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LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor J.W.S. Cassels, Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, England The books in the series listed below are available from booksellers, or, in case of difficulty, from Cambridge University Press. 34 36 39
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London Mathematical Society Lecture Note Series. 170
Manifolds with Singularities and the Adams-Novikov Spectral
Sequence
Boris I. Botvinnik Department of Mathematics and Statistics York University, Ontario
CAMBRIDGE UNIVERSITY PRESS
CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo
Cambridge University Press The Edinburgh Building, Cambridge CB2 2RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521426084
© Cambridge University Press 1992 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1992
A catalogue record for this publication is available from the British Library ISBN-13 978-0-521-42608-4 paperback ISBN-10 0-521-42608-1 paperback Transferred to digital printing 2005
Contents Preface 1
ix
Manifolds with singularities
1
..
1.1
Bordism theories with singularities
1.2
Generalized Bockstein-Sullivan triangle
.
1.3
Some complexes of cobordism theories
.
1.4
The bordism theories MGJ(k)()
1.5
Proof of Theorem 1.4.2
1.6
E-singularities spectral sequence
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2 Product structures
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70
2.1
Multiplication structures
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2.2
Existence of product structure
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2.3
Commutativity for the case of one singularity
2.4
Associativity for the case of one singularity
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2.5
General case
2.6
Product structure in the E-SSS
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3 The Adams-Novikov spectral sequence 3.1
Basic definitions
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81
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82
CONTENTS
vi 3.2
The modified algebraic spectral sequence
3.3
Symplectic cobordism with singularities
3.4
The E-SSS for MSp
3.5
Splitting of the spectrum MSp2)
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104 114
4 First differential of ANSS
123
4.1
Characterization of the Ray elements
4.2
Product structure in MSpE()
4.3
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128
Some relations in the ring MSp*
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136
4.4
Localization of bordism theories
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141
4.5
The generators of the ring MSp;
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152
4.6
Some notes
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166
Bibliography
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171
Logical interdependence of the sections
4.6
vii
Preface The purpose of this book
is to discuss some natural relations
between geometric concepts of Cobordism Theory of manifolds with singularities and the Adams-Novikov spectral sequence.
We begin by motivating this discussion. The central problem of Algebraic Topology has been and continues to be that of obtaining geometrically manageable descriptions of the main algebraic invariants
and constructions. For example, let us note the Steenrod problem on realization of integer homology cycles by manifolds. When cycles are presented in terms of manifolds we are able to deal with the cycles by means of Smooth or Piecewise Linear Topology, to apply surgery, to paste and cut and so on. A nice example of successful combining of algebraic and geometric methods may be found in Sullivan's approach to the Hauptvermutung [105]. In particular, Sullivan discovered new geometric objects of Algebraic Topology, namely manifolds with singularities. The simplest example is a (Z/n)-manifold (see Figure 0.1, where n = 4). (Z/n)-manifolds present homology cycles with Z/n coefficients allowing us to apply the machinery of smooth topology. The corresponding (co)-bordism theory naturally represents ordinary (co)-homology theory with coefficients in Z/n.
ix
Figure 0.1: Z/4-manifold. In the early seventies D.Sullivan [106] and N.Baas [11] defined the bordism and cobordism theories of manifolds with singularities in the
general case. Then it was made evident that many known homology theories may be realized as bordism theories with singularities. The notion of manifolds with singularities allows us to geometrize the complex and real K-theories, Morava K-theories [93], [115], [116], the Conner-Floyd theory W(C, [67] and some others whose cycles were not provided with some geometric structure.
Although the notion of manifold with singularities is very close to the notion of ordinary manifold we have to be much more careful when dealing with the former. For example, the direct product of two manifolds with singularities doesn't possess the structure of such a manifold. Actually there exist some manifolds with singularities which are obstructions to the existence of an admissible product. In particular the problem concerning multiplicativity of the ordinary homology theory with Z/n coefficients may be solved in the same terms. O.K.Mironov [67], [68] was the first to give the general geometric approach to multiplicativity of bordism theories with singularities; see also [9], [16], [70], [116]-[119]. The main constructions pertaining to multiplicativity are x
described in Chapter 2 of this book.
Now manifold with singularities is a common and convenient notion as well as ordinary manifold. The corresponding cobordism and bordism theories have been considered from several viewpoints. The connection between the complex cobordism theory with singularities and Formal Group Theory has been found (see [48], [51], [93], [94], [115]-[121]); now a new interesting application has appeared, namely the Theory of Elliptic Genera. It isn't our purpose to describe all the rich applications of bordism theories with singularities. We would rather concentrate our attention on the connection between geometry of manifolds with singularities and the Adams-Novikov spectral sequence.
Traditionally the Adams-Novikov spectral sequence (ANSS) has been considered as a computational machine allowing us to describe the stable homotopy groups 7r*X of the given spectrum X in algebraic terms of generators and relations. Actually it can be seen that geometric methods are widely used for description and computation of the Adams-Novikov spectral sequences. For example, the concepts of J-homomorphism and Hopf invariant which originally were geometric tools have been transformed to the powerful chromatic machinery; see [35], [48], [61], [82]-[85]. The chromatic technique allows us to subdivide
the Adams-Novikov spectral sequence for a sphere into parts, each of which is determined by the corresponding Morava K-theory k(n)*(.). So the computational problem is reduced to some particular algebraic and homotopy problems. We note that consideration of the Adams-Novikov spectral sequence for spheres is not the subject of this book. For detailed information refer to D.Ravenel's book [84].
The main topological object which is going to be considered here is the symplectic cobordism ring MSp*. Actually we are going to examine the Adams-Novikov spectral sequence for this ring. We shall not regard ANSS as a computational tool only, but as a mathematical object provided with rich algebraic and geometric structures. Particular
attention will be paid to finding and describing the above geometric
xi
structure. V.Vershinin has gone rather far in the computation of this AdamsNovikov spectral sequence [109]-[113]. Our intention is to use his results and constructions widely.
Geometric methods are also very useful to deal with the AdamsNovikov spectral sequence for the ring MSp*. For example, Two-valued Formal Group Theory (see V.Buchstaber [26], [27] ) describes the ring JI* = Hom%MU (MU*(MSp), MU*)
,
which is the zero line of E2'*, in various terms. We emphasize that application of Two-valued Formal Group Theory is of substantial help to compute this spectral sequence; see [109]-[113].
The first necessary step to start dealing with the Adams-Novikov spectral sequence for a spectrum X is to present a particular Adams resolution for this spectrum. Conventionally this is constructed in a standard category of spectra. Sometimes the Adams resolution as well as the Adams-Novikov spectral sequence can be constructed directly from geometric consideration of some notions of cobordism with singularities. To clarify the above statement we consider the following.
Example. The Adams resolution for the spectrum MSU can be constructed as follows. Suppose 01 is the generator of the group MSUI = Z/2, and P is the framed circle presenting the element 01 . Accord-
ing to Mironov [67], the bordism theory with 01-singularity MSU (.) is isomorphic to the Conner-Floyd cohomology theory W(C, 2)*(.) (see Stong [103]). We note that the Bockstein-Sullivan exact sequence
...
MSU
i MSU* (.) B'. MSU* (.)
induces the diagram of the classifying spectra:
MSU
MSU
8/ MSUB'
b
MSU
\\ 8 / MSUB'
b
It is obvious that the diagram (0.1) is an Adams resolution of the spectrum MSU in the cohomology theory MSUI (). In the 2-local x11
local category this diagram also presents a particular Adams resolution in the Brown-Peterson theory BP*(.) since the spectrum MSU91 ) splits into a wedge of the spectra EnBP. A complete description of the bordism ring MSU* in terms of generators and relations was obtained as a result of considering the above Adams-Novikov spectral sequence from geometric and algebraic viewpoints; see [14].
It can be seen now that the Adams-Novikov spectral sequence may be determined by a procedure of resolving singularities.
It is natural to suppose that such a procedure does exist in the case of several singularities as well. Indeed every given bordism theory MG*(.) and sequence E = (Pi, , Pk, ) of closed manifolds naturalwhich are interconnected and are ly determine the theories related to the theories MG*(.), MGE(.) as can be seen in the following diagram: MG*
ry-- (1) MGEr(1)
\
a(1)
-(0) % MGE(1)
MG; r(2)
"
^43-)
...
a(2)
'r(1)
a(1)
MGr(2)
Here the theories MG*E(k)(.) split into the sums of the theories MG,(.)
and the transformations y(k), ir(k), 8(k) will be defined in geometric terms of cutting and gluing the manifolds. For example, the manifold M in the bordism theory is glued out of the blocks yt(M) xP= in the same way as a boundary of a closed manifold with singularities is glued after removing the cones over singularities. The diagram (0.2), being an exact couple, induces the spectral sequence (it will be called the E-singularities spectral sequence). This spectral sequence restores the bordism theory MG*(.) out of the bordism theory with singularities MGI(.). The top line of the diagram (0.2) represents the filtration of the bordism theory MG*(.), which may be considered as a geometric analogy xiii
of the algebraic filtration generated by powers of the ideal
m=
([Pl],...,[Pk]...) C MG*.
The differentials in the E-singularities spectral sequence also have a simple geometric description. The first differential is a direct sum of the Bockstein operators /3k (which are similar to boundary operators on ordinary manifolds). It is very important for our purposes that in several cases the E-singularities spectral sequence may be naturally identified with the corresponding Adams-Novikov spectral sequence. Now let us briefly describe the subject of every chapter. Chapter 1 contains basic geometric constructions of manifolds with singularities. The definition and properties of the E-singularities spectral sequence are given at the end of the chapter. In Chapter 2 we give a geometric construction of a multiplication on bordism theories with singularities. The chapter provides a necessary geometric tool for algebraic considerations. Chapter 3 is concerned with algebra. First of all we deal with definitions of the Adams-Novikov spectral sequence (ANSS) and the Novikov algebraic spectral sequence. Next the symplectic bordism theory with singularities which has been discovered by V.Vershinin [111], is considered. The coefficient ring MSpE of this theory is a polynomi-
al ring and the theory
splits into a direct sum of BrownPeterson theories BP*(.). So we can identify the corresponding AdamsNovikov spectral sequence with the E-singularities spectral sequence.
The proof of Vershinin's theorem concerning the theory given, since its details will be applied later.
is
It can be seen that the first differential in the Adams-Novikov spec-
tral sequence splits into a sum of Bockstein operators. So we have a new way to compute the algebra ExtABP(BP*(MSp), BP*).
Some computation is given in Chapter 4. The product structure in the theory is carefully chosen and the action of Bockstein operators on the generators of the coefficient ring MSpF, is computed. xiv
The result is rather surprising. Indeed, the algebra ExtABP(BP*(MSpe' ), BP*)
has a module structure over the symmetric group, so it may be described in terms of representation theory. So we have tried to come along the way from simple geometric considerations concerning manifolds with singularities up to some computational results describing the structure of the Adams-Novikov spectral sequence.
We use many figures hoping they may be helpful to understand the discussions and arguments. The list of references doesn't claim completeness.
It is a pleasure to acknowledge Victor M. Buchstaber for helpful comments and valuable critical notes and Vladimir V. Vershinin, Vassily G. Gorbunov, Roin G. Nadiradze for fruitful collaboration and discussions on the subject of this book.
Finally the author would like to thank the staff of the Press Syndicate of the University of Cambridge and the Referee for their unflagging patience and cooperation.
Boris Botvinnik
xv
Chapter 1 Manifolds with singularities The purpose of this chapter is to describe a procedure restoring an ordinary bordism theory MG*(.) out of the bordism theory with singularities MGE(.). This geometric procedure looks like resolving the singularities in the manner of Cusp Theory. It is our hope that the constructing of the E-singularities spectral sequence (E-SSS) will not get us bogged down in modern Homological
Algebra. The main objects for consideration will be manifolds with various geometric structures. The initial point here is the following simple observation. The given bordism theory MG*(.) and the sequence E = (P1, , Pa, ) of closed manifolds induce not only the bordism
theory with singularities MGE(.), but also the family of intermediate bordism theories for k = 1,2,... . A manifold M in the theory (it will be called a EI'(k)-manifold) is consistently glued out of the blocks
where a = (al, ... , an, -)is a sequence of nonnegative integers such that al + ... + an + ... = k, and yaM are ordinary manifolds. It is evident that the family of theories the theory MG*(.): MG*(.) 7F 1 MG;r(1)(.)
gives us the filtration of
... 1
Ery(k+l)
...
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
2
This filtration generates the E-SSS. Following O.K.Mironov [67], [68] we begin with the definitions of the
bordism theories with singularities MG; (.) . His constructions seem to have a most clear and obvious form for our purposes. Then we'll define the bordism theories MG:(' from the following diagram in the same manner:
MG* -
MGEr(1)
MG* (1)
p(1)
y(2)
MGEr(2)
7(3)
...
MG; (2)
The transformations y(k), 8(k), ir(k) will also be defined geometrically, by gluing and cutting the manifolds. Actually the main part of the chapter (sections 1.2-1.5) contains only some geometric constructions from elementary Cobordism Theory. Almost all the proofs will be given by means of constructing bordisms joining some manifolds. We'll deal with some elementary homologi-
cal algebra only in section 1.3. We'll define the E-SSS and prove its simplest properties. We note also that the E-SSS was defined by V.V.Vershinin [111] in the one-singularity case, and the general case was described by the present author [15], [16].
1.1
Bordism theories with singularities
Here we define the bordism and cobordism theories with singularities. We wouldn't like to consider an absolutely general case; here we shall be restricted to the following situation.
The starting point is a category of smooth manifolds with a stable G-structure in the stable normal bundle, where G is one of the classic Lie groups. The corresponding bordism and cobordism theories will be denoted by MG*(.) and MG*(.). Our main examples will
1.1. BORDISM THEORIES WITH SINGULARITIES
3
be connected with the cobordism theories MO*(.), MSO*(.), MU*(.), MSU*(.),
Also we suppose that a direct product of the manifolds generates some external product structure in the theories MG*(.) and MG*(.) with the usual properties (see [67], and section 2.1). The classifying Thom spectrum for these theories will denoted by MG. Let us take a sequence E = (P1, ... , P,,, ...) of closed manifolds. It is supposed below that the sequence E is locally finite, i.e. the number sequence {dim has only infinity as a point of condensation. We denote Ek = (Pi, . . . , Pk) f o r every k = 1, 2, .... It is convenient to denote P0 = pt.
Definition 1.1.1 We call a manifold M a Ek-manifold if there are given the following:
(i) the partition 19M=OOMUa1MU...U49kM of its boundary aM into such a union of manifolds that the intersection
aIM = ail M fl ... fl ai9M
is a manifold for every collection I its boundary is equal to
i9} C {0, 1, ... , k} and
0 (aIM) = U (aIM fl a;M) ; (ii) the compatible product structures (i.e. diffeomorphisms preserving the stable G-structure)
0I:8IM- 1 #IMxPI, where I = {il, ... , iq} C {0,1, ... , k},
PI=Pi, x...XPiq.
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
4
Compatibility here means that if I C J and
c:QIM - ) f3jM is the inclusion, then the map
¢I0t00J1:/31MxPJ j31MxPI is identical on the direct factor PI. Note 1.1.1 Now we have defined half-finished manifolds with singularities, to obtain real manifolds with singularities we have to do some identification.
Two points x, y of the Ek-manifold M are equivalent if they belong to the same manifold 8,M for some I C {0, 1, ... , k} and
pr o OI(x) = pr o Oj(y), where
pr:/3,MxPI -) flIM is the projection on the direct factor. The factor-space of the topological space M under this equivalence relation is called the model of the Ekmanifold M and is denoted by ME.
Indeed it is convenient to deal with Ek-manifolds without considering their models. For this we only have to remember consistency of the constructions with the projection
7r.M---,ME. The boundary SM of a Ek-manifold M is the manifold BoM. It is also a Ek-manifold: 81(SM) = a1M fl SM. Manifolds ,QIM are also Ek-manifolds: 0
ifjEI,
e;(/3IM) = l /3{j}uIM x Pj otherwise.
1.1. BORDISM THEORIES WITH SINGULARITIES
5
Here we denote QIM = /3il o (Ni2 0
(... o /i9M) ...) ,
ifI={il,...,iq}C{1,...,k}. The pair (M, f) is a singular Ek-manifold of the space pair (X, Y) if M is a Ek-manifold, and
f : (M, 5M) --- (X, Y) is such a map that for every index subset I = { i 1 , . . . , iy} C {1, ... , k} the map f I aIM has the following decomposition:
fla1M=fiopr o0I Here the map
pr : QIM x PI ---* QIM is the projection on the direct factor as above and the map
f:j37M- 3X is a continuous one.
Note 1.1.2 The map f may be also decomposed: f = fE o 7r; here 7r : M -+ ME is the projection, fE : ME --> X is a continuous map. Let us notice that singular Ek-manifolds of the point coincide with their topological models. 0
So the bordism theory and cobordism theory of Ek-manifolds are well defined. The theories MGE(-) and MG*(.) are determined as direct limits of the theories and respectively.
Theorem 1.1.2 The theories
and MG*(.) are extraordinary
homology and cohomology theories respectively.
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
6
Proof may be given in a standard manner; it is sufficient to verify that the theories MGF-(.) and MG*(-) satisfy the Eilenberg-Steenrod axioms; see [11].
Below we will deal mainly with the bordism theories; all the constructions here have a simple geometric interpretation. Every ordinary manifold may be considered as a Ek-manifold with empty set of singularities and the Ek-manifold may be considered as a for m > k. So the following natural transformations are well defined:
irk : MG*(-) --p
M -+ QkM generates the transformation of degree
-(dimPk + 1) Sk : MG£k (.) --- i MG£k-'
(Note that every manifold QkM is a Ek_1-manifold by definition.)
The transformations irk-1, Sk connect the theories MGEk into the following exact Bockstein-Sullivan triangle:
and
MG*Ek''
MGEk_,
MGEk_1
MGEk
Here we denote the transformation which is generated by direct product (from the right) on the manifold Pk by .[Pk].
1.2
Generalized Bockstein-Sullivan triangle
Now we construct the exact triangle which connects the theories MG, and MGE (.) for every locally-finite sequence E = (F1,. .. , P,ti, ...) of
1.2. GENERALIZED BOCKSTEIN-SULLIVAN TRIANGLE
7
closed manifolds. For this we would define a new bordism theory which is closely connected with the theories and
MG.(.). Definition 1.2.1 The manifold M is called a EkF(1)-manifold if there are given
(i) the partition of the manifolds
a union of manifolds glued along boundaries, i.e. the intersection
MI = Mi, n ... n Mi. is a manifold for every index subset I = {i1, ... , iq} C {1, ... , k} and its boundary is equal to
a(MI)=(MInaM)u
(UMInMj; 7
I
(ii) the compatible product structures
XPI:ryIM -*yIM x PI, where yIM are manifolds, I = { i i , . .. , iq} C {1, ... , k}; compatibility means that the map
TIO10 Til :yIMxPJ --+yIMxPI is the identity map on the direct factor P' for every I C J, where
t:MI is the corresponding inclusion.
It is evident that a EkF(1)-manifold simulates the structure which the part of the boundary of the Ek-manifold M,
aM=a1MU...UakM
8
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
has. Its ordinary boundary OM is the boundary OM of the EFk(1)-
manifold M; it has this structure by the definition. It should be noted that the manifolds yIM as well as the manifolds /31M are Ek-manifolds. The map F : (M, aM) ----> (X, Y)
is called the singular EPk(1)-manifold of the space pair (X, Y) where M is a EI'k(1)-manifold, M = M1 U ... U Mk , such that the map FPM, is decomposed as follows:
FIM,= flopro0, for every index subset I = {il, ... , iq} C {1, ... , k}. Here
pr :y1MxP'--) y1M is the projection on the direct factor as above, and the map
f:/31M-,X is a continuous map.
So the bordism theory MG*" (')(.) is well defined. We define the bordism theory as a direct limit of the theories MG Consider the transformation y(1) :
forgetting the EI'(l)-structure and the transformation
a(1) : MGE(.) -i defined by the formula:
,9(1): [(M,f)]E -' LIaM,.fIaM)]Er(l), where aM = a1M U ... U akM
.
$
4 4
10
CHAPTER I. MANIFOLDS WITH SINGULARITIES
Consider the cylinder (see Figure 1.1)
((M U -V) x I, (f U -G) x Id) = (W, H) as a E-bordism between the E-manifold
(Mx{1},f x{1})=(M,f) and the ordinary manifold
((M U -V) x 101, (f U -G) x 10}). We obtain that [(M, f)]r E Im ir. 2. EXACTNESS OF THE VERTEX MGF(1)(
)
If (L,g) is a singular EI'(1)-manifold and [(L,g)] E Im 0(1), then there exists a singular E-manifold (W, H), such that aW = L, HJav = g; see Figure 1.2. We obtain that y(1) ([(L,g)]) = 0 by considering (W, H) as an ordinary manifold, i.e. Im 9(1) C Ker y(1). The inverse inclusion is obvious. 3. EXACTNESS OF THE VERTEX MG*(.).
Let (M, f) be an ordinary singular manifold, [(M, f)] E Ker 7r; then there exists a singular E-manifold (V, G) with the boundary aoV = M, G1aov = f. We have
(01Vu...uakV)naoV=0 because M is a closed manifold. So (V, G) may be considered as a bordism between (M, f) and the following EI7(1)-manifold
(01VU. .U akV,GIa;vu...uakv) The inclusion Im y(1) C Ker 7r is obvious.
We have the commutative diagram for every space pair (X, Y)
MG*(X,Y)x[] - MG*(X,Y) I Id
6- MG* (X, Y)
MGF(X,Y) j,Id
lW
MGEr(1) (X, Y)
(0. MGE (X, Y) .L
(1.4)
1.2. GENERALIZED BOCKSTEIN-SULLIVAN TRIANGLE
11
L
Figure 1.2: E-bordism (W, H)
when E = (P).
Here the top line is the Bockstein-Sullivan exact sequence and the transformation w is generated by the map
M -+MxP. In particular, the theory is equivalent to the theory MG*(.) if E = (P), where the grading is shifted on dim P.
Corollary 1.2.3 The bordism theory
is an extraordinary
homology theory.
Proof. It is sufficient to verify that the theory satisfies the EilenbergSteenrod axioms. All these axioms are verified in a standard way. The excision axiom may be verified using the exact triangle (1.3) and the five-lemma.
Let us consider several examples of the cobordism and bordism theories with singularities.
1. We take a sequence E = (P1'...' P"'...) of closed manifolds in the complex cobordism theory MU*(.) such that the element [Pk] is a polynomial generator of the ring MU* in dimension -2k, k = 1, 2,... .
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
12
Then the cobordism theory
coincides with the ordinary
cohomology theory H*(.; Z). 2. We assume E = (P) for [P] = 01 E MSU-1 in the cobordism theory MSU*(.), where 01 is a generator of the group MSU-1. O.K.Mironov coincides with [67] noted that the cobordism theory the Conner-Floyd cohomology theory W(C, 2)*(.). This case was considered in detail in [14]. 3.
Let us consider a sequence E = (F1,. .. , P/1, ...) of closed
manifolds in the theory MU*(.) such that the elements [Pk] give a regular system of generators of the ideal Ker Td C MU* (where Td is Toda genera). Then the theory is isomorphic to the connected complex K-theory; see [93]. 4. Let us consider the oriented cobordism theory MSO*(.). We put
E = (P), where [P] = 2. Then the cobordism theory coincides with the wl-spherical cobordism theory W(R, 2)*(.) which splits into a wedge of theories H*(.; Z/2). 5. Now we consider the integral Brown-Peterson cobordism theory and its generalizations. That is, we choose U-manifolds P1, ..., Pk, ..., dim Pk = 2k, whose cobordism classes are polynomial generators of the ring MU* . If dimPk = 2(p' - 1) for a prime p , then p divides all the Chern characteristic numbers of the manifolds Pk. Let's take a prime number p and introduce the notations
E(°1={P; E(n)
where [ P o ] = p, n = 1, 2,
ii(pq-1),q=1,2,...},
I
fpo,p
....
Then the p-localization of the theory P(9)*(.)
coincides with the Brown-Peterson cohomology theory BP*(.): 1'(O)*(-) ® Z(r) = BP*(-).
The cobordism theory
MUU(&) = P(n)*()
1.3. SOME COMPLEXES OF COBORDISM THEORIES
13
has the following coefficient ring:
P(n)* = BP*/(p, vi, ... , vn), where vi are the Hazewinkel generators of the ring BP*; see [8], [9], [51], [67], [68], [117], [118].
1.3
Some complexes of cobordism theories
By definition the Bockstein operator
determines the transformation MG*(.)
,Ok :
of degree -(pk + 1), where pk = dimPk. Definition 1.1. implies the following properties of Bockstein operators: //
(1.5)
QkcQv=(-1)pkn9+loq°Qk.
Qko/39=0, In particular we have the complex MG*
±-k, MG*
MG*
MG*
Q
(1.6)
for every k = 1, 2.... and space pair (X, Y). We now introduce the collection of sequences
% = {a = (al, ... , an, ...) an > 0, all but finitely many an are zero} . We put L(a)3(X, Y)
MGs
E. ac(rj+i)(X Y) = for every a E 2(. So we have that the theory
MG;
isomorphic to up to degree. The Bockstein operators ,Ok induce the transfor-
mations /3k(a) :
L(ak)*(-)
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
14
for every k = 1, 2, .... Here ak = (a,,. . . , ak-1, ak + 1, ak+1.... ) for the
sequence a = (al, ... ) an, ...). We define the number k
ck(a) = E aipipk i=1
for every element a = (al, .... an, ...) E 2t. We denote /3k(a)
The collection of the graduated groups and transformations
{L(a).(X, Y); 01(a), ... , /3k(),.... }aEA may be considered as the lattice of complexes; we denote it by £E(X, Y).
Lemma 1.3.1 The differentials /3k(a) are anticommutative in the lattice of the complexes LE (X, Y).
Proof. Consider some square of the lattice GE(X,Y):
L(a).()
Mot)
L(at)*(.) ps(a
Qs(a) 1
(1.7)
L(as)*(.) at(e)' where at,, = (at)., = (as)t Let s < t; then we have: ,s(a) + tct(as) + lct(a) + Ks(at) s-1
_
t-1
s-1
t-1
aipips + E ajpjpt + E akpkps + E atpipt + pspt = pspt mod 2 i=1
j=1
k=1
1=1
We obtain from (1.5) that Qt(as) o QS(a) _
-,
(at) o Qt (a).
1.3. SOME COMPLEXES OF COBORDISM THEORIES
15
Let's denote the total complex of the lattice £E(X, Y) by TE(X, Y):
MGE(1)() °
... -> MGE(k_1)(.)
MGE(2)(.)
Q(k-1)4
MG;(')(.) - ...
We denote the collection of the partitionings of the nonnegative number k by 00
%k=Sa=(a,,...Ia,,,...) E%
E ai = k
.
i=1
The collections 2(.k will be used below as the index sets for manifolds and
their cobordism classes. Let us agree that M(a) = 0 and [M(a)] = 0 if the sequence a contains a negative element. The next simple lemma describes the complex TE (X, Y) and follows from the above definitions.
Lemma 1.3.2 1) There is an equivalence of the homology theories
MG;(k)(.) = ® aE%A;-1
f o r every k = 1, 2,
....
If the element x has the form
x= ® x(a) E MGs (k) (X, y) aE2(k-1
then x(a) E MGE (X, Y), where
degx(a)=s-k-aipi. 2) The differential Q(k) acts according to the following formula: Q(k)
® x(a) _ ® y(a)
aElak-1
oEak
for every k = 1,2,...; here we put o0
Y(or) = E(-1)"`(a)/3ix(cji ... ) ci_1, ci - 1, Ci+l.... ) i=1
f o r every element a = (c1, ... , c , . . . ) E 2tk. 0
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
16
The local finiteness property of the sequence E implies that the groups MGn (k) (X, Y) are finitely generated if the groups MGm (X, Y) are finitely generated for all m.
The bordism theories
1.4
Here we define the theories MG*"(')(.) to extend the triangle (1.3) up to the diagram MG* ir(°)
y(2)
MGEr(1)
\
\
a(1)
MGEr(2) a(2)
ir(1)
MG; 1
MGE Z 16(1)
-
This diagram induces the E-singularities spectral sequence which restores the bordism theory MG*(.) from the theory MG;(.).
Definition 1.4.1 The manifold M is called a Er(k)-manifold if there are given
(i) the partitions
M = U M(a), am= U SM(a), aE21k
0, E2lk
where M(a) are manifolds, W(a) = M(a) n am, and the equality OM(a) = SM(a) U
U
(M(a) n M(a'))
a,E%A;,a'#a
holds for each M(a), and a boundary of the manifold
MM = n M(a) aEZ
is equal to
am, = U (M(a) n Mme)
1.4. THE BORDISM THEORIES
17
for every subset 23 C 2tk having more than one element;
(ii) compatible product structures, i.e. there are defined diffeomorphisms
0z:MM-+ycMxP!B, preserving the G-structures of the manifolds for every subset B C 2tk; here 7ZM are ordinary manifolds; we have introduced the notations
P1 =Pi1x...xPnnx... for a sequence a = (a1,.. . a,i, ...), and )
P`IZ=P11 x...xPnn x...,
where bi = maZx jai ( ai the i-th term of the sequence a} ; C1E
the compatibility here means that the map
Oe o t o 0%, : y,(M) x P8 -- 7c(M) x PC is the identity map on the direct factor Pc for every subset it C B and the corresponding inclusion t : MM --+ Me.
The boundary of a EF(k)-manifold has the EF(k)-structure by definition. A singular EF(k)-manifold for the pair (X, Y) is defined similarly to the case when k = 1. So the theory is well defined for every k = 1, 2, ... .
Let's define the transformation
a(k) : MGE(')(.) -+ MG:r(k)(.) as follows. Let the element x lie in the group MGE(k) (X, y); then it has the form x = ®I(M-1 g-)1 F a
where
00
dim Ma=s-k-Eaipi i=i
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
18
V
-- #W(0, 1) X P2
/31M(0,1) x P1 132M(1, 0) x P2
L(1,1)
L(0, 2)
xP1
L(2, 0)
X P2
Figure 1.3: EF(k)-manifold (L, F)
for every a = (a1, ... , a,,...).
Let us next take the disconnected union of the ordinary singular manifolds
(L, F) = U (5Ma x Pa, .fa o pr)
,
aE`Ilk-i
where OMa = 01Ma U
U 9,,,Ma as above, and the map
pr :5Ma x P' -+ 5M,, is the projection on the direct factor fa = gaIaMM. 00
L(am) = U (-1)x"(0')/3iM(c1, ... , ci-1, ci - 1, Ci+1, ...) i=1
for every o = (e1,. .. ) ci_1) ci, ci+1, ...) E Qtk. structure of a EF(k)-manifold; see Figure 1.3.
So (L, F) has the
It is simple to verify that this construction defines correctly the transformation 0(k).
1.4. THE BORDISM THEORIES
19
The transformation 7r (k) :
....
is defined f o r every k = 1, 2, That is the E-manifold structure is defined on every manifold yaM for a given EF(k)-manifold M:
M = U M(a),
M(a)t_'-yaM x Pa.
aE2lk
Let us consider the element x = [(M, F)]Er(k) E MGEr(k) (X, y), where FI M(a) = fa o pr o
`Ya,
fa : 'yaM -) X.
We put ir(k)(x) = y, where
y = ® [(yaM, fa)]E . aE2j;
Finally we define the transformation MG*r(k-1) (')
'y(k) :
for k > 2 as follows. We consider the map
which is defined on the element a = (al, ... , as,...) E 21k by the formula Sk(a) = (al, ... , at-1i at - 1, at+1.... ) E %k-1,
where t = min j j I aj > 1}. Let the map
F : (M, 8M) -> (X, Y) be a representative of the element x E MGFr(k) (X, Y), where x = [M],
M = U M(a), M(a)'""yaM x Pa. aE2lk
20
CHAPTER I. MANIFOLDS WITH SINGULARITIES
We note that
yaMxPa=yaMxPt xP°, where t = t(a), Q = k(a). We define the EF(k - 1)-structure on the manifold L = M as follows. We put
L(o) = U yaM aEf-1(o)
for every a E 21k-1 and then let 'Y(k)(x) = [(L,
F)]Er(k-1).
Note 1.4.1 The transformation y(k) depends on the choice of the map For k , which is not symmetric with respect to the manifolds Pk. example, we consider below the case when E = (P1i P2). Suppose M is a EF(2)-manifold: M = M (2, 0) U M (1, 1) U M (0, 2).
(1.8)
By applying the transformation y(k) we obtain the EF(1)-manifold L = L (1, 0) U L (0, 1) = M:
L(1,0) = M(2,0),
L(0,1) = M(1,1) U M(0,2).
We choose another EF(k)-structure on the manifold L as follows: L (1, 0) = M (0, 2),
L(0,1)=M(2,0)UM(1,1).
The bordism joining these two EF(l)-manifolds is displayed in Figure 1.4.
It also can be shown that in the general case the EF(k - 1)-bordism
class of the manifold y(k) (M) doesn't depend on the choice of the EF(k-1)-structure (it need only be compatible with the original EF(k)structure on the manifold M).
1.4.
THE BORDISM THEORIES MG,
21
L=Mx{1}
L= M x {1}
Figure 1.4: Bordism between L and L
Theorem 1.4.2 The following triangle of the bordism theories and transformations is exact: MGEr(k-1) (.)
MG* (k) ( )
for every k = 2,3 ..... The identity ir(k) o ,9(k) = ii(k) also holds.
Proof of this theorem will be given in section 1.5.
E = (P). Then the theory is naturally isomorphic to the theory MG*(-) for every k =
Note 1.4.2 Consider the case when 112,
... by changing the graduation on pk, where p = dim P, and the
triangle (1.9) is isomorphic to the Bockstein-Sullivan triangle (1.2).
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
22
It is not difficult to verify that the theory
satisfies the Eilenberg-Steenrod axioms for every k > 1 as in the case of k = 1. The excision axiom may be verified by the help of triangle (1.9) and the five-lemma.
So we obtain
Corollary 1.4.3 The theory MGFF(k) () is an extraordinary homology theory for every k = 1, 2, ... . 0
The spectra representing the theories MG: r(k) be denoted by MGEr(k), MG£(k) respectively.
MG'(.) will
Proof of Theorem 1.4.2
1.5
The proof will be given by direct constructing of the corresponding bordism manifolds. For simplicity we consider the case when the space
Y is empty. When the space Y isn't empty the proof is quite similar. Let's apply the triangle (1.9) to the space X. 1 EXACTNESS OF THE VERTEX MG*
(k)
.
i) Ker 8(k) C Im ir(k). Let x E Ker 8(k); it has the form
x =aE2(k-1 ® [(M(a), F(a))}E where M(a) are E-manifolds and the maps F(a)jajm(,) are decomposed as follows:
F(a)Ia,M(a) : a,M(a)
Q,M(a) x P"
Q1M(a)
X
.
The definition of the transformation 8(k) implies that there exists a singular EI'(k)-manifold (V, G) with boundary (aV, Glay), such that V
U V(o),
V(v) 'b° y°V x P°,
°E2tk
Here we introduce the following notation:
aV = U SV(o). °E2lk
1.5. PROOF OF THEOREM 1.4.2
23
y(o,1)W
M(0,1) X P2
Figure 1.5: EF(k - 1)-manifold W.
bV(o) = U00 (-1)K;(°)QiM(ci, ... , ci-1)ci - 1, Ci+1, ...) x Pa, i=1 00
GIsv(°) = U (-1)K;(°)fi(cl, ... , ci-1, ci - 1, ci+1, ...) o pr i=1
for every element o = (cl, ... , en) ...) E stk. Let's consider the manifold
V1 = - U M(a) x Pa. aE21k-1
We glue it with the manifold V along the boundary aV = aV1 using the equalities
aiM(a) x P" = /3 M(a) x Pi x P" = (-1)"'(")pjM(a) x P°'i, where ai = (al, ... , ai-1, ai - 1, ai+l,...), if a = (al, ... , ai,...).
CHAPTER I. MANIFOLDS WITH SINGULARITIES
24
We define the Er(k - 1)-structure on the manifold W = V Ua -V1 by putting
W(a) = yaW x Pa for every a = (al, ... , a,,, ...) E %k-1, where
U yoV
yaW = -M(a) U
(61EC-1(ce) see Figure 1.5.
So we obtain the EI'(k - 1)-manifold (W, H), where
HIv = G,
HIM(a)xpa = F(a) o pr,
where the map
pr : M(a) x Pa --> M(a) is the projection on the direct factor. Also we have that
H(a) = HIW(a) = ha o pr o qa, where
ha:yaWX.
pr:yaW We have obtained
ir(k - 1) ([(W, H)]Er(k-1)) = ® [(yaW, ha)]E , aE%-1
by the definition of ir(k - 1). It is clear that the E-manifolds (yaW, ha), (M(a), F(a)) are bordant for every a E 2(k-1 according to the construction of the manifolds yaW.
ii) Im ir(k - 1) C Ker a(k). Let x E Ker ir(k - 1). Let us find the element y, such that ir(k - 1)(y) = x; here y = [(L, H)]:
L= U L(a), aE%A;-1
L(a)'°°yaL
x Pa,
HJL(a) = ha o pr o 0a,
1.5. PROOF OF THEOREM 1.4.2 where
25
pr: yaLxPayaL,
If the element x has the form
x = ® [(M(a), F(a))]E , aE'21k-1
then the next identity is true by the definition of the transformation
ir(k -1): [(M(a), F(a))]E = [(yaL, ha)]E
a E Qtk-1.
,
Now we take some E-bordism (V (a), G(a)) between the E-manifolds
(M(a), F(a)) and (-t,, L, ha) for every a E 2tk-1
Let us glue the cylinder -L x I with the manifold
W=- U V(a)xPa aE2ik-1
along the boundary by identifying the manifolds (see Figure 1.6)
-L(a) x {0}
and yaL x Pa
V(a) X P.
The boundary of the obtained manifold U is decomposed into the union of the next three manifolds: 00
U
OU = -Lx{1}U
U,OiV(a) x Pa x Pi
aE2Ik-1
i=1
U U M(a)xPa. aE2(k-1
We denote the second part of the boundary OW by S and consider it as a singular EF(k)-manifold (S, E) by putting 00 U(
ryas =
l-1)
Ki(O)
QiV(Cl....,ci-lI Ci -
Ci+1,...),
i=1
for every a = (c1, ... , c, ...) E %k, where the map E is defined by the maps G(a), G(a)Ia;v(a) : OjV(a)
(a) /3iV(a) x Pa
#iV(a) s`
X,
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
26
V(1,0)xP1
Figure 1.6: EF(k)-manifold U. as follows:
Ela;v(a)xp4 : 8OV(a) x P" P*. 81V(a) m'(a)°pf /;V(a)
X.
Let's note the next identity: 00
E I S(o) = U ((-1)K'(`)fi(ci,... , Ci-1, ci - 1, ci+1, ...) o Oi(a) o pr) . i=1
We obtain 8(k)(x) = 0 by the definition of the transformation 8(k). 2 EXACTNESS OF THE VERTEX
MG*r(k-1)
i) Ker 7r(k - 1) C Im y(k). Suppose the element [(L, G)] lies in Ker 7r(k - 1), where
L = U L(a),
x P",
aE2lk-1
GIL(a):L(a)-°+yaLxPa -f-r4yaL-9+ X,
1.5. PROOF OF THEOREM 1.4.2
27
and = 0 for all a E 2tk-1 We take a singular E-manifold (V(a), H(a)) for every a E 2tk-1 such that 5V(a) = y,,L, H1 6v(,,,) = g(a).
Then we glue the manifold -V(a) x P" to the top side of the cylinder L x I by identifying the manifolds yaL x P" and -5V(a) x P". We consider the obtained manifold W as the EF(k - 1)-bordism between the EF(k - 1)-manifolds L x {0} and M, where
M= U (91V(a)xPIU...U,6,,,,V(a)xPi)xP". aE2lk_1
In general the EF(k - 1)-manifold M is not the image of some EI'(k)manifold under the action of the transformation y(k). Let us consider the cylinder M x I and define the EF(k)-structure on it as follows. Let's introduce the notation a = (al, .... an, ...) E 2ik, ai = (al, .... ai-1, ai - 1, ai+l.... ) E 2tk-1. We put
(M X I) (a) =
U
(#iV(a) x Pi) X P" X I
a E 2tk-1
ai E -1(a) for every & E 2tk_l. We glue the cylinder M x I along the bottom side with the manifold W by identifying the manifolds M x {0} and M C W. As a result we obtain the next EF(k - 1)-manifold:
U = U U(a), aE2(k-1
where U(a) = W(a) U (M x I) (a); see Figure 1.7.
The correctness of the defined EF(k - 1)-structure on U may be verified directly. The last step is to note that the EF(k)-structure is well defined on the manifold M x I C 8U. That is, we put M(7) = QtV (bk(Q)) X P°
28
CHAPTER I. MANIFOLDS WITH SINGULARITIES
MxI
Figure 1.7: EI'(k - 1)-manifold U. for every o E 21k, where
t=min{ j
I
o=(cl,...,cn,...
c3>1 },
The above construction defines the singular EF(k -1)-manifold (U, F) having as first boundary (L, G) and as second boundary the singular EP(k -1)-manifold which lies in the image of the transformation y(k). ii) Im y(k) C Ker 7r(k).
Let x = [(M, F)] E Im y(k), where
M = U M(a),
M(a)'_°yaM X Pa,
aE%k_1
FlMlal : M(a) ±22 yaM x Pa
fi yaM f°, X.
Then there exists a EF(k - 1)-bordism (W, H), such that
W = U W(a), aE2lk_1
W(a)t="yaW X Pa,
1.5. PROOF OF THEOREM 1.4.2
29
and also 8W=MUL,HIM=F, FIw(a): W(a)
-a,,
Pa*,
yaW x
yaW °+ X,
where (L, HIL) is a singular EI'(k)-manifold:
L = U L(o), L(o)T°yoL x P. oE%k
We consider now the manifold (L, HIL) as a singular EI'(k-1)-manifold as follows according to the definition of the transformation y(k) . We
put
L(a) = U
L(o),
yaL =
U (7aL
x Pt x Pa) ,
oEt-1(a)
oE£-1(01)
f o r every a = (al, ... , an,, ...) E 21k_1, where
t = to = minj j 1aj > 1}. In particular the manifold (yaW, ha) is a singular E-manifold with the boundary b (yaW, ha) = l7aM, fa) This means that x E Ker ir(k - 1). 3 EXACTNESS OF THE VERTEX MG£(k)
i) Im 8(k) C Ker y(k). Let x E [(M, F)] E Im 8(k), where
M = U M(a),
M(a)t--,7,,M X PC"
aE2tk
FIM(a):M(a)
'+yaMxPapryaM*X.
There exists a EI'(k)-bordism (V, G) such that 8V = M U L, GIM = F according to the definition of the transformation 8(k), where the EI'(k)manifold L is defined by the singular E(k)-manifold {(S(a),
H(o))}OE2ik-1
30
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
S(0,1) X P2
S(1, 0) x P1
Figure 1.8: EF(k - 1)-manifold W. as follows:
L = U L(a), L(a)'-°yaL x Pa, aE2lk 00
7aL = U (-1)"t(a)/3=S(al, ... , ai_1, a; - 1, aj+1....
(1.10)
i=1
where a = (al, ... , an, ...) E %k. We glue the manifold V with the manifold
-
U S(o) x PoE'Ak_1
along L using the identity (1.10). The EF(k - 1)-structure on the manifold W is defined as follows:
-S(o,) x P° U
U V (a) 1(0
,
aW = U M(a); aEEk 1(0)
1.5. PROOF OF THEOREM 1.4.2
31
see Figure 1.8.
The above construction defines the singular EF(k - 1)-manifold (W, G), such that its boundary coincides with the EF(k - 1)-manifold (M, F). So we obtain that -y(k)(x) = 0. ii) Ker y(k) C lm ,9(k).
Let x = [(M, F)] E Ker y(k), where
M = U M(a),
M(a)t_"-^/,,M x PC"
aEak
FIM(a) : M(a) - yaM x Pa 14 yaM ° X. The definition of the transformation a(k) implies that the EF(k)manifold (V, H) has the boundary (OV, H1av) such that aV = M and HJav = F, and the identities
SV(a) = U M(a),
V(a)t°yCV x P°
a E %k-1Let's note that the disjoint union
(L, G) = U (OV(a), H!av(o)) oE2 A;-1
is a EF(k)-manifold whose bordism class lies in the image of 0(k). That
is, the identity
a(k) ®
ho)]E
_ [(L, G)]Er(k)
E2ik_3
is true. Let aV(a) = SV(a) U aV(o). We consider the disjoint union of the cylinders
U (aV(a) x I)) oE2(k_1
whose top sides are glued according to the identification between the parts of the boundaries OV(a), v E 2tk_1, of the EF(k - 1)-manifold
32
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
V. Indeed we glue the obtained manifold U to the cylinder L x I, by identifying the manifolds
5V(o)x{0}9 BV, and 5V(o)CLx{1} for every o E 2lk_1. The manifold W is a EI'(k)-bordism between the singular EI'(k)-manifolds M and L, as is verified directly. So we have proved that Ker y(k) C Im 8(k). The identity 7r(k)oa(k) = ,6(k) follows straight from the definitions. Theorem 1.4.2 is proved.
1.6
E-singularities spectral sequence
Let's construct the E-singularities spectral sequence for every locallyfinite sequence E of closed manifolds as above. We take a pair of spaces
(X, Y) and introduce the notations D°t = MGt (X, Y) , Di't = MGEr(') (X, Y) , Ei't = MGE(s) (X, Y) . The transformations ir(k), 8(k), y(k) induce the homomorphisms 7(s)
Di_l,t,
a(s)
Es+t,t+s
I''t : Di't
J''t :
Ds,t
K''t Ei't
8(s)
Di't-'-1.
So we have the exact couple D1
where the maps I*,*, J*'*, (0, -s - 1) respectively.
D1
K*,*
have the bidegrees (0, -1), (1, s),
1.6. E-SINGULARITIES SPECTRAL SEQUENCE
33
Definition 1.6.1 The spectral sequence associated with the exact couple (1.11)
will be called the E-singularities spectral sequence
(E-SSS).
Let's consider the filtration which induces this spectral sequence:
F; (X, Y) = IM (MGF(') (X, Y) ry(1)0
.ory(k),
MG. (X, Y))
MG.(X,Y) = F°(X,Y) D F1(X,Y) D ... D Fk(X,Y) D ... (1.12)
Lemma 1.6.2 The identity
flF;(X,Y)=0 k>1
is true for every finite dimensional CW-complex pair (X, Y).
Proof. We denote
fn = n Fn (A', 1') k>1
Then we fix n and prove that f,, = 0. The locally finite property of the sequence E implies that MGnr(k) = 0
for some sufficiently large number k. The finiteness of the dimension of the CW-pair (X, Y) means that H, (X, Y; Z) = 0 for some sufficiently large number j. The Atiyah-Hirzebruch spectral sequence
H* (X, Y; MG; r(k)) => MG* r(k) (X, y)
instantly gives that MGmr(k) (X, y) = 0 for m > max { k, j }. It follows that Fn = 0. According to Cartan and Eilenberg [31] it follows that the E-singularities spectral sequence converges. The differential d1 of the spectral sequence coincides with the transformation /3(n) : MG(n)(.) --) MGE(n+1)(.),
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
34
which is a sum of the Bockstein operators /3k, k = 1, 2, ..., according to Lemma 1.3.2. The other differentials dT: Es,* -- ET+T,*
are the highest Bockstein homology operations of the r-th order in the homology theory MGE (their algebraic definition was given by Maunder [62]; see also [81], [124]).
The following theorem is the final result of this section.
Theorem 1.6.3 The E-singularities spectral sequence has the properties:
(i) it is natural on the category of pairs of spaces; (ii) it converges when (X, Y) is a finite dimensional CW-pair; if
E = (P), where [P] = p, then it converges to the p-prime component o f the group MG*(X, Y); i f E = (Pl,... , Pk, ...) and dim Pk > 0, k = 1, 2, ..., then it converges to the group MG* (X, Y);
(iii) the differentials d, of E-SSS are prime Bockstein operators (for
r = 1) and higher order ones (for k > 2) in the theory MG; Now let's consider the case E = (P). We denote a projection of the element 0 = [P] into the term Ei'* by 0. It is clear that the element 0 has a degree (l, p+ 1), where p = dim P. We have the following isomorphism:
E" = MG'(s) (X, Y) = MGt+s(r+1) (X, Y) So the line E;'* is the one-dimensional MGF (X, Y)-module with the generator 0' (for s > 1). It is clear that the line E2'* coincides with the s-th homology group of the following complex:
MG; (X, Y) - MG* (X, Y) ->
-+ MG' (X, Y)
-
So we obtain that E2'* = Ker /3 and the line E2'* coincides with the factor Ker fl/Im Q
1.6. E-SINGULARITIES SPECTRAL SEQUENCE (b)
(a)
Y)
2
91Ker ,0/Im 0
91MGF (X, Y)
1
01Ker /3/Im /3
MG; (X, Y)
0
Ker3
92IMGr-(X,
35
Figure 1.9: Terms (a) El'*, (b) E2,*.
with the generator 0' for every s > 1; see Figure 1.9. A geometric interpretation of the spectral sequence is evident. That is, we resolve the singularities (see Figure 1.10, where we show the procedure for the element x = [M]E from the zero line).
Of course, the general case is much more complicated. Let E _ (P1,.. . , Pk, ...). We denote the element [Pk] as well as its projection into the term E l * ' * of the spectral sequence, by Ok for every k = 1, 2, .. .
Their degrees are equal to (1, pk + 1) where Pk = dimPk.
The line El" coincides with the MGE (X, Y)-module with the generators 01022...0,m,
for s > 1, where a, +... + am = s, aj > 0. The line Ei'* is the s-th homology group of the complex
MG; ill (X, Y)
Q(1).
MG£(2) (X, y) -
... - MG; (k) (X, Y)
0(k)+...
So a structure of the term E2'* depends on the action of the Bockstein operators ,0k on MG; (X, Y).
Figure 1.11 shows what happens with the element x = [M]E from the zero line (in the case when E = (P1, P2)).
36
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
N1B1XP1
Figure 1.10: Geometric meaning of E-SSS for one singularity.
Figure 1.11: Geometric sence of E-SSS for two singularities.
1.6. E-SINGULARITIES SPECTRAL SEQUENCE
37
Note 1.6.1 The exact triangles (1.9) induce the following diagram of representing spectra:
MG
MGEr(2) .
MGEr(1) a(2)
a(1)
-(0) \
MGE(1)
ry(3)
W(1)
p(1)
MGr(2)
The above diagram presents also the Adams resolution of the spectrum MG in the theory MG*(-) and the corresponding E-singularities spectral sequence coincides with the Adams-Novikov spectral sequence. We will deal with the subject in Chapter 3.
38
CHAPTER 1. MANIFOLDS WITH SINGULARITIES
Chapter 2
Product structures The E-singularities spectral sequence has provided us with the geometric procedure for restoring the bordism theory MG*(.) out of the bordism theory with singularities MG'(.). To convert the construction into a computation tool we need to have a natural multiplicative structure in this spectral sequence. Even in the simplest cases we can't have any hope of doing computations successfully without a product structure.
The spectral sequence has a product structure of this kind only when the corresponding bordism theory MGE(.) has some admissible multiplicative structure, i.e. a structure which is compatible with the ordinary product structure in the bordism theory MG*(-). If we have only one singularity then a product structure in the E-singularity spectral sequence can be determined by a direct geometric construction; see section 2.6. The problem becomes too complicated in the case of many singularities. Anyway we can do without direct construction. That is, we can identify the E-singularities spectral sequence with the corresponding Adams-Novikov spectral sequence and then make use of the resulting multiplicative properties. Both ways give us the desired product structure determined by a geometric product structure in the bordism theory with singularities. O.K.Mironov has described a geometric construction of the admissible product structure in bordism theories with singularities; see [67], 39
CHAPTER 2. PRODUCT STRUCTURES
40
[68]. We follow some of his ideas here as well as the present author's paper [16]; but as was mentioned above the most general case is not the subject of our consideration; rather we are going to dwell upon a restricted situation sufficient for our purposes.
The fact is that the construction does work well when the dimensions of the manifold-singularities are even or their bordism classes are of order two. Other restrictions besides those mentioned above are not assumed. We hope that the basic construction of the admissible product structure is rather simple and obvious.
2.1
Multiplicative structures
Here the problem of interest is when the bordism theory with singularities MGE(.) admits a product structure compatible with the ordinary one in the theory MG*(.). Let us recall some definitions and constructions.
Definition 2.1.1 The collection of the maps
µ(X,X,;Y,Y,) :h.(X,Xi)®h.(Y,YI)--->h,(X xY,XI xYUX xY1) is called an external product structure in the homology theory h,(.) if they are determined for all pairs of spaces (X, X1), (Y, Y1) and satisfy the following axioms.
10 The maps p(X,X,;Y,Y,) are natural with respect to the maps of the pairs of spaces.
20 The maps p(X,X,;YY,) are compatible with the boundary homomorphisms.
3° There exists a two-sided unit 1 E h(S°). These are minimal requirements which a product structure should satisfy. The most interesting is the case when the product structure y also satisfies the following axioms 4°, 51.
2.1. MULTIPLICATION STRUCTURES
41
41 Commutativity. The diagram h. (X) ® h. (Y)
" - h. (X x Y)
11
x*
h*(X) ® h*(Y) µ is commutative for all spaces X, Y
,
(2.1)
I
J
Y)
where
X:X xY---)X xY is the permutation of the factors, and I is the following canonical isomorphism:
I(x ®y) =
(-1)degxdegyy ®
X.
50 Associativity. The following diagram is commutative for all
X,Y,Z : h*(X x Y) 0 h*(Z)
h*(X) ® h*(Y) ® h*(Z)
I
h*(X) ®h*(Y x Z)
I
)
h*(X x Y x Z)
A direct product of the manifolds induces the external product structure in the bordism theories MG*(.) which satisfies the axioms 10 - 50. The bordism theory with singularities MGF(.) naturally possesses a module structure over the bordism theory MG*(.). This means that there exist the bordism theory pairings PR :
®MG*(.) --
PL :
0
)MG"(.).
Definition 2.1.2 The external product u in the bordism theory MG; is called an admissible product structure when it satisfies the axioms 1° - 30 and is compatible with the pairings PL, PR , i.e. the following diagram is commutative:
CHAPTER 2. PRODUCT STRUCTURES
42
®MG.(.)
'`R
' MGF-(.)
'u
)MGE(.)
1
MG(.) ®MG£(.) T
1
(2.3)
T
MGE(.)
As was already mentioned it is convenient to deal with manifolds whose cones over singularities are omitted, i.e. with E-manifolds. It is to be remembered that all constructions should be compatible with the projections on the models of E-manifolds. A geometric procedure to determine a product structure on the bordism theory may be divided into the following steps. Firstly, we construct the product M M. N for all E-manifolds M, N, i.e. a Emanifold which is bordant to the direct product M x N when one of the factors has empty set of singularities. Secondly we extend the construction up to singular E-manifolds and verify its compatibility with the boundary operator. The above procedure give us a well defined product structure on bordism classes. To formalize this technique we would like to apply a notion of canonical constructions which was introduced by O.K.Mironov [68].
Let us take the original bordism theory MG.(.) and denote the corresponding category of manifolds by J)t. As was mentioned above the sequence E = (P1, ... , Pk, ...) of the closed manifolds is supposed to be locally-finite. Let's consider the sequences Ek = (P1,. .. , Pk) for all k = 1, 2, ... too. Let 9Ak denote a corresponding category of Ek-manifolds.
Definition 2.1.3 A canonical n-linear construction 21 of dimension p is a rule which determines the E-manifold %(M1,..., Mn) of dimension Mn) of its boundary (ml + ... + Mn + p), the submanifold b2l(MJ,... , Mn) of dimension (m,+...+mn+p-1) and the continuous
2.1. MULTIPLICATION STRUCTURES
43
maps 7r(Mi, ... , Mn) : 2i(Mi, ... , Mn) --- A )E X... X (Mn)E
for every collection (M1,.. . , M,-1n) of E-manifolds. The rule should have the following properties:
1. Compatibility with the boundary operator. There exists the decomposition
U 2i(b,A ...,Mn))
b%A'...,Mn) = 93(Mi.... ,Mn)U
Ic{1,...,n}
where the map bl converts the collection (Mi, ... , Mn) into the collection (L1i ... , Ln), which is defined as follows:
_
M1
if
j
E I.
The decomposition has to satisfy the following identities: ' (M1,...,Mn) na(bI(Mi,...,Mn)) = 23(6l(Mi,...,Mn)) Mn)),
.,M)) = 23 (br(M1i ... , Mn))U (U
2z (b.l(Mi, ... , M.)))
.ICI
7r(Mi.... ,Mn)I2l(SI(M1i...,Mn)) = 7r (5I(M1,..., M'n))
2. Compatibility with the direct product on the manifold Lr without singularities. The following identities should be true:
%(M,,..., Mj-i, Mj x L, M;+i, ... , Mn) = (_1)1(mj+l+...+mn)2!(Mi,
M;-,,Mj
..., Mn) x L,
x L,M;+1,...,Mn) (_1)1(mj+1+...+mn)8(M1, ... , Mn) x L,
=
7r(Mi,...,M7-1,Mj x L,M,+1,...,Mn) = 7r(M1,...,Mn) x Id.
CHAPTER 2. PRODUCT STRUCTURES
44
3. Compatibility decomposition
with a gluing. If the manifold M has the
then the following identities should be satisfied:
2i(Ml, ... , Mjf, ... , Mn) U2(M,,...,M;....... Mn) 21(M1, ... , Mj ) ... , Mn),
Mi,...,M,) = (Ml,...,M',...,Mn) U% M
M,,
Mn)
(M,,...,M",...,Mn),
ir(M,,...,M;,...,Mn) = 7f(M1i...,A1,...,Mn)U.r(M1,...,M,,...... Mn)7r (M1,...,Mill,...,Mn).
A canonical construction a2I = (93, 0, 7r I%)
is called a boundary of the canonical construction %. The canonical construction 2i is closed if 93(M1,... , Mn) = 0 for every collection (Ml, ... , Mn) of E-manifolds. Simple examples of canonical constructions are the operator of mul-
tiplying by an ordinary manifold without singularities M -> M x N and the Bockstein operator /3k : M - /3kM. Canonical constructions have the following properties.
1) The result of gluing along the boundaries is also a canonical construction. 2) A composition of canonical constructions is also a canonical construction. The definitions mentioned above imply the following.
2.2. EXISTENCE OF PRODUCT STRUCTURE
45
Lemma 2.1.4 A closed n-linear canonical construction 2( of dimension p determines the following natural MG,-module homomorphisms: MGR (2() : MGm1(Xi) ®... ® MGmn (Xn) -- MG9 (Xl x
... x
Xn ),
which commute with the boundary operators for every collection (X1, ... , Xn)
of CW-complexes, where q = m1 + ... + mn. If the canonical construction 2( is a boundary of some other one then the homomorphisms MGE(2t) must be trivial for every collection (X1,.. . , Xn). So to determine the product structure in the bordism theory MG; () it is sufficient to obtain a corresponding canonical construction. The following lemma is also implied by the above definitions.
Lemma 2.1.5 Let 21 be a canonical bilinear construction of zero dimension on the category of E-manifolds. Then if 2((pt, pt) = pt it determines the admissible product structure in the theory
2.2 Suppose
Existence of product structure is the bordism theory where the sequence of the closed
manifolds E = (P1,. .. , Pk,...) is the locally-finite one as before. Let Ek = (P1, . . . , Pk) be the first part of the sequence for k = 1, 2.... and let the whole sequence E satisfy the following condition:
(*) 2[Pk] = 0 in the group MG*F'-1 if the dimension of the manifold Pk is odd. The condition (*) is sufficient for our purpose.
Let's consider the manifold Pk = Pkl) X Pk2) X I,
CHAPTER 2. PRODUCT STRUCTURES
46
where Pk1), Pk2) are copies of the manifold Pk, as a Ek-manifold in the following way:
akPk=aPk=NkPkXPk QkP = (-1)pk+1Pk1) x {0} U Pk2) x I,}, Pk = dimPk. Note that the Ek-manifold Pk is bordant to a manifold without singularities under the condition (*).
The Ek-manifold Pk is an obstruction to the existence of the admissible product structure µk in the bordism theory M be some closed Ek-manifold. The diagram (2.3) implies that /2k ([M] ®[Pk]) = 0.
The manifold Pk has no E;-singularities for every i < k, so the direct product /3kM x Pk has a well determined Ek-manifold structure.
Lemma 2.2.1 The identity [M X Pk]Ek
= [/3kM x Pk]E,
is true in the group MG£k for every closed Ek-manifold M.
Note 2.2.1 There is another Ek-structure on the manifold M x Pk which is induced by the Ek-structure on the manifold M. The projection of the manifold M x Pk onto its direct factor M restricted to the boundary
akMxPk=/kMXPk1)XPV) results in a Ek-structure on M x Pk
,
which doesn't coincide with the
above one.
Proof of Lemma 2.1.1. Let's consider the cylinder
MxPkx[0,2]=C as a Ek-manifold having the boundary (see Figure 2.1)
2.2. EXISTENCE OF PRODUCT STRUCTURE
47
2
/3kMxPk' 1
C
0
MxPkx{0}
Figure 2.1: Ek-manifold M x Pk x [0, 2] = C.
bC=MxPkx{0}U/3kMxp )xp(2)x[1,2] =MxPkx{0}U/3kMxPk. Also we have
a;C=a;MxPk x [0, 2], j
_ -M x Pk x {2} U (-1)Pk+1 (1kM X Pkl) x [0, 1]) x P,(2).
Let us consider the simplest example which illustrates the situation. We introduce the singularity E0 = (Po) into the bordism theory where [Po] = q. It is simple to verify that + 1)gl, [1 o]Eo - q(q 2
where the element 01 = [P1] is a generator of the group MSU1 = Z/2. Let's consider an SU-manifold W such that OW = 2P1 as a Eo-manifold
48
CHAPTER 2. PRODUCT STRUCTURES
by assuming that QW = P1. Lemma 2.1.1 implies that
q[W]E° - q(q2 1)91. In particular, we get q [W]
0 when q = 4k + 2, so an admissible product structure doesn't exist in the bordism theory Similar examples may be given for the symplectic and framed bordism theories.
Now let us formulate a sufficient condition for the existence of the admissible product structure in the bordism theory MG'k ().
Theorem 2.2.2 Let the equality [P']r; = 0
be true for every i = 1,.. . , k, where Qi are Ei-manifolds such that SQi = P'. Then there exists the admissible product structure pi = µi(Q1i ... , Qk) in the bordism theory MG*' () for every i which makes the following diagram commutative:
MG;'-1(.)
MG;'-1 -1® -1
MG;'
n
I
J®.
MGE'-'
MGr'
(2.5)
MG;'
Here MG;-(-) = MG*() and yo is the original product structure in the bordism theory MG*(-).
Proof. Every object of the category'931k is well determined up to a diffeomorphism preserving the G-structure. So we can suppose that every Ek-manifold M considered has a collar along its boundary OM.
Let's examine the case when k = 1. The direct product M x N of El-manifolds has the following decomposition of its boundary:
U(-1)m(Mx#INxp(2))
2.2. EXISTENCE OF PRODUCT STRUCTURE
(-1)'1
49
/31M x ,8,N x Q1
/31MxP11)xN
Figure 2.2: El-manifold ml(M, N).
We note that the manifold
/31MxPi1) x/31NxPi2) XI is diffeomorphic to the manifold
(-1)nr'/31M x /31N x Pi, which can be bounded by the manifold
(-1)'p'/31M x $1N x Q1. So the pairing m1 :9x1 X9A1 -->9XI
is well defined; see Figure 2.2. To finish the induction now suppose that the bilinear canonical construction m; is determined for every i = 1, ... , k - 1, such that its boundary is equal to the E;-manifold Sm= (M'n, Nn) _ mi (SM', Nn)
U m;(SMm,BNn)
mi (Mm, SNn).
CHAPTER 2. PRODUCT STRUCTURES
50
(-1)"pkmk-1(mk-1(/3kM,
/3k N), Qk)
(-1)mmk-1(M, akN) mk-1(akM, N)
Figure 2.3: Ek-manifold mk(M, N). Let's consider the Ek-manifolds Mm and N"; the boundary of the Ek_1manifold mk_1 (Mm, N") has the following decomposition:
Smk_1(Mm, N") = mk-1(SMm, N") U (Mk-1 (SMt, SN") X I) U(-1)mmk-1 (Mm, SN") U Mk-1 (akMm, N") U (Mk-1 (8kMm,
akN") x I) U (-1)mmk-1 (Mm,
akN")
.
Definition 2.1.3 implies that the bilinear canonical construction mk_1 satisfies the following equality: Mk-1
(Mm, akN")
xI=
(-1)"pkmk-1
(/3kMm,,0kN")
x P.
The bilinear canonical construction Mk is determined in the following way:
Mk (Mm, N") = Mk-1
(Mm,
N")
U (-1)"pkmk-1 (Mk-1 (/3kMm, /3kN") , Qk) ;
2.2. EXISTENCE OF PRODUCT STRUCTURE
51
see Figure 2.3. Verification of the properties 1-3 of Definition 2.1.3 may be done directly. We note also that Mk (Mm, N") = Mk-1 (Mm, N") for
all Ek_l-manifolds M, N. It is also clear that mk(pt, pt) = pt; so the canonical construction Mk determines the admissible product structure The diagram µk = pk(Q1) ... , Qk) in the bordism theory MGF-k (2.5) is commutative according to the above constructions.
The product properties of the Bockstein operators /3k, k = 1, 2, ..., are very important for our purposes. To describe them we should more carefully consider the manifolds Qk bounding the obstructions Pk. The manifold /3kQk is a Ek_1-manifold with the boundary 'I3kQk = (-1)Pk+lPk U P.
Let's define the structure of a Ek-manifold on /3kQk by putting ai(NkQk) =
if 8,(/3 Qk) (-1)Pk+1F 1) U Pk2) if
j # k, 1 j = k. J
We denote this Ek-manifold by /3kQk. The following formula may be proved by simple induction: /3kmk (Mm, N") = (-1)Pk"mk(QkMm, N")
(2.7)
U (-1)Pkn+n+mmk(mk(QkM,QkN),QkQk) U (-1)mmk(M,/3kN)
The corresponding product formula for Bockstein operators /3t is much more complicated when i < k. It depends upon the E;-singularities of the En-manifolds Qn for j < n.
Anyway there exists an important case when these formulas are quite simple. We denote by w= the element
- road E; for every i = 1, ... , k. The following lemma may be proved by simple induction.
CHAPTER 2. PRODUCT STRUCTURES
52
Lemma 2.2.3 Let the Ei-manifolds Qi be such that [QiQiJEk = 0
for all i
j. Then the formula
/3i (µk(x ® y)) = (-1)Pinpk($1x ® y) + (-l)mµk(x ®Qiy)
+(-1)Pin1k (1k(/ix ®Qiy) 0 wi) holds for every i = 1, 2.... where x E MG* k (X, X1), y E MG; k (Y, Yl ).
The dependence of the product structure Uk upon the Ei-manifolds Qi can also be examined for i = 1, .... k - 1. Let us suppose that two different Ei-manifolds Qk1), Qk2) bound the obstruction Pk. We define the element Ck = [-`w k1) U Qk2)] Ek
of the ring
The product structure in the theory MG£k-'
is denoted by l-Ik-1 = µk-1(Q1) ... , Qk-1) and two admissible product
structures in the bordism theory MG*Ek-' () by (1)
(1)
(2)
µk = /Uk(Q1, ... ) Qk-1, Qk ), /1k = Uk(Q1, ... , Qk-1,
Q(2)). k
Theorem 2.2.4 Every two elements
x E MG'k(X,XI),
y E MG;k(Y,Y1)
satisfy the following formula:
Y(kl)(x®y)- 42)(x®y) =
(-1)(deBY)Pkµk-1(/1k-1(/3kx®/3ky)®Ck). (2.9)
Proof may be done by direct comparing of the constructions.
Note 2.2.2 So we can see that the admissible product structure µk = µk(Q1, ... , Qk) can be corrected by some element of the group MG9 k where q = dim Qk = 2 dim Pk + 2. In particular if we take Qk2) _ Qk1) U S where S is some Ek-l-manifold then the product formula for the Bockstein operator /3k should be the same. Product properties of the operator /3i depend on the manifold S in the case i < k.
2.3. COMMUTATIVITY FOR THE CASE OF ONE SINGULARITY 53
Finally we would like to formulate one possible generalization of Theorem 2.1.2 (it is proved by simple induction [16]). There can be a situation when the obstructions [P.']E; are not zero in the groups MGEi f o r i = 1, ... , k - 1, and at the same time they are zero in the group MGEk as well as the obstruction [Pk]Ek, i.e. [P']E,F = 0 for i = 1, ... , k. Then an admissible product structure in the bordism theory MG*Ek may be constructed too.
Theorem 2.2.5 Let the sequence Ek = (P1, ... , Pk) be such that dim Pi # dim P for i i4 j. Then the admissible product structure Pk in the exists if the obstructions theory MG' A; [P1]Ek = 0 , ...
,
[Pk]Ek = 0
are zero.
For example, let us consider the bordism theory where E = (Po, Pi), [Po] = 2, [P,] = 01. We also denote Eo = (Po). As we have seen above, the bordism theory doesn't have an admissible product structure, while the theory MSU; does.
2.3
Commutativity for the case of one singularity
Here we consider the case when E = (P). Let [P']E = 0 and suppose the
E-manifold Q is such that 6Q = P, which determines the admissible product structure p in the bordism theory MGE(.). The involution
T:P'--->P on the manifold P is determined by the formula T(pi,p2,t) = (p2,p1,1 - t),
where (pl, p2i t) are coordinates in the product PM X P(2) X I. So we have the E-manifold
B = Q U -Q. P'=-rP'
CHAPTER 2. PRODUCT STRUCTURES
54
m(M,N)
Figure 2.4: E-manifold 'B(M, N).
Theorem 2.3.1 Every two elements x E MGI(X,XI), y E MG; (Y,Y1) satisfy the following equality: µ(x ® y)
-
(-l),mxdimYX*/-I(y
(p(/3(x)
_
® x)
/ ® [B]E) 0 #(y)),
(2.10)
where X: X x Y ---> Y x X is the factors' permutation.
Proof. Let Mm and Nn be E-manifolds and a = n(p + 1) + 1. Note that we have supposed the manifolds M and N have collars along their boundaries. So we have
m(M,N)=MxNU(-1)"/3Mx /3NxQ=0l,
(-1)m`m(N,M)=MxNU(-1)'7Mx/iNxQ=02i where in the first case
PMxfNxbQ=/3MxJNxP',
2.3. COMMUTATIVITY FOR THE CASE OF ONE SINGULARITY 55
and in the second case
PM xON xSQ=OM xON xrP'. Let us consider two cylinders 01 x I and 02 x I and glue them by the cylinder M x N x 13 as follows:
MxNx{1}=MxNx{0}C01x{O},
MxNx{0}=MxNx{1}C01x{1}. The manifold obtained is denoted by B(X, Y); see Figure 2.4. Its boundary is the manifold
xB1U(-1)"°"NxM, where B1 = Q(1)
U
U
P' X I
6Q=P'x{0}
-Q(2).
6Q=TP'x{1}
The E-manifold B1 is diffeomorphic to the E-manifold B. So the manifold 93(X, Y) determines the bilinear construction 'Xl. According to Lemma 2.1.1 the equality (2.8) holds.
In particular, we get that [B]E = 0 in the group MGE. So this is a sufficient condition for commutativity of the admissible product structure it. The following simple lemma will be very useful for applications.
Lemma 2.3.2 If the number p = dim P is odd then 2[B]E = 0.
The element b = [B]E depends on the choice of the manifold Q, bounding the E-manifold P'. Suppose Q('), Q(2) are two such manifolds, µ(Q(1)), µ(Q(2)) are the corresponding product structures, and b(1), b(2)
are the obstructions to their commutativity. We denote the element from the group MGE by c, which is determined by the E-manifold Q(1) U -Q(2). P1
CHAPTER 2. PRODUCT STRUCTURES
56
We have the next equality according to the definitions of the elements bc'),
b(2), c:
W>
-
b(2)
= 2c.
So the element
k(P) - b mod 2
(2.11)
of the group MGE ®Z/2 doesn't depend on the choice of the E-manifold Q bounding the obstruction P' and is determined only by the bordism class [P]. So we come to the conclusion:
Lemma 2.3.3 If the element k(P) E MGE ®Z/2 is not trivial then the admissible product structure µ(Q) is not commutative for every choice of the E-manifold Q. Now we consider the simple example when it is so. We choose the Then we obtain [P']E0 = 01 = 0 as above. Let us take the disk D2, bounding the circle S1, [S'] = 01. It is clear that there exists a Eo-manifold Q bounding the manifold P such that the corresponding E;-manifold B is bordant to the manifold CP1. So the formula (2.9) gives the equality
singularity Eo = (P), [P] = 2, in the bordism theory
µ(x ® y)
-
(-1)d,mXa;mlX*µ(y
0 x) = p(µ(a(x) ® NO ®[CP1] ).
Note 2.3.1 The obstructions [P']E, [B]E may be computed in terms of the Steenrod-tom Dick operations, as was proved by O.K.Mironov [67], [68].
2.4
Associativity for the case of one singularity
Let us determine the E-manifold whose cobordism class is the obstruc-
tion to associativity of the admissible product structure p(Q) in the bordism theory MG, (when the product structure is commutative)
2.4. ASSOCIATIVITY FOR THE CASE OF ONE SINGULARITY 57
(Qxp)(1)
Figure 2.5: E-manifold A.
as a E-manifold by setting SO = S(°)0 U S(1)0 U S(2)0
[2i + 1)ir (2i + 2)lrl S(i)0 = p(l) x p(2) x p(3) x 3
3
i = 0, 1, 2, J'
QD = (-1)pP(1) x p(2) [0, 3] U p(1) x p(3) x [, 7] 47r 57
U(-1)pP(2) x p(3) x13
)
3
Then we glue the manifold (_l)n(i+l)+1 (Q X P)(i)
to the E-manifold 0 along every boundary component b()0, for i 0,1,2, such that the manifold obtained O1 turns out to be a E-manifold, that is (see Figure 2.5), 6(°)A
= (-1)p (p(1) x p(2) x
137r
2r ' 3
]) x p(3) = (-1)n+16 (Q x p)(0),
58
CHAPTER 2. PRODUCT STRUCTURES
Figure 2.6: E-manifold r.
S(1)0 = p(l) x p(3) x [7, 3 J
x
p(2) = (-1)p+l 8 (Q X I')(1)
6(2)0 = (-1)p (p(2) x p(3) x [ 3 , 27r)) x p(1) = (-1)p+16 (Q x p)(2) . The diffeomorphism 0 : /. ---i Ol is defined in the following way. It is determined on the part 0 of the manifold Al by the formula b(pl, p2) p3, r, q) _ (p3, p1, p2, r,
+ 27),
where (r, 0) are the polar coordinates on the disk D2. Then we put 0(_l)p(=+1)+1 (Q x p)(=) = (_l)p(i+2)+1 (Q x P)(i+1). We are able now to construct the E-manifold r as follows (see Figure 2.6):
F = O1 x I/ (I(Ol x {0}) _ Al x {1}) .
2.4. ASSOCIATIVITY FOR THE CASE OF ONE SINGULARITY 59
Note 2.4.1 Let Ei = (Pi.... , Pi) for i = 1, ... , k; and let the manifolds Qi be Ei-manifolds such that bQi = P'. We note that the above construction determines the Ei-manifolds ri = IF(Pi) for every i =
1,,k.
Theorem 2.4.1 Let [P']E = 0 and Q be a E-manifold such that bQ = P determining the admissible product structure p(Q) in the bordism theory Then if [B]E = 0, [r], = 0, then the product structure p(Q) is commutative and associative. Proof. The product p is commutative by Theorem 2.3.1. We consider some E-manifolds Mn, Nn , Lr. To prove associativity of the product we use commutativity and compare the following E-manifolds: (M -
We are going to construct a corresponding bordism between them in three steps. Here we construct several manifolds required. Let's take the cylinder D2 x I with the coordinates (r, 0, t), where (r, 0) are the polar ones on the disk D2. Then we consider two curves yl, y2 on the side of the cylinder Sl x I which are determined by the equations 1.
7 71 :
27r
57r
3t+q- 3 =0,
72:
3
t+0-
3
=0
(where 0 < t < 1, 0 < _< 2ir), and the family of horizontal intervals between the curves yl and rye:
It=
13-3t<23 -I-23t 10
Then we choose the family of curves J,-, depending smoothly on r on the disk D2 x fl}, such that
i) J0=J,, ii)
J1={r=1,7r<0:5 3,t=1},
CHAPTER 2. PRODUCT STRUCTURES
60
0
2T 3
Figure 2.7: D2 x I.
iii) JT
JT, then 7-
1V) UO
T',
- D2
V) JT(0) = s + (1 - T)37r,
JT(1) = s + (1 - T)3 7r.
See Figure 2.7. Now we glue the E-manifold
(-1)PQ x P x [0, 2] to the cylinder
p(l) X P(2) x p(3) x D2 X I,
by identifying the following manifolds:
(P(3) x p(2) x II) X P(1)
(-')PQ xPx{'}=
(2.12)
(P(3) X P(2) X J2- t) x p(1)
fork <2.
2.4. ASSOCIATIVITY FOR THE CASE OF ONE SINGULARITY 61
Figure 2.8: E-manifold IIo.
The manifold obtained is denoted by Ho (see Figure 2.8) Finally we glue the manifold
QxPx[0,1]UQxi3QU(-1)pQxPx[2,3] to the E-manifold IIo, where
S#Q = p(l) U(-1)pPl2i,
Q X P x {1} = Q X P(l),
QxPx{2}=Qxp(2), as follows: SQ x #Q = (- 1)p/3(Q 1x P) x [1, 2], (P(l) X P(3) X [jr, 3 ] X P(2) X 177}, for 0 < 77 < 2,
SQ x P x {r7} _
(2.13)
(p(2) X P(3)
x [3, 2a x PM x {3 - 71},
for 2<,q <3.
62
CHAPTER 2. PRODUCT STRUCTURES
aMv$NxLxI
Figure 2.9: A boundary of M x N x L. The manifold obtained is denoted by H (see Figure 2.10(i)).
2. We suppose that the E-manifolds M, N, L have collars along their boundaries. (We consider them as ordinary manifolds with nonempty boundaries.) So the boundary of the manifold M x N x L may be decomposed as follows:
a(MxNxL)=OMxNxLUaMxaNx'1,2xL U(-1)mMxaNxLU (-1)mMxONxaLxI2,3 U(-1)m+'MxNxOLU(-1)m+"aMxNxaLx'1,3 UOMxONxOLxD2. A gluing scheme is shown in Figure 2.9.
We glue the manifolds
8MxfNxLxQ, Mx,ONx/LxQ, ,QMxNx,6LxQ
2.4. ASSOCIATIVITY FOR THE CASE OF ONE SINGULARITY 63
to the product M x N x L by identifying
/3Mx/3NxLxSQ, Mx/3Nx/3LxSQ, /3MxNx,QLxSQ with
aMxaNxLxI1,2, MxaNxaLxI2,3, aMxaNxLxI1,3 respectively. Here we suppose that SQ = p(l) x p(2) x I.
Note that the manifold obtained U is a E-manifold with boundary
/3MxON x/3LxA. The manifold (M N) L is obtained by the gluing of the manifolds U with the product
/3MxONx/3LxH along the manifold /3M x ON x /3L x A; see Figure 2.10. A similar procedure gives the manifold (_l)m(n+l)(L
N) M.
That is, we are to use a new manifold Hl (instead of the manifold 1I), which we construct as follows. We define the diffeomorphism
T: P(') x p(2) x p(3) x D2 X I --> p(l) x p(2) x p(3) x D2 X I by the equality T(p1,p2,p3,r,0,t) _ (W(p1ip2ip3ir,q5),t). The next gluing of the manifolds
(-1)PQxPx[2,3], QxPx[0,1]UQx/3QUQxPx[2,3] must be twisted by the diffeomorphism T.
CHAPTER 2. PRODUCT STRUCTURES
64
i)
ii)
Figure 2.10: A gluing of the E-manifold (M N) L.
2.4. ASSOCIATIVITY FOR THE CASE OF ONE SINGULARITY 65
As a result we obtain the diffeomorphism T1 : II1 --> II1 which coincides with the diffeomorphism T : A --> A under restriction to A. 3. We consider two cylinders
Let us glue them by the cylinder U X 13 :
U x {0}
U x {0} = (-1)m(n+l)(L N) M x 10}.
Then the manifolds
QM x ON x QL x II1 x {0} c ,3 ((-1)m(n+1)(N L) M x {0}) should be glued by the diffeomorphism which is determined by the formula
(x, y) z, 6) -+ (x, y, z, Ti()) The manifold obtained A(M, N, L) is a E-manifold with the boundary (M -
x #N x fL x
So if [r]£ = 0, then the 3-linear construction 2((M, N, L) is determined, its boundary equal to the manifold
6% (M, N, L) = (M N) L U (_1)m(n+l)(N L) M. To finish the proof we have to glue to 2((M, N, L) the bilinear construction 23(N L, M). Lemma 2.1.4 gives us the associativity of the product
µ(Q) Further we will use the following simple property of the manifolds
r.
CHAPTER 2. PRODUCT STRUCTURES
66
Lemma 2.4.2 The following equality is true in the group MGR: 3[r]E = 0.
Note 2.4.2 The admissible product structure may be associative without a commutativity condition. In particular it can be so when the obstruction manifolds P', B, and r are bordant to manifolds without singularities. For example, the noncommutative product in the bordism theory where Eo = (P), [P] = 2, is associative; see [67, 68].
2.5
General case
Let Ei = (F1,... , Pi) for i = 1,...J. We suppose as before that [P']Ei = 0. We take the Ei-manifolds Qi such that SQi = P,' which determine the admissible external product structure pi = /ii (Qi, ... , Qi) in the theory MG*' for every i = 1, ... , k. Note that we have already determined the following Ei-manifolds:
B = B(Pi),
r = r(Pi).
Theorem 2.5.1 Let l1k = µk(Q1, ... , Qk) be an admissible product structure in the bordism theory MGFk(.) If the obstructions [Bi]E. = 0,
[]Pi]E;
=0
are trivial for all i = 1, ... , k then the product structure µk in the bordism theory MG;k(.) is commutative and associative.
Proof. We begin by clarifying the scheme of the proof. We have the bilinear constructions and 3-linear construction 2ti on the category 9J1, for every i = 1, ... , k, which have the following properties: 1)
S'.Bi(M, N) = (mi(M, N) U (-1)mnmi(N, M)) U(93i(SM, N) U Zi(M, SN));
2.5. GENERAL CASE 2)
67
S2ti(M, N, L) = mi(mi(M, N), L) U mi(M, mi(N, L)) U 2(i(SM, N, L) U 2ti(M, SN, L) U 2ti(M, N, SL),
where the constructions
2ti(SM, N, L) and 2ti(M, SN, L) are glued along the construction 2ti(SM, SN, L), the constructions 2(i(M, SN, L),
2ti(M, N, SL)
along 2ti(M, SN, SL), and the constructions 2ti(SM, N, L),
2ti(M, N, SL)
along 2(i(SM, N, SL);
3) If M, N, L are E;-manifolds, where j < i, then 2ti(M, N, L) = 2(;(M, N, L),
Bi(M, N) = B;(M, N);
4) If one of the manifolds M, N has no singularities then
B1(M,N)=MxNxI; 5) If one of the manifold M, N, L (N for example) has no singularities then 2(i(M, N, L) = (-1)nlmi(M, L) x N x I; 6) The constructions 93o and 2to on the category931 are defined as
Bo(M,N)=MxNxI, 2to(M,N,L)=MxNxLxI. Let us describe an induction step for defining the construction Bi. Let MM and Nn be Ei-manifolds and put that a(n) = n(pi + 1) + 1, where pi = dim Pi . For simplicity we suppose that SM = 0, SN = 0. Let's apply the construction 93i_1 at the Ei-m_anifolds X, Y; the latter may be considered as the Et_1-manifolds X, Y as follows:
a;X=a;x, a;Y=a;Y, j = 1,...,i-1,
68
CHAPTER 2. PRODUCT STRUCTURES
sx=sxuaix, 6Y=6YuaxY Note that all our manifolds have collars. So we have mi(M, N) = mi-1(M, N) U (-1)a(n)mi-1(mi-1(QiM, 0iN), Qi) Consider two cylinders
C1 = mi(M, N) x Ii,
(-1)mnmi(N, M)
C2 =
x I2.
We glue the cylinders C1 and C2 with m;_1 (M, N) by identifying the manifolds mi_1(M, N) x {0} = m;_1 (M, N) y S'Bi_1(M, N), (_1)mnmi_1(M, N) x {1} =
(-1)mnm:_1(M, N)
S93a_1(N, M).
The construction obtained is denoted by SB(I)(M, N). Note that the boundary of the construction i_1 (M, N) is equal to a-1 (M, N) = ma-1(M, N) U (_1)mnmi_1(N, M) U93i_1(aiM, N) U 93i_1(8 M, AN) x I U 93i_1 (M, aiN)
= mi_1(M, N) U
(-1)mnma_1(N, M)
U `Bi-1(fliM, N) x Pt(1)
USBi-1(61M,/3iN) x P' U 9Bi_1(M,/3iN) X Pi(2).
Then we glue the above construction to T(' )(M, N): ma-1(`Bi-1(/3M,QiN),SQi) _Ti-1(/31M,/.31N) x P' C S'B(l)(M,N); -1(93,_1 (#,M,
/32N), Qi)
_ (-1) "(n)mi-1(`Bi-1(pM, QiN), Qi) x {0} C mi(M, N) x {0} C C1. The resulting bilinear construction is denoted by 'B(2)(M, N); see Figure 2.11. The manifold 'B(2)(M, N) may be considered as a Ei-manifold with the boundary S93(2)(M, N) = mi-1(M, N)U(-1)mnmi_1(N, M)Umi(mi(/9 M, /3 N), Bi),
2.5. GENERAL CASE
69
N,M)xI Qt)
Figure 2.11: Construction 93i (M, N).
where Bi = B(P2) is the corresponding obstruction to commutativity. We take the Ei-manifold Di which bounded the obstruction Bi and then glue the construction mi(mi(31M,,81N), Di)
to 93(2) (M, N). The result is the bilinear construction 93i (M, N), possessing all the above properties. The construction 2(i(M, N, L) may be determined in a similar way using the proof of Theorem 2.4.1.
Note 2.5.1 1) Theorem 2.5.1 provides the sufficient conditions for commutativity and associativity of the admissible product structure. In several cases it is sufficient to verify that the groups MCEk
2pk+2
MGFk 3pk+3
haven't torsion of order two and three respectively.
70
CHAPTER 2. PRODUCT STRUCTURES
2) There exists the problem of describing the operation algebra of a (co)-bordism theory with singularities. The solution seems to be not so simple. Only a handful of examples of solving the problem are known. The results were obtained by purely algebraic methods. In particular, the operation algebras k(n)*(k(n)),
P(n)*(P(n))
have been computed completely for all p and n; see [51], [120], [121]. It would be very interesting to investigate the connection between the operation algebras of the cobordism theories MG*(.) and MG*(.). 3) It is natural to consider bordism theories with deep singularities, i.e. where the manifold-singularities themselves have singularities of the previous level. (K.Nowinsky [78] has given a rather general description of such theories.) It is not difficult to give corresponding definitions and examples; one of them will be considered in Chapter /. The multiplica-
tivity problem in these theories is much more complicated, moreover there are some obstructions to equipping such a theory with deep singularities with module structure over the original bordism MGE(.) theory with the usual singularities. The above obstructions are certainly trivial if a ring structure of the spectrum MGE can be extended up to a Ham-structure. To investigate the problem seems to be interesting. Note also that all the bordism theories from the examples 1-4 (section 1.2) do have commutative and associative admissible product structures. The same is true for the theory P(0)* for every prime p and for the theories for p > 2, n > 1. Nevertheless the theory for p = 2, n > 1 doesn't admit the commutative product structure, there exists only associative product structure; see [67], [68].
2.6
Product structure in the E-SSS
The above geometric construction of a product structure in a bordism theory with singularities may be extended up to a product structure in the corresponding E-singularities spectral sequence. Actually we
2.6. PRODUCT STRUCTURE IN THE E-SSS
71
can do without the direct construction because the product structure in the spectral sequence appears after identifying it with the AdamsNovikov spectral sequence. This product structure is also generated by an admissible one in the bordism theory with singularities. Our plan is the following. First we intend to formulate the general problem of constructing the product structure in the E-SSS and then we are going to explicitly consider the case of one singularity. The EI'(k)-manifolds and the bordism theory of these manifolds as well as the transformations
MG.(.) ") MGFr(')(.) _ ... E- MGEr'(k-1)(.) "E
'4
k)
...
were determined in section 1.4.
The transformations y(k) were determined at the level of the man-
ifolds, in particular every EP(k)-manifold M has the structure of a EI'(n)-manifold for every n < k. Two EF(n)- and EI'(k)-structures on the given manifold M (for n < k) are compatible if the EF(n)-structure induced by the EF(k)structure coincides with the first one.
Let p < q be integers.
Definition 2.6.1 The manifold M is called a EF(p, q)-manifold if (i) it has the EF(p)-structure, (ii) its boundary aM has the partition am = MOW U a(1)M,
such that a(a(°)M) = -a(0(1)M), also the manifold a(1)M has a EF(q)structure which is compatible with the original EF(p)-structure.
The manifold a(°)M possesses the induced EI'(p, q)-structure by definition and is called the boundary of the EI'(p, q)-manifold M. Note that if p > p', q > q', then the EF(p, q)-manifold M has the structure of a EF(p', q')-manifold. The notion of bordism of such manifolds may be determined in a standard way. Thus we have the corresponding bordism theory
MGRr(n'e)
()-
CHAPTER 2. PRODUCT STRUCTURES
72
Note 2.6.1 These bordism theories are natural generalizations of the bordism theories with ordinary singularities. In particular, it is clear that and the theory MG*r(°'p)(.) coincides (where EP is the sequence of the pwith the bordism theory fold products of the manifolds Pi, i = 1, 2, ...). The theory MGF may be considered as the bordism theory of Er(p)-manifolds with Eqsingularities.
The following transformation is defined for all numbers p > p',
q>q': ,q
MGEr(p',q')(.).
: MGEr(p,9)(.)
(2.14)
It forgets EF(p, q)-structure up to EF(p', q')-structure. By taking the singularity manifold a(1)M of the given EF(p, q)-manifold M we get the transformation d:
MGEr(p,q)(.)
-+ MGEr(q)(.).
To formulate the multiplicativity problems in the E-SSS we are going to use the terminology of spectral systems (see [38]). For this we determine the bordism theory pairings 01
MGEr(p,p+r) (. ) ®MGEr(9,9+r)(,)
,
MGEr(p+q,p+9+r)(.)
for all p, q, r, which have to be compatible with the transformations 77 and d. The general case of many singularities leads us to a very complicated construction. Here we prefer to restrict attention to the case when E = (P); we shall further assume either that the number k = dim P is even, or that 2[P] = 0 in the group MG*. We have the isomorphism MGnr(p,p+r) (X, y)
MGnr kp (X, Y)
where Er = (Pr). Here it is convenient to decompose a EF(p, p + r)manifold into the direct product M x Pp, where M is Er-manifold.
If p > p', r > r', then the transformation 71:
MGEr(p,p+r) (.)
-1 MG* r(p'p'+r) (. )
2.6. PRODUCT STRUCTURE IN THE E-SSS
73
is induced by the map which translates the direct product M x PP (where M is a Er-manifold) to the direct product M x PP' (where the manifold M should be considered as a Erg-manifold).
The transformation b : MGnr(p,p+r) (.) -) MGEr(p+r,p+r+r')
is induced by the map which translates the direct product M x PP (where M is a Er-manifold) to the direct product /3(r)M x Pp+r,
where /3(r)M is considered as a Erg-manifold.
Now we are going to construct the pairings Or : MGm- kP(X) ® MGnrkq(Xl) -- MGm+m_k(p+9)(X AX1),
which can be determined when the bordism theories MG;" admissible product structures.
possess
It is the bordism class of the Er-manifold
(Pr)' = (Pr)(1) X (Pr)(2) X I that is the obstruction to the existence of the admissible product structure in the theory
Lemma 2.6.2 If [P']E = 0 in the group MG£ then the element [(Pr)']E is also zero in the group MGE for every r = 1, 2,.... Proof. According to the above definitions we have
(Pr)' = p(l) x ... X P(r) x p(r+l) x ... X P(2r) X I. Let us choose the partition of the interval I = [0, 1] into r parts,
CHAPTER 2. PRODUCT STRUCTURES
74
(_1)k(r-1)p(1) x ... X p(=) x ... X p(r+i) x ... X p(2r) x Q(i)
[0,
p(1) x ... X p(2r) x I x [0,1] Figure 2.12: Q(r).
Now let's take some E-manifold Q , such that SQ = P' = p(l) x p(2) x I. Then we take r copies of this manifold denoting them by Q(1), ... , Q(r) respectively. We suppose that SQ(i) = p(`) X p(r+i)
Ii
X
r
1, rI .
Then the Er-manifold r
Q(r)
= U(-1)k(r-1)P(1) x ... X P(i) x ... X p(i+r) x ... X p(2r) x Q(i) i=1
bounds the obstruction S(r)Q(r) = (pr)'. 0
Theorem 2.2.2 implies that the bordism theory MG*r(.) has the admissible product structure 0r. Let us examine the Er-manifold Q(r) more carefully.
2.6. PRODUCT STRUCTURE IN THE E-SSS
Lemma 2.6.3 If [P']E = 0 in the group MG' then the
75
element
/j(r)Q(') is also zero in the group MG; F-' where Q(r) is the above '-manifold.
Proof. Let's construct the manifold which is diffeomorphic to the manifold Q('). We consider the partition of the interval I = [0, 1] into
2r - 1 parts,
2r-1
f I= U L2r-1'2r-1l J
and take the cylinder
P(1)x...xP(r)xP(r+l)x...xp(2r)xIx[0,1]. We glue to its top side the manifolds (-1)k(r-1)P(1) x ... X P(') x ... X P(i+r) x ... X P(2r) X Q(=)> SQ(t) = P(1) x
...
X P(2r) x
[2(i_1) 2i
11
2r-1 2r-1
(where i = 1, . . . , r), by identifying the manifolds
(-l)k(r-1)P(1) x
... X P(t) x ...
X P('+r) x
... X P(2r) X 6Q(i)
[2(i - 1) 2i - 11 = P(1) x ... X P(2r) x x 10}.
2r-1 2r-11
We denote the resulting Er-manifold by Q('); see Figure 2.12. It is clear
that Q(r) is diffeomorphic to Q(r). Now we consider the E'-manifold /9(r)Q(r) According to the above constructions we have r-1
(r)Q(r)
= U (-1)k(r-1) (P(1) x ... x P(i-1) x ... i=1
X P(`+r) x U
(U -1
...
(-1)k(r+1)(i-1)p(1) X... x P(=-1) x ...
2i - 1
X P(2r) x 12r
2i
-1'2r-1 J I J
x P(`+r) x ... x P(2r) X QQ(' J
76
CHAPTER 2. PRODUCT STRUCTURES
Q(k)Q(r) X J
p(l) x ... X P(i-2) x p(r+i+i) x ... X p(2r) x Q(i) Figure 2.13: /3(r)Q(r) x J
where P(°) is the point. Let us take the cylinder g(r)Q(r)
x J, j = [0, 1],
and glue the following manifolds to its bottom side:
p(l) x ... X P(i-2) x p(r+i+l) x ... X p(2r) X Q(i) have where i = 2, ... , r, and P(2r+1) is the point. Then we proceed to the following identification: P(l) x ... X p(i-2) X P('+r+l) X ... p(2r) x
No
= P(l) x ... X P(i-l) X P(i+r) x ... X p(2r) x
[2i_32i_2]
2r-12r-
1
using the fact that the direct product
{2i_3 2i - 2l P(l) x ... X P(s-1) X P(i+r) x ... X P(2r) x
2r - 1'2r - 11
'
2.6. PRODUCT STRUCTURE IN THE E-SSS
77
is a part of fl (r)Q(r). The manifold obtained is to be a E'-1-manifold with the boundary /j(r)Q(r); see Figure 2.13. Let us formulate Douady's conditions for the pairings 0, in terms of the new geometric interpretation (see [38]). These conditions guarantee the existence of the product structure in the E-SSS. 1.
If p > p', q > q', r > r', then the following diagram must be
commutative:
MGmrkp (X) ® MGnr kq (Y)
Or
MGm+n-k(P+q) (X A Y) (2.15)
MGEr, kP'(X) 0 MGn'Er' kq'(Y)
MG.I+n'_k(P'+q')(X A Y) '3r
where m' = m + k(p' - p), n'=n+k(p'-p). 2. All the homomorphisms of the diagram
MGEr kP (X) ® s®n
(Y)
MGmr kP (X)
MGm+n-k(P+9) (X A Y)
MGm' kP (X) 0 MGnrkq (Y)
o MGn' kq (Y)
m
(2.16)
MGM+n_k(P+9)-1 (X A Y)
satisfy the equality
60 Or = 01 0 (6®rl)+01 0 (7I ®
for all numbers p, q > 0 and r > 1 and spaces X, Y. Let us verify the first condition. Suppose M, N are Er-manifolds. According to the definitions of the transformations q, Or we have q o Or ([M] ® [N]) = [M N Or'(11® rl) ([M] ® [N]) = [(M x
X PP-P')
PP+9-P'-9'],
. (N x P9-°')],
CHAPTER 2. PRODUCT STRUCTURES
78
where the point means the product in the theories MGEr(.) and MGEr/ (.). The module structure of these theories over the theory MG.(-) implies commutativity of (2.15).
Let us verify the second condition. According to the definition of the transformations b, 77 we get b o cr ([M] ® [N]) = il[Qlr)(M ' N)].
Note that 10(r)(M ' N] Er =
[p(r)M N] [M. #(r)N] Er ' Er +
+(_l)k(m-k)
[(f(*)M '
,Q(r) '
N) ' $(r)Q(r)] Er
.
Lemma 2.6.3 implies that [$(r)Q(r)] Er = 0.
So we have the equality 't [ (r)(M '
N)] Er/ _
"t
( [(#(") M) ' N]
Er) +
"t
[M ' (#(*)N)]E
_ ((4i o (S®'l)+ (0i o (77 ®5))([M] ® [N]). Thus we have proved the following.
Theorem 2.6.4 Let the manifold P have even dimension or the element [P] have order two in the group MG., and the obstruction [P']E E MGF- to the existence of the admissible product structure in the bordism theory E = (P), be trivial. Then the E-singularities spectral sequence possesses a multiplicative structure. The differential dr satisfies the following formula for every r > 1:
dr(a, b) = dr(a) b+ (_1)degaa
d,. (b).
Note 2.6.2 1) The differential dl, which coincides here with the Bockstein operator
0 : MGE(.) --- MG£(.),
2.6. PRODUCT STRUCTURE IN THE E-SSS
79
satisfies the formula
di(a,b) _
di(a),b+(_1)dega[
.((di(a).di(b)).[QQ])+(_1)degaa,di(b),
where a division by [P] is justified by the structure of the algebra El*,*; see section 1.6.
2) If the admissible product structure in the bordism theory with singularities is commutative and associative then so is the product structure in the E-singularities spectral sequence.
Chapter 3 The Adams-Novikov spectral sequence Now it is high time to dwell upon the Adams-Novikov spectral sequence.
The first thing is to define all the features of the Adams-Novikov spectral sequence. Though the homology version is much more popular and has a nice algebraic description (see [4], [84], [108]) we are going to deal with the cohomology version of the Adams-Novikov spectral sequence, as it is very convenient for our purposes. Besides, we make use of the Novikov algebraic spectral sequence, which converges to the algebra
Ext", (BP*(X), BP*) . The latter is the second term E2'* of the Adams-Novikov spectral sequence. In addition we touch upon the results obtained by V.Vershinin and V.Gorbunov [43], [109]-[113] concerning the structure of these spectral sequences for the symplectic cobordism case.
The second thing is to apply the Adams-Novikov spectral sequence to the cobordism theory of symplectic manifolds with singularities. The purpose here is to present some of Vershinin's results [111] (Theorems 3.3.3 and 3.3.5) pertaining to a subject of interest. 81
82 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE The last thing is to interpret the Adams-Novikov spectral sequence for the symplectic cobordism ring from a geometric point of view. That is, we identify the above spectral sequence with the E-singularities spectral sequence (Theorem 3.4.1, Corollary 3.5.4). Then we formulate algebraic results concerning the Adams-Novikov spectral sequence in the geometric terms of manifolds with singularities (Corollary 3.5.5).
And finally we come to the geometric structure of the AdamsNovikov spectral sequence for the symplectic cobordism ring. This geometric description gives us the possibility of using the computations of the spectral sequences for investigating some geometric properties of the Bockstein operators so as to compute the first Adams-Novikov differential.
3.1
Basic definitions
There are certain constructions related to the Adams-Novikov spectral sequence. We deal with the cohomology version of the constructions, as they are much more convenient for our purposes, while the most general case is out of the scope of our interest. Anyway all the necessary cases will be considered. Now some notes. We mean by spectra the objects of stable Boardman category [4] or of some version of it; see [10], [108]. Ring spectra are also required; see [4], [108]. The classifying spectrum of a multiplicative cohomology theory serves as an example of a ring spectrum.
Let h be the ring spectrum which classifies the multiplicative cohomology theory h*(.). The corresponding Steenrod algebra of operations in the cohomology theory h*(.) is denoted by Ah = h*(h).
We suppose that the algebra Ah is provided with the structure of a Hopf algebra in the standard manner; see [4], [108].
Note 3.1.1 In the cohomology case the natural skeletal filtration provides the algebra Ah and the Ah-modules h*(X) with a topology. So the cohomology theory h*(.) transforms the category of spectra to the category of topologized modules over the topological algebra Ah . We
3.1. BASIC DEFINITIONS
83
are not going to go into details here; for information on the homological algebra, refer to [4], [10], [108].
Now let us describe the spectral sequence which converges to the graded set [X, Y]* of the homotopy classes of the maps between the (-1)-connected spectra. The category of spectra is quite suitable for constructing such spectral sequences. Let us consider the following exact couple in this category:
X=Xo
it
jp
Zl
4\/
Xl __ ... y Xn-1
In
Xn .- .. . / kn
jn_1
Zn
where the maps iq, jq have zero degree and the map kq has degree (-1). Applying the functor [Y, -]* to this diagram we obtain the exact couple which induces the spectral sequence converging to [X, Y]* (under some
restrictions). We need some additional assumptions on the diagram (3.1) to efficiently describe this spectral sequence in terms of the given cohomology theory h*(.).
Definition 3.1.1 The diagram (3.1) is called the Adams resolution of the spectrum X in the cohomology theory h*(.), if (i) the homomorphisms h*(iq) are trivial for all q = 1, 2, ..., (ii) the induced complex h*(X) E
h*(Z1) E
... <
h*(Zn) (- h*(Zn+1) E
...
is the projective resolution of the Ah-module h*(h) (in the corresponding category above) where e = h*(jo), dq = h*(kq 0 jq), q = 1, 2,....
Note 3.1.2 The condition (ii) is always true in the case when the spectra Zn are the wedges of the spectra E8ih, where the sequence {sj} has no finite condensation points.
84 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
Definition 3.1.2 The spectral sequence which is induced by the exact couple [Y, X*]*
t.
[Y, X*]*
[Y,Z*]*
is called the Adams-Novikov spectral sequence for the spectrum X in the cohomology theory h* () when the diagram (3.1) is the Adams resolution.
When it is clear what spectrum and cohomology theory are meant, the above spectral sequence is called the Adams-Novikov spectral sequence (ANSS).
According to Definition 3.1.1 we have the following isomorphism:
Ei'*
[Y, Z*].
HomAh(h*((Z*), h*(Y))
The term E2coincides with the homology algebra of the complex
Hom*h(h*(Zl),h*(Y)) +(-dl). Hom*h(h*(ZI),h*(Y)) E1d2)*
...
So we obtain the isomorphism E2,* ^' Ext*,*(h*(X), h*(Y)).
(3.2)
Note 3.1.3 1) The problems of existence of the Adams resolution and of convergence of the ANSS are not so simple; see [4], [10], [38], [69], [77]. When possible we are not going to touch upon them as they do have solutions in all necessary cases. 2) The Adams resolution may be obtained in different ways, while the corresponding spectral sequences are isomorphic beginning with the second terms E2'*.
3.1. BASIC DEFINITIONS
85
3) Let the spectrum X be a ring spectrum and Y = S be the sphere spectrum; then the set [S, X]* = zr*X has the structure of a commutative ring. This ring structure is also recoverable from the AdamsNovikov spectral sequence (under some assumptions about the spectrum X) when the cohomology theory h*(.) is multiplicative. Moreover the Adams-Novikov spectral sequence is a spectral sequence of commutative
and associative algebras, and the product in the term E2,* coincides with the algebraic one which is determined for the functor ExtAh(-, h*);
see [63], [113]. We intend to use the above multiplication structure in the cases to be considered. The most popular cohomology theories used for the Adams-Novikov spectral sequence are the complex cobordism theory MU*(.) and its p-
local version, the Brown-Peterson theory BP*(.). Now let us take a fixed prime p and consider the case h*(.) = BP*(.) explicitly. We restrict to the case when Y = S, so that [S, X]* = 7r*X are the homotopy groups of the spectrum X . Besides we suppose that the following condition is true:
(* *) The spectrum X is a (-1)-connected spectrum such that for every m the group Hm(X; Z(p)) is finitely generated and torsion free.
Theorem 3.1.3 [10], [69], [77], [84], [113] Let the spectrum X satisfy the condition (* *); then the Adams-Novikov spectral sequence for the spectrum X in the theory BP*(-) converges to the groups r*X ® Z(p). If the spectrum X is the ring spectrum then the Adams-Novikov spectral sequence possesses a multiplication structure. O
We have seen already that the second term of the classic AdamsNovikov spectral sequence has the form E2'* ?' ExtABP (BP*(X ), BP*).
(3.3)
The computation of this algebra may be rather difficult for many spectra X, because in many cases it is not possible to efficiently describe
the structure of the ABP-module BP*(X).
86 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE Anyway there exist several constructions which allow us to compute the algebra (3.2). One of them is the Novikov algebraic spectral sequence and its generalization proposed by V.Vershinin [111].
Suppose Vk, k = 112, .., are standard Hazevinkel generators of the ring BP*: BP* = Z(p) [vl, ... , vn, ...] .
Let m = (p, vl, ... , vn, ...) be the maximal prime ideal of the ring BP*; we consider the following multiplicative filtration of the ring BP*:
BP*=FoDF1J...DFnD....
(3.4)
where Fl = m, p E Fk (under given k > 1) and v;
F2 for every i = 1, 2, .... We denote the bigraded ring associated to the ring BP* with respect to the filtration (3.4) by BP*. Let 91 be a projective resolution of the ABP-module BP*(X):
BP*(X) i---- Rl E-- ...
(
Rn (
Rn+l F
...
The filtration (3.4) induces the filtration of the complex (E = HomA*BP (9R, BP*)
as follows: Fn = Hom* BP (91, Fn).
(3.5)
As usual the filtration (3.5) determines some spectral sequence which converges to the algebra (3.3) (under some assumptions).
Definition 3.1.4 The spectral sequence induced by the filtration (3.5) is called the Novikov algebraic spectral sequence (NaSS) for the spectrum
X. It isn't difficult to determine the first term of this spectral sequence. Note that BP*/Fl ^_' Z/p. Let Ap denote the ordinary mod p Steenrod algebra. Let Ap = Ap/(Qo) be its quotient algebra by the two-sided ideal generated by the Bockstein operator Qo E Ap. The next isomorphism is well known (see [231):
3.1. BASIC DEFINITIONS
ABP/ (m . ABP)
87
= A.
So the algebra A' acts naturally on the ring BP*. It may be verified directly that the graded differential module
? = 1:(Fn/Fn_i) n>O
is isomorphic to the module HomABPt(m.ABP)(9i/(m 9t), BP*).
Note the groups H*(X; Z(p)) are torsion free under condition (* *). In particular this means that the cohomology H*(X; Z(p)) possesses Apmodule structure. The factorization by the ideal m turns the projective resolution 91 of the ABP-module BP*(X) into a projective resolution of the A'-module9 1/(m 91): H*(X; Z(r))
Rl/(m . R1)
... E---- Rn/(m . Rn) (.. .
According to the above definitions the homology of this complex coincides with the first term of the Novikov algebraic spectral sequence. We obtain the isomorphism of trigraded algebras: El*'*'*
ExtA,(H*(X; Z/(p)), BP*).
(3.6)
One of the statements of the next theorem is thus proved. Theorem 3.1.5 [109], [111]-[113] The Novikov algebraic spectral sequence is natural on the category of spectra. If the spectrum X satisfies the condition (**), then the spectral sequence converges to the algebra E2,*
ExtABP(BP*(X), BP*),
and its initial term is isomorphic to the algebra (3.6). If the spectrum X is the ring spectrum then the Novikov algebraic spectral sequence is the multiplicative one. 0
88 CHAPTER S. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
Note 3.1.4 The classic Novikov algebraic spectral sequence [77] can be obtained by putting k = 1 and Fn = m'. We emphasize the case when k = 2 with the corresponding spectral sequence being called the modified Novikov algebraic spectral sequence (after V. Vershinin [111]). Then the abbreviations are maSS for the spectral sequence and maSSfiltration for the generating filtration.
The modified algebraic spectral sequence is a point of great interest. In this case the bigraded ring BP* is isomorphic to the following polynomial ring: Z/p [h0i hl, ... , hn,...]
where deg h0 = (2, 0), deg hn = (-1, -2(pn - 1)). The algebra A' acts on the ring BP* as follows: Pp'hn
_
h -1
- { 0n
j=n-1>0, otherwise,
where P' are the Steenrod powers which generate the algebra A,. The formula (3.7) follows from the well known action of the Quillen algebra ABP; see [109].
Note that the maSS-filtration of the ring BP* ®Z/p is induced, the filtration being generated by the powers of the ideal m = (vi.... , v,,. .. The corresponding adjoined ring (Zlp) [hl, ... , hn, ...]
,
is denoted by BP*/p. It is also the A,-module. The isomorphism
Ext *(H*(X; Z/p), BP*) Ext*,,*(H*(X; Zlp),BP*lp) ® (Z/p)[ho] P
(3.8)
is based on the formula (3.7). Now we are going to apply the above constructions for the symplectic cobordism case.
3.2. THE MODIFIED ALGEBRAIC SPECTRAL SEQUENCE
3.2
89
The modified algebraic spectral sequence
The symplectic cobordism ring MSp* is known to have no p-primary torsion for the odd primes. We will consider the modified algebraic spectral sequence for p = 2. According to Theorem 3.1.5 its initial term is isomorphic to the algebra ExtA2(H*(MSp; Z/2), BP*/2) 0 (Z/2)[ho].
(3.9)
It isn't difficult to compute the algebra (3.9); to do this we have to recall some facts on the structure of the module H*(MSp; Z/2) over the Steenrod algebra A2. We consider the images of the elements Sq2°i E A2 in the quotient algebra A' = A2/(Sql) which will be denoted by Sq2°i, where
Ai = (0
1011,01 ...)7
and the elements Sq°i are from the Milnor basis of the Steenrod algebra A2. Let 13 be the subalgebra of the algebra A' which is generated by the elements Sg2°i, i = 1, 2, .... Note the following properties: (i) the algebra 13 is a normal subalgebra of the algebra A'2;
(ii) the quotient algebra A2//13 is isomorphic to the quotient algebra A2/(Sq2) of the algebra A2 by a two-sided ideal generated by the operator Sq2; see [49], [59], [76].
Let A" be the algebra A'//13
A'/(Sq2). We use the following
isomorphism (see Novikov's paper [76] for instance):
H*(MSp, Z/2) ^-' ® A"o,, w
where o,,, are free generators and the summing is given over all collections w = ( i 1 , . .. , iq), where it 2m - 1, deg o,,, = 4(ii + ... + iq).
Let us consider now the bigraded polynomial algebra C. = (Z/2) [c2, ... , cn, ...]
1
90 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE where n = 2,415, ..., n
2'n - 1, deg cn = (0, 0, 4n).
The following isomorphism is an easy consequence of the formula (3.7):
Ext*' '*(H*(MSp; Z/2), BP*/2) 0 (Z/2)[ho]
Ext*' (A", BP*/2) 0 (Z/2) [ho] 0 C.
(3.10)
The formula (3.7) implies that Sg2h = 0, so the algebra A'2 acts on the ring BP* trivially. Let the ring
BP* = (Z/2) [ho, hl, ... , hn, ...]
be dual to BP*, where deg ho = (0, 0, 2), deg hn = (1, 0, 2(2n - 1)), n=1,2, ... We obtain the isomorphism Ext*' (A'2//'.2i, BP* /2) 0 (Z/2)[ho] 0 C* ExtA2 (A'2, Z/2) ®BP* ®C*.
Finally we need the isomorphism
Ext*, (A2, Z/2) ?' (Z/2) [u1,... ,u,...],
(3.11)
where deg uj = (0, 1, 2(2j - 1)), j = 1, 2, ...; see [109], [111], [112]. The first degree comes from the filtration's degree and is equal to zero according to the isomorphism (3.10). The isomorphisms (3.9)-(3.11) imply the proof of the following lemma.
Lemma 3.2.1 There is a trigraded algebra isomorphism El'*'* = ExtA2(H*(MSp; Z/2), BP*) ti (Z/2) [c2,
.
cn, .... ul, ... , uj, ... 7 ho, hi I ... I hi, ...]
7
(3.12)
where deg cn = (0,0,4n), n = 2,4,5,..., n # 2'n - 1, deg uj = (0, 1, 2(2j - 1)), deg ho = (2, 0, 0), deg hi = (1, 0, 2(2i - 1)), i, j = 172'.... 11
3.2. THE MODIFIED ALGEBRAIC SPECTRAL SEQUENCE
91
Note 3.2.1 The proof of Lemma 3.2.1 is based on the properties of the maSS-filtration. The latter differs a little from the filtration generated by powers o f the maximal proper ideal m = (vo,... , v , . . . ) C BP*. The classic Novikov algebraic spectral sequence is determined by that filtration. Its initial term is much more complicated in comparison with the algebra (3.12); it was computed by V. Vershinin [109]. Now we summarize the results of V.Vershinin and V.Gorbunov [21], [112] concerning the first differential of the modified algebraic spectral sequence for the symplectic cobordism ring.
But first some preliminary considerations. The Landweber-Novikov algebra AMSp of operations in the symplectic cobordism theory has the form AMSp ,,, MSp*®S,
where a free basis of the algebra S consists of the operations sw; see [49], [53], [77]. The Landweber-Novikov algebra AMSp acts on all the objects which naturally depend upon the spectrum MSp and its automorphisms. In particular, it acts on the homology and cohomology groups of the spectrum MSp and on the terms E, of every algebraic spectral sequence. The action is very important for computations of spectral sequences (see [109]-[112]), but we do not intend to describe the action of the algebra AMSp in detail. It is sufficient to mention only some relevant facts which allow us to carefully formulate the results. Nigel Ray determined the elements Oi E MSp4i_3 which are indecomposable for i = 1, 2, 4, 6, ...; they are of order two and are closed with respect to the action of the algebra AMSp; see [86]. The Ray elements are very important for a comprehension of the structure of the Adams-Novikov spectral sequence. Let us recall some properties of the Ray elements and consider their projection into the term E2'* of the Adams-Novikov spectral sequence and into the initial term of the modified algebraic spectral sequence. Let cpi be the elements 92i for i = 1, 2, ..., and the element 01 be a standard generator of the group MSp1 = Z/2. We project the element
01 into the term E2'* of the Adams-Novikov spectral sequence, then into the term Ei'*'* of the modified spectral sequence. As a result we
92 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE have that the image of these projections coincides with the element u1 for dimensional reasons. So the element u1 is a cycle of all differentials in the maSS and uniquely determines (also for dimensional reasons) the element ul E ExtABp(BP*(MSp), BP*) which is also a cycle of all differentials in the ANSS. We consider now some Adams resolution of the spectrum MSp in the theory BP*(.):
MSp \,io
Al .- ... . Xn-1 t 1k,
/
in
X.
\'n_1 / kn
The corresponding Adams filtration of the ring MSp* MSp* <-- T7i E--- ...
(
, n_l --_ In <-- .. .
where Fn = i*Xn, n = 1, 2, ..., is also invariant with respect to the action of the algebra AMSP. We have seen earlier that 91 E .F1, but 01 V I. The element cpi lies in the group F1 and cpi V F2 because since the element cpi is of order two. S(2t_i) i = 91i then cpt V We note that the line EE* of the Adams-Novikov spectral sequence is monomorphically imbedded into the line E2'*. So we obtain that the projection of the element cpi into the term E2'* lies in the first line
Ext"p(BP*(MSp), BP*). The projection of the elements cpi into the term El'*'* of the modified algebraic spectral sequence is also denoted by cp=. For dimensional reasons it follows that the differentials cannot act on the groups Eo'l'* So the line E°'1'* is monomorphically imbedded into the line E o,',*, and 00 so it is a free C*-module with the generators u1, ... , u ... according to Lemma 3.2.1, where
C* = (Z/2) [cl,... , cn, ...]
,
where n = 214, 5, ..., n 2q -1. The projection of the element cpi into the line E°'1'* is also denoted by 'p2.
3.2. THE MODIFIED ALGEBRAIC SPECTRAL SEQUENCE
ho
93
ujho
hi
cn, ci,J
uihj + ujhi
ui
Figure 3.1: The algebra Ei'*'*.
Note 3.2.2 The modified algebraic spectral sequence for the symplectic cobordism ring has surprising properties. That is, the first differential is a unique nontrivial one and its action on the generators of the algebra El'*'* depends only on the binary decomposition of their indexes.
Let us introduce the following notation.
Let ci,j denote cn if n = 2i-1 + 2j-1 - 1, 1 < i < j. In the case n = 2m - 1 where m = 2" -2 + ... + 2',, -2 is a binary decomposition of the number m (here 2 < i1 < ... < i9, q > 3), the element cn is denoted by cil..... i,,. The corresponding Ray elements are denoted by
cpi...... iq, and their projection to the term E;'*'* by -ti,...,iq di Al
The following theorem describes the action of the first differential of the modified algebraic spectral sequence for MSp*.
Theorem 3.2.2 There exist polynomial generators of the algebra E1*'*'*
' Z/2 [c2i . . . , cn, . . . , u1,. . . , u,
.. . , hOi hli ... , hsl
]
such that the following statements are true. 10 The differential dl' acts on the generators cn as follows:
(a) if n is even and is not a power of 2, then the element cn is a cycle of all differentials of the modified algebraic spectral sequence;
94 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
(b) if n = 2'-'+ 2j-'- 1, 1 < i < j, then di'(c1,j) = uihj + ujhi;
(c) if
(3.13)
n=211'1+...+2iq-1-1, 23, then
dl , (Cii ,....iq) = E (ui, hi, + ui, hi, )c1,i, ...
C1,iq; (3.14)
1<s
(d) there is an isomorphism E2'*'o =- E * , 00 *,'.
2° The generators uj are the cycles of all differentials, and the following equalities are true in the algebra Ei'*'*:
ul = 01, ui = uic1,j + ujcl,i + >
(12*-2,
i = 2, 3, ... ,
1 < i < j,
J
where 4j,j is a projection of cp2;_1+2;_1, and if q > 3 the Ray element J°i1,...,iq projects to q
ui=C1,i1 ... C1,it ... C1,iq
'Oil,...,iq = u1Cil,...,iq +
+E
(3.15)
J
i=1
where the elements cJ E E°'o'* above are the cycles of all the differentials. 3° The formula
d,'hj = houj.
(3.16)
holds for every j = 1, 2.... Note 3.2.3 The statements 1°(a), 1°(d), 3° were proved by V. Vershinin [109], [110], the other by V.Vershinin & V.Gorbunov [21], [112]. These proofs are essentially based on the known module structure over the Landweber-Novikov algebra as well as on Buchstaber's results concerning Two-valued Formal Group Theory (the statements 1°(a), 1°(d)).
3.2. THE MODIFIED ALGEBRAIC SPECTRAL SEQUENCE
95
The algebra El'*'* and the action of the first differential dl t are shown in Figure 3.1.
Here the statements of Theorem 3.2.2 are cited in their original form. For our purposes it is convenient to take some other generators Ci1,...,iq for q > 3.
Lemma 3.2.3 There exist generators ci,..... iq, where q > 3, such that the following formulas are true: q
q
ul E hiaC+ h 1 E uiaCal,...,il,...,:y
dl (Ci1,...,ae)
t=1
(3.17)
t=1
where the elements ci,j coincide with the generators from point 1°(b) of Theorem 3.2.2.
Proof. Let us apply induction with respect to q > 3 . We put Ci,j,k = Ci,jCk,1 + Ci,kCj,l + Cj,kCi,1 + Ci,j,k,
for the element ci,j,k taken from the formula (3.17), where 2 < i < j < k. Hence we have dl `Ci,j,k = hl(ukCi,1 + UJCi,k + uiCj,k) + ul(hkC1,j + hjCi,k + hiCj,k)
Let us make the induction step. We write Z(1)
E Cis,iaCl,i1 ... Cl,i, ... Cl,ia ... C1,in 1<s
for all n and define the following elements: Z(k) a1 ,...,in
1:
Cial ...,itk cl,i1 ... C1,ia1 ... Ci,iak ... C1,in
1
for all n < q, k = 1, 2, .... It is clear that q-1 Z(k)..,iy
Al
dCil,...,iy+1 + k=1 q
= ul E hiac' t=1
q
.I
+ hl E uiac
->
O
t=1
Later we'll use the chosen generators Cil,...,iq from Lemma 3.2.3.
96 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
3.3
Symplectic cobordism with singularities
Here we will discuss some results obtained by V.Vershinin [111]. For clarity we need to present some of them together with their proofs. Let us consider the following sequence of closed symplectic manifolds:
E _ (Pi, ... , Pk.... ), where [Pi] = 01, [Pk] = T2k_2 for k > 2, cps being the Ray elements. As before we make use of the notation Ek = (P1, ... , Pk) for k = 1, 2, .... Let be the corresponding bordism theories with singularities and MSpE, MSp£k their classifying spectra respectively. Sometimes the spectrum MSp can be alternatively denoted by MSp'0. Note that the integer cohomology groups of the spectrum MSp are nonzero only in dimensions - 0 mod 4, and are torsion free.
Lemma 3.3.1 The cohomology groups H*(MSpl;k; Z)
are nontrivial only in even dimensions, are finitely generated, and are torsion f r e e f o r every k = 1, 2,
....
Proof. The Bockstein-Sullivan triangle (1.2) induces the cofibration: EPkMS,pEk-'
xlpk
MSpEk-1
MSpEk
(3.18)
where pk = dim Pk = 2k+1 - 3. Induction with respect to k allows us to conclude that corresponding cohomology exact sequences (with integer and mod p coefficients) split into the following short ones: 0 <- Hm(MSpEk-1) El k-1)* Hm(MSpEk) & Hm-Pk-1(MSpEk-1) 4- 0
These short exact sequences immediately imply the desired statement.
0
3.3. SYMPLECTIC COBORDISM WITH SINGULARITIES
97
The spectrum MSp£k satisfies the condition (* *) from section 3.1 for every k, hence there exists a corresponding Adams-Novikov spectral sequence as well as a modified algebraic spectral sequence. Now let us consider the case when p = 2 and the corresponding Brown-Peterson cohomology theory BP*(.). The odd prime case is trivial due to the following lemma.
Lemma 3.3.2 The groups MSp'k are p-torsion free for all k = 1, 2, .. . and every odd prime p.
Proof. Induction with respect to k. We begin by recalling the fact that the Bockstein-Sullivan exact sequence
... -
MSpEk-1 x` --1% MSpEk-1 :k-1+ MSp£k k) MJpEk-'
-- ...
tensor multiplied on Z(,) (where p > 2) splits into 0
MSpEk-1
xlPkl
MSp; k-1 ®Z(p)
®Z(n)
-kk_1i MSp;k
0 Z(n) -4 0
The induction assumption implies that the groups MSp£k are p-torsion free for all p > 2. The morphism of the initial terms of the modified algebraic spectral sequences for p = 2,
irk : El'*'*(MSp) -- E,'*'*(MSp'k), is induced by the spectrum map Irk
.
MSp ---1 MSpEk.
The algebra E1'*'*(MSp) is a polynomial algebra over Z/2 with the generators c,,, u=, h;. The images of these generators in the algebras Ei'*'*(MSpEk) are also denoted by c,,, ui, h;.
98 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
ukho
ho
h1uki... hk-luk
hk
h1,. .. , hk_l
hlc.,,...
cl k, ... ck-1 k it = k, ci...... i,
i
.....i , Uk
Uk
Figure 3.2: The algebra
Ei'*'*(MSpEk-1)
Theorem 3.3.3 There exists a commutative and associative admissible product structure µk in the symplectic bordism theory with the singularities such that the coefficient ring is isomorphic to the following polynomial ring up to dimension 2k+2 - 3:
Z [wl,...,Wk,...,x2i...,xn,... 11 where the elements wk can be presented by the symplectic manifolds Wk
such that OWk = 2Pk, deg w i = 2(2i - 1), i = 1, 2, ..., n = 21415, ..., 2"a - 1. The initial term of the modified algebraic spectral sequence for the spectrum MSpEk for k > 1 is isomorphic to the algebra n
(Z/2) [c2, ... , cn, ... , uk+1, ... , u;, ... , ho, hl, ... , hi, ...J .
In addition ak(u) = 0 for q= 1
(3.19)
k Uq E E*'*'*(MSp).
Proof. Induction with respect to k We assume that all the statements are proved for the bordism theory MSp* Now let us consider the maSS for MSpFk-1 more explicitly. The algebra Ei'*'*(MSpEk-1) is shown in Figure 3.2 in homotopy dimensions up to 2k+2 -3.
Figure 3.2 shows the generators and the action of the first differential. This action is determined by the formulas for the differential dl t (Theorem 3.2.2). The induction assumption implies that the spectrum MSp'k-1 is a ring spectrum, so the spectral sequence possesses multiplicative properties.
3.3. SYMPLECTIC COBORDISM WITH SINGULARITIES
99
So the cycles (of given dimensions) of all the differentials lying in the algebra E1",;
*'*(MSpEk_1 ),
s > 1,
have the form cuk, where c E E°'*'*(MSpEk-1) is a cycle as well.
Hence elements (of the same homotopy dimension) of the first line E2'*(MSpEk-1) of the Adams-Novikov spectral sequence have the form y cp2k_2, where y E E2'*. The ANSS possesses a product structure, so we have
Lemma 3.3.4 The torsion elements of Tors MSpEk of dimensions up to 2k+2 - 3 have the form z So2k_2, where z E MSp*Ek .
Note 3.3.1 The above arguments are true when k > 2; in the case k = 1, the statement of the lemma is obvious. Now we are going to prove the existence of an admissible product
structure. Here the results of Chapter 2 would come in handy. The obstruction to existence of the admissible product structure pk is of dimension 2k+2 - 5. The Ray element cp2k_2 is of order two, so the manifold-obstruction Pk is Ek-bordant to a manifold without singularities. In particular, we have
[Pk] Ek E Im (Ir k-1 : MSp; k - MSpEk-1 . According to Lemma 3.3.4 every preimage of the element [Pk]Ek in the group MSp*Ek-1 has the form Z cp2k_2. Hence [Pk]Ek = 0 due to the Bockstein-Sullivan exact sequence.
Now we prove commutativity of the product structure µk, which is determined by the product structure µk_1 in the theory and by some Ek-manifold Qk which bounds the obstruction Pk. An obstruction to commutativity is the Ek-manifold Bk = Qk U, -Qk;
see section 2.3. Suppose bk = [Bk]Ek. According to Lemma 2.3.2 the element bk is of order two in the group MSp2 +2_4. Let us consider the
100 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE Bockstein-Sullivan exact sequence £k_1
Wk-1
Sk
Ek
Ek-1
.W2k-2
MSp2;1 _8 is torsion free, so bk E Im irk-1. Let k be some preimage of the element bk. Lemma 3.3.4 implies that the element bk lies in the image of the homomorphism 'W2k-2. Hence bk = 0 due to exactness, so the product structure Pk is commutative (Theorem 2.3.1). Associativity of the product structure Pk is a consequence of Lemma 2.4.2 since the corresponding obstruction is of order three.
Now let us prove the statement concerning the structure of the initial term of the modified algebraic spectral sequence. Lemma 3.3.1 implies the following short exact sequence with coefficients in Z/2: E(,,k-1 )*
Hm (MSpEk) Esk Hm-Pk-1(MSpEk-1) E- 0
0 i- Hm (MSp£k-1)
The sequence can be considered as an exact sequence of A2-modules. Note that it corresponds to the cofibration (3.19). We apply the functor ExtA2 ( - , BP*)
to this exact sequence to obtain the following exact triangle: (MSPEk-1
Wk
1 E*'*'* (MSpEk-1)
(3.20)
1 E*'*'* (MSp£k )
where wk is a connecting homomorphism. The natural properties of maSS imply that the homomorphism wk is adjoined to the homomorphism generated by a multiplication by the element V2k-2 E MSp*k-1 Thus the homomorphism wk coincides with a multiplication by the element Uk being the projection of the element cp2k-2. According to the induction assumptions the element Uk is a generator of the polynomial algebra. So the triangle (3.20) implies the isomorphism Ei'*'*(MSpF'k) - Ei'*'*(MSpEk-1
)/E°'*'*(MSp :k-1)
uk.
3.3. SYMPLECTIC COBORDISM WITH SINGULARITIES
hk, ho
hk, ho
houk
101
houk
'uk hk
hk
Cn
Uk
Cn
'ak, ho
0
hk
0
Cn
0
Uk
Figure 3.3: The triangle (3.20)
In particular we can see that the groups Ei'''*(MSpEk) are zero for s > 1 for homotopy dimensions up to 2k+2 - 3. It is obvious that the cells of the line E2,* of the Adams-Novikov spectral sequence are also
trivial in the same dimensions for all s > 1. So the ring MSpFk is torsion free in the same dimensions.
Finally, we have to prove the polynomiality of the ring MSpEk in dimensions up to 2k+2 - 3. We note that the ring homomorphism k-1
: MSpEk-1 -- MSp; k
is an isomorphism in dimensions up to 2k+1- 3 since dim Pk = 2k+1- 3. Now let us consider carefully the triangle (3.20); see Figure 3.3. We have the isomorphism E*,*,* CV 2
E*,*,* oo
in dimension 2(2k - 1). At the level of the terms E,* *, we get (ok)*hk = ho.
102 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
Due to the Bockstein-Sullivan exact sequence
...
2k-2
irk-1
Ek_1 E E MSp2(2k_1) --) MSp2k2k_,) -46k MSpOk_1
2
k-. ...
the element wk adjoined to the element hk has to be such that hk(wk) _
2. The element wk may be determined by an Sp-manifold Wk, such that aWk = 2Pk. The element hk is adjoined to the element wk = µ(wk ® wk) by multiplicativity. This proves the polynomiality of the ring MSp*Ek ®Z(2) up to dimension 2k+2 - 3. The last thing is that the element wk cannot be divided by an odd number in the ring MSpEk.
We assume that wk = (4q + e) y, where e = ±1. Theorem 2.2.4 allows us to make some correction to the product µk. Suppose the manifold S is some Ek-manifold having dimension dim Sk = 2k+2 - 4; then the manifold Q° = Qk U S also bounds the obstruction Pk. Let [S]Ek = q y. The new product structure (which is determined by the product structure µk_1 and the manifold Qk) will be denoted by k. Due to formula (2.9) we get µk(Wk 0 Wk) = µk(Wk 0 Wk)
- µk(µk(I3kWk 0 /3kwk) 0 q ' y)
So we have the product structure µk, such that the ring MSp* k is polynomial up to dimension 2k+2 - 3. The theorem is proved. Note 3.3.2 We emphasize that the product structure µk does exist for every manifold Wk and is always commutative and associative. Taking a limit over k we get the following result.
Theorem 3.3.5 There exists a commutative and associative admissible product structure µ in the symplectic bordism theory with singularities MSp; such that the coefficient ring MSp' coincides with the polynomial ring:
MSp; '-'Z[wl,...,Wk,...,x2i...,x,i,...],
(3.21)
where the generators wk may be presented by any Sp-manifolds Wk,
such that Wk = 2Pk,deg wk = 2(2k - 1), k = 1, 2, ..., deg x = 4n,
n=2,4,5,..., n
2m-1.
3.3. SYMPLECTIC COBORDISM WITH SINGULARITIES
103
The cohomology groups of the spectra MSpEk and MSpS can also be computed. As we have seen earlier the cohomology groups H*(MSpEk; Z/p),
H*(MSpE; Z/p)
are free modules over the algebra A, = Ap/(Qo) for p > 2.
We consider the case p = 2. Let 13k be the subalgebra of the Steenrod algebra A2, whose basis consists of the elements Sq', where J = {il, , iin, n ...} are admissible sequences such that it < 1, if t < k, and it _> 3, if t > k. It isn't difficult to prove that '3k is a normal subalgebra of the algebra A2 for each k. - -
The following theorem was proved by V.Vershinin; see [111].
Theorem 3.3.6 The (Z/2)-cohomology of the spectra MSp£k is a free module over the algebra A2//8k for every k and that of the spectrum MSpE is a free module over the algebra A'2. 0 Finally we consider the cohomology ring H*(MSpE; Z).
The groups H*(MU; Z) are computed by means of Chern classes or Formal Group Theory. In our case it is simpler to apply the classic mod p Adams spectral sequence for the spectrum MSpE A K(Z). Following the speculations of [76], we have the following result.
Corollary 3.3.7 There is an isomorphism H*(MSpE, Z) N .
Z [1711,
... , Mk, ... , C2, ... , Cn.... I I
where deg Mk = 2(2k - 1), k = 1, 2, ..., deg cn = 4n, n = 2,4,5,..., n 2n`-1. D Now we can come back to the Adams-Novikov spectral sequence for MSp*.
104 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
The E-SSS for MSp
3.4
The above considerations allow us to conclude that the bordism theory with singularities is quite suitable for constructing the Adams-Novikov spectral sequence. Now we intend to present a particular Adams resolution for the spectrum MSp. Let us consider a corresponding E-singularities spectral sequence (it is called the E-SSS for symplectic cobordism). According to section 1.6, we have the following exact couple: MSp, (.)
...
MSpEr(1)()-.....-MSpEr(n-1)(.)ILn)MSpEr(n)()-
(1)
Ir()
a(n 1)
el
MSp* (I) ()
MSp*(n) ()
The diagram determines the following diagram of classifying spectra:
MSp
MSpEr(i)
1(1)
/a,
Ir(o)
MS
... r- MSpEr(n-1) 'Y(n) MSpEr(n)
...
ir(n 1
E(I)
p
MSpE(n)
(3.22)
Theorem 3.4.1 The diagram (3.22) is an Adams resolution of the spectrum MSp in the cohomology theory
Proof. The above definition dictates the necessity of proving the following statements: (i) the homomorphism MSp*(y(k)) is trivial for every k; (ii) the complex
MSp*(MSp) 42 MSp*(MSpE(1))
MSp*(MSpE(n)) F-- ... (3.23)
is a projective resolution of the AE-module MSp*(MSp), where ej MSp*(ir(0)), dk = MSp* (ir (k) o 8(k)), k = 1, 2, ....
3.4. THE E-SSS FOR MSP
105
Here AE is the operation algebra in the cohomology theory MSpE
It should be mentioned that the structure of this algebra is unknown. The spectra MSpE(k) split into the wedges of suspensions of the spectrum MSp5. So the A5-modules MSp* (MSpE(k)) are free for all k. The dimensions of the manifolds Pk increase: dim Pk = 2k+2 - 3. So it is sufficient to prove exactness of the complex (3.23). To do this we should examine the cohomology homomorphisms, which are induced by the maps y(k), 7r(k) and 0(k).
We will use the notation MSp£k where Ek = (F1,. .. , Pk) for every k = 1,2.... as above. We remark that there are defined Bockstein operators acting in every bordism theory MSp; k #(k) : MSp£k
i MSp£k
k = 1, 2, ... ,
which uniquely determine the maps of the classifying spectra, 'ilk) : MSpEk ---> MSp5k
up to homotopy. The corresponding homomorphisms in integral cohomology are H*(MSpEk). H*(MSpEk) It is clear that the homomorphisms (/3(k))* satisfy the following equalities (see section 1.3): lQjk))* 0 (3 k))* = 0,
($ k))*
(Qjk))* _ (Qjk))* o (N(k))*.
In particular, Im
J
C Ker (13(k))* J
Lemma 3.4.2 There is the inclusion Im (f3(k))* D Ker (,3(k))* J for every k,j, 1 < j < k.
(3.24)
106 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
Proof. 10 Let j = k. Then the map tion
#k(k)
is homotopic to the composik-1
MSpEk -i EPk+l ]I f CpEk-1 k
EPk+l MSpr-k .
From the corresponding cohomology exact sequence (Lemma 3.3.1) it follows that (irk-1)* is an epimorphism and Sk is a monomorphism. So we get the equalities Ker (Qkk))* = Ker (5k)* = Im (Qkk))*.
2° The lemma is supposed to be proved for all spectra MSpE^ when
n _< k - 1. The statement is proved in 1° for j = k. Let j < k. Now the point of departure is the diagram 3(.k))
H*(MSpEk)
H*(MSpEk)
H*(MSpEk)
(,rjk)) r
k)
fr
H*(MSPE) s
cas))
H*(MSpE)
0
' ')),
)
H*(MSpEi)
0
0
(3.25)
Let x E Ker
C H*(MSpEk). The induction implies that there exists the element y E H*(MSpEk), such that -(3(k))*(y) + x E Ker (7r (k))*.
Also if xl = -(flk))*(y) + x, then (#(k))*(xl) = -(Njk))* 0 (N(k))*(y) + (/3jk))*(x) = 0 (,8(k))* o since = 0. The induction implies that there exist the elements x;, y; E H* (MSpEk ), such that
3.4. THE E-SSS FOR MSP a)
b) c)
107
xi = -(#jk))*(yi) + xi E Ker (WJ+1)*, (#(k)
)*(Yi) = 0,
xo=x1,yo=y1
It is convenient to proceed with the diagram 0
0
0
r
(9(k-1) *
k-1))* *
(0(.3
H*(MSpEk-1) H*(MSpEk-1)
H*(MSp£k-1)
bk k) r
("jk))*
ak T
(.k))*-
bk
H*(MSpxk)
H*(MSpEk)
(3.26)
(
("jk))*
(-"jk)). I
(3(k-1))*
(P(k-1))*
H*(MSpEk-1)
H*(MSp;k-1)
H*(MSpEk-1)
0
0
0
The equality (1rj(k))*(xk_j_1) = 0 gives that there exists the element z E H*(MSpEk-1) such that S*(z) = xk- j-1. The exactness of the rows of the diagram (3.27) and induction assumptions imply that there exists the element Yk-j-1 E H*(MSpEk) such that (3(k))*(yk_j-1) = xk-j-1. The latter is denoted by y = yo + ... + yk-j-1. Then we have (Q;k))*(y)
= (/3jk))*(yo) +... + (#jk))*(yk-.i-1)
= (x0 - x1) + ... + (xk-.7-2 + xk-J-1) + xk-7-1 = x0.
Lemma 3.4.2. is proved.
Note also that the Bockstein operators #(k): MSpEk() --> MSgk()
108 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
all together induce the transformation ,3(k)(m) :
and the corresponding map of the classifying spectra MSpEk(m+l)
13(k)(m) : MSpEk(m)
Lemma 3.4.3 The sequence H*(MSpr-k(1)) EQ(k)(1) H*(MJpEk(2)) 1'O(k)(2) H*(MJpEk(3))
E0(k)(3)
is a total complex of the lattice of complexes H*(Cr k) =
N(k)(a)}aE`2((k)
where
H* k (a) =
H-(s-Ek
1
ai(ni+1))(MSpEk
)
for every collection a = (al, ... , ak) E Qt(k) C 21.
Proof. Recall that the lattice L*r have been defined in section 1.3. Let us take the following sublattice of the lattice £ k :
$k)(.), ...
{LEk
,C; k
/3kk-1)(y)}oE2(k-1)
where we set LEk (Q) (A, Y)
= MSp sk
ki
(Pi+1)
(ni Y)
for every collection o = (Cl, ..., ck_1) E %(k-1) C 21(k). We note that
the transformations /3j(v) anticommute as in Lemma 1.3.2. Suppose is the total complex of the lattice £k:
MJ p*E k(1)( ) d( )4
(.)
.
of the lattice £k(.)
A total complex Q
MSpEk(2)(.)
_
... , MSpEk(m)()
Q(m)4
...
3.4. THE E-SSS FOR MSP
109
coincides with the total complex of the double complex
Ek(j)() MSp*
d(l)
r
Ek(2)
MSp*
) r ... r
&l
&I
QkI
MSp*Ek(m)( m)
Ek(1) _ Ek(2)(.) MSp*
MSpEk(m)(.)
MSp*
&I
r )...
Qkl
&I
(3.27)
IEk(1)( MSP*k(2)( )
MSp* Qkl
Pk1
... r ak1
Here the transformations /3k are induced by the Bockstein operators Qk (with the corresponding signs; see Lemma 1.3.2):
Qk :
Let MSpEk(m) be a classifying spectrum of the bordism theory (which splits into a direct sum of the theories As a result the complex (3.27) coincides with the total complex of
110 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE the following double complex: H*(MSpEk(1))
"'*
H*(MpEk(2))
..... k
Qk
Qk
H*(MSpEk(2))
.....
H*(MSpEk(m)d().
i
i
I
Qk
H*
(MpE*k(1) )
d(1)'
E k (2) )
H*(MSp*
H*(MSp*k (
"`) )
.4 ...
(3.28)
The rows of the diagram (3.28) are exact due to the induction, the first step of which is provided by Lemma 3.4.2. The latter also implies exactness of the columns. Hence the exactness of the complex (3.28) follows from the spectral sequence for the double complex. 0 We note that the homomorphism 7r(0)* in the diagram
H*(MSp)
-r(1)'
H*(MSpEr(1))
H*(MSpE(1))
is an epimorphism (as a direct limit of epimorphisms). So y(1)* = 0
3.4.
THE E-SSS FOR MSP
111
and the cohomology groups of the spectrum MSpEr(1) are torsion free and are nontrivial only in even dimensions.
Let us have a look at the diagram
H*(MSp)
H*(MSpEr(1))
(3.29)
H*(MSpE(1))
H*(MSpE(2))
Lemma 3.4.4 The homomorphism a(1)* is epimorphism.
Proof. The homomorphism a(1)* is known to be a monomorphism, so it suffices to show that
Ker 7r(1)* C Im /3(1)*.
Let x E H*(MSpE), ir(1)*x = 0 . We take k, such that xk = irk(x) 7rk-1(x) = 0, where the map
7rk : MSpEk --4 MSpE
is a direct limit: Irk = 1irn(ak+1 o
... o Irk+i+i )
0,
112 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
Then we need the following diagram: 0
rk
H*(MSpE)
H*(MSpE-) -- 0 6,r
Alt
H*(MSpE)
Wk
H*(MSpEk-') - 0
(3.30)
H* (MSpEk )
It follows from exactness of the right column that there exists the element y E H*(MSpE), such that x-,3k(y) E Ker irk. Let x1 = x-Qk(y); then we take kl > k such that irk1(xl) = 0, 'rk1 _1(xi) 0. Repeating the procedure gives that there exists the element yl E H*(MSpE), such that x1 - ,ak1(yl) E Ker 7rk1. We recall that the homomorphism 7rn is an isomorphism in dimensions up to 2n+2 - 3, so repeating the above procedure gives the elements xo = x, x1,. .. , x1, Yo = y, y1, ... , yi, such
that x= = xi-1 - Qk;-1(yi-1) E Ker
irks-1
Pk,(yl) = x1 = XI-1 - Qk,-1(Yi-1).
The definition of the transformation Q(1) implies that r
9(1)*(yo ®... ®yi)
EQki(y=) = x. 0 :=o
3.4. THE E-SSS FOR MSP
113
Now we need the following commutative diagram: H* (MS pErk_, )
7(k)' _ H*(MSpEr(k)) (3.31)
H*(MSpE(k) )
H*(MSpE(k))
Lemma 3.4.5 The homomorphism Tr(k)* is epimorphism.
Proof. The exactness of the triangle of (3.31) implies that 19(k)* is monomorphism and the inclusion Ker 0(k - 1)* C Im ir(k - 1)*. So ir(k)* is epimorphism if Im /3(k)* D Im a(k)*. Lemmas 3.4.2, 3.4.3 and exactness of the above triangle imply the equality
Im /3(k)* = Ker j3(k - 1)* = Ker 7r(k - 1)* = Im a(k)*.
In particular, the homomorphisms ry(k)* are proved to be trivial. As noted above, the cohomology groups of the spectra MSpEr(k), MS PE(k)
are torsion free and nontrivial only in even dimensions. The AtiyahHirzebruch spectral sequence gives the following isomorphisms: MSp* (MSpEr(k))
MSp* ® H*(MSPEr(k)),
MSp* (MSpE(k))
MSp* ® H*(MSpE(k)).
(3.32)
So the homomorphisms MSp*(y(k)) are trivial and the complex (3.24) is a projective resolution (its exactness follows from Lemma 3.4.2 and isomorphisms (3.33)). Theorem 3.4.1 is proved.
Corollary 3.4.6 The E-singularities spectral sequence for the symplectic cobordism ring coincides with the Adams-Novikov spectral sequence for the spectrum MSp in the cohomology theory
The cobordism theory MSp* possesses some commutative and associative product structure. Therefore the Adams-Novikov spectral
114 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
sequence also possesses a product structure generated by the admissible product structure in the cobordism theory
In particular, we have the Adams filtration MSp*
" - MSp; r(1) F- ... E- MSpIT(k)
ryE
(k)
...
It follows that the isomorphism
MSp*/Tors
MSp,/MSp1r(1)
is naturally implied by the definition of the filtration.
Corollary 3.4.7 Every element x E Tors MSp* can be represented by a EF(k)-manifold for some k > 1. Note 3.4.1 By now (March, 1991) all the torsion elements of the ring MSp* are known to be generated by the Ray elements and Massey products containing them. It seems convenient to interpret EP(k)-manifolds as nonlinear Massey brackets.
So we have
Corollary 3.4.8 The torsion Tors MSp* is generated by the elements 01, p1,...,
2k-2,...
and by the nonlinear Massey brackets containing them.
3.5
Splitting of the spectrum MSp(2)
According to Boardman [13] the spectrum MSp2) splits into the wedge
of suspensions of the spectrum BP. We are to take a particular splitting, since it is necessary to identify the E-singularities spectral sequence with the Adams-Novikov one for MSp in the Brown-Peterson cohomology theory BP*(.).
3.5. SPLITTING OF THE SPECTRUM MSP(2)
115
Let us examine the Hurewicz homomorphism ---I H*(MSpE(2); Z)
hE :
which is clearly monomorphism. So the ring (MSp*')(2) can be identified with its image in the ring H*(MSp2); Z).
Theorem 3.5.1 There exist multiplicative generators mi E H2(2$-1)(MSp2); Z)
of the polynomial ring H*(MSp(2); Z)
Z [ml, ... , mk, .... Y2, ... , ym, ...]
satisfying Hazewinkel's formula (see [4] k-1
Vk = 2mk -
m= vk` 1,
(3.33)
where vi = .iwi + qi, Ai E Z(2) are invertible elements, and qi E (MSp*I;)(2) are decomposable ones.
Proof. Suppose h£ : MSp2) --1 MSp2) A K(Z) is the spectrum map which generates the Hurewicz homomorphism. We consider the classic Adams spectral sequences for the spectra MSp2) and MSp2) A K(Z):
ExtA2(H*(MSp2); Z/2), Z/2) = (MSp£)(2), ExtA2(H*(MSp(2) A K(Z); Z/2), Z/2) =
(MSp(2) A K(Z))*.
Theorem 3.3.3 implies the isomorphisms ExtA2(H*(MSp2); Z/2), Z/2) ^_' Ext*' (A'2, Z/2) ®Z [C2.... , cn ... , ] ,
116 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
Ext
2(H*(MSp') A K(Z); Z/2), Z/2)
Ext *'*(A'2 ® H*(K(Z); Z/2), Z/2)0 Z [c2, ..-,C,, ... , ] , A where deg c, = 4n, n = 2,4,5,..., n 2m - 1. We consider the inclusion of A'2-modules A''
H*(MSp2); Z/2)
®A'2 Sw W
into the direct summand with the generator 1.
Lemma 3.5.2 There exists a spectrum map
a : BP --> MSp2), such that the A'2-module homomorphism
A2 `- H*(MSpE(2);Z/2)
H*(BP; Z/2)
is the identity map.
Proof. The map a : BP ->
determines the element a E
(MSp2))*(BP). We note that (MSp(2))*(BP)
HomABP(BP*(MSp(2)),BP*(BP)).
(3.34)
We consider the filtration of the ring BP* generated by powers of the maximal proper ideal in = (2, v1, ... , vk, ...) C BP*. As we have seen in section 3.1, this filtration induces the filtration of the algebra (3.35), and the corresponding adjoint object is equal to Hom*,
Z/2), A2 ®BP*),
(3.35)
where BP* is the ring adjoined to the ring BP* with respect to this filtration. Consider the homomorphism Z/2)
A2 = A'2 ®1 --* A'2 0 BP*,
3.5. SPLITTING OF THE SPECTRUM MSP(2)
117
where pr is a projection on the direct summand with the generator 1, and i is a standard inclusion. The element & of the algebra (3.36) is covered by some element Q E HomABP(BP*(MSp2)),BP*(BP)) which determines the element & E (MSp2))*(BP) due to isomorphism (3.35). It is clear that every map of the class u E Q gives the desired cohomology homomorphism.
To continue the proof of the theorem we need the commutative diagram
H*(MSp2))
hE
H*(MSp2) A K(Z)) cAId
hBP
(3.36)
BP A K(Z)
where o is the map from Lemma 3.5.2, hBP is the map corresponding to the Hurewicz homomorphism in the Brown-Peterson theory BP*(.). The homomorphism
(hBP)* : H*(BP; Z/2) -- H*(BP A K(Z); Z/2) was examined in several papers; see, for example, S.Novikov [76]. Suppose ExtA2(2`-1)(A'2i Z/2) hi E
is a projection of the generators vi E BP2(2'-1), and qi E ExtA2,2`-1)(A'2 ® H*(K(Z); Z/2), Z/2)
is a projection of the standard generators mi E H*(BP; Z/2) (and qo is a projection of 2,deggo = (0,1)). According to [76] (see also [103, chapters 4-5]) the induced homomorphism (h BP)* : Ext'A22(2`-1)(A'', Z/2), Z/2) 2 Z/2) -+ Ext' 2(2'-1)(A'2®H*(K(Z); A2
118 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
converts the generators hi into the elements gogi for all i = 1, 2, .... Commutativity of the diagram (3.36) implies that the homomorphism ExtA2(2`-1) (H*(MSpE; Z/2), Z/2) h*
+.
ExtA2(2`-1)(A'
0 H*(MSpE A K(Z); Z/2), Z/2)
of the initial terms of the Adams spectral sequences behaves as (h BP)*. Recall that the elements w, E MSp, are adjoined to the elements o*(hi) by definition (see the proof of Lemma 3.3.3), where o* : Ext,14 2`-1>(A2, Z/2) -+ Ext 4 2`-11 (H*(MSpE; Z/2), Z/2)
is the homomorphism which is induced by the map o. So the homotopy group homomorphism
o* : BP* -p converts the elements vi into the elements wi up to decomposables. It can be supposed that o*(vi) = )iwi + qi, where Ai E Z(2) is an invertible scalar, qi is a decomposable element of the ring (MSp*E)(2). Commutativity of diagram (3.36) allows us to take mi E H2(2$_1)(MSpr(2); Z)
such that the Hazewinkel formulas hold.
Corollary 3.5.3 There exists a multiplicative projector ire :
(MSp(E2))*( )
such that its image is the Brown-Peterson cohomology theory BP*(.). In addition, the standard Hazewinkel generators vi of the ring BP* satisfy the following formula:
70(vi)=)twt+qi, i=1,2,..., where )i E Z(2) are invertible scalars and qi are decomposable elements of the ring (MSp*)(2).
3.5. SPLITTING OF THE SPECTRUM MSP(2)
119
Proof. Let gE(z) = z +
00
mi z2' E H*(MSpa) A K(Z); Z/2) [[z]]
be a formal series, where mi are the elements of formula (3.34). The series gE(z) determines the first Chern class cE in the cohomology theory Suppose Fr (x, y) = g-l(gE(x) + gr(y)) is a corresponding formal group. Universality of the first Chern class gives that the class cE is determined by some map:
Q : BP -p The standard results of Formal Group Theory give that the coefficients of the formal group FE(x, y) generate the ring BP* = Z(2)
[v1, ... , vk,
...]
where vi = Q*(Aiwi + qi). Boardman's Theorem B [13] implies that the first Chern class determines the splitting MSp(2) = BP A M(G),
where G = Z(2) [x2i ... , xn, ...], n = 27475, ... , n
2m -1, deg xn = 4n,
M(G) is a graded Moore space.
Note 3.5.1 So the theory BP*(.) is represented as a direct summand of the cobordism theory It would seem that we can find a certain splitting of the theory into the sum of the theories BP*(.) such that the generators wi would be exactly covered by the elements vi of the ring BP*. However we cannot; see V.Gorbunov [42]. His proof is essentially based on a description of the Ray elements in terms of the Two-valued Formal Group Theory (V.Buchstaber [26, Theorem 23.11]).
According to Theorem 3.1.3 we have
120 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
Corollary 3.5.4 A 2-localization of the diagram (3.22) is an Adams resolution of the spectrum MSp in the theory BP*(.), and a localized E-singularities spectral sequence for the symplectic cobordism ring coincides with the Adams-Novikov spectral sequence for the spectrum
MSp in the theory BP*(.). O A few remarks concerning E-singularities spectral sequence would come in handy.
Let ul, U2.... , uk, ... be the projections of the elements 01, (p1, ..., cp2k, ... into the first term El*'* of our spectral sequence. As in section 1.6, we have the isomorphism
El = (MSp* )(2) [ul, u2.... , uk, ...) . The line El'* = (MSp£("+1))(2) is a free module over the ring (MSpE)(2)
with the generators a2 ak ul ,u2 ,...,uk ,..., where al > 0, al + ... + ak + ... = s + 1. Corollary 3.4.5 implies that al
the algebra Ext*'** p(BP*(MSp2)), BP*)
coincides with the homology algebra of the complex (MSp*E(1))(2)
16(1)
(MSpE(2))(2)
(MSp£(2))(k)
...
---
(3.37)
Recall also that the differentials (3(k) are determined by the Bockstein operators /3i as follows: /3(k) (x
U1
u 2 2 , ... , ukk ,
...)
(3.38)
00
((-1)E`(a)/3ix (u71
... ua-1'ua`+lui+1 ...ukk
))
,
{.1
where a = (al, ... , ai, ...) and the sign ei(a) was determined in section 1.3.
Now we are able to translate the above results concerning the modified algebraic spectral sequence into geometric language. The projector
3.5. SPLITTING OF THE SPECTRUM MSP(2)
121
70 from Corollary 3.5.3 actually converts maSS-filtration of the ring BP* into some filtration of the ring (MSp*F)(2) (which is also called maSS-filtration). Recall that 7°(vi) = Aiwi + qi, where Ai E Z(2) are invertible elements and qi are decomposable ones of the ring (MSpE)(2). So the maSS-filtration coincides with the filtration determined by the graduation wE for the generators of the ring (MSpE)(2): wF- (2) = 2,
w'(wi) = 1, i = 1,2,...,wE(xn) = 0, n 54 2' - 1.
Let (M pE)(2) be the ring which is associated to the ring (MSpE)(2) with respect to maSS-filtration. We have the isomorphism E
(MSp*)(2) _ (Z/2) [ho, hl, ... , hk, ... , C2, ... , cn,...] ,
where ho is a class of two, hk is a class containing the element wk, k = 1,21 ... and cn is a class of the element xn, n = 214,51 ..., n * 2' - 1. Suppose U1,. .. ,Zen, ... are a projection of the elements u1, ... , un, .. . into the term Ei'*'* of maSS. So we get the isomorphism (MSpE)(2) [ U1,
... , U, ...] .
Suppose ,(3k is the operator which is associated to the Bockstein
operator ,3k with respect to maSS-filtration for every k = 1, 2, ... Then the complex (MSpE(1))(2) a(1>
pE(2))(2)
(M
, ...
)
( pE(k))(2)
a
.
...
associated to the complex (3.38) has the differentials /3(k) acting as follows: ,Q(k) 00
_ J:(-1)E'(a)/3;x
(x- "1',U22,...,''kk,...) (7/al
(3.39)
...Ua''ua'+1u+;, ...U, ...) . aA;
i=1
Now it is very useful to compare (3.40) with the formula for the first differential dal of the maSS of Theorem 3.2.2. We note that
/3iwi=2,
/3;wi=O, i
j,
122 CHAPTER 3. THE ADAMS-NOVIKOV SPECTRAL SEQUENCE
by definition of the elements wi E MSp,E.
So it is not surprising that ,Q1hi = ho, ,(3jhi = 0 for i use the notation of section 3.2 to formulate
j
.
Here we
Corollary 3.5.5 There exist the polynomial generators c,d of the ring
(MSp.
1
) (2 )
-N (Z/2) [ho, h1, ... , hk, ... ,C2, ... cn, ...]
such that the operators Qk act on them as follows:
1. if n = 2i-1 + 2j-1 - 1, then
h;
ifk=j, ifk=i,
0
if k
hi Qk(ci,.i)
2.
_
ifn=211-1-}-...+21q-1
(3.40)
i, j;
-1, q>3, then
if k = it, >i-1 hitci...,tq ifk = 1, ifk # 1, ill ... I iq; 0 h1Cil,...,ie,...,iq
Qk(Ci1..... 1q) _
(3.41)
3. if n is even and isn't a power of two then k(cl) = 0 for all k = 1,2,....
Note 3.5.2 The formula for the first differential of the maSS partly describes the action of the Bockstein operators Qk on the generators of
the ring MSp;. It is the starting point for correcting the admissible product structure in the cobordism theory and for choosing generators of the ring MSpE in Chapter 4. As a result the exact formulas for the action of the Bockstein operator Qk will be obtained. This means that we'll have a complete description of the first differential in the Adams-Novikov spectral sequence.
Chapter 4 First differential of ANSS The purpose of this chapter is to combine the geometric machinery of the E-SSS with the algebra of the ANSS for the spectrum MSp.
The first stage is to choose an admissible product structure in the theory such that the multiplication formulas for the Bockstein operators have the simplest form. We emphasize that the choice will be based on specific properties of symplectic cobordism theory, and in particular depends essentially on known information concerning Ray
elements. To achieve the desired product structure we take a relevant algebraic result of Two-valued Formal Group Theory. Then a localization procedure is applied to compute the action of the Bockstein operators fl k on the generators of the ring MSpE (Theorem 4.4.1). The computations of the maSS are also used in an essential way.
Finally we get a complete description of the first differential of the ANSS. At the end we discuss the simplest application of the results achieved of the algebra
Ext
(BP*(MSpe' ), BP*).
The main conclusion we come to is that the algebra possesses a module
structure over the symmetric group S,, hence it may be completely described in terms of the representation theory of the symmetric group. 123
124
4.1
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
Characterization of the Ray elements
As noted above the Ray elements are very important for a description of the ANSS. We begin by recalling some definitions. be the bordism theory of U-manifolds with fixed Let M(U, Sp-structure on the boundary. We have the following exact sequence: 0 --1 MU4k_2 P*) M(U, Sp)4k_2 d-*- M(U, Sp)4k_2 -* 0.
(4.1)
J.Alexander [6] showed that the elements v1 E M(U, Sp)2,
v2k E M(U, Sp)8k_3i
such that d*(vl) = 01, d*(v2k) = Vk, have infinite order. The equalities
201 = 0, 2Wk = 0 imply that there exist the elements Sl, b2k of the group MU*, such that p*(Sl) = 2v1i p*(S2k) = 2v2k. It is clear that the elements Sl, b2k are uniquely determined in the group MU* 0 (Z/2).
Note 4.1.1 The elements °2k+1 are trivial for every k > 1 (Roush [91]). So the elements v2k+1, b2k+1 are assumed to be zero.
The exact sequence (4.1) induces the following exact triangle of the classifying spectra:
MSp
d
M(U, Sp)
MU where 7rp' is a standard map of the classifying spectra. The AtiyahHirzebruch spectral sequence gives MU*(d) = 0. So the triangle (4.2) may be considered as the first stage in setting up the Adams resolution of the spectrum MSp in the theory MU*(.). The triangle (4.2) may be
4.1. CHARACTERIZATION OF THE RAY ELEMENTS
125
extended up to a certain Adams resolution:
MSp d M(U, Sp)
d11)
X(2)
.. .
MU The part of the above diagram, d(2)
M(U, Sp)
d(3)
X(s)
X(2)
...
Z(s)
Z(2)
is also the Adams resolution of the spectrum M(U, Sp) in the theory MU*(.). The elements v1, v2k have infinite order, so their images in the group it*Z(2) are not trivial. By definition the elements 01 = Fr(1)(v1), 'pk = Ir(1)(v2k) coincide with a projection of the Ray elements into the first line E1'* of the initial term of the ANSS. The first differential acts on these elements as follows: d1(bt)
° P*)(b+)
f
201
if i = 1,
l
l 20k ifi = 2k. J
So the elements b1, b2k are of order two in the line E2'*. The elements b1i b2k were computed by V.Buchstaber in terms of Two-valued Formal
Group Theory. Now we recall some of his results.
Let F(u1i u2) be a formal group in complex cobordism theory, and u E MU* [[u]] be a series which is the solution of the equation F(u, u) _ 0. Then we have G(x) = u + u E MU* [[x]] ,
where x = uu. Let's take the formal series
X+ = F(u1, u2) F(91,Q, X- = F(ul, u2) F(u1, u2), and introduce the following notation:
01(x1, x2) = X+ + X-,
e2(x1, x2) = X+ X-,
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
126
where xi = ui ui, i = 1, 2, and the series O1(xl, x2), O(xl, x2) belong to the ring MU* [[xl, x2]]; see [30]. These series determine the two-valued formal group: 2 - 01(x1, x2)13 + 02(x1, x2) = 0.
The series O1(xl,x2), G(x) are related as follows: 01(x1, x2) = 2Ao(xl, x2) + A1(xl, x2)G(xl)G(x2).
It is convenient to set
A(x) = Al(x, 0),
8(x) = G(x)A(x).
Theorem 4.1.1 (V.Buchstaber [26], [27]) The coefficients of the series 8(x) E MU* [[x]] coincide with the elements Si modulo two for all i =
1,2,.... Note 4.1.2 We know that 62i+1 = 0 for all i = 1,2,.... So we can identify the elements 6k E MU* with the corresponding coefficients of the series 6(x).
Note 4.1.3 Let us take Sp-manifolds Nk, such that [N1] = 01, [Nk] = 'Pk-1, k = 2, 3, ..., and Sp-manifolds Vk which bounds two copies of the manifold Nk. The Sp-manifolds Vk may be chosen arbitrarily. Indeed let V k ' be different manifolds, such that Wk' = 2Nk, k = 1, 2, .... The exact sequence (4.1) implies that there exist (US p)-manifolds Dk bounding the manifolds Nk, such that bordism classes of the manifolds [Dk U -VV U D'k]MU
coincide with the elements 6k in the group MU* for every k > 1. Following V.Buchstaber we denote the ring E2'* = Hom*A mu (MU*(MSp), MU*) C MU*
by JI* to formulate the following technical lemma.
4.1. CHARACTERIZATION OF THE RAY ELEMENTS
127
Lemma 4.1.2 There exist elements ai in the ring MU. such that the elements S ; + 4ai belong to the ring JI. for every i = 1, 2, ....
Proof. The series 01(x1, x2), 01(x1, x2) have the following form:
01(x1, x2) = 2x1 + 2x2 + E ai,jxix27 i+j>1
02(x1, x2) = (xl - x2)2 + E Ji,jxlx2 i+j>1
The discriminant of the two-valued formal group in the complex cobordism theory has the form O1(xl, x2) - 402(x1, x2) = H(xi, x2) g(x1) . q(x2),
(4.7)
where
H(x1, x2) = A( () 1 A( )1),
q(x) = 52(x) - 4A(x).
By differentiating (4.7) with respect to x1 and supposing x2 = 0 we get
q(x) = -4x + 1:(#I,i - a1,i_1)xi.
(4.8)
i>2
The equality (4.8) implies 2
(x) i
s
- 4x A(x) E JI. [[x11,
(4.9)
1
because the coefficients of the series 01(x1, x2), 02(x1, x2) belong to the ring JI.. Setting i>2
we rewrite the inclusion (4.9) as follows: i-1
5i2 + 2 E Sj_iSi + 4ci E JI.. j=1
(4.10)
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
128
Formula (4.6)) gives the equality 01(x1, x2) = 6(x1) 6(x2) + 2C(xi, x2). H(xl, X2)
The coefficients of the series 01(x1, x2), H(xl, x2) belong to the ring II., so there exist elements bi,3 E MU. such that the following inclusion holds: 6,6i -1- 2bij E JI..
(4.11)
The inclusions (4.10), (4.11) imply that there exist elements ai E JL, such that 6; + 4ai E 1I.. 0
4.2
Product structure in
that the admissible product structure in the theory MGE(.) depends on the choice of the manifolds Qk, bounding Recall (Chapter 2)
the obstructions Pk, k = 1, 2, .... It is clear that the existence of manifolds Qk with a minimal number of singularities is determined by specific properties of the cobordism theory MG.(.) and the manifolds Pk. Such manifolds Qk are assumed to exist in the theory To prove this we shall resort to identifying the ANSS and the E-SSS for the symplectic cobordism ring, and make use of section 4.1 as well as the following geometric consideration.
Let Nk be Sp-manifolds such that [N1] = 01, [Nk] = cpk_1 when k = 2,3,..., where W, are the Ray elements. See [88], [90] for their description. Let us take Sp-manifolds Vi, such that aV = N(l) U N,(2) where Ni(') are copies of the manifold Ni, t = 1, 2. There are the following Sp-manifolds:
for every i = 1, 2, .... Next we take (U, Sp)-manifolds Di, bounding the manifolds Ni : Mi = Ni, i = 1, 2,.... Also we take the following U-manifold:
Mi=D;1)U-VUD'l).
4.2. PRODUCT STRUCTURE IN
129
Figure 4.1: Manifold Mi. All the above manifolds are supposed to have sufficiently large collars; see Figure 4.1. Here aD=1) = N(1) aD(2) = Now we need some (U, Sp)-manifold bounding the manifold R. We set U= = D(1) x D(2) U -V U Di,
where we identify the following submanifolds of the boundaries: D(1) x N. (2)
-N(1) x D(2)
O(DO) x v 2))
a(V(1) x D(2))
N(1) x D,2)
-D(1) x N(2)
(4.12)
Thus aUi = R. Finally we take the (U, Sp)-manifold
Ti=DixNixS1, where the Sp-structure on the circle S1 is such that [S']sp = 01, i.e. [OTi]Sp = c 181, 2
- 2.
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
130
-01)
D(1'1) X D(2'2)
D(1'2) XD (2t2)
X D(2'2)
3
-V(1) X Di2'1)
X
-D,1'2) X U(2)
D(1'2) X Di2'1)
Figure 4.2: Mi x Mi x { I} Lemma 4.2.1 There is the equality in the cobordism group M(U, Sp)., [M?]
sp = 4 [Ui]vsp + [Ti]v,sp .
Proof. Knowing that the manifold Mi has a sufficiently large collar we examine the bottom side of the cylinder Mi x M2 x I. This manifold is glued together as in Figure 4.2.
The manifold corresponding to the central part of the figure is an Sp-manifold. The manifolds corresponding to its numbered parts are diffeomorphic to the manifold
N(')xN(2)xI1x72,
4.2. PRODUCT STRUCTURE IN
Q
131
,<
\N;xNixI
Figure 4.3: A corner of M; x M; x { 1}. preserving Sp-structure.
Now we glue together the manifolds having the same numbers by means of the following diffeomorphism: (ni, n2, ti, t2) M
i (n2i ni, ti, t2),
where n, E t, E I s = 1, 2. This diffeomorhism preserves the Spstructure as well. The corner part of Figure 4.2 is shown in Figure 4.3. The manifolds corresponding to the hatched strips are diffeomorphic to the manifold D= x Ni x Io. As a consequence every boundary Ni x Ni x Io of the manifold
DixN1xII is glued together with the manifold Ni x Ni x Io. So it is possible to
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
132
identify the manifolds corresponding to the identically numbered strips; see Figure 4.2.
In particular the gluing procedure gives the manifold
T, = D, x N1 x St, where the circle Sl has the desired frame as a result of 4-fold inverting
of the factors in the product Ni x Ni. The cylinder Mi x Mi x I is the required manifold which establishes (U, Sp)-bordism between the manifold Mi x Mi and the disjoint union of the U-manifolds 4Ui U Ti.
0 It is an easy exercise to prove
Lemma 4.2.2 There is the equality in the bordism group M(U, Sp)*, 2 [Ti]a,sp = 0,
i = 1, 2,
...
.
Let ai = [Ri] be the corresponding elements of the ring MSp*. For dimensional reasons ai E Tors MSp*, so the element ai belongs to the image of the homomorphism d* of the Adams filtration for every
i = 1,2,...: MSp* 2 -. M(U, Sp)*
d(2>- X (2) Ed(3)' X!3)
E-- .. .
which is induced by the diagram (4.3). According to the definition of the manifolds Ui the elements ui = [Ui]usp of the group M(U, Sp)* are such that d*ui = ai.
Lemma 4.2.3 The projection of the elements ai into the first line ExtAMu(MU*(MSp), MU*) of the second term E2'* of the Adams-Novikov spectral sequence is equal to zero f o r all i = 1, 2, ....
Proof. We have the two following possibilities for the element ui E M(U, Sp)*:
.4.2. PRODUCT STRUCTURE IN
133
1° the element ui is equal to zero or has finite order;
2° the element ui isn't zero and has infinite order. To see what happens in the first case we consider the Adams filtration induced by the diagram (4.4): M(U, Sp)*
dtd(2)0
X(2) td(3)* X(3) Ed(4)* X(4) _ .. .
If the element ui E M(U, Sp)* has finite order then ui E Im
(d(2)
--> M(U, Sp)*)
.
The diagram MSp*
d' M(U, Sp)* (4.13)
MU*
Z(2)
gives that Fr(1)*(ui) = 0, i.e. the projection of the element ai into the first line E1'* of the ANSS is trivial. In the second case the image of the element ui with respect to the homomorphism M(U, Sp)* ---p HomAMu(MU*(M(U, Sp)), MU*)
is not trivial. For simplicity we will denote this image by the same ui. We note that the projection of the element [Ti]M(U sp) E M(U, Sp)*
into the ring HomAMu (MU*(M(U, Sp)), MU*)
is zero since 2 [T]Usp = 0 due to Lemma 4.2.2. The cofibration
MSp A MU -°* M(U, Sp) induces the exact sequence 0 --1 JI*
U (nsP)*
MU* Z HomAMU(MU*(M(U, Sp)), MU*) --; (4.14)
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
134
Lemma 2.2.1 gives the equality p*(S?) = 4u;. On the other hand, there is the element ai E MU*, such that b? + 4ai E A. (see Lemma 4.1.2). Exactness of the sequence (4.14) gives p.(-ai) = ui. Therefore the first Adams differential kills the projection i(1).(ui) of the element ai into the line Ei,* (see the diagram (4.13)):
dl d(-ai) _ (1). o p.(-ai) = Fr(1).(ui). So the image of ai is always zero in the line ExtA cu(MU*(MSp), MU*). 0 Now we can come back to the theory (see section 3.3). Here we intend to make some extensions to Vershinin's Theorem 3.3.5.
Theorem 4.2.4 There exists an admissible product structure it in the symplectic cobordism theory with singularities MSp; such that 1° the coefficient ring MSpE is isomorphic to the polynomial ring
Z [wl,...,wk,...,x2i...,x,,,,...I i where the generators Wk may be presented by any Sp-manifolds Wk, such that a W k = 2Pk, deg wk = 2(2k - 1), deg x,, = 4n, k = 1, 2, ...,
n=2,4,5,...,n 2m-1,
2° for every k = 1, 2.... the Bockstein operator Qk satisfies the product formula Nk(x - y) = Qk(x) - y + x . #k(Y) - Wk - Qk(x)/3k(y), where x, y E MSp*E.
Proof. In terms of the sequence E = (F1,. .. , Pk, ...) we have
Pi=N1, Pk=N2k-2, k>2. Denoting W1 = V1,
Wk = V2k-2,
k>2,
(4.15)
4.2. PRODUCT STRUCTURE IN
135
Lk
-WkXPk
Figure 4.4: Ek-manifold Qk.
we consider the Sp-manifolds
P,=R1, Pk=N2k_2, k>2. We obtain that the projection of the elements
[Pt] SP, k > 1, into the
line ExtAMU(MU*(MSp), MU*).
is equal to zero due to Lemma 4.2.3. We consider the filtration of the ring MSpE MSp* Ery(1) MSpF r(1) 11(2)' MSp; r(2) E (3)' .. .
(4.16)
which induces the E-SSS.
The elements [R&p, k > 1, belong to the image of the homomorphism'y(1)* o y(2)*: E
(Mspr2
Any Sp-manifold Qk, joining the manifold Pk with some EF(2)-manifold Lk, may be considered as a Ek-manifold with boundary Pk (see Figure 4.4).
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
136
Then
8Qk=(PPU-Wk xPk)ULk. A EF(1)-structure on the manifold Lk is determined by the function Sk; see section 1.2. Dimensional reasons give /3 Qk = 0 when i > k. Thus we have [Q'Qk] E
Wk ifi=k, if i#k, 0
(4.17)
according to the definition of the manifold Qk.
It follows from Lemma 2.2.3 that formula (4.15) holds for all k = 112.... (here we use the fact that the numbers pk = dim Pk are odd and that the product structure pk is commutative and associative). As in Theorem 3.3.4, we suppose that for a given product structure p the equality wk = (4q + e)y holds (here e = ±1). By formula (2.8) we get Nkwk = 2Wk + 2Wk - 4Wk = 0.
So /3i y = 0 for every i = 1, 2, .... Let S be a Ek-manifold such that [S]Ek = q y. We Put Qk = Qk U S; then a new product structure µk determined by the product structure pk_1 and the manifold Qk is a polynomial one (see the proof of Theorem 3.3.4). Then we have [/3iQk]E = [/3;Qk]E for every i = 1, 2,.... So formula (4.15) holds.
4.3
Some relations in the ring MSp*
Here we begin to deal with the generators in the ring MSpE. For this we should take some relations between the Ray elements in the symplectic cobordism ring into consideration. We are going to prove only the results that are not published yet. Theorem 4.3.1 (V.Gorbunov [40], [41]) The Ray elements satisfy the following relations in the ring MSp : jcp==0,
e2
Oicowi=0,
i,j=1,2,....
(4.18)
4.3. SOME RELATIONS IN THE RING MSP*
137
we should recall that Concerning the cobordism theory the ring MSpe1 is isomorphic to the ring Z [w1] down to dimension -4; here deg wl = -2 (see Theorem 3.3.15). Lemma 4.3.2 The module MSp0I (CP2) is free over the ring MSpe1
Proof. The second term E2'* of the Atiyah-Hirzebruch spectral sequence for MSpe1(CP2) is isomorphic to the algebra E2,* ,.' H*(CP2; Z) 0 MSpe1.
For dimensional reasons it is clear that all the differentials in the spectral sequence are trivial. So we get the isomorphism MSp91
(CP2) ' H*(CP2; Z) 0 MSpe1.
Let rln be a standard linear bundle over CPn. We denote the bundle in ® in ---4 CPn
by A. Now recall the definition. The k-dimensional real bundle C over X is called symplectic orientable if there exists a Thom class A E MSpk(TC) (here TC' is the Thom space of the bundle (), such that its restriction to
the sphere Sk y T( is a generator of a free MSp*-module MSp*(Sk). In this case we have the Thom isomorphism
,t( : MSpn(X+) --) MSpk+n(TC),
where X+ = X U pt. Let i : X -* T( be a zero section. The element i*(0) E MSpk(X) is called the Euler class of the bundle C. According to [40], [41] the bundle An is symplectic orientable. In addition, its spherical fibration Si bn. RP2n+1 4 = CPn
138
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
is associated to the principal U(1)-fibration Stn+1 ---I CPn,
where the action of the group U(l) on the circle S' is determined by the homeomorphism Si ^_' U(1)/(Z/2).
The Euler class of the bundle An is denoted by P,, E MSp2(CP2). According to [40, Theorem 5], [41, Corollary 4.8] the equalities
Pn - V= O,
>1,
(4.19)
hold for the MSp*-module MSp*(CPn). We denote the generator of the MSpe,-module MSpe,(CP2) by x. As noted above the element x2 E MSpe, (CP2) is a generator as well. These groups are connected by the following triangle: MSp*(CP2)
AMSp*(CP2) (4.20)
MSpe, (CP2)
The proof of the following result is due to V. Gorbunov.
Theorem 4.3.3 The equality irl(P2) = 2x
- wlx2
(4.21)
holds in the MSpe, -module MSpe, (CP2).
Proof. Suppose y is a generator of the group MSp4(CP2). The AtiyahHirzebruch spectral sequence converging to MSp*(CP2) gives y.O = 0; see [43]. The triangle (4.20) gives Sly = x. For dimensional reasons we have Sl(x2) = 0. The Atiyah-Hirzebruch spectral sequences for MSp*(CP2) and MSpe, (CP2) give the equality 7rl(P2) = 2x + axe,
(4.22)
4.3. SOME RELATIONS IN THE RING MSP*
139
where a E MSpel. Applying the homomorphism bl, we have the equality 0 = 2y + Sl(a)y. This holds only for the element -wl, i.e.
a = -wl. Corollary 4.3.4 The relation Vi wl
0 holds for the MSp*-module
Mspe, .
Proof. The relations (4.27) imply the equality 7f1(Vi) 7r1(P2) = cpi(2x + wlx2) = 0.
Lemma 4.3.2 gives the desired relation.
Now we come to particular generators wk of the ring MSp*
Z[w1,...,wk,...,x2i...,xn....II
to give a geometric representation of the elements x2;. According to Corollary 4.3.4 the Massey triple product (p2k_2, 2, 91) contains a zero for every k > 2. By the definition of the Massey product there exist Sp-manifolds Wik), Wk, Yk,l, such that aYk,1 = Wik) X Pk U P1 X Wk.
We have to compare the Sp-manifolds Wlk) and W(') for different numbers k, m. The group MSp2 contains a unique nontrivial element 61, hence the above manifolds are Sp-bordant with respect to their boundaries up to a manifold lying in the bordism class of 91.
Note 4.3.1 If the manifolds W1 are considered as 81-manifolds then they are 01-bordant for different k. 0 There is an important generalization of the above results; the proof below is due to V.Gorbunov.
Theorem 4.3.5 (V.Gorbunov, 1990) For i = 24'1 , j > i each Massey product (qi, 2, ci) contains zero in the ring MSp£°.
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
140
P, XPm
W. X P1
Figure 4.5: Ek-manifold X,.,1
Proof. It suffices to prove that qjwi = 0. This was proved for i = 1 above. Let us assume that it's true for j = n - 1. We consider the diagram n_1(CP2"+1_2)
MSPE
According to [41] there exists an element d E MSp2(CP°°) which annihilates all elements 0_. We denote by do the element 7rn 7r,(d). By the main result of [41] MSpEn(CPZn+'-2) is free over MSp£n, so the element d may be written down as a sum E=>o a=z`, where a; E MSp*EnAs above we conclude that ao = 0, a2n+1 _2 = wn. All other coefficients ak, 1 < k < 2n+1 - 2, have the degree 4k + 2, so they depend on w;
for i < n. Since the qj are of order 2, the statement of the theorem follows. 0
4.4. LOCALIZATION OF BORDISM THEORIES
141
The manifold X.,1 = Ym,l U -Vm X P. U S.
is shown in Figure 4.5. Here the manifold Sm bounds the manifold p12
X Pm (Sm exists due to Theorem 4.3.1).
The elements of the ring
which are determined by the E-
manifolds Wi21, Wm, Xm,l, will be denoted by w1, wm, Xm,1 respectively.
4.4
Localization of bordism theories
Here we focus on the localization procedure of bordism theory with singularities. We'd like to describe it in the form which will be convenient for our purposes, putting aside its more general versions. The procedure itself will be applied in the next section.
We start with the sequence of the closed manifolds E = (P,,..., Pk.... ) in the bordism theory MG.(.). As before we suppose that the sequence E is locally finite, Ek = (P1, . .
.,
Pk) for every k = 1, 2, ..., and
is the corresponding bordism theory with singularities. All theories MGEk are supposed to possess admissible commutative and associative product structures compatible for different k (in the sense of Theorem 2.2.2). The pairing of the Ek-manifolds which generates the above product structure in the bordism theory will be denoted by mk. We use the notation MGEk
Mn = mk(M, mk(M, mk(...) ...)) n times
for all Ek-manifolds M. In addition, we take also the E1-manifold W, dim W = a > 0, which may be considered as a Ek-manifold for every k > 2. The element [W]£k is denoted by w for every k > 1. Finally the product homomorphism MG511 * MGR+an
is supposed to be nontrivial for all numbers k and n.
(4.23)
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
142
Definition 4.4.1 The Ek-manifold M is called a 6k,n-manifold if we have
(i) a partition of the boundary SM = DoM U DnM of the Ekmanifold M into the union of Ek-manifolds such that
6DoM=6DnM=DoMf1DnM holds,
(ii) a decomposition into the product of Ek-manifolds on the submanifold DnM of the boundary SM : DnM --1 mk(Wn, 6nM) (i.e.
is a diffeomorphism preserving the G-structure).
Definition 4.4.2 A singular CSk,n-manifold of the pair (X, Y) is a pair (M, Do), where M is the 6k,n -manifold, and the map
f : (M, DoM) -' (X, Y) is a singular Ek-manifold such that its restriction f I the following composition:
coincides with
mk(Wn, bnM) ?'. (bnM)E _L X.
Here pr is a projection on the model of the Ek-manifold bnM, f is a continuous map.
Note 4.4.1 The above notions are an obvious generalization of manifolds with singularities. According to R.Stong [103] we have that the bordism theory
of these manifolds is well defined.
Note 4.4.2 The bordism theory MG'k,^(.) obviously is provided with a module structure over the theory MG.(.). This module structure will mean a left module structure. In general it is not clear that the theory has a module structure over Such module structure certainly does exist if a ring structure of the spectrum MGEk may
4.4. LOCALIZATION OF BORDISM THEORIES
143
be extended up to Ham-structure. To examine the problem seems to be interesting. Further we do not use a module structure over MG*k (); in our case it is sufficient to know that a multiplication by wk commutes with certain transformations. 0
Now it is convenient to consider the Bockstein-Sullivan sequence, exactness of which may be proved in the same way as in section 1.2: n
MG*-
MG*k ()-^+ MG?kn (.) -"+ MG*k (.) -, .. . (4.24)
We obtain the following commutative diagram whose lines are exact:
MG*-(.)
... _'
I'
MG*^(.)
Id
MG!1.1() 6i W1
W1
1
MG?1 .2 (.)
62
.. .
'W1
Id
.wl
Id I
MG*^(.)
wn
W1
1
w1
.wi
Id I
I
MG?'n(.) W1 I
'k+' +MG6 1,n+1 (.)
MG*^()
b,+1
...
I
Id
W1
w1 I
(4.25)
Note that the transformation wn at the level of manifolds is the multiplication by the manifold Wn on the left. We have a direct limit of the diagram (4.25),
MG
0.
wi 1MG*^(.)
where
MG*^(.)
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
144
MG; k
lim(MG!'''
Z MG!' 2 (.)
'
...
_ ...).
is called a w-localization of the bordism The theory wi theory MGEk(). The homology theories
may be considered as ordinary bordism theories. To give a complete picture we give the definitions.
Definition 4.4.3 An ordinary Ek-manifold M is called a manifold in the theory w11MG* n the manifolds M and N are bordant in this theory if there exists a number i such that the Ek-manifolds M M. W' and
N W' are Ek-bordant. 0 Definition 4.4.4 A 6q,k-manifold M of dimension m + aq is called a When N is some manifold of dimension m in the theory other 6q',k-manifold of dimension m + aq', then M and N are bordant in the theory if there exists a number j such that the 6j,k-manifolds Wj _q M and Wj_q N are 6;,k-bordant. O .
Other attributes of the bordism theories
wi
and
may be defined in a standard way.
Now we need some more properties of the E-SSS. The Ekr(n)manifold M is glued out of the blocks y,,M x P°` (see Definition 1.4.1), where
Pa =P1 x...xPk al
ak
al-}-...-}-ak=n,
for every number collection a = (al, . . . , ak). A similar notion may be introduced in the case when the original bordism theory is the bordism theory with Em-singularities, where m < k (instead of the theory MG.(.)). In other words a Em-manifold M is called a Ek,mP(n)manifold, which is glued out of the blocks y,,,M x P« , where 7,,M are Em-manifolds,
p,
X pa,;
al+m + ... + ak = n,
4.4. LOCALIZATION OF BORDISM THEORIES
145
for every collection a = (am+1, ... , ak). Compatibility conditions are required here similarly to Definition 1.4.1. The complexes of the theories
... _, MGEk(.) 04 ...
MGEk(.)
(4.26)
form the lattice of the complexes (where i = k + 1, k + 2,. . ., signs of Pi are as in section 1.3). Suppose Tk is a total complex of this lattice: MG+Ek,m(1)(.)
MGEk,m(2)(,)
R
Qk,m(2).
Qk.m (3) MG£k,m(3)(.)
...
As before we have the transformations ?km(n) : 7rk,m(n) : MGEk,mr(n)(.)
ak,m(n) :
MGEk,m(n)(.)
MGEk,m(n+1)(.),
i
--
MGEk,mr(n)(.).
Finally we come to the following theorem, the proof of which is similar to that of Theorem 1.4.2.
Theorem 4.4.5 There is the following exact triangle of bordism theories: 'Yk,m(n+1)
MGE k,mr(n)(.)
r(n+1) (.)
(4.27) MGEk,m (n) (. )
f o r every n = 1, 2, ...; also the equality 7rk,m(n) o ak,m(n) = Qk,m(n) holds.
So we have the bigraded exact couple MGEk . 'Yk.m(1)
n,/Ek,mr(l)
IYl G*
'Ykm (2)
MGrF'k,mr(2)
...
ak,m(1) ,/ \ ak,m(2) lrk,m(1)
MGEk,m (1)
- MGEk,m (2)
Qk,m (1)
,3k.m(2)
(4.28)
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
146
Recall that the E-manifold W is such that /3tW = 0 for all i = 2,3,.... So the transformations
MG;-(.) for m > k + 1 may be extended up to the transformation of the exact triangles rykm ( )
MG; k
MG*Ek. mr(1)
Yk,m(2)
MG*Ek mr(2)
Yk,m(2)
MG*Ek mr(2)
ak,m(1) irk,m (1)
7k,m (0) MGEk,m (1)
w
w
MGEk
MG* k,mr(1)
7rk,m (0)
w
(4.29)
Setting a direct limit we obtain the exact couple w-1MGEk Yk,m(1)
ak,m(1) / Irk,
w-1 MGEk,m (1) *
\
w-1MG;k,mr(1)
Yk)w-1MGEk,mr(2) ak,m(2)
-. w-1 MGEk,m (2) Qk,m(1)
Qk,m (2)
(4.30)
4.4. LOCALIZATION OF BORDISM THEORIES
147
The following simple lemma holds.
Lemma 4.4.6 If the spectral sequence associated with the exact couple (4.28) converges to MGEm (X, Y) then the spectral sequence associated with the exact couple (4.30) converges to w-1 MGEm (X, Y).
In other words the operation of w-localization extends up to the operator of w-localization of the E-SSS spectral sequence and preserves its convergence.
Note 4.4.3 The operator of introducing 6n,k-singularities may be extended up to the operator of the spectral sequences with the same properties.
Now we would like to apply the above constructions to the symplectic cobordism case.
We consider the exact couple (1)
MSp;1
MSPE1,.r(l)
ai,00 (1)
W1,00(e) \
\
y'
MSpE1,.r(2)
lr1,00 (1)
MSpE1.. (1)
MSpE1,°°(2)
Q1,.(2)
(4.31)
which induces the E-SSS for the ring MSp;1. In particular we have the complex (MSpE1.°°(1))(2) Q'=(1 (MSpE',°°(2))(2) a' l
(2) i
(MSpE'.°°(3))(2) -+
...
(4.32)
which will be denoted by M. Now we recall some of its properties.
1. The homology of the complex M coincides with the algebra ExtABp (BP*(MSpe' ), BP*).
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
148
2. The complex. is a total complex of the lattice the lines of which are the complexes
(MSp*)(2) k' (MSp=)(2)
ak
(4.33)
(MSp£)(2) - ...
for k = 2,3.... (here signs of the homomorphisms /3k are as before).
To apply the w-localization operator we put w = w1, where the element w1 is as in the section 4.3. At first we apply the w-localization operator to the theory MSp£ gives the isomorphisms (wi 1MSpE)
wi 1MSp£,
(4.34)
(MSpe/wi°) ^_' MSpF-/wi°.
(4.35)
So in this case the exact Bockstein-Sullivan sequence has the following form:
0 -> MSp; -* wi 1MSpE -- MSpE/w.--+0. Now let us consider the cobordism theory wi'MSpe' Bockstein-Sullivan exact sequence
(4.36)
We have the
... -> MSpe' - (wi 1MSpe' )* ') (MSpe' /wi°)* -> ...
(4.37)
which according to the above extends to the exact sequence of complexes:
MSpe' 1
0 --> MSpE'.°°(1) Q1,.(1) 1
in
(wi 1MSpe' )* 1
wi 1MSp;'.°°(') 01.°°(1) 1
0 -* MSpE'.°° (2) -=, wi 1 MSp;'.°° (2) Ql.°o(2)
/31,°°(2) 1
(MSpe' /wi°)* 1
MSpE',°°(1)/w10° -40 01.°°(1) 1
MSp;'.°° (2)/w1o° - 0 /31,00(2) 1
(4.38)
4.4. LOCALIZATION OF BORDISM THEORIES
149
We can rewrite diagram (4.38) without the first line:
0 ---> M --p wi 1 M --+ M /wi° --1 0.
(4.39)
Then we have the long exact sequence
0--+ Ho(.M)--pHo(wi1M)-i
Let M. = MSpE. Recall that we already have constructed the generators x2k_2 = xk,l of the ring MSp*E for every k > 2.
Lemma 4.4.7 The complex wi 1M is acyclic, i.e. H;(wi 1M) = 0 for every i > 1.
Proof. Note that the complex wi 1M is a total complex of the lattice the lines of which are the complexes £Q'):
wi 1M
W1-IM - ... - wi 1M
...
As before it is convenient to present the complex f(k) as a differential graded algebra W1 1M, [Uk] with a differential determined as follows: Nk(x - uk) _ I3k(x)
Uk'+1,
n = 0, 1, 2, ... .
Here Uk are generators of dimension 2(2k - 1), x E w-1 M.. The lattice of the complexes the lines of which are the complexes C(k), k = 2,3,. .. , n, is denoted by C(n), and ,C(n) is its total complex.
Lemma 4.4.8 The complex ,C(k) is acyclic for every k = 2,3,....
Proof. We have the equality
=1 13k(x!2) W1 in the ring w11 M. for every k > 2, i.e. the equality Nk
w'kuk =uk 1, n=0,1,... 1
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
150
holds for the complex ,C(k). Let x E Ker /3k, and we obtain
n>1. Lemma 4.4.9 The complex £(n) is acyclic for every n = 2,3,.... Proof. The case n = 2 follows from Lemma 4.4.8. Let us make the induction step. The complex £(n) may be considered as a total complex
of the double complex made up of the complexes C(n - 1) and L(n). The second term of the double complex spectral sequence is E2,* ti H1 H11
Here H11 is a homology of £(n - 1), HI is a homology of £(n). By induction we have EZ,* = 0 for every s = 1, 2.... and the line E2,* coincides with a homology of the complex Ho (.C(n - 1)) -a2 Ho (,C(n - 1)) -n-+ Ho (,C(n - 1)) ---> ...
(4.40)
The element
lies in the group Ho(L(n-1)) C wi 1 M*. Actually the ring Ho(C(n-1)) is a subring of the ring wi 1 M* by induction: n-1
Ho(,C( n-1))= n Ker (wi4M .&wi1M*). k=1
According to the properties of x1,k we have Nkxl,n = 0 for every k So if
1, n.
x E Ho(,C(n-1))flKer13n, then we get the equality
/
/3
1
q=1,2,...,
so the complex (4.40) is acyclic. Thus Lemmas 4.4.9, 4.4.7 are proved.
4.4. LOCALIZATION OF BORDISM THEORIES
151
Corollary 4.4.10 The coefficient group of the bordism theory wi 1 is torsion free, and there is a ring isomorphism
n Ker (wr1Mspw1MSp)
(wi 1
.
O
(4.41)
k>2
has an admissible Note 4.4.4 The bordism theory wi product structure, and in particular the ring (wi 1 MSp°1)* is a direct limit of the rings. The computation of the ring (wi 1MSpe1)* is to be dealt with in section 4.5. 0
Corollary 4.4.11 There exist the exact sequence
0 -> Ho(M) - Ho(wi 1M) -'-i Ho (M/w1°) -L H1(M) - 0 and the isomorphisms
Hi (M/wr) =-' Hi+1(,M), i = 1, 2, ...
.
0
Note 4.4.5 We consider the 01-manifold X1,i,
X1,14
in the bordism theory MSp*6',for every number collection I = {i1, ... , iq}. Here the manifolds X1,i, represent the elements x1,i,, and denotes their 01-product. The equality 4
b (x1,11 ... x1 iq) = r uit (x1 it ... x1,it ... x1,{,) t=1
holds due to the definition of X1,i,. The projection of b (x1,11 ... x1,iq)
into the term E2'* of the E-SSS for the bordism theory
in the zero line, i.e. it maps into x1 it ... x1,ig W1
E MSpE/wi°.
lies
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
152
All higher differentials act trivially on this element since it is originally Therefore the represented by a manifold in the theory MSp?''°° element 9
B(il,... , 2q) = E ui, (x1,=1 ... 2l ii ... xl tq) t=1
being represented by a manifold with 01-singularity, is a cycle of all differentials in the E-SSS for the ring MSp;' (the same is true for the corresponding ANSS).
Though quite obvious the above observation is very important for us.
1) The element B(il,... , iq) coincides with the Ray element S02'1-2+...+2'e-2
up to decomposable elements (we mean coincidence in the ring MSp;' ). The details will be clarified in the section 4.5.
2) The elements B(il,... , iq) are decomposed into the Massey triple product by definition. That is, we have B(i, j) E
w1, V2s-2)
B(il,...,iq) E (B(il,...Iiq-1),w17V2a-2) is entirely suitable to detect 3) So the bordism theory MSp*5''°° torsion elements of the ring MSp;'. Many problems dealing with torsion may be adequately formulated in terms of this bordism theory, and interpreted as a simple version of the chromatic machinery of Ravenel; see [841. O
4.5
The generators of the ring MSp*
Recall that we have defined the maSS-filtration in the ring M, _ (MSp*E)(2) which induces the maSS; see sections 3.3-3.5. Suppose
M. = (MSp*)(2)
4.5. THE GENERATORS OF THE RING MSP;
153
is a ring associated with the ring M. (with respect to the maS filtration), ,(3k is an operator associated with the Bockstein operator /3k for every k = 1, 2, .... The polynomial generators c,, E V. were defined in Theorem 3.2.2, the operators /3k act as described in formulas (3.13)-(3.17).
Now we propose to define generators of the ring M., such that the action of the genuine Bockstein operators on them is similar. We need to recall some notation.
Let n = 2m-1 and m = 2ii-2+...+2i9-2 be a binary decomposition of the number m, where 2 < i1 < ... < iq, q > 3; then the generator xn is denoted by xi...... i4; if n = 2i-1 + 2j-1 - 1, for 1 < i < j, then the generator xn is denoted by xij.
Note 4.5.1 If the number n is even and isn't equal to a power of 2, then the generators en E M. are cycles of all differentials in the maSS. So the class cn contains the element xn E M. such that the first Adams differential acts trivially on it. This means that /3kxn = 0 for every
k = 1,2,.... The following theorem is the main result of this section.
Theorem 4.5.1 There exist polynomial generators xn, where deg xn =
4n, n = 2,4,5,..., n# 2'z - 1, in the ring M. = (MSp*F)(2), such that for all k > 2 the Bockstein operators /.3k act on them as follows:
10 if n = 2i-1 +2j-' -1, 1 < i < j, then Nk(xij)
2° if n = 2'1-1 + ... + Qk(xi1..... iq) =
2i9-1
wi
Wi ifk=i, ifk i,j; 0
(4.42)
- 1, 2 < i1 < ... < iq, q > 3, then
w1 - xi, ,... ,s,,...,iq 0
ifk = j,
if k = it,
ifk
i1i...,iq;
(4.43)
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
154
3° if n is an even number and is not a power of 2, then Qkxn = 0,
for all k = 2,3,...
.
(4.44)
Note 4.5.2 In particular the statement 1° of the theorem means that there exist generators x1,k, such that ,Qkxl,k = w1, $ixl,k = 0 for every i k > 2. Elements of this kind were determined in section 4.3. Notice that the elements cl,k = c2k+1 are such that Qlel,k = hk,
QkCl,k = h1.
While computing the ring (MSp£)(2) (see Theorem 3.3.4) we used an arbitrary representation of the elements cn as polynomial generators. So we can assume that the elements X1,k are polynomial generators of
the ring M. Proof. The induction assumption is
(+) The generators x,a of the ring M. are determined for every
n
Note 4.5.3 We are going to deduce several corollaries from the assumption (+). The statements depending on the induction assumption will be marked by the sign (f), for example: Lemma 4.5.2M. First we examine the ring w-1 M up to the corresponding dimension. That is, we consider the generators xij, xi1..... i9 of the ring M. whose multi-indexes satisfy the inequalities 2i1-1
+... + 2i4-1 - 1 < N,
2i-1 + 2'-1 < N.
Lemma 4.5.2 M The elements Xi = 2 W1 - wi, (4.45)
Xi,3 = xi,7(xij - wiw1),
V = w2, i = 2,31 ... ,
4.5. THE GENERATORS OF THE RING MSP
155
q-2
Xil,. 49
(-1) k
xil,...,iq +
X1
k=2
1
4
(-1)q_1
-}
,it, ... xl itkxi 1...i tl...'k... q St
(ti witxl,il ... i1 it ... xl ig
- 2x1 {1 ... xl ig
wl
where q > 3, 1 < it <
(4.46)
w2
< iq, belong to the ring
Ho(wi'M.) ' Ker $
C wi 1M..
Proof. The statement concerning the elements Xi, Y, V1 is obvious and doesn't depend on the induction assumption (+). The statement concerning the elements X1,3 is also obvious but it depends on (+). As for the element Xi,,,,,,iq E Ho(wi 1M.) the proof will be the following. We put
Xii,k)..,iq =
E
1
xl,itl ... xl itkxil,...,=a ,...,4--i')
where k = 1, ... , q - 2. Due to formula (4.43) we have the equality
E
Q(1)X k)..,iq = w1 1
xl itl ... xl,itk A(il, ...
itk, ... , iq)
1
+W1
xl it1 ... x1,itk_1 A(ii, . . . , 2t1, ... , ttk_1, ... , 2q), 1
where s.1
It remains then to prove that
r
x1 it ... xl itl ... xl it2 ... xl,igA(it1 , it,)
1
Q\1)
(t=i9 witxl it ... X1 it ... xl ig wl
- 2x1 it ... x1,iq W1
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
156
Indeed we have due to formula (4.43)
NO () = E xl i1 ... xl it, ... x1 i" ... x1,igA(it1, it2) 1<-t1
q
+2w1 1 E wiixl,il ... xl i, ... x1,14 - 2wl 1 E wiixl,il ... xl io ... xl,iq. t=1
t=1
So we have /3(1)Xi1..... iq = 0.
Note 4.5.4 We have the relation
X2=4V+V.
(4.47)
We consider the following subring of the ring M.:
C.=7(2)[xn Ineven, n# So we have the isomorphism 11
wi 1 M.
wl 1Z(2) [wl, ... , wk, ... , Xi, ... , Xi,.t, ... , Xi1,...,iq, ...J ® C*,
which holds for the same dimensions as implied by (+). Suppose A. is the polynomial ring A. = Z(2)
[X2,...,Xk,...,Y2,...,Yn,...].
We note that the elements V lie in the ring A. since
V =X?-4V, i=2,3..... So we can conclude:
Corollary 4.5.3 (+) The isomorphism Ho(wi 1M)
n Ker (wr1M* - wi 1M.) k>2
Z(2) [wl, w11 ] ®Z(2) [Xi,j, ... , Xi1,...,iq,
holds in the same dimensions as implied by (+).
...] ®C. ®A.
(4.48)
4.5. THE GENERATORS OF THE RING MSP£
157
Now we take k, such that 2k+1 < N. We can change the variables in the ring M. as follows:
1) x() = xi,j; .7
2) i f k
{i1
, . .. , iq}, then we put x21, .,iq
3) if k # it, then we put (k)
xil,...,tq = xil,-..,iq
- xl,itx...,tq.
Lemma 4.5.4 (+) For q > 3 the generators x(k.. iq satisfy the formula i1 I. Ptn(x(k) ,iq) = 1 wl x(k)
t .....
0
Proof. Let m = it
q
ifm=2t#k, if m=k,m#il,...,iq.
k = i3... Formula (4.43) gives that ,
Ntt xti' ..,iq = I3it('xii,...,iq - xl+itxii,...,it,...,iq)
. - wlx.
= wlx.
wlx(k) .
If m=k= it, then,
pitxji' /=
. - wlx.
wlx.
^
.=0
.
Note 4.5.5 We note that the action of the Bockstein operator Ni on the generator x(,)k differs from the action implied in formula (4.49), i. e.
Lemma 4.5.5 (+) For all numbers i, j, il, . . . , iq, satisfying 1 < i < j, and 2 < it < ... < iq, q > 3, there exist elements Yj, Y...... jq such that Q(1)Y,; = 2(uiwj + ujwi) = 2A(i,j),
(4.50)
q
ui,xi......i...... i = 2A(il,
fi(1)Y1,...,iq = s=1
... , iq).
(4.51)
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
158
Proof. We put Y,j = wi wj; this case doesn't depend on (+). In the case q = 2n > 2 we consider the sum
E
(4.52)
xl,itl ... xl itnxil,...,itl...,t:n,...,t2n
1
It is clear that for every set of numbers {t1, ... , to}, 1 _< ti < to < 2n, there exists the following complementary set:
... <
{T1i...,Tn} _ {1,...,tl,...,t2,...,tn,...2n}. So the summand xitl ,...,itn - x=Tl ,...,iln
enters into the sum (4.52) together with the summand xirl --iln - xit1 ...,itn
We denote S
E- (-1)nw1 2
1
X1,41
... xl'ttnxtl.... ,i..t
1
.
,...,i.
t n ,...,i2n'
It is clear that the element S
2n
Yl,...,tq 8=1
n
E
w E(-1)k k=2
w,,xil,....IS-.,i2n -
x..
E
.. X.
1
satisfies formula (4.51). Let us examine the case when q is odd. It is obvious that the element Y,j,k = ((xi,jwk + xi,kwj + xj,kwi) + wiwjwk) satisfies formula (4.51)
4.5. THE GENERATORS OF THE RING MSP
159
Let q = 2n+1 > 3, I = {il, ... , iq} and p = iq. The generators x(p) i9 are taken from Lemma 4.5.4(+). By their definition we get the equality 2!11 A(il,... i2n,p)=Es=uisxi1+uPxil...,izn
2n 3-1 ui sx(P)
_
tt
(4.53)
....12n,P
2n
(P) 3-1 utaxil+ upxi(P)1, ,t2n
x1,P
So the element to be bounded has the following form:
2A(il, ... , iq) = 2 _2n1
,uiax(P) ^ i1
,P
(4.54)
+2x1,P
Y:n1 uiax(P) 7
+ 2upx(P)
i
We put Z(1)
2n
L
s=1
Z(k)
= wl
Wiax(P)
(4.55)
it
P
E 'x'til ,...,ttk xi] 1
(P)
where k = 2, ... , 2n - 1, 2n-1
7i(O)
= z 1) + > (-1)k-l-1(k). k=2
Formulas (4.53)-(4.55) imply Q(1)Z(o)
=
- wix1,P
(
2
wl
1
\xituia + xiauit)x
LI (xituia + xiauit)xil.... ,i2n
1
2n
+2w1 E uitx(P) i] t--tt,...,t2n,P t=1
(4.56)
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
160
Let's take 2n Z(*)
= xl,P E wtexwpx:l,...,i2n7 s=1
2n
xia,Px't l ,...,i a ,..42n
w1 s=1
Then let's compute the action of /3(1):
ui,xtl ,...,ia,...,i2n
3(1)Z(') = 2x1,P s=1 22nn
(4.57)
(xitut8 + xs,utt)x
+ wix1,P 1
2n
2n
wl (t wi°xil.... ,te,...,t2n) u + 2xtl...,i2n UP + w1wP 1: uit xtl s=1
t=1
/
2n
0(1)Z(*,*)
Euitxl+ wl - w1wP t=1
12n s=1
wiaxuP (4.58)
+wl
1
(xituia + xi,uit)x
By comparing the equalities (4.55)-(4.58) we have
NO (Z(°) -
Z(*)
- Z(*'*)) =
2A(ii,
... , i2n+1).
El
Note 4.5.6 The elements 2xi1..... i9 - wlil..... ig E M*
belong to the ring JI* I = HomABp (BP*(MSpel ), BP*),
in the same dimensions as implied by (+). These elements are adjoined to the elements h°ci,..... iq in the term El'*'* of the maSS for the spectrum MSp°I.
4.5. THE GENERATORS OF THE RING MSPF-
161
The action of the Bockstein operators Nk and the corresponding adjoint operators Qk are similar, as illustrated by
ifk=i,
Qk(hi)
ifk 7i,
0ho
hi
ifk=j,
h,,
if k = i,
0 (Ci,,...,
_
i,j,
ifk
i) =
r h1 c.
fk(wt)
2 ifk=i, 0
ifk # i,
(4.59)
wi ifk=j, /3 (xij) =
w3
0 1
if k = i,
ifk#i,j,
if k = it,
ifk=it, 0
(4.60)
(4.61)
(4.62)
The complex associated with the complex M with respect to the maSS-filtration is denoted by .M. Comparing formulas (4.60)-(4.62) gives
Corollary 4.5.6 (+l The homology algebra H*(M), which is isomorphic to the algebra ExtABP (BP*(MSpe' ), BP*)
in the dimensions implied by (+), is associated to the homology algebra H*(3t), which is the second term of the maSS for MSpe'. In other words, all higher differentials are trivial in the maSS for MSpe'.
Note 4.5.7 The above corollaries of (+) describe the structure of the ring wi 1MSp;' in the corresponding dimensions, but do not provide enough information to complete the induction. We can only conclude that there exists a new generator xi,..... iq such that
f3(1)xt,,...,iq = A(il,... , iq) + D. So we need some additional geometric information.
162
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
We know from section 4.3 that the elements q
B(ii,... , 2q) = E ui,xil,l ... 2it,1 ... xtq 1 t=1
belong to the ring MSp;1(as they are represented by 81-manifolds). Besides they belong to
H1(M) ^' ExtABp(BP*(MSpe1), BP*) by definition of the E-SSS. Let (pm be a Ray element, where m = 2i1-1 + ...+2iq-1, where q > 3. According to V.Vershinin and V.Gorbunov (see
Theorem 3.3.2), the projection 4tm of the element w'm into the initial term of the maSS has the form q
m = E'itCil,l ... Cit,l ... Ciq,l + t=1
CJ . 4)J,
(4.63)
J
where q > 3, ui is a projection of the elements 01, j = 1, and cp2;_l, j > 2, -DJ are products of the elements %, the elements cJ belong to the ring Ho(M) and are cycles of all differentials in the maSS. We recall (Theorem 3.2.2) that the element cpij projects to 4 i,A = uicl,7 + u cl i + E CJ-Dj. J
Let xJ be elements adjoined to the elements cJ, xJ E JI;1. We obtain
that the element q
B(il,... , iq) + E xJ -ti E ExtABP (BP*(MSpo1), BP*)
(4.64)
t=1
is the image of the projection of the Ray element cpm into the algebra ExtABp(BP*(MSpo1), BP*)
which is the second term E2 of the ANSS. The fact that the Ray elements have order 2 and formula (4.63) allow us to conclude:
4.5. THE GENERATORS OF THE RING MSP
163
Lemma 4.5.7 The equality
2B(il,... , iq) = 0 holds in the group H1(M). Now we examine the ring J1 ' in dimensions 4k + 2.
Lemma 4.5.8 Every element x E J14;+2 has the form x = x'w1, where e1 xEJl4*.
Proof. The cofibration
E2MSp-14 MSp '+ MSp°' gives the exact sequence
0 -+ JI* `+ jig' a'> JI*
ExtABP(BP*(MSp), BP*) ->
(4.65)
We remark that J14*+2 = 0. Let x E JI41*; then Qkx = 0 for k = 2,3.... due to the definitions. For the element y = 2x - w1Q1x we have
thy=2/31x-2/31x=0,
/3ky=2#kx-w1QAx=w1Ql/3kx=0, k>2. So y E J14*+2 = 0 and 2x = w1Q1x. This equality holds in the polynomial ring M, so x = x'wl. Also wl/3kx' = 0, if k >_ 2, i.e. /3kx' = 0. So x' E J14;.
Now we are able to make a final induction step.
Suppose xN+1 E M. We already have the generators for the case when N + 1 is even.
Let N + 1 = 2''-' +... + 2'9-1 - 1. Then we take the element of the ring wi 1 M*,
y=
2x1,11
... x1,ie W1
9
- E wioxil 1 ... 1 xitj ... xiq 1 t=1
CHAPTER .¢. FIRST DIFFERENTIAL OF ANSS
164
q-2
+w1
r Lam(-1)k-q k=2
E 1
xI ssi
..
x1'=tkxsi,...,itl,...,isk ....,sq
Lemma 4.5.2(+) implies that this element has boundary
/x(1)1' = (-1)gwiA(i1.... , iq). On the other hand the homomorphism wi 1 MSp ---* MSp£ /w00°
maps the element Y into 2x1,1, ... x1,ig w1
which lies in Ho(M/wi°). We consider the exact sequence
0 - Ho(M) -) Ho(wi'M) -'-> Ho(M/wi°) 6i HI(M) -* 0 The equality
.
(2t, ... x1,ig
/
wi
holds by definition of the element B(i1, ... , iq). Lemma 4.5.7 implies that there exists an element z E M, such that
/3(1)z = 2B(ii,... , iq). So the element
z* = Y -
2x1,1, ... x1,ig Wi
-zEM
has the boundary #(1)z* = (-1)gw2IA(i1, ... , iq). Now we take the element Y,,,,,,iq from Lemma 4.3.5(+). The element q
2
2z*
4.5. THE GENERATORS OF THE RING MSP
165
has a zero boundary, i.e. it lies in the ring J1401+2. Lemma 4.5.8 implies
(-1)qw21Y- 2z* = w1R, where R E JI4;. This equality holds for the polynomial ring, so the element z* is also divided by w1, i.e. z* = w1S. We have wi11,...,iq = 2(-1)qS E JI4*,
so there exists the element w1A(il, ... , iq).
E M, such that Q(1)Xi,,...,iq =
Now we are through with Theorem 4.5.1.
Note 4.5.8 Existence of the elements xij, Xi,j,k may be proved independently of the above constructions. The proof is based on the results of V.Buchstaber [26], [27], R.Nadiradze [73] and L.Ivanovskii [46, 47]. So we come to the following corollaries of Theorem 4.3.1.
Corollary 4.5.9 There are isomorphisms
\
n Ker (wr1M* - wi 1M* I k>2 / Z(2) [w1, W-1] ® Z(2) [Xi,,, ... , Xi,,...,iq, ...] ® C. ®A*, (7x*wi 1 MSpe')(2)
where A. and C. are the following polynomial rings: A* = Z(2) [X2, ... , Xk, .... Y2i ... , Yn, ...] C* = Z(2) [xi
I
even, i
23]
.
Note 4.5.9 We note again that the generators Vi lie in the ring A*
Vi =X?-4Y, i=2,3,...
.
Corollary 4.5.10 The higher differentials of the maSS for MSpe' are trivial.
Note 4.5.10 The statement of Corollary 4.5.10 was conjectured by V.
Vershinin (for the spectrum MSp). The same statement is true for every MSpEk when k > 1.
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
166
4.6
Some notes
Not once has it been observed that the element 01 E MSp1 possesses some specific properties in the ring MSp*. So it seems reasonable to `split' the problem of computing the symplectic cobordism ring. That is, we take the bordism theories with singularities MSpI' (), where E = E\ {P,}, to obtain the commutative square
(4.66) it
Two vertices of the square are known: these are the theories MSpE(), (the latter, being very similar to the bordism theory was computed in [14]). So all the main problems of the ring MSp* are hidden in the ring MSpI'. The problem still remains of gluing the ring MSp* out of the rings MSpI' and MSp; . We have described the first Adams differential for MSp°'. Now let's examine the result obtained more carefully. We consider the following subring of the ring M* = (MSp£)(2): M(0)* = Z(2) [W1.... , Wk, ... , Z1,21 ... , xi,.9 i ... , Xii,...,iy....]
We note that the ring M(0)* is invariant with respect to the action of the Bockstein operators Nk for every k = 2,3,.... We have the isomorphism M* ^_' M(0)* ®C*1
where C* is the ring defined in section 4.5. According to Theorem 4.5.1 the Bockstein operators act trivially on the ring C*. Now we consider the complex M of the previous section as a functor depending on the ring M* with the Bockstein operators Qk, k = 2 , 3, ...,
4.6. SOME NOTES
167
acting on it. Then the complex corresponding to the ring M(0)* is denoted by M (0), and the trivial complex C.-o*C* 0 i...-0iC*-0 +C*-01 . by C*. It is clear that
M ' M(0) ®C*.
(4.67)
In addition we have the isomorphisms ExtABP(BP*(MSpe' ), BP*) ^_' H* (,M)
H. (M(0)) ®C*.
Suppose
Sf00 = lim Sq _q is the infinite symmetric group. Every group Sq is assumed to act on the finite set {2, 3, ... , q + 11, the action on other numbers being the identity. Thus the group Sq acts on the ring M(0)*, the action being determined by changing indices of generators and extended over the entire ring by multiplicativity. It is clear that the direct limit Sf of these groups acts on the ring M(0)* turning it into an Sam-module. We note that the product formula for Bockstein operators is invariant with respect to the action. This allows us to consider the complex M (0) as a complex of Sf-modules and to define St-module structure in the homology algebra H* (M(0)). So the algebra ExtABp(BP*(MSpe1), BP*) ^_' H. (.M(0)) ®C*
may be described completely in terms of the Representation Theory
of the symmetric group S. The problem is still to be solved. One more point of interest is to figure out a geometric meaning for this Sf.module structure, i.e. to describe it in terms of characteristic classes, Two-valued Formal Groups Theory and so on.
Now let us come back to earth. We consider the first line ExtABp(BP*(MSpe1), BP*)
H1(M)
taken from the exact sequence
(.M)-+Ho (wi1M)->Ho (Mlwi°) H,(M)->0.
168
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
We determine the elements
Y(J; I
I1 ... xI, ..... I,,,) = ",al ... x1,AgX w1
of the M*-module M./w-, where J = {i1, ... , iq}, and the collections It contain more than two elements, it n I8 = 0, It n J = 0 for all s It is clear that the elements Y(J; I,, ... , In) lie in the module Ho (M/wr).
A simple computation gives
Lemma 4.6.1 The element A(J; I1, ... , I,1) = 8 (Y(J; Il, ... , In))
has order 2n, if J 54 0, and order 211+1, if J = 0. It is an easy job to compute the topological dimension of the element
A(J; I,, . . . , In) having order 2n. It equals d(n) = 23n+1 - (4n + 7). For example, d(2) = 113, d(3) = 1005, d(4) = 8169, d(5) = 65509 and so on. A first such element having order 4 is A113
=
b
(x1,5x2,3,4) W1
J
We consider the exact sequence
0 -> JI* `) JI;' a1 JI* a, ExtA,(BP*(MSp), BP*) generated by the Bockstein-Sullivan exact sequence
... -, MSp* -14 MSpe1 -4 MSp* 81+ MSp*
It is clear that A113 E (x1,5, w1, x2,3,4),
B111 =
(A 113) E (08, 2, A49),
4.6. SOME NOTES
169
where A49 E MSp49 is the element of order two, and this element of the ring MSp;I has the decomposition A49 = X2,3U4 + x3,4'u2 + x2,9u3
at the level of the second term of the ANSS. In other words,
-
A49 = 6 (x2,3,41 `
wlJ
Recently (September, 1989, Soviet-Japan symposium on Topology) V.Vershinin & A.Anisimov have announced the following results. (i) The Massey product (A49, 2, 91) contains zero.
(ii) The elements B103 E (A49, 2, cp7) C MSp103,
Bill E (A49,2,8) C MSplll
are indecomposable and have order 4.
Result (i) means that the element x2,3,4 can be represented by a 01-manifold X2,3,4i such that 6(X2,3,4) = (X2,3u4 + X3,4u2 + X2,4u3) - W1
where Xt,2 is a 01-manifold as well. To use the above geometric terms the result (ii) means that the element A113 E (A49, wl, cp8) has order 4 and may be represented by a The same is true for the element Bill = 6(A113). 01-manifold.
The element B103 being the first known element of order 4 has a more complicated decomposition in the ANSS. The final result may be presented as follows (it is proved by elemen-
tary tools).
Theorem 4.6.2 1. The first line ExtABP (BP*(MSpe' ), BP*)
170
CHAPTER 4. FIRST DIFFERENTIAL OF ANSS
of the second term of the Adams-Novikov spectral sequence has elements
of order 2" for every n; the topological dimension of the first element of order 2n is less than or equal to d(n) = 23n+1 - (4n + 7). 2. The algebra
ExtABp(BP*(MSpe' ), BP*) is multiplicatively generated by its zero and first lines.
0
Note 4.6.1 September, 1991. Recently the present author [18] has constructed higher torsion elements of order 2k in the ring MSpE2 for each order 2k. The first known element of order 8 in MSp;2 has dimension 501. The proof is based on the above geometric description of the Adams-Novikov spectral sequence for MSpE2 by interpretation of V.Gorbunov's result (Theorem 4.8.5) that certain Massey products of Ray elements contain zero.
The conclusion is that torsion elements of order 2' do exist in the symplectic cobordism ring MSp* for all k > 2.
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