. (1998) Floer homology and Arnold conjecture, J. Diﬀerential Geom. 49, 1 – 74. Liu, G.and Tian, G. (1999) On the equivalence of multiplicative structures in Floer homology and quantum homology, Acta Math. Sinica (Engl. Ser.) 15, 53 – 80. McDuﬀ, D. and Salamon, D. (2004) J-Holomorphic Curves and Symplectic Topology, Vol. 6 of Univ. Lecture Ser., Providence, RI, Amer. Math. Soc.

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McDuﬀ, D. and Slimowitz, J. (2001) Hofer – Zehnder capacity and length minimizing Hamiltonian paths, Geom. Topol. 5, 799 – 830. Milnor, J. W. and Stasheﬀ, J. D. (1974) Characteristic Classes, Vol. 76 of Ann. of Math. Stud., Princeton, NJ, Princeton Univ. Press. Novikov, S. P. (1981) Multivalued functions and functionals. An analogue of the Morse theory, Dokl. Akad. Nauk SSSR 260, 31 – 35, Russian. Novikov, S. P. (1982) The Hamiltonian formalism and a multivalued analogue of Morse theory, Uspekhi Mat. Nauk 37, 3 – 49, Russian. Oh, Y.-G. (1992) Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions, Comm. Pure Appl. Math. 45, 121 – 139. Oh, Y.-G. (1997) Symplectic topology as the geometry of action functional. I, J. Diﬀerential Geom. 46, 499 – 577. Oh, Y.-G. (1999) Symplectic topology as the geometry of action functional. II, Comm. Anal. Geom. 7, 1 – 55. Oh, Y.-G. (2002) Chain level Floer theory and Hofer’s geometry of the Hamiltonian diﬀeomorphism group, Asian J. Math. 6, 579 – 624; Erratum, 7, 447 – 448. Oh, Y.-G. (2004)a Floer mini-max theory, the Cerf diagram, and the spectral invariants, arXiv: math.SG/0406449. Oh, Y.-G. (2004)b The group of Hamiltonian homeomorphisms and C 0 -symplectic topology, arXiv: math.SG/0402210. Oh, Y.-G. (2005)a Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, In The Breadth of Symplectic and Poisson Geometry, Vol. 232 of Progr. Math, pp. 525–570, Boston, MA, Birkh¨auser. Oh, Y.-G. (2005)b Normalization of the Hamiltonian and the action spectrum, J. Korean Math. Soc. 42, 65 – 83. Oh, Y.-G. (2005)c Spectral invariants, analysis of the Floer moduli space and geometry of Hamiltonian diﬀeomorphisms, Duke Math. J., to appear; arXiv:math.SG/0403083. Oh, Y.-G. (2005)d Spectral invariants and length minimizing property of Hamiltonian paths, Asian J. Math. 9, 1 – 18. Ostrover, Y. (2003) A comparison of Hofer’s metrics on Hamiltonian diﬀeomorphisms and Lagrangian submanifolds, Commun.Contemp. Math. 5, 803 – 812. Piunikhin, S., Salamon, D., and Schwarz, M. (1996) Symplectic Floer – Donaldson theory and quantum cohomology, In C. B. Thomas (ed.), Contact and Symplectic Geometry, Vol. 8 of Publ. Newton Inst., Cambridge, 1994, pp. 171–200, Cambridge, Cambridge Univ. Press. Polterovich, L. (1993) Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems 13, 357 – 367. Polterovich, L. (1998) Geometry on the group of Hamiltonian diﬀeomorphisms, Doc. Math. 1998 Extra Vol. II, 401 – 410. Polterovich, L. (2001) The Geometry of the Group of Symplectic Diﬀeomorphisms, Lectures Math. ETH Zurich, Basel, Birkh¨auser. Polterovich, L. (2005) Floer homology, dynamics and groups, in this volume. Rabinowitz, P. H. (1978) Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31, 157 – 184. Sacks, J. and Uhlenbeck, K. (1981) The existence of minimal immersions of 2-spheres, Ann. of Math. (2) 113, 1 – 24. Salamon, D. and Zehnder, E. (1992) Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45, 1303 – 1360. Schwarz, M. (2000) On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193, 419 – 461.

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Seidel, P. (1997) π1 of symplectic diﬀeomorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7, 1046 – 1095. Ustilovsky, I. (1996) Conjugate points on geodesics of Hofer’s metric, Diﬀerential Geom. Appl. 6, 327 – 342. Viterbo, C. (1992) Symplectic topology as the geometry of generating functions, Math. Ann. 292, 685 – 710. Weinstein, A. (1978) Bifurcations and Hamilton’s principle, Math. Z. 159, 235 – 248.

FLOER HOMOLOGY, DYNAMICS AND GROUPS LEONID POLTEROVICH Tel Aviv University

Abstract. We discuss some recent results on algebraic properties of the group of Hamiltonian diﬀeomorphisms of a symplectic manifold. We focus on two topics which lie at the interface between Floer theory and dynamics: 1. Restrictions on Hamiltonian actions of finitely generated groups, including a Hamiltonian version of the Zimmer program dealing with actions of lattices; 2. Quasi-morphisms on the group of Hamiltonian diﬀeomorphisms. The unifying theme is the study of distortion of cyclic and one-parameter subgroups with respect to various metrics on the group of Hamiltonian diﬀeomorphisms.

In the present lectures we discuss some recent results on algebraic properties of the group of Hamiltonian diﬀeomorphisms Ham(M, ω) of a smooth connected symplectic manifold (M 2m , ω). We focus on two topics which lie at the interface between Floer theory and dynamics, where by dynamics we mean the study of asymptotic behavior of Hamiltonian diﬀeomorphisms under iterations: − Restrictions on Hamiltonian actions of finitely generated groups, and in particular a Hamiltonian version of the Zimmer program dealing with actions of lattices (Polterovich, 2002); − Quasi-morphisms on Ham, including the Calabi quasi-morphism introduced in a joint work with Michael Entov (Entov and Polterovich, 2003). The unifying theme is the study of distortion of cyclic and one-parameter subgroups with respect to various metrics on the group of Hamiltonian diffeomorphisms. We refer to Hofer and Zehnder (1994), McDuﬀ and Salamon (1995; 2004), and Polterovich (2001) for symplectic preliminaries. 1. Hamiltonian actions of finitely generated groups 1.1. THE GROUP OF HAMILTONIAN DIFFEOMORPHISMS

Recall that symplectic manifolds appear as phase spaces of classical mechanics. An important principle of classical mechanics is that the energy of a system de417 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 417–438. © 2006 Springer. Printed in the Netherlands.

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termines its evolution. The energy (or Hamiltonian function) Ft (x) := F(x, t) is a smooth function on M × R. Here t is the time coordinate. Define the timedependent Hamiltonian vector field sgrad Ft by the point-wise linear equation isgrad F ω = −dF. The evolution of the system is described by the flow ft on M generated by the Hamiltonian vector field sgrad Ft . We always assume that the union of the supports of Ft , t ∈ R, is contained in a compact subset of M. This guarantees that the evolution is well defined. We will refer to the time-one-map f1 of this flow as to the Hamiltonian diﬀeomorphism generated by F and denote it by φF . Hamiltonian diﬀeomorphisms form a group which is denoted by Ham(M, ω) and which is the main object of our study. We start with the following problem. Let (M, ω) be a closed symplectic manifold. PROBLEM 1.1. Find restrictions on Hamiltonian actions of finitely generated groups on M, and, in particular, on finitely generated subgroups of Ham(M, ω). Polterovich (2002) develops an approach to this problem for some symplectic manifolds with π2 = 0 which is based on Floer theory. Below we discuss (with an outline of proofs) some sample results in this direction. The selection was made with the idea to avoid as much as possible the use of sophisticated algebraic machinery (the only exception is the Margulis finiteness theorem). 1.2. THE NO-TORSION THEOREM

THEOREM 1.2. Let (M, ω) be a closed symplectic manifold with π2 = 0. Then the group Ham(M, ω) has no torsion. The proof is given in Section 2.2 below. Note that the assumption on π2 is essential: the 2-sphere admits isometries (rotations) of finite order. As an immediate consequence of the theorem, we get the following result. COROLLARY 1.3. Let (M, ω) be a closed symplectic manifold with π2 = 0. Let Γ be any group generated by elements of finite order. Then every homomorphism Φ: Γ → Ham(M, ω) is trivial: Φ ≡ 1. A classical example of a group generated by elements of finite order is the group SL(k, Z) with k ≥ 2. Indeed, it is shown in Newman, (1972, Theorem VII.3) that SL(k, Z) is generated by the matrix of the transformation (x1 , . . . , xk ) → (x2 , . . . , xk , (−1)k−1 x1 ) which is clearly of finite order, and the matrix A ⊕ 1k−2 with 1 1 A= . 0 1

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The matrix A, in turn, can be represented as the product of finite order matrices: 0 −1 0 1 A= · . 1 0 −1 −1 Hence SL(k, Z) with k ≥ 2 is generated by elements of finite order. Corollary 1.3 yields the following result. COROLLARY 1.4. Every Hamiltonian action of SL(k, Z) on a closed symplectic manifold with π2 = 0 is trivial. Consider now the case when Γ ⊂ SL(k, Z) is a normal subgroup of finite index. What can one say about Hamiltonian actions of Γ? In other words, we ask whether the phenomenon presented in Corollary 1.4 is robust from the viewpoint of group theory. Each such Γ is finitely generated (see de la Harpe, 2000). We will need also the following important result which is a particular case of the Margulis finiteness theorem. THEOREM 1.5 (Margulis, 1991; Zimmer, 1984). Let Γ be an infinite normal subgroup of SL(k, Z) for k ≥ 3. Then Γ is of finite index in SL(k, Z). Moreover, every infinite normal subgroup of Γ is of finite index in Γ. In contrast to this, the group SL(2, Z) has infinite normal subgroups of infinite index. Let Γ ⊂ SL(k, Z), k ≥ 2, be a normal subgroup of finite index. In general it may happen that all elements of Γ have infinite order, and hence our argument used in the proof of Corollary 1.4 does not work anymore. For instance, take any integer l ≥ 2 and define the principal congruence subgroup Γl ⊂ SL(k, Z) as the kernel of the natural homomorphism SL(k, Z) → SL(k, Z/lZ). Clearly, Γl is a normal subgroup of finite index. It turns out that Γl has no torsion for l ≥ 3 (see Witte, 2001, Section 5.I). Another example is given by the commutator subgroup of SL(2, Z) which is isomorphic to the free group F2 with 2 generators (see Newman, 1972) and hence has no torsion. Now we come to a well known and quite important point: the cases k = 2 (the rank-one case) and k ≥ 3 (the higher-rank case) are dramatically diﬀerent. The free group F2 admits a monomorphism to Ham(M, ω) for any symplectic manifold (M, ω). Indeed, take two Hamiltonian diﬀeomorphisms f and g with a common fixed point x ∈ M. It is easy to arrange that the diﬀerentials d x f and d x g generate a free subgroup of linear transformations of T x M. Hence the subgroup generated by f and g is free. In contrast to this, in the case k ≥ 3, there exist obstructions to Hamiltonian actions of infinite normal subgroups of SL(k, Z). We will focus for simplicity on

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the following special class of symplectic manifolds. Consider a closed symplectic of manifold (M, ω) with π2 (M) = 0. Then the lift ω ˜ of ω to the universal cover M = dλ. We say that (M, ω) is symplectically hyperbolic if π2 (M) = 0 M is exact: ω admits a primitive λ which is bounded with respect to a Riemannian metric and ω coming from M. It is an easy exercise in hyperbolic geometry to show that on M surfaces of genus ≥ 2 endowed with the hyperbolic area form are symplectically hyperbolic. The same is true for their direct products. This class of symplectic manifolds is a counterpart of K¨ahler hyperbolic manifolds considered in complex geometry. THEOREM 1.6. Let Γ ⊂ SL(k, Z), k ≥ 3, be an infinite normal subgroup. Let (M, ω) be a closed symplectically hyperbolic manifold. Then every homomorphism Φ: Γ → Ham(M, ω) is trivial: Φ ≡ 1. For the proof, we have to introduce the notion of distortion, which in a sense is a unifying theme for various topics discussed in these lectures. 1.3. DISTORTION IN NORMED GROUPS

Let G be a group endowed with a norm g, g ∈ G. The axioms of a norm are as follows: − g > 0 if g 1, and 1 = 0; − g−1 = g; − gh ≤ g + h for all g, h ∈ G. We say that an element g ∈ G is distorted if gn = 0. n→+∞ n lim

Otherwise, g is called undistorted. Informally speaking, one can think of a cyclic subgroup generated by an undistorted element as of a minimal geodesic in G. We will return to the notion of distortion many times throughout these lectures. For instance, let Γ be a finitely generated group. Let S be a symmetric finite generating set of Γ. This means that s ∈ S ⇐⇒ s−1 ∈ S and every element g ∈ Γ can be written as g = s1 · · · · · s N ,

si ∈ S .

(1)

Define the word norm g as the minimum of N over all decompositions (1). Note that the word norms associated to diﬀerent finite generating sets are mutually

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equivalent, hence the property of g to be distorted/undistorted with respect to the word norm is well-defined. Let us illustrate this notion in the following important example. The Heisenberg group H is the group with three generators f , g, h which satisfy the following relations: h = [ f, g] := f g f −1 g−1 , [ f, h] = [g, h] = 1. It is not hard to check that 2 hmn = [ f m , gn ] for all m, n ∈ N. This yields hn ≤ const · n for all n ∈ N. In particular, the element h is a distorted element of infinite order in H. The Heisenberg group can be considered as a subgroup of SL(3, Z) (see, e.g., de la Harpe, 2000, IV.A.8): the map ı: H → SL(3, Z) with       1 0 0 1 0 0 1 0 0       ı( f ) = 0 1 0 , ı(g) = 1 1 0 , ı(h) = 0 1 0       0 1 1 0 0 1 1 0 1 is a monomorphism. It follows that ı(h) is a distorted element of infinite order in SL(3, Z). Note that SL(3, Z) naturally lies in SL(k, Z) for all k ≥ 3: we identify a matrix A ∈ SL(3, Z) with A ⊕ 1k−3 . Hence the same conclusion holds true for SL(k, Z) with k ≥ 3. We will need the following proposition which is a particular case of a much stronger theorem by Lubotzky et al. (2000). PROPOSITION 1.7. Let Γ be an infinite normal subgroup of SL(k, Z) with k ≥ 3. Then Γ has a distorted element. Proof. Let ı: H → SL(k, Z) be the monomorphism described above. Put φ = ı( f ), ψ = ı(g) and θ = ı(h). Since Γ is a normal subgroup of finite index, there exists N ∈ N so that φN and ψN lie in Γ. It was already mentioned that [φnN , ψnN ] = 2 2 2 θn N for all n ∈ N. This yields that the element θ N ∈ Γ is distorted: 2

2

(θ N )n ≤ const · n.

1.4. THE NO-DISTORTION THEOREM

The next result provides a restriction on Hamiltonian actions of finitely generated groups on a closed symplectically hyperbolic manifold in terms of distortion. THEOREM 1.8. Assume that (M, ω) is a closed symplectically hyperbolic manifold. Consider any finitely generated group Γ. Let g ∈ Γ be an element which is distorted with respect to the word norm on Γ. Then Φ(g) = 1 for any homomorphism Φ: Γ → Ham(M, ω). In particular, if Γ is a finitely generated subgroup of Ham(M, ω), every element of Γ \ {1} is undistorted with respect to the word norm on Γ.

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The proof based on Floer theory will be given in Section 2.4 below. Proof of Theorem 1.6. Apply the theorem above to a distorted (with respect to the word norm) element g of infinite order lying in Γ. Such an element exists due to Proposition 1.7. We see that g lies in the kernel of any homomorphism Φ: Γ → Ham(M, ω). Note that Ker(Φ) is a normal subgroup of Γ. It is infinite since g has infinite order. The Margulis finiteness theorem (see Theorem 1.5 above) guarantees that Ker(Φ) has finite index in Γ. Hence the quotient Γ/ Ker(Φ) is finite, so Φ has finite image. Applying No-Torsion Theorem 1.2 we conclude that Φ ≡ 1.

1.5. THE ZIMMER PROGRAM

Infinite normal subgroups of SL(k, Z) are basic examples of much more general finitely generated groups called lattices (see Witte, 2001, for an introduction to lattices). The problems formulated below are already highly nontrivial for the following class of lattices. Consider the semisimple Lie group G = SL(k1 , R) × · · · × SL(kd , R) ) where ki ≥ 2 and d ∈ N. The number di=1 (k1 − 1) is called the real rank of G. A discrete subgroup Γ ⊂ G is a lattice if the Haar measure of G/Γ is finite. The proof of the fact that SL(k, Z) ⊂ SL(k, R) is a lattice is quite involved, see, e.g., Feres (1998, Appendix A.1) for a transparent exposition. A lattice Γ is reducible if there exists a decomposition G = G1 × G2 and lattices Γi ⊂ Gi such that the intersection Γ ∩ (Γ1 × Γ2 ) of the two subgroups has finite index in both of them. Otherwise, Γ is irreducible. Every lattice in SL(k, R) is irreducible. A lattice is called uniform if G/Γ is compact, and nonuniform otherwise. It is not hard to see that SL(k, Z) ⊂ SL(k, R) is nonuniform. The same holds for its infinite normal subgroups since they are of finite index by Margulis theorem. An important property of nonuniform lattices is the existence of “strongly” distorted elements of infinite order (see Section 2.5). This diﬃcult theorem was established by Lubotzky, Mozes and Raghunathan in Lubotzky et al. (2000). Proposition 1.7 above is an elementary manifestation of this fact. Problem 1.1 goes back to Zimmer (1987) which contains a number of exciting conjectures about actions of lattices on manifolds. Roughly speaking, the Zimmer program can be formulated as follows: Assume that the real rank of G is at least two. Let Γ be an irreducible lattice in G. The Zimmer conjecture states that the image of every homomorphism of Γ to the group of (volume-preserving) diﬀeomorphisms of a closed connected manifold of suﬃciently small dimension is finite. The Zimmer program gave rise to many remarkable developments, see, e.g., Burger and Monod (1999; 2002), Farb and Shalen (1999), and Ghys (1999). Ghys

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(1999) gave a highly nontrivial proof of the Zimmer conjecture in the case of actions of lattices on the circle by (not necessarily volume-preserving) diﬀeomorphisms. The two-dimensional case of the conjecture is an active field, see (Polterovich, 2002) and Franks and Handel (2003; 2004). We refer to Fisher (2003) for a discussion of recent progress in this direction. Below we address a Hamiltonian version of the Zimmer program which deals with actions of lattices by Hamiltonian diﬀeomorphisms of closed symplectic manifolds. PROBLEM 1.9. Find restrictions on Hamiltonian actions of lattices on closed symplectic manifolds. Theorem 1.6 should be considered as a step in this direction. We refer the reader to Polterovich (2002) and Franks and Handel (2003) for various generalizations, and to Section 2.5 for a further discussion. 2. Floer theory in action 2.1. A BRIEF SKETCH OF FLOER THEORY

Here we present a very brief sketch of Floer theory (Floer, 1988, 1989a, 1989b) with an emphasize on the main tool we are going to use — spectral invariants of Hamiltonian diﬀeomorphisms coming from filtered Floer homology. These invariants were introduced and studied by Schwarz (2000) in the case when the cohomology class of the symplectic form vanishes on π2 of the symplectic manifold, and by Oh (2002b) in the general case (see also Viterbo, 1992; Oh, 1997, 1999, 2002a; Entov and Polterovich, 2003). A more detailed exposition of this theory can be found in Oh (2005) and McDuﬀ and Salamon (2004). Let (M 2m , ω) be a closed symplectic manifold.1 Consider the space Λ of all smooth contractible loops x: S 1 = R/Z → M. Denote by F the space of all smooth Hamiltonian functions F : M × S 1 → R which satisfy the following , normalization condition: M F(·, t) ωm = 0 for any t ∈ S 1 . Define the action functional AF (x, u) :=

S1

F(x(t), t) dt −

ω, u

where x ∈ Λ and u is a disc spanning x in M. In general, the action functional is a multi-valued function on Λ — it depends on the homotopy class with fixed boundary of the disc u. However, it becomes single-valued on a suitable covering ˜ of Λ, and its diﬀerential dAF is a well-defined closed 1-form on Λ itself. A Λ 1 There is a general belief that the picture presented below is valid for all closed symplectic manifolds. At the moment, it is confirmed under some additional assumptions on (M, ω), see (McDuﬀ and Salamon, 2004).

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crucial fact is that the critical points of this 1-form correspond to contractible 1-periodic orbits of the Hamiltonian flow { ft } generated by F. Floer homology is Morse – Novikov homology of the 1-form dAF . To develop Morse theory in this context start with a loop {Jt }, t ∈ S 1 , of ω-compatible almost complex structures and define a Riemannian metric on Λ by 1 (ξ1 , ξ2 ) = ω ξ1 (t), Jt ξ2 (t) dt, 0

˜ and consider the negative gradient flow of where ξ1 , ξ2 ∈ T Λ. Lift this metric to Λ the action functional AF . For a generic choice of the Hamiltonian F and the loop {Jt } the count of isolated gradient trajectories connecting critical points of AF ˜ Each connecting gives rise in a standard way to the Morse complex of AF on Λ. trajectory is a path in the space of loops and hence forms a 2-dimensional cylinder. An important feature of the gradient flow is that the connecting trajectories are solutions of a deformed Cauchy – Riemann equation, and hence they can be studied with the Gromov theory of pseudo-holomorphic curves. Floer homology is defined as the homology of the above-mentioned Morse complex with coeﬃcients in an appropriate Novikov ring. This homology is independent of the Hamiltonian F and is canonically isomorphic to the quantum homology QH(M, ω) — a suitable deformation of the homology ring of M (see Piunikhin et al., 1996). Let φ˜ F be the natural lift of the time-1-map f1 = φF of the flow to the universal * cover Ham(M, ω) of the group of Hamiltonian diﬀeomorphisms of M. It turns out ˜ depend only on that the Floer homologies of the sublevel sets {AF < α} ⊂ Λ ˜ the element φF but not on the specific choice of a generating Hamiltonian F. Hence they give rise to a bunch of invariants of φ˜ F . These homologies form a rich * algebraic object canonically associated to the group Ham(M, ω), and carry some interesting information about the group. The natural inclusion of sublevel sets ˜ {AF < α} → {AF < +∞} = Λ induces a morphism in the corresponding Floer homologies: ˜ = QH(M, ω). iα : HFloer ({AF < α}) → HFloer (Λ) For a quantum homology class a ∈ QH(M, ω) set c(a, φ˜ F ) = inf{α ∈ R | a ∈ Image iα }.

(2)

This number is called a spectral invariant of the (lifted) Hamiltonian dif* feomorphism φ˜ F ∈ Ham(M, ω). Intuitively speaking, spectral invariants are homologically essential critical values of the action functional AF . They play an important role in symplectic dynamics.

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Suppose now that π2 (M) = 0. In this case the action AF is a single-valued functional on Λ. Introduce the width of the Hamiltonian diﬀeomorphism φF as the maximal diﬀerence between the critical values of AF . Schwarz (2000) proved that this quantity does not depend on the specific choice of a Hamiltonian function generating φF , and therefore gives rise to an invariant of the Hamiltonian diﬀeomorphism φF . It is not hard to see that width(φF ) is always finite. We will need the following deep result of Floer theory: THEOREM 2.1 (Schwarz, 2000). Assume that φF 1. Then the action functional AF has at least two distinct critical values. In particular, width(φF ) > 0. On the other hand, width(1) = 0. Another, this time very simple, property of the width is its nice behavior under iterations of a Hamiltonian diﬀeomorphism: width( f n ) ≥ n · width( f )

for all f ∈ Ham(M, ω).

(3)

2.2. WIDTH AND TORSION

As an immediate application of the width, we prove the No-Torsion Theorem. Proof of Theorem 1.2. Take any f ∈ Ham(M, ω) \ {1}. Then width( f n ) > 0 for all n ∈ N by Theorem 2.1 and formula (3). Thus f n 1, so f is of infinite order. 2.3. A GEOMETRY ON Ham(M, ω)

Assume that (M, ω) is closed and symplectically hyperbolic. Fix any Riemannian metric ρ on M. Given a path { ft }, t ∈ [0; 1], of Hamiltonian diﬀeomorphisms of M, denote by F(x, t) the corresponding Hamiltonian. By adding a time-dependent , constant const(t) to F we can achieve the following normalization: M Ft ωn = 0 for all t ∈ [0; 1]. Define 1 max | sgrad Ft (x)|ρ + max |Ft (x)| dt. length{ ft } = 0

x∈M

x∈M

Introduce a norm ν on Ham(M, ω) by ν( f ) = inf length{ ft }, where the infimum is taken over all paths { ft } of Hamiltonian diﬀeomorphisms joining the identity with f . It is easy to check (and we leave this as an exercise) that ν satisfies the axioms of a norm presented in Section 1.3 above. Our next result relates this norm with the width introduced in Section 2.1.

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THEOREM 2.2 (Geometric inequality). There exists a constant C > 0 so that ν( f ) ≥ C · width( f ) for all Hamiltonian diﬀeomorphisms f 1. The proof is based on a simple argument of diﬀerential-geometric flavor, see Polterovich (2002). 2.4. WIDTH AND DISTORTION

THEOREM 2.3. Every element f ∈ Ham(M, ω) \ {1} is undistorted with respect to the norm ν. Proof. Combining Theorem 2.2 with formula (3) we get ν( f n ) ≥ C · width( f n ) ≥ Cn · width( f ). But, crucially, Theorem 2.1 yields width( f ) > 0. Thus f is undistorted.

Now we are ready to prove the No-Distortion Theorem. Proof of Theorem 1.8. Take any symmetric finite generating set S ⊂ Γ, and put C = max ν Φ(s) . s∈S

Then ν Φ(g) ≤ C · g, and hence ν(Φ(g)n ) gn ≤ C · lim = 0. n→+∞ n→+∞ n n lim

We get that Φ(g) is distorted with respect to ν. By Theorem 2.3, Φ(g) = 1.

2.5. MORE REMARKS ON THE ZIMMER PROGRAM

We conclude our presentation of the Hamiltonian version of the Zimmer program with some remarks and open problems. The geometric inequality given in Theorem 2.2 above can be extended in a weaker form to any closed symplectic manifold with π2 = 0. In fact, one gets a lower bound on the growth type of the sequence ν( f n ), f 1 in terms of the symplectic filling function of (M, ω), see Polterovich (2002). This function measures the “minimal growth” of a primitive of M. of the symplectic form on the universal cover M n The lower bound on ν( f ), f 1 yields in the same way an analogue of Theorem 1.8: one can show that if Γ is a finitely generated group, and gn /n

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decays suﬃciently fast for some element g ∈ Γ, this element must lie in the kernel of any homomorphism Γ → Ham(M, ω). admits a primitive which grows Assume, for instance, that the lift of ω to M not faster than R , > 0, on balls of radius R. A good example is given by the standard symplectic torus T2n , where one can take = 1. Let Γ be a finitely generated group, and suppose that lim inf

log gn =0 log n

(4)

for some g ∈ Γ. Then g lies in the kernel of any homomorphism Γ → Ham(M, ω). The Lubotzky – Mozes – Raghunathan theorem (Lubotzky et al., 2000) states that every nonuniform lattice of rank ≥ 2 has an element of infinite order which satisfies equation (4). Arguing as in the proof of Theorem 1.6 one can show that every Hamiltonian action of an irreducible nonuniform lattice of rank ≥ 2 on such a symplectic manifold is trivial. Some of these results extend to the identity component of the group of all symplectic (not necessarily Hamiltonian) diﬀeomorphisms of (M, ω), and in the case of closed surfaces — to the group of all area preserving diﬀeomorphisms. It is, of course, a challenging problem to handle the remaining case of uniform (that is, co-compact) lattices. Let us mention that our method does not work for symplectic manifolds with π2 0: The notion of width of a Hamiltonian diﬀeomorphism, which is crucial for our approach in the case when π2 (M) = 0, does not make sense anymore, say, for M = CPm . The reason is that the action functional becomes multi-valued on the loop space Λ. It is an interesting problem of Floer theory to find an appropriate modification of Theorem 2.1 and inequality (3) in the case when π2 (M) 0. My feeling is that it should involve not only symplectic actions of closed orbits of Hamiltonian flows, but also the Conley – Zehnder indices. On the other hand, for manifolds with π2 0 the filtered Floer homology form a richer object than in the case π2 = 0 due to the existence of “quantum eﬀects.” Hopefully, one can use the full strength of this rich structure to get information about actions of lattices. Recently some restrictions of symplectic actions of lattices on the 2-sphere were found by Franks and Handel by using sophisticated tools of 2-dimensional dynamics. For instance they proved the following extension of Theorem 1.6. THEOREM 2.4 (Franks and Handel, 2003). Let Γ ⊂ SL(k, Z), k ≥ 3, be an infinite normal subgroup. Then every homomorphism Φ: Γ → Ham(S 2 ) has finite image. It would be interesting to reprove and extend their results, for instance to complex projective spaces of higher dimensions, by using Floer theory.

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3. The Calabi quasi-morphism and related topics 3.1. EXTENDING THE CALABI HOMOMORPHISM

Until recently, as far as I know, the only purely algebraic results on the structure of the group Ham were obtained by Banyaga (1978). They split into two cases: C I: M  . Then Ham(M, ω) is simple, that is it has no nontrivial normal subgroups. C II: M    : ω = dλ. In this case Ham(M, ω) admits the Calabi homomorphism Cal M : Ham(M, ω) → R whose kernel coincides with the commutator subgroup of Ham(M, ω). Moreover, this commutator subgroup turns out to be a simple group. The Calabi homomorphism is defined as “the average energy required in order to generate a Hamiltonian diﬀeomorphism.” More precisely, 1 Cal M (φF ) = F(x, t) ωm dt. (5) 0

M

One can show that this map is well defined (that is, its value depends on φF but not on a specific F) and is indeed a homomorphism. Return now to the case when (M, ω) is a closed symplectic manifold. Cover M by suﬃciently small open discs Uα and consider the collection of homomorphisms CalUα : Ham(Uα ) → R. The Calabi homomorphisms obviously agree on overlaps. Hence a natural question is whether it is possible to extend this collection to a global homomorphism of Ham(M, ω). The answer is of course negative since the group is simple by Banyaga’s above-mentioned theorem (Banyaga, 1978). It turns out, however, that for certain M’s one can perform such an extension with a bounded error (Entov and Polterovich, 2003). The formalism of “homomorphisms up to a bounded error” is given by the notion of a quasi-morphism which originated in the works by Gromov (1982) and Brooks (1981) on bounded cohomology of groups and recently became quite popular in group theory and dynamics (see, e.g., the beautiful short survey by Kotschick, 2004). The rest of these notes is dedicated to an outline of this construction, its applications to the study of distortion with respect to Hofer’s norm on Ham(M, ω) and discussion of some related topics. Another set of applications, to symplectic intersections, lies outside the scope of the present notes, see Biran et al. (2004) and Entov and Polterovich (2005).

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3.2. INTRODUCING QUASI-MORPHISMS

A quasi-morphism on a group G is a function r: G → R which satisfies the homomorphism equation up to a bounded error: there exists C > 0 such that |r( f g) − r( f ) − r(g)| ≤ C for all f, g ∈ G. A quasi-morphism r is called homogeneous if r(gn ) = nr(g) for all g ∈ G and n ∈ Z. Given a quasi-morphism r, one defines its homogenization rh by r(gn ) . rh (g) = lim n→+∞ n One can show that rh is a homogeneous quasi-morphism and the diﬀerence |rh − r| is a bounded function on G. Homogeneous quasi-morphisms are invariant under conjugations in G. They play an important role in the study of groups. We refer the reader to Bavard (1991) and Kotschick (2004) for an introduction to quasimorphisms. EXAMPLE 3.1 (The Barge – Ghys quasi-morphism; Barge and Ghys, 1988). Consider the isometric action of a discrete group G ⊂ PSL(2, R) on the upper half-plane H equipped with the hyperbolic metric. To construct quasi-morphisms of G, start with a smooth G-invariant one-form α such that |dα/Ω| ≤ C for some C > 0, where Ω is the hyperbolic area form. Given such α and a base-point z ∈ H, set r(g) = α, (z,gz)

where (z, w) denotes the geodesic segment between two points z, w ∈ H. Take any f, g ∈ G. Let ∆ be the hyperbolic triangle with vertices z, f z, and f gz. Clearly, |r( f g) − r( f ) − r(g)| = α = dα ≤ Cπ ∂∆

∆

since the area of any hyperbolic triangle does not exceed π. Thus r is a quasimorphism. Assume, for example, that G is the fundamental group of a closed oriented surface of genus ≥ 2. Applying this construction to various forms α one can show that homogeneous quasi-morphisms on G form a linear space of dimension continuum. Homogeneous quasi-morphisms, when they exist, serve as a substitute of homomorphisms in some interesting situations. For instance, using quasi-morphisms one can prove that certain elements are undistorted with respect to some meaningful metrics on the group. As an illustration, consider the commutator subgroup G of a group G. Every element g ∈ G can be written as the product of a finite

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number of simple commutators of the form aba−1 b−1 with a, b ∈ G. Define the commutator norm of g as the minimal number of simple commutators required to represent g. It is an easy exercise to show that if r(g) > 0 for some homogeneous quasi-morphism r on G, the element g is undistorted in the sense of the commutator norm. Surprisingly, the converse statement is true as well. Its proof, which is due to Bavard (1991), is nonconstructive — it uses the Hahn – Banach theorem. 3.3. QUASI-MORPHISMS ON Ham(M, ω)

Now we focus on the case when G is either the group Ham(M, ω) of Hamiltonian * diﬀeomorphisms of a symplectic manifold, or its universal cover Ham(M, ω). It was shown by Banyaga (1978) that, when M is closed, such a group is perfect (that is, it coincides with its commutator subgroup) and therefore admits no nontrivial homomorphisms to R. Sometimes, however, these groups admit nontrivial homogeneous quasi-morphisms. Existence of such quasi-morphisms on the group of Hamiltonian diﬀeomorphisms and/or its universal cover is known for the following classes of closed symplectic manifolds: − orientable surfaces (Gambaudo and Ghys, 2004); − closed manifolds whose fundamental group has trivial center and admits nontrivial quasi-morphisms (see Section 3.7 below); − manifolds with c1 = 0 (Barge and Ghys, 1992; Entov, 2004); − complex projective spaces (see Givental , 1990; Entov and Polterovich, 2003) and, more generally, spherically monotone symplectic manifolds with semisimple even quantum homology algebra (Entov and Polterovich, 2003). The quasi-morphism constructed in Entov and Polterovich (2003) for spherically monotone symplectic manifolds with semi-simple even quantum homology algebra comes from Floer theory. The list of such manifolds includes for instance CPm , S 2 × S 2 and CP2 blown up at one point. For simplicity, let us concentrate on the case when M = CPm . Our quasi-morphism is defined as follows. Take any f ∈ Ham(M, ω). Let f˜ be its lift to the universal cover H am(M, ω) associated to any Hamiltonian flow generating f . Put c([M], f˜n ) , n→∞ n

µ( f ) = − Volume(M) · lim

where c is the spectral number defined by formula (2) and [M] is the fundamental class of M which corresponds to the unity in the quantum homology ring. One can show that µ( f ) is well defined (in particular, it does not depend on the specific lift f˜) and the map µ: Ham(M) → R is a quasi-morphism. In addition, it has the following peculiar Calabi property: Let U ⊂ M be an open displaceable subset, which means that φ(U) ∩ Closure(U) = ∅ for some Hamiltonian diﬀeomorphism

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φ ∈ Ham(M, ω). Assume that the restriction of the symplectic form ω to U is exact. Consider the subgroup Ham(U) ⊂ Ham(M) consisting of all Hamiltonian diﬀeomorphisms generated by Hamiltonian functions vanishing outside U. Then the restriction of µ to Ham(U) coincides with the Calabi homomorphism CalU given by equation (5). Hence we fulfilled the promise given in Section 3.1. We call a homogeneous quasi-morphism with the Calabi property a Calabi quasimorphism. 3.4. DISTORTION IN HOFER’S NORM ON Ham(M, ω)

Here we present a geometric application of the Calabi quasi-morphism. Let (M, ω) be a symplectic manifold. Given a path { ft }, t ∈ [0; 1], of Hamiltonian diffeomorphisms of M, denote by F(x, t) = Ft (x) the corresponding Hamiltonian. Define 1 max Ft (x) − min Ft (x) dt. lengthH { ft } = 0

x∈M

x∈M

Define Hofer’s norm νH on Ham(M, ω) by νH ( f ) = inf length{ ft }, where the infimum is taken over all paths { ft } of Hamiltonian diﬀeomorphisms joining the identity with f . This norm was introduced by Hofer (1990). A deep fact2 is that νH is a genuine norm on Ham(M, ω), that is νH ( f ) > 0 for f 1. This was proved in Hofer (1990) and Polterovich (1993) for some classes of symplectic manifolds, and by Lalonde and McDuﬀ (1995) in full generality. The metric ρH ( f, g) := νH ( f g−1 ) on Ham(M, ω) is called Hofer’s metric. It was an object of intensive study since its discovery in 1990. We refer the reader to Polterovich (2001) for preliminaries on Hofer’s geometry. Any compactly supported smooth function F on (M, ω) generates a one parameter subgroup { ft } of Hamiltonian diﬀeomorphisms. Define its distortion d(F) = lim

t→+∞

νH ( ft ) . t

(6)

We say that a one-parameter subgroup is undistorted if d(F) > 0. It was shown by Sikorav (1990) that when M = R2n every one parameter subgroup of Ham(M, ω) remains a bounded distance from the identity. This result was extended in Polterovich and Siburg (2000) as follows: THEOREM 3.2 (Dichotomy theorem). Let (M, ω) be an open surface of infinite area. Every one parameter subgroup of Hamiltonian diﬀeomorphisms is either undistorted, or remains a bounded distance from the identity. 2

In contrast to the nondegeneracy of the norm ν defined in Section 2.3 above.

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QUESTION 3.3. Does the dichotomy above remain true on any symplectic manifold? The answer is not known even on closed oriented surfaces. However one can state the following weaker result. THEOREM 3.4. Let M be a closed oriented surface equipped with an area form. The one-parameter subgroup of Ham(M) generated by a generic Hamiltonian function on M is undistorted with respect to Hofer’s norm. Here by a generic Hamiltonian function we mean a function from an open dense set in C ∞ (M). We believe that this holds true on every closed symplectic manifold, though at the moment such a result seems to be out of reach. In the case of the 2-torus the proof is given in Polterovich (2001), and the same argument settles the case of surfaces of higher genus. For the 2-sphere, the problem remained open until the appearance of the Calabi quasi-morphism. Let us explain the argument in this case (see Entov and Polterovich, 2003). First of all, it turns out that the Calabi quasi-morphism µ introduced above is Lipschitz with respect to Hofer’s metric. Assume without loss of generality that Volume(M) = 1. Then |µ( f )| ≤ νH ( f ) (7) for every f ∈ Ham(M, ω). Let us focus on the case, when M is the 2-sphere of total area 1, and F is a Morse function on M with S 2 F · ω = 0. Look at connected components of the level sets of F. A simple combinatorial argument shows that there exists a unique (maybe singular) component, say γ, with the following property: S 2 \ γ = U1 ( · · · ( Uk , where Ui are open topological discs of area ≤ 12 . Recall that φF stands for the Hamiltonian diﬀeomorphism generated by a function F. The next lemma was proved in Entov and Polterovich (2003) (see Entov and Polterovich, 2005, for various generalizations of this result): LEMMA 3.5. Let µ: Ham(S 2 ) → R be any Calabi quasi-morphism continuous in Hofer’s metric. Then µ(φF ) = −F(γ). Proof. After a small C 0 -perturbation (which is a legitimate operation in view of the continuity of µ) we can assume that F ≡ F(γ) in a neighborhood of γ. Put H := F − F(γ). We see that H decomposes as H = H1 + · · · + Hk ,

support(Hi ) ⊂ Ui .

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Note that φH = φH1 ◦ · · · ◦ φHk and all φHi ’s pair-wise commute. Now we use a simple algebraic property of any homogeneous quasi-morphism on any group: its restriction to an abelian subgroup is a genuine homomorphism. Hence µ(φF ) = µ(φH ) =

k

µ(φHk ).

(8)

i=1

Note that since the area of Ui is ≤ 12 , each function Hi has displaceable support. Therefore, the Calabi property guarantees that Hi · ω µ(φHi ) = S2

for all i = 1, . . . , k. Substituting this to equation (8) we get that µ(φF ) =

k i=1

S2

Hi · ω = −F(γ),

as required.

Proof of Theorem 3.4 for S 2 . It suﬃces to prove the theorem for functions with zero mean (otherwise, add a constant to achieve this). Choose such a function, say, F to be Morse and assume in addition the following generic property: the level set {F = 0} is regular and does not contain a connected component which divides S 2 into two discs of equal areas. Therefore F(γ) 0 and hence |µ(φF )| > 0. Let { ft } be the one parameter subgroup generated by F. With our notation, f1 = φF . Combining the Lipschitz property of the Calabi quasi-morphism with Lemma 3.5, we get that νH ( ft ) ≥ |µ( ft )| = t|µ( f1 )| = t|F(γ)|, and hence

νH ( ft ) ≥ |F(γ)| > 0, t→+∞ t

d(F) = lim as required.

3.5. EXISTENCE AND UNIQUENESS OF CALABI QUASI-MORPHISMS

Here are some basic questions which we are unable to answer at the moment. QUESTION 3.6. Which symplectic manifolds admit a Calabi quasi-morphism? This problem is open already for all closed surfaces of genus ≥ 1. Recently Ostrover (2005) extended the results of Entov and Polterovich (2003) to nonmonotone symplectic forms on S 2 × S 2 and CP2 blown up at one point.

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QUESTION 3.7. Is a Calabi quasi-morphism unique? This is unknown even for the 2-sphere. Note that Lemma 3.5 shows that all Calabi quasi-morphisms which are continuous in Hofer’s metric must coincide on the set of Hamiltonian diﬀeomorphisms of S 2 generated by autonomous Hamiltonians. The situation changes when one considers groups of compactly supported Hamiltonian diﬀeomorphisms of open symplectic manifolds. It is proved in Biran et al. (2004) that for the standard symplectic ball the Calabi quasi-morphisms which are continuous in Hofer’s metric form an aﬃne space of dimension at least continuum. In fact, one can obtain a family of continuum cardinality of linearly independent Calabi quasi-morphisms on Ham(B2n ) by considering suitable conformally symplectic embeddings B2n → CPn and pulling back the Calabi quasi-morphism µ defined above. 3.6. “HYPERBOLIC” FEATURES OF Ham(M, ω)?

The above-mentioned ampleness of the space of quasi-morphisms was known earlier for certain discrete subgroups of hyperbolic isometries (and more general Gromov-hyperbolic groups), see Example 3.1 above. The Dichotomy Theorem 3.2 has the following counterpart for discrete subgroups G of the M¨obius group PSL(2, R) endowed with the commutator norm: given an element f ∈ [G, G] of infinite order, either f is undistorted or the cyclic subgroup generated by f remains a bounded distance from the identity. This result readily follows from the fact (Polterovich and Rudnick, 2004; Epstein and Fujiwara, 1997; Bestvina and Fujiwara, 2002) that either there exists a homogeneous quasi-morphism of G which does not vanish on f , or f is conjugate to its inverse in G. (The proof of this fact given in Polterovich and Rudnick, 2004 is based on the Barge-Ghys construction in Example 3.1 above combined with some elementary hyperbolic geometry.) In the first case f is undistorted. If f is conjugate to its inverse we have the following: f = g f −1 g−1 =⇒ f n = g f −n g−1 =⇒ f 2n = [ f n , g]. Thus, writing for the commutator norm we have that f 2n = 1 and f 2n+1 ≤ f 2n + f ≤ 1 + f for all n ∈ N. This proves the desired dichotomy. The list of common features of the symplectic and hyperbolic worlds can be continued. I do not know whether it is a superficial coincidence, or there exists some deeper reason for that.

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3.7. FROM π1 (M) TO Diﬀ 0 (M, Ω)

We complete these lectures with the remark that quasi-morphisms on the fundamental group π1 (M) can be, under certain assumptions, canonically lifted to quasi-morphisms on the group of volume-preserving diﬀeomorphisms of M. For instance, for closed symplectic manifolds with hyperbolic fundamental group (e.g., for surfaces of higher genus) this enables us to produce plenty of quasimorphisms on Ham (compare with the last paragraph of the previous section.) The construction below is implicitly contained in Gambaudo and Ghys (2004) which presents various beautiful quasi-morphisms on groups of area-preserving diﬀeomorphisms of surfaces. Let M be a closed connected manifold endowed with a volume form Ω. Let G = Diﬀ 0 (M, Ω) be the identity component of the group of volume-preserving diﬀeomorphisms of M. Suppose for simplicity that Γ := π1 (M, z) has no center (for instance, M is a closed oriented surface of higher genus). For every x ∈ M choose a path a x between x and z. The paths are assumed to be “short,” that is their lengths are bounded in some Riemannian metric on M. Moreover, we assume that the path a x depends on x continuously outside a closed set of measure 0 on M. One can easily produce such a system of paths as follows: Triangulate M by suﬃciently small simplices and choose a x to depend continuously on x in the interior of each simplex. Choose any auxiliary Riemannian metric on M. Take any isotopy { ft } with f0 = id and f1 = f . For x ∈ M consider the loop ({ ft }, x) ⊂ M based at z, consisting of 3 pieces: − the path a x oriented from z to x; − the trajectory of x under { ft }; − a f (x) oriented from f (x) to z. Denote by σ( f, x) ∈ Γ the homotopy class of this loop. We claim that σ( f, x) does not depend on the choice of the isotopy { ft }. Indeed, any two such isotopies diﬀer by a loop, say, {ht }, t ∈ [0; 1] of diﬀeomorphisms with f0 = f1 = 1. It suﬃces to show that the loop D := {ht x}t∈[0;1] is contractible. Take any element γ ∈ π1 (M, x) and represent it by a closed curve C ⊂ M. Look at the torus t ht (C). We see that the loops D and C lie on the torus and hence represent commuting homotopy classes. The conclusion is that D belongs to the center of π1 (M, x), and hence is contractible by our assumption. The claim follows. Observe now that σ: G × M → Γ is a cocycle, that is σ( f g, x) = σ( f, gx) · σ(g, x)

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for all f, g ∈ G and x ∈ M. Let r: Γ → R be any quasi-morphism. Define σ∗ r: G → R by ∗ σ r( f ) = r σ( f, x) · Ω. M

Note that for a fixed isotopy between 1 and f , the loops ({ ft }, x) have uniformly (with respect to x ∈ M) bounded length. Hence the function σ( f, ·): M → Γ has finite image. By our assumption on the system of paths a x , the function σ( f, x) is locally constant on an open set of full measure. Thus the integral above exists. Further, by the cocycle property, ∗ σ r( f g) = r σ( f, gx) · σ(g, x) · Ω = r σ( f, gx) · Ω + r σ(g, x) · Ω + Q, M

M

M

where |Q| ≤ K := Volume(M) · sup |r(uv) − r(u) − r(v)|. u,v∈Γ

Making the change of variable y = gx and using that g is volume-preserving we get that r σ( f, gx) · Ω = r σ( f, y) · Ω. M

M

Thus σ∗ r is a quasi-morphism of G. It is easy to see that the quasi-morphisms σ∗ r obtained in this way are nontrivial. Acknowledgements I owe a lot to Marc Burger for his generous help with my paper (Polterovich, 2002) on the Zimmer program. I am grateful to Paul Biran and Michael Entov for sharing with me their insight on various aspects of the Calabi quasi-morphism. I thank Yaron Ostrover and Felix Schlenk for carefully reading the manuscript and pointing out numerous mistakes. Finally, I thank Octav Cornea and Franc¸ois Lalonde for their warm hospitality in Montreal where these lectures have been delivered.

References Banyaga, A. (1978) Sur la structure du groupe des diﬀ´eomorphismes qui pr´eservent une forme symplectique, Comment. Math. Helv. 53, 174 – 227. ´ (1988) Surfaces et cohomologie born´ee, Invent. Math. 92, 509 – 526. Barge, J. and Ghys, E. ´ (1992) Cocycles d’Euler et de Maslov, Math. Ann. 294, 235–265. Barge, J. and Ghys, E. Bavard, C. (1991) Longueur stable des commutateurs, Enseign. Math. (2) 37, 109 – 150.

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Bestvina, M. and Fujiwara, K. (2002) Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6, 69 – 89. Biran, P., Entov, M., and Polterovich, L. (2004) Calabi quasimorphisms for the symplectic ball, Commun.Contemp. Math. 6, 793 – 802. Brooks, R. (1981) Some remarks on bounded cohomology, In I. Kra and B. Maskit (eds.), Riemann Surfaces and Related Topics: Proceedings of the 1978 Stony Brook Conference, Vol. 97 of Ann. of Math. Stud., Stony Brook, NY, 1978, pp. 53–63, Princeton, NJ, Princeton Univ. Press. Burger, M. and Monod, N. (1999) Bounded cohomology of lattices in higher rank Lie groups, J. Eur. Math. Soc. (JEMS) 1, 199 – 235. Burger, M. and Monod, N. (2002) Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal. 12, 219 – 280. de la Harpe, P. (2000) Topics in Geometric Group Theory, Chicago Lectures in Math., Chicago, IL, Univ. Chicago Press. Entov, M. (2004) Commutator length of symplectomorphisms, Comment. Math. Helv. 79, 58 – 104. Entov, M. and Polterovich, L. (2003) Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 30, 1635 – 1676. Entov, M. and Polterovich, L. (2005) Quasi-states and symplectic intersections, Comment. Math. Helv., to appear; arXiv:math.SG/0410338. Epstein, D. B. A. and Fujiwara, K. (1997) The second bounded cohomology of word-hyperbolic groups, Topology 36, 1275 – 1289. Farb, B. and Shalen, P. (1999) Real-analytic actions of lattices, Invent. Math. 135, 273 – 296. Feres, R. (1998) Dynamical Systems and Semisimple Groups: An Introduction, Vol. 126 of Cambridge Tracts in Math., Cambridge, Cambridge Univ. Press. Fisher, D. (2003) Nonlinear representation theory of finitely generated groups, available at http: //comet.lehman.cuny.edu/fisher/. Floer, A. (1988) Morse theory for Lagrangian intersections, J. Diﬀerential Geom. 28, 513 – 547. Floer, A. (1989)a Cuplength estimates on Lagrangian intersections, Comm. Pure Appl. Math. 42, 335 – 356. Floer, A. (1989)b Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120, 575 – 611. Franks, J. and Handel, M. (2003) Area preserving group actions on surfaces, Geom. Topol. 7, 757 – 771. Franks, J. and Handel, M. (2004) Distortion elements in group actions on surfaces, arXiv:math.DS/ 0404532. ´ (2004) Commutators and diﬀeomorphisms of surfaces, Ergodic Gambaudo, J.-M. and Ghys, E. Theory Dynam. Systems 24, 1591 – 1617. Ghys, E. (1999) Actions de r´eseaux sur le cercle, Invent. Math. 137, 199 – 231. Givental , A. B. (1990) Nonlinear generalization of the Maslov index, In Theory of Singularities and its Applications, Vol. 1 of Adv. Soviet Math., pp. 71–103, Providence, RI, Amer. Math. Soc. ´ Gromov, M. (1982) Volume and bounded cohomology, Inst. Hautes Etudes Sci. Publ. Math. 56, 5–99. Hofer, H. (1990) On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115, 25 – 28. Hofer, H. and Zehnder, E. (1994) Symplectic Invariants and Hamiltonian Dynamics, Birkhauser Adv. Texts Basler Lehrbucher, Basel, Birkh¨auser. Kotschick, D. (2004) What is . . . a quasi-morphism?, Notices Amer. Math. Soc. 51, 208 – 209. Lalonde, F. and McDuﬀ, D. (1995) The geometry of symplectic energy, Ann. of Math. (2) 141, 349–371.

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Lubotzky, A., Mozes, S., and Raghunathan, M. S. (2000) The word and Riemannian metrics on ´ lattices in semisimple groups, Inst. Hautes Etudes Sci. Publ. Math. 91, 5 – 53. Margulis, G. A. (1991) Discrete Subgroups of Semisimple Lie Groups, Vol. 17 of Ergeb. Math. Grenzgeb. (3), Berlin, Springer. McDuﬀ, D. and Salamon, D. (1995) Introduction to Symplectic Topology, Oxford Math. Monogr., Oxford, Oxford Univ. Press. McDuﬀ, D. and Salamon, D. (2004) J-Holomorphic Curves and Symplectic Topology, Vol. 52 of Amer. Math. Soc. Colloq. Publ, Providence, RI, Amer. Math. Soc. Newman, M. (1972) Integral Matrices, Vol. 45 of Pure Appl. Math., New York, Academic Press. Oh, Y.-G. (1997) Symplectic topology as the geometry of action functional. I. Relative Floer theory on the cotangent bundle, J. Diﬀerential Geom. 46, 499 – 577. Oh, Y.-G. (1999) Symplectic topology as the geometry of action functional. II. Pants product and cohomological invariants, Comm. Anal. Geom. 7, 1 – 54. Oh, Y.-G. (2002)a Chain level Floer theory and Hofer’s geometry of the Hamiltonian diﬀeomorphism group, Asian J. Math. 6, 579–624. Oh, Y.-G. (2002)b Mini-max theory, spectral invariants and geometry of the Hamiltonian diﬀeomorphism group, arXiv:math.SG/0206092. Oh, Y.-G. (2005) Lectures on Floer theory and spectral invariants of Hamiltonian flows, in this volume. Ostrover, Y. (2005), Ph.D. thesis, Tel Aviv University, in preparation. Piunikhin, S., Salamon, D., and Schwarz, M. (1996) Symplectic Floer – Donaldson theory and quantum cohomology, In Contact and Symplectic Geometry, Vol. 8 of Publ. Newton Inst., pp. 171–200, Cambridge, Cambridge Univ. Press. Polterovich, L. (1993) Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems 13, 357 – 367. Polterovich, L. (2001) The Geometry of the Group of Symplectic Diﬀeomorphisms, Lectures Math. ETH Zurich, Basel, Birkh¨auser. Polterovich, L. (2002) Growth of maps, distortion in groups and symplectic geometry, Invent. Math. 150, 655–686. Polterovich, L. and Rudnick, Z. (2004) Stable mixing for cat maps and quasi-morphisms of the modular group, Ergodic Theory Dynam. Systems 24, 609 – 619. Polterovich, L. and Siburg, K. F. (2000) On the asymptotic geometry of area-preserving maps, Math. Res. Lett. 7, 233–243. Schwarz, M. (2000) On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193, 419–461. Sikorav, J.-C. (1990) Syst`emes hamiltoniens et topologie symplectique, Quaderni del Dottorato, Pisa, Universit`a di Pisa. Viterbo, C. (1992) Symplectic topology as the geometry of generating functions, Math. Ann. 292, 685 – 710. Witte, D. (2001) Introduction to arithmetic groups, arXiv:math.DG/0106063. Zimmer, R. (1984) Ergodic Theory and Semisimple Groups, Vol. 81 of Monogr. Math., Basel, Birkh¨auser. Zimmer, R. (1987) Actions of semisimple groups and discrete subgroups, In Proceedings of the International Congress of Mathematicians, Vol. 1, 2, Berkeley, 1986, pp. 1247–1258, Providence, RI, Amer. Math. Soc.

SYMPLECTIC TOPOLOGY AND HAMILTON – JACOBI EQUATIONS CLAUDE VITERBO ´ Ecole Polytechnique, Palaiseau

Abstract. We introduce the current theory of Generating functions for Lagrangian submanifolds, and apply it, first to prove some basic theorems in symplectic topology, and then to the theory of Hamilton – Jacobi equations. Finally we explain how the variational solutions thus introduced can be usefully applied to solve multi-time Hamilton – Jacobi equations.

1. Introduction to symplectic geometry and generating functions Consider the cotangent bundle of a compact manifold N, endowed with its canonical symplectic form: T ∗ N = {(q, p) | p ∈ T q∗ N}, ω = d p ∧ dq, where d p ∧ dq means that in local coordinates q1 , . . . , qn on N and dual coordinates p1 , . . . , pn , ) ω = nj=1 d p j ∧ dq j . It is also useful to introduce the Liouville form λ = p dq, in ) local coordinates λ = nj=1 p j dq j . Even though we shall only work in cotangent bundles, there is of course a general definition of a symplectic form on a manifold as a nondegenerate closed two-form. A Lagrangian submanifold L is a submanifold of dimension n such that ω vanishes on L. This is equivalent to the condition that the restriction of λ to L is closed. If moreover this restriction is exact, we shall say that the submanifold is an exact Lagrangian. Lagrangian submanifolds which are graphs of sections of T ∗ N → N, i.e.

graphs of 1-forms on N, are easy to characterize. Setting Lα = q, α(q) | q ∈ N , we have that λLα = α, hence ω|Lα = dα. Thus L is Lagrangian if and only if α is closed, and exact if and only if α is exact. Note that the zeros of α correspond to the intersection points of Lα with the zero section 0N = {(q, 0) | q ∈ N}. Since for α = d f both Morse theory and Lusternik – Schnirelman theory give us lower bounds for the number of critical points of a function f on N, we have lower bounds for the number of intersection points of Ld f with 0N . Arnold conjectured that such lower bounds are still valid for general exact Lagrangian submanifolds. The original proof of the Arnold conjectures (or at least a slightly weaker version of them) is due to Hofer (1985). Shortly after a simpler proof was given by Laudenbach and Sikorav (1985), it used the concept of generating functions, originally used in the theory of Fourier integral operators (H¨ormander, 1971). 439 P. Biran et al. (eds.), Morse Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, 439–459. © 2006 Springer. Printed in the Netherlands.

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DEFINITION 1.1. A generating function for a Lagrangian manifold L ⊂ T ∗ N is a smooth function S : N × Rk → R such that (a) (x, ξ) → ∂S (x, ξ)/∂ξ has zero as a regular value, (b) L = LS = {(x, ∂S (x, ξ)/∂x) | ∂S (x, ξ)/∂ξ = 0}. The condition that (x, ξ) → ∂S (x, ξ)/∂ξ has zero as a regular value guarantees that ΣS = {(x, ξ) | ∂S (x, ξ)/∂ξ = 0} is a submanifold and that the map iS : ΣS → T ∗ N is an immersion, where iS (x, ξ) = (x, ∂S (x, ξ)/∂x). Proving that LS is Lagrangian involves two concepts: symplectic reduction (see Weinstein, 1977) and automatic transversality (an idea I owe to F. Laudenbach). To define symplectic reduction we need the notion of coisotropic submanifold. For a vector subspace V in a symplectic vector space (E, ω), we denote by V ω the symplectic orthogonal of V, V ω = {y | ∀x ∈ V, ω(x, y) = 0}. DEFINITION 1.2. A submanifold K ⊂ T ∗ N is coisotropic if and only if at each point z ∈ K we have (T z K)ω ⊂ T z K. The distribution (T z K)ω on K is integrable, and defines the characteristic foliation K. If the quotient K/K is a manifold, it has a natural symplectic structure induced by ω, since the symplectic product of two vectors of T z K only depends on their projection along (T z K)ω . PROPOSITION 1.3. Let L be a Lagrangian manifold, and K a coisotropic manifold in the symplectic manifold (M, ω). If L is transverse to K, then the projection of L ∩ K on K/K is a Lagrangian immersion. It is called the symplectic reduction of L. Proof. The crucial ingredient is the following remark: if L is a Lagrangian vector subspace of the symplectic space (E, ω), if K is a coisotropic vector subspace, and if K and L are transverse, that is K + L = E, then the projection of K ∩ L parallel to K ω is injective, that is (K ∩L)∩K ω = {0}. Indeed, since Lω = L we have K ω ∩ L = {0}, and since K ω ⊂ K this means (K ∩ L)∩ K ω = {0}. If L is Lagrangian, ω vanishes on L ∩ K and it is of course still zero after projecting on K/K. That LS is exact Lagrangian follows obviously from the fact that i∗S (λ) = d(S |ΣS ). We will mainly use two cases: ∗ N = {(q, p) ∈ T ∗ N | q ∈ V} is the (a) V is a submanifold of N, and K = T |V ∗ restriction of T N to V. The foliation K is given by Kq = {(q, p) | p|T q V = 0}. In this case the quotient K/K can be identified with T ∗ V. Thus symplectic reduction sends Lagrangian submanifolds of T ∗ N to Lagrangian submanifolds of T ∗ V.

(b) π: E → B is a fibration, and L is Lagrangian in T ∗ E. Let K = {(x, p) ∈ T ∗ E | p restricted to T x Fπ(x) is 0}.

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Then the leaves of K are given by the fibers of π, and the quotient K/K may be identified with T ∗ B. REMARK 1.4. (a) If S is a function satisfying the condition (a), then (b) defines an immersed exact Lagrangian submanifold LS . (b) LS is then the reduction of the graph of dS in T ∗ (N × Rk ) by the coisotropic submanifold associated in the example (b) above to the projection N × Rk → N. (c) If S is a generating function of L in T ∗ M and V is a submanifold of M, then the reduction in the example (a) above yields a Lagrangian submanifold LV with the generating function S = S |V×Rk . (d) There are diﬀerent definitions of generating functions. A useful generalization considers the case of functions on a nontrivial vector bundle E over N. Since to every vector bundle we may associate another bundle F such that E ⊕ F is trivial, LS is also generated by S (x, ξ) ⊕ |η|2 where | · | is any norm on F. The intersection points of LS with the zero section are in one-to-one correspondence with the critical points of S . This has no practical use for the moment, since a function on N × Rk need not have any critical point. However, it does have critical points if we prescribe some behavior at infinity. This leads us to the next definition. DEFINITION 1.5. A generating function is quadratic at infinity (GFQI) if there is a nondegenerate quadratic form Q such that S (x, ξ) = Q(ξ) for (x, ξ) outside a compact set. Note that this definition, which will mostly be used for N compact, also makes sense for general N. There are equivalent conditions. PROPOSITION 1.6. Assume there is a constant C such that (a) |∇(S − Q)|C 0 < C, (b) sup{|S − Q| | x ∈ N, |ξ| ≤ r} ≤ Cr. Then LS has a generating function quadratic at infinity. Proof. Consider a smooth decreasing function ρ: R+ → R+ such that ρ = 1 on [0, A], ρ = 0 on [B, +∞[ and −ε ≤ ρ ≤ 0. We set S 0 = S , S 1 (x, ξ) = 1 − ρ(|ξ|) Q(ξ) + ρ(|ξ|)S (x, ξ).

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Then S 1 = S 0 for |ξ| ≤ A and S 1 is exactly quadratic at infinity. Moreover it is easy to see that LS 1 = LS 0 , since there are no critical points for S 0 , S 1 for |ξ| ≥ A. Consider the family S t (x, ξ) = (1 − t)S (x, ξ) + tS 1 (ξ). We claim that under the assumptions of the proposition, there is an isotopy ϕt such that S t ϕt (x, ξ) = S (x, ξ). Since S 0 = S it is enough to check that d t S t ϕ (x, ξ) = 0, dt that is, setting R(x, ξ) = (S 1 − S 0 )(x, ξ) = ρ(ξ)(S − Q)(x, ξ) F(x, ξ) = (S 0 − Q)(x, ξ) and dϕt (x, ξ)/dt = X t, ϕt (x, ξ) , the following quantity must vanish: d R ϕt (x, ξ) + dS t ϕt (x, ξ) ϕt (x, ξ) dt t = R ϕ (x, ξ) + dQ ϕt (x, ξ) X ϕt (x, ξ) + (1 − t)dF ϕt (x, ξ) X ϕt (x, ξ) + tdR ϕt (x, ξ) X ϕt (x, ξ) . Setting (y, η) = ϕt (x, ξ) and X(t, y, η) = ut (y, η), σt (y, η) , we must solve R(y, η)+ < η, σt (y, η) > +(1 − t)dF(y, η)X(t, y, η) + tdR(y, η)X(t, y, η) = 0, or in other words ∂ ∂ (1) R(y, η) + ut (y, η) (1 − t) F(y, η) + t R(y, η) ∂y ∂y < = ∂ ∂ + η + (1 − t) F(y, η) + t R(y, η), σt (y, η) = 0. (2) ∂η ∂η Now by assumption R vanishes for |η| ≤ A and ∂ R(y, η) = t|dS (y, η) − dS (y, η)| 1 0 ∂η = d ρ(|η|) S (y, η) − Q(η)

∂ ≤ −ρ (|η|) · |(S − Q)(y, η)| + ρ(|η|) · (S − Q)(y, η) . ∂η

Since |(S − Q)(x, η)| ≤ C|η| and |∂F(y, η)/∂η| = |∂(S − Q)(x, η)/∂η| ≤ C, we have that line (2) is bounded by −ρ (|η|)C|η| + Cρ(|η|) + C ≤ Cε|η|

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provided A is large enough. Thus for |η| ≥ A, η + (1 − t) ∂ F(y, η) + t ∂ R(y, η) ≥ (1 − Cε)|η|. ∂η ∂η We can then choose u = 0 (this means the diﬀeomorphism is fiber preserving) and σt (y, η) satisfying the equation < = ∂ ∂ R(y, η) + η + (1 − t) F(y, η) + t R(y, η), σ(y, η) = 0. ∂η ∂η The inequality C 1 |R(y, η)| ≤ |η| (1 − Cε)|η| (1 − Cε) implies that the vector field X(t, y, η) = 0, σt (y, η) is complete. |σt (y, η)| ≤

The existence of generating functions quadratic at infinity is given by the following THEOREM 1.7 (Sikorav). If there is a Hamiltonian isotopy ϕt such that ϕt (L1 ) = L2 and if L1 has a GFQI, then L2 also has a GFQI. As a special case, if L = ϕt (0N ) then L has a GFQI. REMARK 1.8. In fact, if G is the set of GFQI, and L is the set of Lagrangian submanifolds, then the map G → L is a Serre fibration, where a continuous map P → L is by definition induced by a continuous family of Hamiltonian isotopies. It is unknown whether this map is surjective. Proof. We follow the proof in Brunella (1991), the original proof can be found in Sikorav (1987). We shall first prove the proposition for Hamiltonian flows C 1 close to the identity; it is enough to prove it for ϕ1 C 1 -close to the identity. We are going to start with a very special case. Assume first that N = Rn . Let h: N × N → R be a smooth map. We may associate to it a symplectic map  ∂    p = h(x , x ),    1 ∂x1 1 2 ϕ(x1 , p1 ) = (x2 , p2 ) ⇐⇒   ∂    h(x1 , x2 ).  p2 = − ∂x2 Note that the existence of h is equivalent to the condition that the graph of ϕ, the

Lagrangian submanifold Γ(ϕ) = (x1 , p1 ), ϕ(x1 , p1 ) ⊂ T ∗ (N ×N) (endowed with

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C. VITERBO

the Liouville form p1 dx1 − p2 dx2 ) can be considered as a graph over N × N. Note that h is then uniquely determined up to a constant. For h(x1 , x2 ) = 12 |x1 − x2 |2 defined on Rn × Rn , the corresponding map is ϕ(x1 , p1 ) = (x1 − p1 , p1 ). Since the property of being a graph is open in the C 1 topology, any map ψ C 1 close to ϕ is associated to a function k(x1 , x2 ), which is also close to h(x1 , x2 ). Moreover if ψ = ϕ outside a compact set, we will have that k = h outside a compact set. Note also that the inverse of ϕ is associated to the function h(x1 , x2 ) = − 12 |x1 − x2 |2 . Assume that S (x, ξ) generates L, L coincides with the zero section outside a compact set, and S (x, ξ) is equal to a Q(ξ) outside a compact set. Then ϕ(L) is generated by S(x, ξ, η) = h(x, η) + S (η, ξ) where η ∈ Rn . Indeed, ∂ ∂ S (x, ξ, η) = S (η, ξ) ∂ξ ∂ξ ∂ ∂ ∂ h(x, η) + S (η, ξ) S (x, ξ, η) = ∂η ∂x2 ∂x ∂ ∂ h(x, η). S (x, ξ, η) = ∂x ∂x1 We have that (x, ξ, η) satisfies the following two equations ∂ S (x, ξ, η) = 0 ∂ξ ∂ S (x, ξ, η) = 0 ∂η if and only if

η,

∂ S (η, ξ) ∈ L ∂x

and ∂ ∂ ∂ ∂ h(x, η) = x, h(x, η) = x, S(x, ξ, η) . ϕ η, S (η, ξ) = ϕ η, − ∂x ∂x2 ∂x1 ∂x Thus ϕ(L) is generated by S. Note that if h and S are quadratic outside a compact set, then S satisfies the assumptions of Proposition 1.6, so we can find a true GFQI generating ϕ(L).

SYMPLECTIC TOPOLOGY AND HAMILTON – JACOBI EQUATIONS

445

Thus we proved that for any L having a GFQI and for any ψ close to ϕ(x, p) = (x − p, p), ψ(L) has a GFQI. The same holds for ψ close to ϕ−1 given by (x, p) → (x + p, p). Now any Hamiltonian diﬀeomorphism f close to the identity is the composition of a diﬀeomorphism close to ϕ and one close to ϕ−1 . Therefore we proved that if L has a GFQI, then f (L) has a GFQI, and by composition this will be true for any time one map of an isotopy, since 1/N ψ10 = ψ1(N−1)/N ◦ · · · ◦ ψk/N (k−1)/N ◦ · · · ◦ ψ0 .

Finally we consider the case of a general manifold N. We can view N as a submanifold of Rq , and get a symplectic embedding T ∗ N → T ∗ Rq as follows: use the standard Euclidean metric on Rq to extend any linear form p on the subspace T q N ⊂ Rq to a linear form p¯ on Rq . Then the map (q, p) → (q, p) ¯ is a symplectic embedding (it is easy to see that it preserves the Liouville forms). Now let L be Lagrangian in T ∗ Rq and assume it is transverse to the coisotropic manifold N × (Rq )∗ . Then the symplectic reduction LN of L, is a Lagrangian submanifold of T ∗ N, and has generating function S N×Rk (see Remark 1.4(c)). Now assume that ϕt is a Hamiltonian isotopy of T ∗ N, then we may extend it to an isotopy of T ∗ Rq . Indeed if in local coordinates N is given by u = 0, we take local coordinates near N given by (u, x), and in a neighborhood of T N∗ (N × Rk ) in T ∗ (N × Rk ) given by (u, x, v, p) with Liouville form p dx + v du and T ∗ N given by the equations u = 0, v = 0. Now the Hamiltonian H(t, x, p) on T ∗ N extends to the u, x, v, p) = χ(u)H(t, x, p) where Hamiltonian H(t,

χ(u) = 1 near u = 0 χ(u) = 0 outside a neighborhood of u = 0.

preserves u = 0 and induces the flow of H Since ∂H/∂v = 0, the flow of H on u = 0. Therefore if L is the reduction of L˜ isotopic to the zero section by a compactly supported isotopy, and if ϕt is a Hamiltonian isotopy on T ∗ N, then it ˜ has a GFQI (since has an extension ϕt to T ∗ (Rk ) preserving T N∗ (Rk ). Now, ϕt (L) ∗ k ˜ this is the case for L), hence its reduction by T N (R ), that is ϕt (L), has a GFQI. This argument does not need that L is the reduction of a Lagrangian isotopic to the zero section. Indeed, let S : N × Rk → R be a GFQI, we may consider L˜ as the graph of dS . Since S is quadratic at infinity, L˜ coincides with the zero section at infinity. Of course, L = LS is the reduction of L˜ by v = 0, where (x, u, p, v) are the coordinates on T ∗ (N × Rk ). If we consider a Hamiltonian isotopy ϕt of ∗ k t t t ˜ will have ϕ (L) as reduction T (N × R ) with reduction ϕ then, as above, ϕ (L) t ˜ by v = 0. Then since ϕ (L) has a GFQI, S (x, u, η), defined on N × Rk × Rq , is quadratic for (u, η) outside a compact set, this means that S seen as a function on N × Rk × Rq is quadratic at infinity, and thus yields a GFQI for ϕt (L).

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C. VITERBO

1.1. UNIQUENESS AND FIRST SYMPLECTIC INVARIANTS

Let L be a Lagrangian submanifold in T ∗ N, with a generating function S : N × Rk → R. Can one describe the other generating functions of L? Notice first that there are three standard operations on generating functions which preserve the set LS (a) let (x, ξ) → x, ϕ(x, ξ) be a fiber preserving diﬀeomorphism from N × Rk to itself. Then S(x, ξ) = S x, ϕ(x, ξ) is a generating function for the same manifold. (b) let q(η) be a nondegenerate quadratic form on Rl , then S(x, ξ, η) = S (x, ξ) + q(η) is again a generating function for LS . (c) let C be any real number. Then S(x, ξ) = S (x, ξ)+C is once more a generating function for LS . DEFINITION 1.9. Two generating functions S 1 , S 2 are equivalent if there are two sequences of the above operations (a), (b), (c) beginning respectively with S 1 and S 2 and ending in the same function. Clearly, two equivalent generating functions generate the same Lagrangian submanifold. Note that the operation (a) does not necessarily preserve the property of being a GFQI, however this will not matter. THEOREM 1.10 (Viterbo, 1992; Th´eret, 1999). If L is Hamiltonian isotopic to 0N , then any two GFQI for L are equivalent. → L from the set G of GFQI The idea of the proof is to use that the map G modulo equivalence to the set of Lagrangian submanifolds is a Serre fibration. Therefore the fiber over two points of the same connected component has the same number of connected components. One first shows that any two GFQI for the zero section are equivalent. Then we have that if two GFQI for the same Lagrangian submanifold L are connected through a path of GFQI for L then they are equivalent. REMARK 1.11. (a) This extends to the case of Legendrian immersions in J 1 (N, R), where J 1 (N, R) is the space of 1-jets of functions on N which can be identified with R × T ∗ N; here the contact form is α = dz − p dq (where z ∈ R, q ∈ N, p ∈ T q∗ N). (b) The theorem implies that the modules H ∗ (S b , S a ), where S λ = {(x, ξ) ∈ N × Rk | S (x, ξ) ≤ λ}

SYMPLECTIC TOPOLOGY AND HAMILTON – JACOBI EQUATIONS

447

are independent of (a, b) up to a global shift of levels and index. In other words, if S 1 , S 2 are GFQI for L, then there are constants k, c such that for all (a, b) we have H ∗ (S 1b , S 1a ) = H ∗−k (S 2b−c , S 2a−c ). 2. The calculus of critical level sets Let f ∈ C 2 (X, R) be a function on the manifold X. We shall look for critical points of f , and for this we shall need a compactness condition on the flow. There are many versions of such a condition, but the simplest one is (PS) Any sequence xn such that d f (xn ) → 0 and f (xn ) → c has a converging subsequence. Of course if x∞ is the limit of such a sequence, we have that d f (x∞ ) = 0. We denote by i∗c the natural map i∗c : H ∗ ( f b , f a ) → H ∗ ( f c , f a ), and by (ic )∗ the natural map (ic )∗ : H∗ ( f c , f a ) → H∗ ( f b , f a ). Now let α be a nonzero cohomology class in H ∗ ( f b , f a ). DEFINITION 2.1. c(α, f ) = sup{c | i∗c (α) = 0}. Similarly for ω ∈ H∗ ( f b , f a ) DEFINITION 2.2. c(ω, f ) = inf{c | ω ∈ Im((ic )∗ )}. There are diﬀerent ways of associating to one or more cohomology classes a new class. We shall try to understand the relation between the numbers c(α, f ), c(ω, f ). Of course, the main reason for studying these numbers is the THEOREM 2.3 (Min-max theorem of Birkhoﬀ – Morse). Assume that f satisfies condition (PS). Then the numbers c(α, f ) (resp. c(ω, f )) are critical values of f , that is there are points x ∈ X such that d f (x) = 0, f (x) = c(α, f ) (resp. f (x) = c(ω, f )). Proof. Note that the (PS) condition implies that the set of critical values is closed. Thus if c is a regular (i.e. noncritical) level, there are no critical levels in [c − ε, c + ε] for ε small enough. Moreover for the same reason, |∇ f (x)| ≥ δ0 on f c+ε − f c−ε = f −1 ([c − ε, c + ε]). Let ϕ s be the flow of a vector field ξ equal to −∇ f (x)/|∇ f (x)|2 in a neighborhood of f c+ε − f c−ε . Then we have d s f ϕ (x) = −1, ds hence for 0 ≤ s ≤ 2ε

448

C. VITERBO

(a) ϕ s ( f c+ε ) ⊂ f c+ε (b) ϕ s ( f a ) ⊂ f a (c) ϕ2ε ( f c+ε ) ⊂ f c−ε . Thus (ϕ2ε )∗ induces an isomorphism H ∗ ( f c+ε , f a ) → H ∗ ( f c−ε , f a ) As a result i∗c+ε (α) = 0 if and only if i∗c−ε (α) = 0, and we see that c(α, f ) cannot be equal to c. If X is R-oriented (R = Z or R = Z/2Z), there is a Poincar´e duality between homology and cohomology with coeﬃcients in R. If α ∈ H ∗ (X), ω ∈ Hn−∗ (X), then they are Poincar´e dual to each other if and only if the maps ∪α: H n−∗ (X) → H n (X) R given by β → α ∪ β and the integration map Iω : H n−∗ (X) → R given by Iω (β) = β, ω coincide. Alexander duality is a generalization of Poincar´e duality (Spanier, 1966), associating to α ∈ H ∗ ( f b , f a ) a class ω in Hn−∗ (X \ f a , X \ f b ). Moreover for a < c < b we have a commutative diagram AD

/ H n−∗ (X \ f a , X \ f c )

AD

/ H n−∗ (X \ f a , X \ f b )

AD

/ H n−∗ (X \ f c , X \ f b )

H∗ ( f c , f a ) H∗ ( f b , f a ) H∗ ( f b , f c )

where the vertical arrows are those of the exact homology (resp. cohomology) sequence of the triple ( f b , f c , f a ) (resp. (X \ f a , X \ f c , X \ f b )). Now if ω ∈ H∗ ( f b , f a ) is in the image of H∗ ( f c , f a ) → H∗ ( f b , f a ), that is c ≥ c(ω, f ), then AD(ω) will be in the image of a class in H n−∗ (X\ f a , X\ f c ), hence its image in H n−∗ (X \ f c , X \ f b ) is zero. If we notice that X \ f c = (− f )−c , we get that −c ≤ c(α, − f ) that is c ≥ −c(α, − f ). We just proved that c(ω, f ) ≥ −c(α, − f ). A similar argument (remember, all horizontal arrows are isomorphisms) shows the reverse inequality. PROPOSITION 2.4. If α ∈ H ∗ ( f b , f a ) and ω ∈ Hn−∗ (X \ f a , X \ f b ) = Hn−∗ (− f )−a , (− f )−b are Poincar´e dual to each other, then we have c(α, f ) = −c(ω, − f ). COROLLARY 2.5. Let µ ∈ H n (X) be a generator, and 1 be the generator of H 0 (X). Then c(µ, f ) = −c(1, − f )

SYMPLECTIC TOPOLOGY AND HAMILTON – JACOBI EQUATIONS

449

Proof. We refer to (Viterbo, 1992).

The next theorem relates c(α, f ) and c(α ∪ β, f ). If c is a critical value, we denote by Kc the set of critical points in f −1 (c). THEOREM 2.6 (Lusternik – Shnirelman). Let α ∈ H ∗ ( f b , f a ) and β ∈ H ∗ (X). Then we have that α ∪ β ∈ H ∗ ( f b , f a ) and c(α ∪ β, f ) ≥ c(α, f ). ∗ If the above inequality is an equality, then β 0 in H¯ ∗ (Kc ), where H is the Alexander – Spanier cohomology. In other words, for any neighborhood U of Kc we have that β 0 in H ∗ (U). Proof. First of all, by the (PS) condition, Kc is compact. Then for any neighborhood U of Kc , we have that for ε small enough, there are no critical points in ( f c+ε \ f c−ε ) ∩ U

Indeed, otherwise there would be a sequence xn in U such that limn f (xn ) = c, but an accumulation point of xn has to be in Kc , a contradiction. Now replacing U by a smaller set and possibly reducing ε, we may assume that U ∩ ( f c+ε \ f c−ε ) is a union of pieces of trajectories of the gradient flow of f intercepted by ( f c+ε \ f c−ε ). As a result, the flow ϕ s of the vector field ξ used in the proof of theorem 2.3 sends U in U ∪ f c−ε . Consequently we get an isotopy from f c+ε to U ∪ f c−ε , and this induces an isomorphism in cohomology. Now if c ≤ c(α, f ), the cohomology class of α vanishes on f c−ε . Slightly changing ε, we may assume that there is a form ρ defined on X such that α − dρ vanishes on f c−ε . Since α is cohomologous to α − dρ, we may assume that α actually vanishes on f c−ε . Similarly if the cohomology class of β vanishes on U, we may replace β by a cohomologous class vanishing on U. But then α ∪ β will vanish on f c−ε ∪ U f c+ε . But this implies that c + ε ≤ c(α ∪ β, f ), that is c(α, f ) + ε ≤ c(α ∪ β, f ) and there cannot be equality.

COROLLARY 2.7. Denote by cl(X) the number cl(X) = sup{k + 1 | ∃α j ∈ H ∗ (X) \ H 0 (X), j ∈ {1, . . . , k}, α1 ∪ · · · ∪ αk 0}. Then any function on X has at least cl(X) critical points. Proof. Assume the set of critical points is finite (otherwise the inequality is obvious). Then for each c we may find U such that Kc ⊂ U and H j (U) = 0 for j 0. Indeed, take U to be the union of disjoint balls centered at points

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C. VITERBO

of Kc . Therefore if β ∈ H ∗ (X) \ H 0 (X), according to the theorem we have that c(α ∪ β) > c(α, f ). Thus we have the inequalities c(1, f ) < c(α1 , f ) < c(α1 ∪ α2 , f ) < · · · < c(α1 ∪ · · · ∪ αk , f ) and we get cl(X) critical levels, hence at least cl(X) critical points.

2.1. THE CASE OF GFQI

The theory in the case of a GFQI, S (x, ξ), follows from the above, once we make some simple remarks. First, it follows from elementary Morse theory (Milnor, 1963), that since Q has no critical levels away from zero, the critical levels of S must be bounded. Therefore the sets S b are all isotopic for b large enough (in fact as soon as there are no critical points above level b), and similarly, for a small enough, the sets S a are all isotopic. We will denote by S ∞ , S −∞ any of these sets. Also since S = Q at infinity, we see that (S ∞ , S −∞ ) is diﬀeomorphic to (Q∞ , Q−∞ ); according to elementary Morse theory, (Q∞ , Q−∞ ) N × (D− , ∂D− ). Now H ∗ (S ∞ , S −∞ ) = H ∗ (Q∞ , Q−∞ ) = H ∗ (N) ⊗ H ∗ (D− , ∂D− ). Thus if ϕ is the generator of H ∗ (D− , ∂D− ), the cup product yields an isomorphism H ∗ (N) → H ∗ (S ∞ , S −∞ ) by α → ϕ ⊗ α. Thus for any cohomology class α in H ∗ (N) we may consider c(ϕ ∪ α, S ) for the cohomology class ϕ ∪ α. We shall denote this number by c(α, S ) since no confusion is possible. Now let S 1 (x, ξ), S 2 (x, η) be two GFQI. We denote by S 1 # S 2 the GFQI S 1 # S 2 (x, ξ, η) = S 1 (x, ξ) + S 2 (x, η). THEOREM 2.8. c(α ∪ β, S 1 # S 2 ) ≥ c(α, S 1 ) + c(β, S 2 ). Proof. See Viterbo (1992).

Now if S is a GFQI for L = ϕ1 (0N ), where ϕt is a Hamiltonian isotopy, then by uniqueness c(α, S ) only depends on L (and not on the choice of S ), up to a constant which is the same for all the cohomology classes. We may thus set DEFINITION 2.9. γ(L) = c(µ, L) − c(1, L) where µ is the generator of H n (N) and 1 the generator of H 0 (N). Note also that if S is normalized in any way, this allows us to define c(α, L) as c(α, S ). We also define

SYMPLECTIC TOPOLOGY AND HAMILTON – JACOBI EQUATIONS

DEFINITION 2.10. zero section. Set

451

Let L1 , L2 be obtained by a Hamiltonian isotopy from the

γ(L1 , L2 ) = c µ, S 1 # (−S 2 ) − c 1, S 1 #(−S 2 ) . Note that S 1 # (−S 2 ) usually does not satisfy the transversality condition, but if we apply to it the formal construction of generating functions, we get the following set of points (which is usually not a submanifold) L1 − L2 = {(x, p1 − p2 ) | (x, p1 ) ∈ L1 , (x, p2 ) ∈ L2 }. Thus, we shall often denote γ(L1 , L2 ) = γ(L1 − L2 ). In particular the set of critical points of S 1 # (−S 2 ) corresponds to {(q, 0) | ∃p1 , p2 ∈ T q∗ , (q, p1 ) ∈ L1 , (q, p2 ) ∈ L2 }. PROPOSITION 2.11. γ defines a distance on the set of Lagrangian submanifolds Hamiltonian isotopic to the zero section. Proof. We first check that γ(L1 , L2 ) = 0 if and only if L1 = L2 . Indeed, start with the case L2 = 0N . Then if γ(L1 ) = 0, according to Lusternik and Schnirelman theorem we have that µ is non zero on any neighborhood of the set of critical points of S . But the set of such critical points is in one to one correspondence with L∩0N . Now the fundamental class of N is cohomologous to zero on any proper subset of N, thus 0N ⊂ L ∩ 0N , and since L is embedded, this implies L = 0N . In the general case we use the following fact: according to Viterbo (1992, Proposition 3.5 p. 695), if L2 = ρ−1 (0N ), we have γ(L1 − L2 ) = γ L1 − ρ−1 (0N ) = γ ρ(L1 ) . Thus γ(L1 − L2 ) = 0 implies that γ ρ(L1 ) = 0 hence ρ(L1 ) = 0N , and thus L1 = ρ−1 (0N ) = L2 . The triangle inequality, according to the same argument, needs to be proved only for the case L2 = 0N . Thus we have to show that γ(L1 − L3 ) ≤ γ(L1 ) + γ(L3 ). But this follows from the two properties c(1, −S ) = −c(µ, S ) and c(u ∪ v, S 1 # S 2 ) ≥ c(u, S 1 ) + c(v, S 2 ).

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C. VITERBO

Indeed, we have c(1, S 1 − S 3 ) ≥ c(1, S 1 ) + c(1, −S 3 ) = c(1, S 1 ) − c(µ, S 3 ) c(1, S 3 − S 1 ) ≥ c(1, S 3 ) + c(1, −S 1 ) = c(1, S 3 ) − c(µ, S 1 ). Using that c(1, S 3 − S 1 ) = −c(µ, S 1 − S 3 ) this can be rewritten as −c(1, S 1 − S 3 ) ≤ c(µ, S 3 ) − c(1, S 1 ) c(µ, S 1 − S 3 ) ≤ c(µ, S 1 ) − c(1, S 3 ), and adding the inequalities we get γ(S 1 − S 3 ) ≤ γ(S 1 ) + γ(S 3 ). Finally the property γ(L1 −L2 ) = γ(L2 −L1 ) follows from the property γ(−S ) = γ(S ) and is a straightforward consequence of the equality c(1, −S ) = −c(µ, S ).

We can also define a norm on the set of symplectomorphism, and in fact several of them. One is given by DEFINITION 2.12. If ψ is the time one map of a Hamiltonian isotopy, we define γ(ψ) = sup{γ(ψ(L) − L) | L ∈ L} With such a definition we have PROPOSITION 2.13. γ defines a metric on the group of Hamiltonian diﬀeomorphisms H(T ∗ N). Proof. Indeed, assume γ(ϕ) = 0. This implies that for any Lagrangian in L, we have γ(ϕ(L), L) = 0 hence ϕ(L) = L. But this obviously implies ϕ = Id since if for some z ∈ T ∗ N, ϕ(z) z, there is a Lagrangian in L through z avoiding ϕ(z). Then we cannot have ϕ(L) = L. c

DEFINITION 2.14. We shall say that ϕn c-converges to ϕ and shall write ϕn → − ϕ if limn→∞ γ(ϕn ϕ−1 ) = 0. Since the notion of γ-convergence is already defined, we avoided the notion of γ-convergence. PROPOSITION 2.15. If ϕ is the time one flow associated to the Hamiltonian H(t, z), we have γ(ϕ) ≤ |H|C 0 .

SYMPLECTIC TOPOLOGY AND HAMILTON – JACOBI EQUATIONS

Proof. The proof is similar to the one for γ(ψ) in Viterbo (1992).

453

2.2. APPLICATIONS

Many classical applications can be found in Hofer (1990), Viterbo (1992), and Polterovich (2001), as well as extensions of such metrics to more general symplectic manifolds. Here we shall prove THEOREM 2.16 (Hofer, 1985). Let L be Hamiltonian isotopic to the zero section. Then L ∩ 0N contains at least cl(N) points. This follows immediately from Corollary 2.7 and Theorem 2.8. C0

PROPOSITION 2.17. Assume that Hn −−→ H where Hn , H are smooth Hamiltoc − ϕt . nians. Then if ϕtn , ϕt are the associated flows, we have that ϕtn → Proof. Indeed, the flow of Hn t, ϕt (z) − H t, ϕt (z) is given by ϕ−t ϕtn , thus according to Proposition 2.15, we have γ(ϕ−t ϕtn ) → 0. PROPOSITION 2.18 (Eliashberg, 1987; Gromov, 1985). Let ϕn be a sequence of symplectic maps isotopic to the identity. If ϕn → ϕ in the C 0 topology, then ϕ is also symplectic. Proof. We first assume dim(N) ≥ 2 and χ(N) = 0. The main point of the proof is that if for any Lagrangian submanifold L isotopic to the zero section, ϕ(L) is Lagrangian, then ϕ is conformal (i.e., ϕ∗ ω = cω). We first prove this in the linear case: we need to prove that if ω(x, y) 0 then ω ϕ(x), ϕ(y) = c · ω(x, y). Now, given x, y, let z be linearly independent from x, y such that ω(x, z) 0. Then ω ϕ(x), ϕ(z) = cz · ω(x, z), and since ω(x, ω(x, y)z − ω(x, z)y) = 0, x and ω(x, y)z − ω(x, z)y are in a coisotropic plane that can be completed into a Lagrangian plane. Hence ω ϕ(x), ω(x, y)ϕ(z) − ω(x, z)ϕ(y) = 0, ω(x, y)cz ω(x, z) − ω(x, z)cy ω(x, y) = 0, and thus cy = cz . This can be extended to the non-linear case by noticing that through any point (x, p) and Lagrangian subspace of the tangent space to T ∗ N, we can find a Lagrangian submanifold isotopic to the zero section and tangent to this subspace. Assume that ϕ is not conformal. Then for some Lagrangian submanifold L, ϕ(L) is not Lagrangian. Since we assumed χ(N) = 0, according to Laudenbach and Sikorav (1994), there is a Hamiltonian H whose flow moves ϕ(L) away from itself, that is ψt ϕ(L) ∩ L = ∅. Now if ϕn C 0 -converges to ϕ, and if ϕn is Hamiltonian,

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C. VITERBO

we have for n large enough that ψt ϕn (L) ∩L = ∅, but this is impossible according to Theorem 2.16. To remove the assumption χ(N) = 0 and to show that ϕ must be not only conformal but symplectic, we apply the above result to N × T 2 and ϕn × Id. Since dim(N × T 2 ) ≥ 2, ϕn × Id converges to ϕ × Id and ϕ × Id can only be conformal if ϕ is symplectic, we get that ϕ must be symplectic. We may now prove the C 0 -closedness of Poisson commutation. c

c

− H, Kn → − K and PROPOSITION 2.19 (Cardin and Viterbo, 2005). Assume Hn → C0

{Hn , Kn } −−→ 0, where Hn , H, Kn , K are smooth. Then {H, K} = 0. Proof. Note that here Hn , Kn are time independent, with flows ϕtn , ψtn , and so are H, K with flows ϕt , ψt . Then the flow of the Hamiltonian isotopy −s t → ϕtn ψns ϕ−t n ψn

is generated by the Hamiltonian Lns (t, z) = Hn (z) − Hn ψns ϕtn (z) .

Now Lns (t, z)

s

= 0

∂ n L (t, z) ds = ∂s s

s

{Hn , Kn } ψσn ϕtn (z) dσ,

0 n |L s (t, z)|

so that if |{Hn , KN }| goes to zero, then follows that −s ϕtn ψns ϕ−t n ψn

also goes to zero. From this it

c-converges to Id. On the other hand, according to Proposition 2.17, we have that ϕtn → ϕt , and t −s t s −t −s ψn → ψt , so that ϕtn ψns ϕ−t n ψn converges to ϕ ψ ϕ ψ . By uniqueness of the limit, we have ϕt ψ s ϕ−t ψ−s = Id for all s, t, therefore the flows ϕt and ψ s commute, and {H, K} must be constant. Since H, K are compactly supported, we have {H, K} = 0.

3. Hamilton – Jacobi equations and generating functions We consider the equation ∂ ∂    u(t, x) + H t, x, u(t, x) = 0,  ∂t ∂x    u(0, x) = f (x).

(3)

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We first explain how the method of characteristics should be translated into symplectic terms, and then how, following Chaperon and Sikorav, we may use generating functions to find general solutions of such equations. Instead of the function u(t, x) let us consider the graph of its diﬀerential in T ∗ (R × N). This will be an exact Lagrangian submanifold L, and any exact Lagrangian submanifold which is a graph is the graph of the diﬀerential of a function. We require that L satisfies the two conditions Λ f ⊂ L ⊂ ΣH where

∂ f ∂ f Λ f = (0, x, −H 0, x, , ) x ∈ N ∂x ∂x ΣH = {(t, x, τ, p) | τ + H(t, x, p) = 0}.

Note that ΣH is a hypersurface, and Λ f is an isotropic submanifold (contained in ΣH ). Now let Φ s be the flow of the Hamiltonian τ + H(t, x, p) on T ∗ (R × N). We then have that (a) Φ s preserves the hypersurface ΣH (b) L = s∈R Φ s (Λ f ) is a Lagrangian submanifold. PROPOSITION 3.1. L defined above is properly Hamiltonian isotopic to the zero section in T ∗ (R × N). As a result, we can find a GFQI for L, and uL (t, x) = c(1(t,x) , S ) is uniquely defined (up to a constant). We claim that PROPOSITION 3.2. The function uL is a solution of (3) almost everywhere Proof. See Viterbo (1996), Viterbo and Ottolenghi (1995).

Thus, given (H, f ) we get a Lagrangian manifold L f,H and then a solution of Hamilton – Jacobi’s equation u f,H . We now have THEOREM 3.3. Let us denote by J the map J(H, f ) = u f,H from C 1 (N, R) × C 1,1 (R × T ∗ N) to C 0,1 (R × N). Then this map has an extension to a continuous map from C 0 (N) × C 0,1 (R × T ∗ N) to C 0,1 (R × N). Proof. In Viterbo (1996) we proved that |uH1 , f − uH2 , f |C 0 ([0,T ]×N) ≤ |H1 − H2 |C 0 ([0,T ]×T ∗ N) The fact that we have also the inequality |uH, f − uH,g |C 0 ([0,T ]×T ∗ N) ≤ T | f − g|C 0 (N)

456

C. VITERBO

may be proved as follows. Consider the submanifolds Λ f , Λg . We claim that there is a Hamiltonian isotopy ψt such that Ψ1 : Λ f → Λg and γ(Ψ) ≤ | f − g|C 0 . Indeed Λ f , Λg are contained in the coisotropic manifold {t = 0}. We can

consider a map sending Λ0f = 0, x, 0, d f (x) | x ∈ N to Λ0g = 0, x, 0, dg(x) |

x ∈ N using the Hamiltonian H(t, x, τ, p) = g(x)− f (x). Now we have to move Λ0f to Λ f , and this is achieved by moving along the characteristics of {t = 0}. Now, it will be enough to prove that one can move along the characteristics with arbitrarily small Hamiltonians. Indeed, the translation (0, x, τ, p) → (0, x, τ + u(τ, x, p), p) is realized by any Hamiltonian such that ∂H(0, x, τ, p)/∂t = u(τ, x, p), but this can obviously be realized with |H|C 0 arbitrarily small. Now there is another notion of a solution defined for continuous data, namely the viscosity solution. We refer to Crandall and Lions (1983) and Barles (1994) for the definition. To avoid confusion we shall call the solutions we just defined as variational solutions. Variational and viscosity solutions are sometimes identical as the following theorem shows (Joukovskaia, 1993). THEOREM 3.4 (T. Joukovskaia). Consider the equation ∂ ∂     u(t, x) + H t, x, u(t, x) = 0 ∂t ∂x    u(0, x) = f (x) where H is convex in p. Then viscosity and variational solutions coincide. One of the features of viscosity solution, is that they are Markovian. This means that if we denote by J[t1 ,t2 ] the map sending f (x) to u(t2 , x) where u(t, x) is the solution of ∂ ∂     u(t, x) + H t, x, u(t, x) = 0 ∂t ∂x    u(t , x) = f (x) 1 we have J[t1 ,t2 ] ◦ J[0,t1 ] = J[0,t2 ] . This needs not be the case for variational solutions, which therefore exhibit some hysteresis, as was proved in Viterbo and Ottolenghi (1995). 4. Coupled Hamilton – Jacobi equations This section describes the work from Barles and Tourin (2001), and Cardin and Viterbo (2005). Some related work is due to Rampazzo (2003).

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The problem of considering coupled Hamilton – Jacobi equations, of the type:  ∂ ∂    u(t , . . . , t , x) + H , . . . , t , x, , . . . , t , x) = 0, t u(t 1 d 1 1 d 1 d    ∂t1 ∂x    .     .. (MHJ)    ∂ ∂    u(t , . . . , t , x) + H , . . . , t , x, , . . . , t , x) = 0, t u(t  1 d d 1 d 1 d   ∂td ∂x    u(0, . . . , 0, x) = f (x), goes back to some questions in economics first asked by Jean-Charles Rochet (Rochet, 1985; Lions and Rochet, 1986). We shall only explain here the case d = 2, since the general case is in all respects analogous. The local theory is quite obvious from the Lagrangian viewpoint. Indeed, we have now an isotropic manifold,

Λ f = 0, 0, x, −H1 0, 0, x, d f (x) , −H2 0, 0, x, d f (x) , d f (x) | x ∈ N and two Hamiltonian vector fields, X1 , X2 , associated to K1 (t1 , t2 , x, τ1 , τ2 , p) = τ1 + H1 (t1 , t2 , x, p) and K2 (t1 , t2 , x, τ1 , τ2 , p) = τ2 + H2 (t1 , t2 , x, p). We need to construct a Lagrangian manifold contained in ΣH1 ,H2 = {(t1 , t2 , x, τ1 , τ2 , p) | τ1 + H1 (t1 , t2 , x, p) = 0, τ1 + H1 (t1 , t2 , x, p) = 0}. For this we need not only that X1 , X2 commute, but also that they preserve ΣH1 ,H2 , and this exactly means that we need {K1 , K2 } = 0, where {·, ·} is the Poisson bracket in T ∗ (R2 × N). Indeed we define L= Ψ1s1 Ψ2s2 (Λ f ) (s1 ,s2 )∈R2

and one can prove that L is Hamiltonian isotopic to the zero section, thus the function uL (t1 , t2 , x) is a solution of (MHJ). The connection with the work of Barles and Tourin is through the result of T. Joukovskaia. Indeed, even though their theorem is stated for N = Rn and for the equations not depending on t1 , t2 , we may rephrase their results as THEOREM 4.1 (Barles – Tourin). Assume H1 , H2 are C 1 and convex in p. Assume moreover that we have {K1 , K2 } = 0. Then the equation (3) has a solution u which is a viscosity solution of each of the single equations of (MHJ). If we wish to extend this result to the case of C 0 Hamiltonians, we need to first define the commutation condition in this setting. This is exactly what Proposition 2.19 does. We then have the following generalization of the Barles – Tourin theorem:

458

C. VITERBO

THEOREM 4.2. Assume that the Hamiltonians H1 (t1 , . . . , td , x, p), . . . , Hd (t1 , . . . , td , x, p) are Lipschitz and C 0 -commute. Then the equation (MHJ) has a unique solution which is a variational solution of each of the equations. If the H j are convex in p, then u is a viscosity solution of each individual equation. We refer to Cardin and Viterbo (2005) for the proof. The diﬃculty here is that we have to start with Hamiltonians K1 , K2 such that {K1 , K2 } is C 0 -small, and construct out of these two Lagrangians L1,2 and L2,1 and two associated functions u1,2 and u2,1 so that they are variational solutions of respectively  ∂ ∂     v(t1 , t2 , x) + H1 (t1 , t2 , x, v(t1 , t2 , x) = 0 ∂t ∂x  1   v(0, t , x) = w(t , x), 2 2 where w is the variational solution of the equation  ∂ ∂     w(t2 , x) + H2 (0, t2 , x, w(t2 , x) = 0 ∂t ∂x    2  w(0, x) = f (x), and

 ∂ ∂     v(t1 , t2 , x) + H2 (t1 , t2 , x, v(t1 , t2 , x) = 0 ∂t2 ∂x    v(t , 0, x) = w(t , x), 1

1

where w is a variational solution of  ∂ ∂    w(t1 , x) + H1 (t1 , 0, x, w(t1 , x) = 0  ∂t1 ∂x     w(0, x) = f (x). We then have to show that u1,2 − u2,1 is C 0 -small with {K1 , K2 }. For this we find a symplectic diﬀeomorphism with γ(Ψ) small such that Ψ(L1,2 ) = L2,1 . References Barles, G. (1994) Solutions de viscosit´e des e´ quations de Hamilton – Jacobi, Math. Appl. (Berlin), Paris, Springer. Barles, G. and Tourin, A. (2001) Commutation properties of semigroups for first-order Hamilton – Jacobi equations and application to multi-time equations, Indiana Univ. Math. J. 50, 1523 – 1544. Brunella, M. (1991) On a theorem of Sikorav, Enseign. Math. (2) 37, 83 – 87. Cardin, F. and Viterbo, C. (2005) Commuting Hamiltonians and Hamilton – Jacobi equations, ´ preprint, Centre de Math´ematiques Laurent Schwartz, Ecole Polytechnique, Palaiseau. Crandall, M. G. and Lions, P.-L. (1983) Viscosity solutions of Hamilton – Jacobi equations, Trans. Amer. Math. Soc. 277, 1 – 42.

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Eliashberg, Y. M. (1987) A theorem on the structure of wave fronts and its applications in symplectic topology, Funktsional. Anal. i Prilozhen. 21, 65 – 72, (Russian); English transl. in Funct. Anal. Appl. 21, 227 – 232. Gromov, M. (1985) Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82, 307 – 347. Hofer, H. (1985) Lagrangian embeddings and critical point theory, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 2, 407 – 462. Hofer, H. (1990) On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115, 25 – 38. H¨ormander, L. (1971) Fourier integral operators. I, Acta Math. 127, 79 – 183. Joukovskaia, T. (1993) Singularit´es de minimax et solutions faibles d’´equations aux d´eriv´ees partielles, Ph.D. thesis, Universit´e de Paris VII. Laudenbach, F. and Sikorav, J.-C. (1985) Persistance d’intersection avec la section nulle au cours d’une isotopie hamiltonienne dans un fibr´e cotangent, Invent. Math. 82, 349 – 357. Laudenbach, F. and Sikorav, J.-C. (1994) Hamiltonian disjunction and limits of Lagrangian submanifolds, Internat. Math. Res. Notices 4, 161 – 168. Lions, P.-L. and Rochet, J.-C. (1986) Hopf formula and multitime Hamilton – Jacobi equations, Proc. Amer. Math. Soc. 96, 79 – 84. Milnor, J. (1963) Morse Theory, Vol. 51 of Ann. of Math. Stud., Princeton, NJ, Princeton Univ. Press. Polterovich, L. (2001) The Geometry of the Group of Symplectic Diﬀeomorphisms, Lectures Math. ETH Zurich, Basel, Birkh¨auser. Rampazzo, F. (2003), private communication on joint work with M. Motta. Rochet, J.-C. (1985) The taxation principle and multitime Hamilton – Jacobi equations, J. Math. Econom. 14, 113 – 128. Sikorav, J.-C. (1987) Probl`emes d’intersections et de points fixes en g´eom´etrie hamiltonienne, Comment. Math. Helv. 62, 62 – 73. Spanier, E. H. (1966) Algebraic Topology, New York, McGraw-Hill. Th´eret, D. (1999) A complete proof of Viterbo’s uniqueness theorem on generating functions, Topology Appl. 96, 249 – 266. Viterbo, C. (1992) Symplectic topology as the geometry of generating functions, Math. Ann. 292, 685 – 710. Viterbo, C. (1996) Solutions of Hamilton – Jacobi equations and symplectic geometry. Addendum ´ to: S´eminaire sur les Equations aux D´eriv´ees Partielles. 1994 – 1995, In S´eminaire sur les ´ ´ ´ Equations aux D´eriv´ees Partielles, 1995 – 1996, S´emin. Equ. D´eriv. Partielles, Palaiseau, Ecole Polytechnique. Viterbo, C. and Ottolenghi, A. (1995) Variational solutions of Hamilton – Jacobi equations, avalaible from http://math.polytechnique.fr/˜viterbo. Weinstein, A. (1977) Lectures on Symplectic Manifolds, Vol. 29 of CBMS Reg. Conf. Ser. Math., Providence, RI, Amer. Math. Soc.

INDEX

bounded orbit, 127

Lagrangian immersion, 233 Local (un)stable manifold theorem, 11 local flow, 282 localization, 136 LS-homotopy index, 283 Lyapunov function, 19

Calabi quasi-morphism, 428 chain level Floer theory, 321 Chas – Sullivan loop product, 166 configuration space, 187 Conley index LS-homotopy G-equivariant, 299 Conley – Zehnder index, 339

manifold stable, 19 unstable, 19 map completely continuous, 283 Morse decomposition, 294 motion planning algorithm, 195 motion planning problem, 195

D-brane, 172 de Rham complex A∞ deformation, 254 determinant bundle, 59 distorted, 420

no-distorsion theorem, 421 Novikov – Floer chain, 338

essential Grassmannian, 57 essential subbundle, 57 extended Morse complex, 121

oriented graph, 152

family of G-LS-vector fields, 297 fat graph, 152, 169 framed bordism class, 115 framed cobordism class, 113 Fredholm pair orientation, 60

pair-of-pants product, 85 Palais – Smale sequence, 21 path-loop fibration, 121 polygonal linkage, 189 quasi-morphism, 429

generating function quadratic at infinity, 441 graded Hopf algebra, 78 graph combinatorial map, 153 graph flow, 150 moduli space, 157 Grobman – Hartman theorem, 12

relative Hopf invariant, 117 rest point hyperbolic, 9 Schwarz genus, 199 Serre spectral sequence, 119 simultaneous control, 209 spectral invariant, 321, 362 spectrality axiom, 367 string topology, 149 submanifold exact Lagrangian, 439 Lagrangian, 439 aspherical, 266

Hamilton – Jacobi equation, 454 Hamiltonian function admissible class, 83 hyperbolic operator, 2 Jacobian, 9

461

462 topological complexity, 198 undistorted, 420 vector field positively complete, 9 Zimmer program, 422

INDEX