Foliations: Geometry and Dynamics

Diff(Af), and M is isomorphic to X x N. Here X is the universal covering space. Recall p

138

HlTOSHI MORIYOSHI

Example 1.1 of the Kronecker foliation. The holonomy groupoid is then given by G=(IxIxJV)/

%

where (x,y,t) ~ (xg~1,yg~1, p(g)t) e X x X x N, g £ ^\{X). Then we choose a volume form dfi induced from X and an arbitrary volume form dv on N. When the foliation von Neumann algebra is of type III, there is no invariant volume form (measure) on N with the action of p. On the other hand we choose a volume form on M and take the pullback to X x N which we denoted by dX. We then obtain a function

^

=

dfi x dv

-^x—

on X x N. It is the Radon-Nykodim derivative of two measures. We then define the automorphisms (at)ten by

for

= e"tT

(t€R).

We further construct the crossed product W*(M, J7) xCT R x^ R. Due to the Takesaki duality theorem it follows that W* (M, T) x a R x ^ R is isomorphic to W*(M,J-) ® C, where C denotes the algebra of all bounded operators on the Hilbert space. Therefore, we can reduce the investigation of type III von Neumann algebras to that of type II von Neumann algebras, which are easier to handle due to the existence of trace. Thus the crossed product W* (M, T) x a M is the first clue to study type III von Neumann algebras. We claim here that it is still realized as a foliation von Neumann algebra. Let TMjT be the normal bundle of the

O P E R A T O R ALGEBRAS AND THE INDEX THEOREM

139

foliation J- and Q denote the principal K + -bundle of the determinant line bundle of TMjT. It then follows that . dim N

~

Q = Xx

A

TN+,

p

where f\dimNTN+ denotes the principal R + -bundle of /\dimNTN This is also a foliated bundle and the foliation is denoted by Ta- The codimension of T0 is dim A'' + 1. The conclusion is that W*(M, T) MCT R is isomorphic to the foliation von Neumann algebra W*(Q,Jr0). Furthermore the dual action a on W*(Q, T0) is nothing but the principal action of K+ on Q. The trace r on W*(M, T) xCT M is also identified. Let (£1,^2, . . . , t ? ) be a local coordinate of N, and tq+i denote the coordinate in the fibre direction of the bundle f\dimNTN+. Then the {q + l)-form Utldt2---dtq\

=

\

U+i

J

on A l m TN+ is invariant under the induced action of Diff(Ar). Thus we obtain a transverse invariant measure on (Q,f0) and the trace T on W*(Q,J70) from dv. The relative invariance of r follows from straightforward computation. 2

Analysis on the leaf space M/J-

In this section we explain how to develop Differential Geometry on the leaf space. More precisely we discuss the Index Theorem of differential operators on the foliated manifold (M, F). As mentioned in §1 the Ktheory is suitably generalized on the leaf space. It is then reasonable to ask what are natural geometrical objects in the K-theory. In the following we explain that the index of longitudinal elliptic differential operator is realized as an element in the X-group of the foliation C*-algebra. 2.1

The index in K-group

Let M be a compact manifold and T an elliptic differential operator on M. Then it is a Fredholm operator and the index is defined by Ind T = dim ker T - dim cokerT. Here we explain that the index is considered as an element of the K-group of the C*-algebra K, of compact operators. Choose a Riemannian metric on M and denote by TL the Hilbert space of L2-functions. Since M is

140

HlTOSHI MORIYOSHI

complete, a formal selfadjoint differential operator on M extends uniquely to a selfadjoint operator D on H. Due to the Stone Theorem we then obtain a 1-parameter family of unitary operators {eltD}teRThe fundamental result is the following [40, p. 63] [45]: Proposition 2.1 Let M be a closed manifold and D an elliptic differential operator on M. 1) Given a rapidly decreasing function / on R, the operator

f(D) = [ f(t)eitDdt JR

is a smoothing operator and admits a C°°-kernel function k : M x M —> C; 2) The above construction extends to a homomorphism p : C 0 (R) -» /C so that p(f) = f(D). Here K. is the C*-algebra of compact operators on H. Proposition 2.1 holds also for complete manifolds with bounded geometry. We refer the reader to [40, p. 63], [45] with the details. Given an elliptic differential operator D and an involution e such that eD + De = 0. The Dirac operators on even-dimensional manifolds satisfies the condition. Let Z2 be the cyclic group of order 2 and Z2 act on R by the reflection e{x) = — x (x e R). Let a = foUe + f\U€ be an element of the crossed product C 0 (R) xi Z2. Here Ue and Ut are the formal unitary elements corresponding to the identity e and the generator e G Z2. We often suppress the formal unitaries and denote them by 1 and e. We then obtain a homomorphism: p : C 0 (R) x Z 2 ^ K such that p(foUe + hUe) = f0(D) +

f1(D)e.

Definition 2.2 We call the homomorphisms p : C 0 (R) -» K and p : C 0 (R) xi Z 2 -»/C the index homomorphism determined by the elliptic differential operator D. Exploiting the index homomorphism p, we can generalize the index for longitudinal elliptic operator on foliated manifolds. But before proceeding further we explain the relation between the Fredholm index and the homomorphism p.

141

O P E R A T O R ALGEBRAS AND THE INDEX THEOREM

We first recall that Kt,(C0(R) x Z2) is isomorphic to the if-group of a point. In fact the generator of KQ(C0(R) XI Z2) = Z is given by the formal difference of projections ex and ey. eI-e1eif0(Co(K)xZ2)=Z,

(1+x)2 1-x2 e x = ^ , ; 22 . ++ - n , 22>e, 2(l+z ) 2(l+2 )f

Also recall that the generator of Ki(C0(R)) u-l€#i(C

0 v

_ _ 1 ei =

= Z is given by : x—i

„,

x +1 Definition 2.3 1) For an elliptic differential operator D with an involution epsilon such that eD + De = 0, the (even) index of D is: Ind D = p(ex) - p(ei) € KQ(K) = Z.

(1)

2) For an elliptic differential operator D the (odd) index of D is: Ind D = p{u) - p{\) e Ki(K).

(2)

Remark. For /C we have K\ (K.) = 0 and nothing is interesting with the odd index. However, we can consider various C*-algebras in various geometric situations. For instance, if we consider a T-covering space M, the C*algebra of T-invariant operators on M appears. As is explained later, if we consider a foliated manifold (M, J 7 ), the foliation C*-algebra appears. In each case, given a suitable differential operator D, it yields the index homomorphism form C0(R) x Z2 or CD(R) to a suitable C*-algebra A. Accordingly the index of D is defined as an element of KQ(A) or K\(A) following Definition 2.3. Since Ki(A) could be nontrivial, the odd index is not necessarily vanish in general. Now we examine Definition 2.3 is compatible with the Fredholm index for elliptic operators. First we introduce the following C*-algebra:

m

A = < I

,

e C „ ( R ) ® M 2 ( C ) : a, d-even, c, d-odd functions

We have an isomorphism a : C 0 (R) xi Z2 —> A given by CT

(/) = [fodd

fev J .

CT £

()

= (0 _1

Here f™(x) = (f(x) + f(-x))/2 and f°dd(x) = (f(x) - f(-x))/2 for / e C 0 (K). We then choose continuous functions ip, ip o n ® such that: i) if is an even function and ip is an odd function;

142

HlTOSHI MORIYOSHI


ii) 0 <
Via the isomorphism sigma, the generator in KQ(A) corresponding to ex — ei e K0(Co{R) xi Z 2 ) is given by:

p« = ( 1 7 $ ) . * = (2!)-

po-piew),

(3)

T h u s we obtain Ind D = p{po)-p{pi)Suppose t h a t s u p p ( l - y ) is contained in a small neighbourhood of 0 e R. We then have p(po)

=

(o l)

=

P(P^

on the eigenspaces of D with eigenvalues A ^ 0, and

on k e r l ? . Since KQ(IC) obtain

is identified with Z by the rank of projections, we

p(po) ~ p(pi) = Tr[p(po)\ ker D - p{p\)\ k e r D ] = d i m k e r £ > + - d i m k e r D " . Therefore the Fredholm index is recaptured from our definition of the index. 2.2

Cyclic cohomology

group

Next we shall explain the cyclic cohomology theory, which plays the role of the de R h a m theory to the ordinary manifolds. We t h e n introduce t h e pairing between the if-group and the cyclic cohomology group. T h e pairing t u r n s out to be a quite strong method to detect the index of operators in the i
We have t h e coboundary m a p b : C™ („4) —> C*(A)

(a t e A). such t h a t

n-l

br(a0,ai,---

,an)

= ^ ( - l ) / c T ( a 0 , • • • ,akak+i,---

,an)

+ (-l)™r(ana0,ai,-• • ,a„_i). It is easy to see t h a t b2 = 0.

143

O P E R A T O R ALGEBRAS AND THE INDEX THEOREM

Definition 2.4 The cyclic cohomology H^(A) is the cohomology group of the cochain complex (C^(A), b). Example 2.5 Let T : A —> C be a trace on A. Since it satisfies the condition br(a,b) = r(ab) — r(ba) = 0, it is a cyclic 0-cocycle on A. From T we further obtain a cyclic 2m-cocycle r on the matrix algebra Mn(A) with components in A. Let a(fe) = ( a ^ ) £ Mn(A), where a^fc) € X denotes the (i, j')-component of the matrix a^k\ We then define f to be

r(a(01,»(11,-,.(J"»)=

£

'"(-a.^l.-.-K)-

Example 2.6 We present a cyclic 2-cocycle on C°°(T2). Since T 2 is an abelian Lie group, it admits invariant vector fields X and Y with [X, Y] = 0, and the dual 1-forms dx and dy. The exterior differentiation is given by d = dxdx + dydy, where dx and 9y denote the differentiation along X and y , respectively. We then have a cyclic 2-cocycle r ( a 0 , a i , a 2 ) = / 2 a0daida2 JT

dxdy[ao(dxai)(dYa2)

-

ao(dYai)(dxa2)}

JT JTr2<

on C ° ° ( T 2 ) .

E x a m p l e 2.7 Put A = CC°°(E x R). Let fc G .A act on L2(1R) with k a kernel function. Let x and — act on L 2 (K) such that a; is the multiplication dx by x and that — is the differentiation in x. We then define derivations d\ dx and 02 on .4 such that (<9ifc)£=[x,/c]£ = (zfc-fcx^ i dx i dx i dx where k e A and £ e L (R). Note that di and 9 2 commute with each other. We then have a cyclic 2-cocycle 2

r ( a 0 , a i , a 2 ) = Tr (a0(diai)(d2a2)

- a0{d2a1)(dia2))

Here Tr is given by Tr(fc) = /

k(x,x)dx.

•

144

2.3

HlTOSHI MORIYOSHI

The pairing between K-theory and cyclic cohomology

Let A be an algebra with unit. We shall discuss the pairing between if* (.4) and HC* (A) obtained basically from the evaluation of elements in A with cyclic cocycles. and HCj(A)

Definition 2.8 The pairing ( , ) between K^A) mod 2) is defined in such a way that: i) Given e G K0(A) and r G HC2m{A),

(i = j

it is

(e,r) = r(e,e, ••• ,e); ii) Given u G KX{A) and r G HC2m+1(A), (U,T)

=

T(U~1

it is

— 1, w — 1, • • • , u

— l , u — 1).

It is remarkable that the values of evaluation above depend only on the classes of e, u and r. Remark. In the above definition we omitted the normalizing constant appeared in Connes [16]. The constant will be important when we introduce the stabilizing operator S. Example 2.9 Let A be C°°(T2) and r the cyclic 2-cocycle on Mk(A) such that r ( a o , a i , a 2 ) = / 2 dxdyTi

(a0(dxai)(dYa2)

-

ao(dYai)(dxa2))

JT

for ai € Mk(A). Here we consider that a^ is a smooth function on T2 with values in Mfc(C). The derivations <9x and dy are extended naturally to act on such elements, and Tr means taking the trace at each point of T 2 . We also consider that a^ are operators acting on the space C°°(T 2 ,C ) of all Cfc-valued smooth functions on T2. The derivations are then given by the following form: (dxa)£ = (Xa - aX)£,

(dya^

= (Ya - aY)£

for a e Mk(A) and £ G C°°(T 2 ,C A: ), where X and Y denote the differentiations considered as operators acting on C°°(T2,Ck). In other words, we have d\a = \X,a\,

dya = [Y, a].

145

O P E R A T O R ALGEBRAS AND THE INDEX THEOREM

Let e e Mk{A) be an idempotent. It then follows that T(e,e,e)= /

JT2

dxdy Tr (e[X,e][Y,e] - e[Y,e][X,e])

= [

dxdyTx(e(Xe-eX)(Ye-eY)-e(Ye-eY)(Xe-eX))

JT2

=

dxdyTr([eXe,eYe]

-e[X,Y}e).

JT2

When we put eZe = V'z for vector fields Z on T 2 , we finally obtain r(e,e,e)= /

JT2

dxdy Tr ([Vx, Vy] - V [ x ,y])

= f2

dxdyTi(RXtY),

JT

where R denotes the curvature tensor with respect to the covariant differentiation V = ede on the vector bundle E on T2 which is given by the image of e. Recall that the first Chern class c\(E) is represented by Tr (^-R). Thus, the above equality amounts to ( e > r ) = — T (e,e,e) = / 2

2TI- V

'

'

2TT

ci(E).

JT

E x a m p l e 2.10 Let A and r be the algebra and the cyclic cocycle in Example 2.7. Let e be a projection in A of finite rank. Recall that T is given by r ( a o , a i , a 2 ) = Tr {a0[x,a1}[-

— ,a2\ - ao[- —

,ai}[x,a2}

By a similar computation to the previous example, we obtain / , _ /. Id r(e, e, e) = Tr \exe, e-—e\ V i dx

. I d . — e\x, - — \e i dx

Note that Tr ([exe,el-^e]) — 0 since exe,e\-^e € A. Observing the canonical commutation relation [x, \j^.] = i, we thus obtain •r(e, e, e) = Tr (—ie) = —irk (e). If we pursue the analogy of the previous example, we can consider that r(e, e, e) evaluates the noncommutative curvature with the projection e.

146

2.4

HlTOSHI MOMYOSHI

The Counts index theorem for longitudinal elliptic operators

Let M be a closed manifold and J- a foliation on M of dimension p. Suppose that there exists an invariant transverse measure v on {M,T). Due to Ruelle-Sullivan [42] we obtain the current CRS of degree p such that:

CRS

u> = > / av 1 ii JTi JViX{t} JTi JViX{t}

pito.

for a differential p-form w on M. Here we take the foliation chart U = {Ui = Vi x Ti}j 6 / such that Tj is a transversal to T, and choose the partition of unity {pi)i£i subordinate to U. Then JVxrt\ denotes the integration along the leaves Vi x {t} (t e Tj), and JT dv is the integration with respect to v. Since v is holonomy invariant, the current CRS is closed. Moreover CRS induces a linear map: C R S - . ^ ^ ^ R

for the complex fi(^-") of longitudinal differential forms. It also satisfies that CRs(d'u>) = 0, where d' denotes the exterior differential along the leaves. Suppose that (M, !F) admits an invariant transverse measure v. We then obtain the trace T : C*{M,F) -> C on C*{M,T) exploiting the RuelleSullivan current CRS and the leafwise measure dp,: r

du

(
JTi

/

PUfdvjr. JVixit]

Here ip is a continuous function on the holonomy groupoid G with compact support. E x a m p l e 2.11 Consider the Kronecker foliation f e o n M = ( l x K/Z)/Z, (x,t) ~ (x + l,t + 6). Recall that the holonomy groupoid is given by G = ( l x l x R/Z)/Z. Note that the standard volume form dt on R / Z yields an invariant transverse measure for (M,Tg). We then identify C*(M,Te) with the C*completion of the algebra of kernel functions such that: i) k : R x R x R/Z -> C is continuous; ii) k(x + l,y+l,t

+ 6) = h(x,y,t),

{x,y,t) e R x R x R / Z and

iii) fc has compact support when it is considered as a function on (R x R x

147

O P E R A T O R ALGEBRAS AND THE INDEX THEOREM

Then the trace is defined by r(fc) = / dt I Jm/z Jo

k(x,x,t)dx.

Take a closed foliated manifold (M, J-) of even-dimensional leaves. Suppose that there exists a family of longitudinal Dirac operators on (M, J7). Let D = {Di)LeM/jr denote the family of longitudinal Dirac operators Di that is lifted to the holonomy covering L from each leaf L. Then we can apply the construction in Proposition 2.1 to D to obtain the index homomorphism

p:C0(R)

xZ2-^C*(M,f).

Hence we define the index of the longitudinal Dirac operator D to be I n d D = p(ex) - p(ei) €

K0(C*{M,F)).

It is completely similar to Definition 2.3. Then the index theorem due to Connes for the longitudinal Dirac operator on (M, !F) is stated as follows: Theorem 2.12 (Connes index theorem on foliated manifolds [15]) Let M be a closed manifold and T a foliation on M. Suppose that there exists an invariant transverse measure v on (M, T). It then follows that:

Here f

denotes the Ruelle-Sullivan current associated to v, and

A (zj^Rr) € ^(-T7) is ^e A-form associated to the Riemannian ture form Ryr along the leaves. 2.5

curva-

The Toeplitz index theorem for longitudinal elliptic operators

In the above theorem we assumed that the leave are of even dimension. There exists as well an index theorem on foliated manifolds of odd-dimensional leaves. In the following we shall describe the index theorem on the Kronecker foliation. Let TB be the Kronecker foliation on M = (R x R/Z)/Z. Here 9 6 R \ Q . Recall that C*(M,Jrg) is identified with the C*-completion of the algebra of kernel functions k : M x R x E / Z -> C in Example 2.11. Note that with t 6 R/Z fixed such a function k(x,y,t) is considered as the kernel function of a operator kt that acts on the Hilbert space L 2 (R x {t}). Let

148

HlTOSHI MORIYOSHI

D — (Dt)tem/z be the longitudinal elliptic operator on (T2,Te)

such that:

i dx on the holonomy covering R x {i}. Furthermore, we define a family F = (Ft) such that Ft is the Hilbert transformation on each L 2 (R x {£}): £—0 IT

J |x-y|>£

ay) dy.

x-y

' 1 + Ft Thus F is constant in t e R/Z. Let P — ( — - — ) be a family of projections. Here we take a C°°-function if : M = (R x R / Z ) / Z -> C \ {0}. Then we make the pullback of ip to R x R/Z and let

P

yields an element of KQ{C*(M, To)). It is called the index of Toeplitz operator Tv. We have the following theorem (see also [21]). Theorem 2.13 Let {M,Te) be the Kronecker foliation. We take a exjunction C\{0} and consider the index of Toeplitz operator Tv ip-ipip-PeKoiC'iM,^)). It then follows that T^Ptp

- P) = —- /

2iri Jrp2

ip~ldipdt

where r is the trace on C*{M,Te) defined previously. We shall illustrate the theorem to calculate the index for ip{x,t) = 2 2-ni(6x-t) e W e first o b s e r v e ^ f ) = ip(x + l,t + 9). On each L (R x {t}) we have

[(^PV-P)flOO = lim£ / \x-y\>e

^ « ( M ) 4

O P E R A T O R ALGEBRAS AND THE INDEX THEOREM

149

Thus the kernel function k(x, y, t) of

,v _

*
2-K

Since

I JR/Z

dt f 9dx = 9. JO

On the other hand we can verify 2ni

dipdt = 9. JT2

Strictly speaking, the element (p~lPip — P belongs to not C*(T2, Tg) but the foliation von Neumann algebra W*(T2,J:e)- However, we can modify the argument and construct an element in C*(T2,Te) that yields the index of Toeplitz operator Tv. 3

Toward noncommutative geometry on foliated manifolds

In this section we discuss a few topics related to noncommutative geometry due to A. Connes. 3.1

The Godbillon- Vey cyclic cocycle and the modular automorphisms

Let X be a closed Riemann surface _of genus > 2. We consider X as the quotient of the universal covering X that is the hyperbolic space. Put T = m(X). Then the group T acts on X isometrically. Let T also act on S 1 via orientation preserving diffeomorphisms. The diagonal action of T on X x S1 then define a foliated bundle M = (X x 5 1 ) / r . The holonomy groupoid is given by G = {X x X x

Sl)/Y.

Recall that the foliation C*-algebra C*(M,J-) can be constructed from functions k : X x X x S1 —>C such that: i) k : X x X x Sl —> C is continuous; ii) k(x-y,yy,t-y) = k(x,y,t)

for 7 6 T ;

HlTOSHI MORIYOSHl

150

iii) k has compact support when it is considered as a function on (X x

x x s^/r. Let w : C* (TV, f ) - > C a weight given by }{k) = / w{

k{x,x,t)dtdx.

JM(V) lM(V)

Here M(T) denotes the fundamental domain with respect to the T-action o n X x S 1 . Then we choose the hyperbolic volume form dfi on X and the standard volume form dt on Sl. We also choose a volume form on M and take the pullback to X x TV. It is denoted by dX. We then obtain a function dfx x dt on X x TV. The modular automorphisms (o~t)tem is then defined by for (p G CC{G). We can verify that the modular automorphisms (extern a l s o preserves the foliation C*-algebra C*{M,T). We then define a cyclic 2-cocycle that is associated to he Godbillon-Vey class. For the details we refer to Moriyoshi-Natsume [34]. Definition 3.1 The Godbillon-Vey cyclic cocycle on C*{N,F) is T(a0,ai,a2)

= /

dxdt (a o [0,ai][0,a 2 ] - a0[,ai][<£,a2]),

JM(T)

where <j> = logi/> and = d(j)/dt. Let D = {DI)L
as is observed in §2. We then have the following: T h e o r e m 3.2 ( M o r i y o s h i - N a t s u m e [34]) Letr the Godbillon-Vey cyclic cocycle defined as above. We denote by gv the Godbillon-Vey class of the foliated bundle M. It then follows that (e,r) = — / gv,

**Jx

where (e, T) is the pairing between the K-theory and the cyclic cohomology group.

151

O P E R A T O R ALGEBRAS AND THE INDEX THEOREM

We observe the similarity of the form of the Godbillon-Vey cocycle and the usual curvature given by the covariant derivative. Thus the above theorem exhibits that the Godbillon-Vey number of the foliated bundle of M can be understood as the noncommutative curvature with respect to two derivations derived from the modular automorphisms on C* (M, F). 3.2

Noncommutative geometry for singular foliations

Let us review briefly the definition of quantum SU(2) due to Woronowicz [46]. Recall that

The quantum SU(2) is a deformation of the function algebra C(SU(2)). Precisely that is the universal C*-algebra C(SMC/(2)) generated by two elements a and 7 satisfying the following equations: 2

/i7a = a7

„

a a + \x 77 = 1 a*a + 7*7 = 1

^7* a = aj*

(4)

7*7 = 77* for - 1 < \x < 1. When fi = 1, C(S^U{2)) is exactly isomorphic to the function algebra C(SU(2)). There is another way to construct C{SfJJ{2)). Let C(T) be the function algebra of 1-dimensional torus T and K. denote the C*-algebra of compact operators. It is then obtained as an extension given by the following exact sequence: 0

• C{T)®K,

> C{SllU{2))

> C(T)

• 0.

(5)

According to Sheu [43] the C*-algebra C(5Mf7(2)) is considered as a deformation quantization whose infinitesimal deformation is the Poisson structure on SU{2). He also understand the structure of C(5 M f/(2)) from the viewpoint of singular foliation induced by the Poisson structure. Roughly speaking it is stated as follows. We first consider SU{2) as S 3 , and the join of S1: S3 = (5 X x [0,1] x S 1 ) / - , where (x,0,y) ~ (x',0,y') if y = y', and {x,l,y) ~ (x',l,y') if x = x'. There is no equivalence relations otherwise. Let D be the unit disk {z € C : \z\ < 1}. Then 5 3 is also realized as the quotient of (D x S1)/ ~,

152

HlTOSHI MORIYOSHI

where (z,y) ~ (z',y') if z = z' € 3D. Thus S3 is foliated with leaves Ly = {(z,y) : \z\ < ljjygs 1 outside the singular set T = {(z,y) : |z| = 1 } / ~. From the construction we easily obtain the following short exact seaquake of function algebras: • C0(D x S1)

0

> C(S3)

• C{T)

• 0.

(6)

o

where D = {z £
In fact, outside T we have nonsingular foliation on S3 \ T = D x S1. Equip S3 \T with leafwise complete metric, we can develop the index theorem for longitudinal operators. Even in the case of a single leaf, the index theorem turns out to be the Atiyah-Patodi-Singer index theorem [3] for manifolds with cylindrical ends. The eta invariant then appears as the evaluation of the index with a relative cyclic cocycle derived from the Heaviside function with cutoff on the boundary. The index theorem for general singular foliations is thus considered as a family version of the Atiyah-Patodi-Singer index theorem. References 1. M. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Asterisque, 32 (1976), 43-72. 2. M. Atiyah, R. Bott and V.K. Patodi, On the heat equation and the index theorem, Invent. Math., 19 (1973), 279-330. 3. M. Atiyah, V.K. Patodi and I.M. Singer, Spectral asymmetry and Riemannian geometry I, Cambr. Phil. Soc, 77 (1975), 43-69; II, 78 (1975), 405-432; III, 79 (1976), 71-99. 4. M. Atiyah and W. Schmid, A geometric construction of discrete series for semisimple Lie groups, Invent. Math., 42 (1977), 1-62. 5. M. Atiyah and G. Segal, The index of Elliptic operators II, Ann. of Math., 87 (1968), 531-545. 6. M. Atiyah and I. M. Singer, The index of elliptic operators on compact manifolds, Bull. Amer. Math. Soc, 69 (1963), 422-433. 7. M. Atiyah and I. M. Singer, The index of Elliptic operators I, Ann. of Math., 87 (1968), 484-530; III, 87 (1968), 546-604; IV, 93 (1971), 119-138; V, 93 (1971), 139-149. 8. M. Atiyah and I. M. Singer, Index theory for skew-adjoint Fredholm operators, Publ. Math. IHES, 37 (1969), 305-326.

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153

9. P. Baum and A. Connes, Leafwise homotopy equivalence and rational Pontriagin classes, in Foliations, (Tokyo 1983), Adv. Stud. Pure Math., 5, North-Holland, Amsterdam, 1985, 1-14. 10. P. Baum and A. Connes, Chern character for discrete groups, in A Fete of Topology, Academic Press, Boston, 1988, 163-232. 11. P. Baum, A. Connes and N. Higson, Classifying space for proper actions and K-theory of group C* -algebras, Contemporary Math., 167 (1994), 241-291. 12. P. Baum and R. Douglas, K-homology and index theory, Proc. Symp. Pure Math., 38 part 1 (1982), 521-628. 13. N. Berline, E. Getzler and M. Vergne, Heat kernels and Dirac operators, Springer-Verlag, New York-Berlin, 1992. 14. B. Blackadar, ^-theory for operator algebras, Mathematical Sciences Research Institute Publications 5, Springer-Verlag, New York-Berlin, 1986. 15. A. Connes, A survey of foliations and operator algebras, Proc. Symp. Pure Math., 38 (1982), 521-628. 16. A. Connes, Noncommutative differential geometry I, II, Publ. Math. IHES, 62 (1986), 257-360. 17. A. Connes, Cyclic cohomology and the transversal fundamental class of a foliation, in Geometric Method in Operator Algebras (H. Araki and E. G. Effros, eds.), Pitman Research Notes in Math. Series, 123 (1986), Longman Scientific and Technical, 52-144. 18. A. Connes, Noncommutative Geometry, Academic Press, 1994. 19. A. Connes and H. Moscovici, The I?-index theorem for homogeneous spaces of Lie groups, Ann. of Math., 115 (1982), 291-330. 20. A. Connes and H. Moscovici, Cyclic cohomology, Novikov conjecture and hyperbolic groups, Topology, 29 (1990), 345-388. 21. R.G. Douglas, S.Hurder and J. Kaminker, The longitudinal cocycle and the index of Toeplitz operators, J. Func. Anal., 101 (1991), 120-144. 22. E. Getzler, Pseudo-differential operators on supermanifolds and the Atiyah-Singer index theorem, Comm. Math. Phys., 92 (1983), 163-178. 23. P.B. Gilkey, Invariance theory the heat equation, and the AtiyahSinger index theorem, 2nd ed., CRC Press, Boca Raton-Ann ArborLondon-Tokyo, 1995. 24. I.Q. Gohberg and M.G. Krein, The basic propositions on defect numbers, root numbers and indices of linear operators , Amer. Math. Soc. Transl., 13 (1960), 185-264. 25. P. Green, Equivariant K-theory and crossed products, Proc. Symp.

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Pure Math., 38 part 1, (1981), 337-338. 26. N. Higson, An approach to Z/k-index theory, Inter. J. Math., 1 (1990), 189-210. 27. M. Hilsum and G. Skandalis, Stabilite des C* -algebres de feuilletages , Ann. Inst. Fourier, (Grenoble) 33 no. 3, (1983), 201-208. 28. F. Hirzebruch, Topological methods in algebraic geometry, Springer, 1966. 29. S. Hurder, Eta invariants and the odd index theorem for coverings, Contemp. Math., 105 (1990), 47-82. 30. G.G. Kasparov, The operator K-functor and extensions of C* -algebras, Math. USSR Izv., 16 (1981), 513-572. 31. G.G. Kasparov, Lorentz group: K-theory of unitary representations and crossed products, Sov. Math. Dokl, 29 (1984), 252-260. 32. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry 1,11, Interscience Publ., 1963, 1969. 33. H.B. Lawson and M-L. Michelsohn, Spin Geometry, Princeton, 1989. 34. H. Moriyoshi and T. Natsume, The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators, Pacific J. Math., 172 (1996), 483-539. 35. R.S. Palais, Seminar on the Atiyah-Singer index theorem, Ann. of Math. Studies 57, Princeton, 1965. 36. V.K. Patodi, An analytic proof of the Riemann-Roch-Hirzebruch theorem for K'ahler manifolds, J. Diff. Geom., 5 (1971), 251-283. 37. G. Pedersen, C*-algebras and their automorphism groups, Academic Press, 1979. 38. M. Rieffel, Applications of strong Morita equivalence to transformation group C*-algebras. Operator , Proc. Symp. Pure Math., 38 part 1, (1981), 299-310. 39. M. Rieffel, Morita equivalence for operator algebras , Proc. Symp. Pure Math., 38 parti, (1981), 285-298. 40. J. Roe, Elliptic operators, topology and asymptotic methods, Longman, 1988. 41. J. Roe, Coarse cohomology and index theory on complete Riemannian manifolds, Memoires of AMS 497, 1993. 42. D. Ruelle and D. Sullivan, Currents, flows and diffeomorphisms, Topology, 14 (1975), 319-327. 43. A.J-L. Sheu, Quantization of the Poison SU(2) and its Poisson homogenous space - The 2-sphere, Commun. Math. Phys., 135 (1991), 217-232. 44. I.M. Singer, Some remarks on operator theory and index theory , Lect. Notes in Math., 575 (1977), 128-138.

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45. M. Taylor, Pseudo-differential operators, Princeton, 1982. 46. S.L. Woronowicz, Twisted SU(2) group: An example of noncommutative differential calculus, Publ. R.I.M.S., 23 (1987), 117-181.

Received May 7, 2001.

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RESEARCH PAPERS

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Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2002 pp. 159-183

D I S T R I B U T I O N A L B E T T I N U M B E R S OF T R A N S I T I V E FOLIATIONS OF CODIMENSION ONE J E S U S A. ALVAREZ L O P E Z Departamento de Xeometria e Topoloxia, Facultade Universidade de Santiago de Compostela, 15706 Santiago e-mail: [email protected]

de Matemdticas, de Compostela, Spain,

Y U R I A. K O R D Y U K O V Department

of Mathematics, Ufa State Aviation Technical 12 K. Marx str., 450025 Ufa, Russia, e-mail: [email protected]

University

Let T be a transitive foliation of codimension one on a closed manifold M. This means that there is an infinitesimal transformation X of (M, F) transverse to the leaves. The flow of X induces an M-action on the reduced leafwise cohomology H(JF). By using leafwise Hodge theory, the trace of this action on each H {J7) can be defined as a distribution 0\. on R, which is called distributional Betti number dis

'

because it is kind of a finite measure of the "size" of H (J7). So the corresponding distributional Euler characteristic, Xdis(-^r)i ' s a distribution on K too. This is relevant because H(JF) may be of infinite dimension, even when the leaves are dense, and its Euler characteristic makes no sense in general. The singularity at 0 of Xdisf-^) ' s expressed in terms of the Connes' A-Euler characteristic, where A is the holonomy invariant transverse measure of T induced by the volume form dt on E. Moreover the whole of Xdis(^) ' s computed by showing a dynamical Lefschetz formula.

1

Introduction

Let M be a closed manifold and J- a smooth foliation on M of codimension one. As usual, let 3£(M, T) C X(M) denote the Lie subalgebra of infinitesimal transformations of (M, J 7 ), and 3L(T) C 3E(M, J") the ideal of vector fields tangent to the leaves. For any X € X(M, J7), the corresponding flow maps leaves to leaves, and will be denoted by Xt : (M, J-) —> (M, J ) , t 6 R. 159

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KORDYUKOV

The foliation J- is called transitive when TXM = {X(x) | X G X(M,F)}

.

Since

TXT = {r(i) I y e x(^)}, we get that T is transitive if and only if there is some X G X{M,!F) transverse to the leaves; i.e., TXM =

RX(x)®Txf

for all x G M. Then the orbits of Xt, t G M., are non-singular and transverse to the leaves. Note that X(M,T)/X(T)^R

(1)

if the leaves are dense, which is the most interesting case. The leafwise de Rham complex of T, {Q,(T),djr), is the the restriction of the de Rham complex of M to the leaves; i.e., it is given by the smooth sections of the exterior vector bundle f\ TT* over M. Its cohomology is called the leafwise cohomology of T, and will be denoted by H{T). Moreover (ft(.F), dp) is a topological complex with the C°° topology, and H(T) is a topological vector space with the induced topology. It is well known that H(F) may not be Hausdorff [14]. So it is interesting to consider its quotient over the closure of the trivial subspace, which is called the reduced leafwise cohomology of T, and is denoted by H{Jr) in this paper. Consider a Riemannian metric on M such that X is of norm one and orthogonal to the leaves. So all flow orbits are geodesies of speed one orthogonal to the leaves. This is what is called a bundle-like metric on M. Consider the induced Riemannian structure on the leaves, and let djr, Ayr be the leafwise coderivative and leafwise Laplacian on fi(jF), which are the restrictions to £l(F) of the coderivative and Laplacian on the leaves. The kernel H(f) of Ayr is the space of harmonic forms on the leaves that are smooth on M. The L2 inner product on M induces a Hilbert space structure in the space L2Q.(T) of square integrable leafwise differential forms on M. Consider Ayr as an unbounded operator in L2Sl(T) with domain Q.{F), and let Ajr be its closure. It is well known that A^r is symmetric on M when the metric is bundle-like (see, for instance, [5, 16]), so A^r is a selfadjoint operator. Let LI be the orthogonal projection L2fl(!F) —> ker A^r. By [3], LI has the restriction H : fl(.F) —» H(f), and there is an orthogonal decomposition n(F) = H{T) © imdjr © im<^ ,

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which can be called a leafwise Hodge decomposition. In particular, the inclusion H(!F) C kerdjr induces an isomorphism H{F) % H{T) ,

(2)

whose inverse is induced by the orthogonal projection II : kerdjr —> Ti(T). For any function / e C£°(R), define an operator Af on Q(T) by the formula

Af = n o f x; • f{t)dtou, Jw and let Aj denote its restriction to Q,l(T). Our first main result is the following. T h e o r e m 1.1 For any function f G C£°(R), the operator Af is of trace class, and the functional f ^Tr [Af j defines a distribution /3^is(T) on R for each i. The distributions Pdis{F) depend only on T and the class of X in X(M, T)IX{!F) (Lemma 2.3); thus, when the leaves are dense, they depend only on T up to linear isomorphisms of R by (1). The usual dimension of the spaces H (T) can be infinite even when the leaves are dense [1, 2, 3]. So the Euler characteristic of H(F) can not be denned, and thus a leafwise Gauss-Bonnet theorem makes no sense in the usual way. This is surely a reason of the poor role played by the reduced leafwise cohomology in foliation theory, which should be similar to the important role played by de Rham cohomology of closed manifolds. To have finite leafwise Betti numbers, they must be defined in another way, by using the another kind of dimension ("exotic dimension"). A solution was given by Connes for foliations with a holonomy invariant transverse measure A [8, 9]. In our case, A is the transverse Riemannian volume element, which corresponds to dt on R. This A is used to make kind of an average on M of the "local dimension" of the space of square integrable harmonic forms on the leaves at each degree i, giving the finite A-Betti numbers /3 A (^), and thus a A-Euler characteristic XA (•?"")• The technical difficulties of this idea are solved by using the noncommutative integration theory of Connes. But, if the leaves are not compact, the forms of our space ~H(J-) are not square integrable on the leaves because they are smooth on M. So, a priori, the A-Betti numbers are not directly related with the reduced leafwise cohomology. Now we give another "exotic" solution to the above problem. Observe that, for / € C£°(R) supported around 0, the operator Af is kind of a

162

J.A. ALVAREZ L O P E Z AND Y.A. KORDYUKOV

diffusion of the orthogonal projection II : fi(.F) —> W(J r ). So the germ of /3jis(jT) at 0 can be considered as a finite measure of the size of Til(T), and thus of H (T) as well. For this reason, the germs at 0 of the distributions PdisiJ7) could be called distributional Betti numbers. But, for the sake of simplicity, the whole distributions PldiS{^) will be called the distributional Betti numbers of J7, even though they should be better considered as Lefschetz numbers away from 0. We also define the distributional Euler characteristic of T by the formula

xdis(^) = E t - i y / ^ m • i

The following theorem describes the singularity of Xdis (•?"") at 0 in terms of Connes' A-Euler characteristic XK(J-)- SO Connes' A-Betti numbers are really strongly related with the reduced leafwise cohomology. The similar result was obtained in [18] when the flow is isometric. Theorem 1.2 In some neighbourhood ofO in R, we have Xdis(^") = X A ( ^ ) • <5o , where 8Q denotes the Dirac measure at 0. Recall that a closed orbit c of length I of the flow Xt on (M, T) is called simple when det(id - XI • TXT* -» TXT*) ^ 0 for any x e c. The following theorem proves, for this type of foliations, a conjecture stated by Deninger in [10]. Under some additional assumptions, it was proved in [11, 18]. Theorem 1.3 Assume that all closed orbits of the flow Xt on (M,T) are simple. Then we have oo

XdUF) = ]T^(c) ]>^ sign det (id - X*(c) : TXT* -> TxT*j • 5ki{c) c

fe=l

on M.+ , where c runs over all primitive closed orbits of the flow Xt, 1(c) denotes the length of c, and x is an arbitrary point of c. Of course, in Theorem 1.3, a symmetric formula for Xdis(-^r) also holds inK_. Observe that, if dimW i (.F) = &(?) < oo, then /3^S(J") is a smooth measure whose value at 0 is 0l(F) dt. On the other hand, when Tt(!F) is of finite dimension, its Euler characteristic can be defined :

DISTRIBUTIONAL B E T T I NUMBERS OF TRANSITIVE FOLIATIONS

163

But, by Theorems 1.2 and 1.3, the distributional Euler characteristic Xdisl-T7) is trivial if it is smooth, obtaining the following. Corollary 1.4 / / dim H( T) < oo, then Xd\s(J~), Xh{f) and x{F) vanish. Theorem 1.3 also has the following consequence. Corollary 1.5 Assume that all closed orbits of the flow Xt on (M,T) are simple. If dim H' (J7) < oo, then, for any I G R,

V

- ^ r sign det (id -Xf:

TXF -+ TXF) = 0 ,

where c runs over all closed orbits of the flow Xt of period I, /J,(C) denotes the multiplicity of c, and x £ c is an arbitrary point. When the dimension of T is two and the leaves are dense, it is possible to relate directly each distributional Betti number with the corresponding A-Betti number because we obviously have Pi(F) = P\{F) = 0 ,

dimW°(J-) = l ,

dimW2(J-)
So Theorem 1.2 has the following consequence. Corollary 1.6 Assume that T is of dimension two with dense leaves. Then the singular part o//3j is (.F) around 0 is PXiF) • <5o for each degree i. It is possible that the statement of Corollary 1.6 holds in general. Indeed, a proof could be given by finding appropriate heat kernel estimates on the leaves. So we propose the following. Question 1.7 For each degree i, is it true that the singular part o//3^is(^r) around 0 is /3\(7") • 60? If this question has an affirmative answer, then dim'W(Jc) = oo whenever j3\{T) ^ 0. This would mean that the existence of non-trivial square integrable harmonic i-forms on the leaves implies the existence of non-trivial harmonic i-forms on the leaves that are smooth on M. Similar results were shown in [1, 3], where integrable harmonic i-forms on the leaves are used instead of square integrable ones, which are much easier to find. Let RL be the curvature of the leafwise metric, and Pf(i?/,/27r) £ QP(J-) the leafwise Euler form, p = dim J?7. The product Pf(i?£,/27r) A A is a differential form of top degree on M. In particular, Pf(i?L/27r) A A = i K?{x) UJM{X) lux if T is of dimension 2, where Kjr is the Gauss curvature of the leaves and UJM is the volume form on M. Then Theorem 1.2 and the foliation Gauss-Bonnet theorem from [8], which computes XAC-^OI have the following consequence.

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KORDYUKOV

Corollary 1.8 We have Xdis(-F) = 50- [ Pf(i?L/27T) A A JM

around 0. In particular, if J- is of dimension two, then Xdis(^) = So • — / 2TI" JM

Kjr{x)uM(x)

around 0. Corollary 1.8 seems to be a powerful tool to produce examples of foliations with dense leaves on closed Riemannian manifolds with dimH l (J-) = oo; specially, if Question 1.7 has an affirmative answer. There are obvious versions of these results with general coefficients, which were not considered here for the sake of simplicity. This type of foliations are just Lie foliations of codimension one. So this is a particular case of our work on distributional Betti numbers for arbitrary Lie foliations [4]. It is worth to explain this particular case here because the arguments are much easier to understand, and moreover the codimension one case is relevant for Deninger's approach to Riemann Hypothesis [10,11]. Finally, let us mention that our results are somehow related with the study of transversely elliptic operators for Lie group actions [6, 23, 20, 9, 15, 17]. 2 2.1

Distributional Betti numbers Leafwise homotopies

A C°° foliation map / : (M,T) —> (M, T) induces a homomorphism of topological complexes, / * : £l{T) —> fl^J7), by pulling-back differential forms. Then it also induces homomorphisms of graded topological vector spaces, / * : H(T) -+ H{T) and / * : H{T) -> H(T). Two maps C°° maps / , / ' : (M, F) —> (M, T) are said to be leafwise homotopic if there is a C°° homotopy between them, hs : (M, J-) —> (M, T), s £ / = [0,1], such that each curve s H-» hs(x), x G M, is contained in a leaf. Such a homotopy is called an leafwise homotopy, and the notation / ~F f will be used. Then the usual construction of an homotopy of de Rham complexes produces a linear continuous map k : 0(J r ) —> fl(J-), homogeneous of degree —1, such that / * - / ' * = k o dT + d? o k.

(3)

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Moreover k depends continuously on the homotopy hs with respect to the C°° topology. We get

f~

f/^

7

t r = r* •• K(F) ^

H(T) \ /* = /'* : H{T) -> H(T) ,

,

and thus / ^ / ' ^ n o f = n o / ' * : H(T) -> H(T)

(4)

by the isomorphism (2). 2.2

Smoothing operators

Let LJM denote the Riemannian volume element of M, and LJJ: the Riemannian volume element of T. A smoothing operator on fl(T) is a linear map P : Q(T) —> £l(T), continuous with respect to the C°° topology, given by (Pa){x) = /

k(x,y)a(y)ujM{y)

,

a € tt(T) ,

JM

where k € C°°{/\TT* k(x,y)e/\TT*®/\TTy

M f\TT)

is called the smoothing kernel of P. So

= Horn ( / \ T ^ * , / \ T ^ )

,

(a:,y) e M x M .

Any smoothing operator P is of trace class, and we have T r P = / trk(x,x)u>M(x)

,

(5)

where k is its smoothing kernel. Let £l(T)' be the dual space of Q(T); i.e., the space of continuous linear functionals £l(T) —* M, equipped with the weak dual topology (or topology of pointwise convergence). Let C(Q(T)', Q{T)) denote the space of continuous linear operators fl(T)' —> £l(T), equipped with the topology of bounded convergence. Consider also the C°° topology on C°° (A TT* IE1 A TT). The following result is well known. L e m m a 2.1 A continuous operator in 0,(T) is smoothing if and only if it extends to a bounded linear operator Q,(T)' —> £l(T). Furthermore the map £(n(T)',n(T))^C°°(/\TT*M/\TT^

,

which assigns its kernel to each operator, is an isomorphism of topological vector spaces.

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J.A. ALVAREZ LOPEZ AND Y.A. KORDYUKOV

Of course, Lemma 2.1 can be stated in terms of Sobolev spaces Wk£l(F) of leafwise differential forms; in particular, a continuous operator in VL(F) is smoothing if and only if it extends to a bounded operator WkQ,(T) —> Wl£l{F) for a l l M . A special type of smoothing operators on fi(.F) can be constructed as follows. A subspace V C X(M) is called transitive if TXM = {Z{x)

\ZeV}

for all x € M. Since M is compact, it easily follows that there exists a finite dimensional subspace W C X{T) such that TXT

= {Z{x)

\ZGW}.

Then V — W © RX C X(M, T) is a finite dimensional transitive subspace. Fix an Euclidean metric on V so that X has norm one and is orthogonal to W. Then the following result was shown in [22]. L e m m a 2.2 (Sarkaria) For any f € C£°(V), the operator P=

f Z{-f{Z)dZ Jv on tt(J-) is smoothing, and its smoothing kernel depends continuously on f (with respect to the C°° topologies). 2.3

Proof of Theorem 1.1

For each Z £ W and t e K , the maps ((1 - s)X)t o(Z + stX)i : (M, F)-+(M,F),

sel,

define a leafwise homotopy between Xt o Z\ = (tX)\ o Z\ and (Z + tX)i. So

n o z{ o x; = n o (z + tx)\ •. n(f)

-»n(f)

by (4). Now take any / e CC°°(M) and any g e C™(W) with Jwg(Z)dZ and let Bf=f

= 1,

Z\ • h(Z) dZ : fi(^) -» fi(^) , Jv where h € C™{V) is given by h(Z + tX) = f(t)-g(Z) for Z e W and t e R. By Lemma 2.2, such a Bj is a smoothing operator whose smoothing kernel depends continuously on /i, and thus on / . We also have Af=Uo

I [ Z{ o X* • f{t) • g(Z) dtdZon Jw JR

=

UoBfoU.

DISTRIBUTIONAL B E T T I NUMBERS OF TRANSITIVE FOLIATIONS

167

On the other hand, it was proved by the authors in [3] that II : Q(F) —> f2(J7) is continuous, and has an extension to a bounded linear operator on every Sobolev space of leafwise differential forms, and thus to Q(^ 7 )'. So, by Lemma 2.1, the operator Af = II o Bf o II is smoothing. Moreover its smoothing kernel depends continuously on Af, and thus on Bf. In turn, Bf depends continuously on its smoothing kernel, and thus on / . So the smoothing kernel of Af depends continuously on / . It follows that Af is a trace class operator, as well as each A, , and their traces depend continuously on / by (5). Therefore each /3^is is a distribution. 2-4

The dependence of the distributional Betti numbers

L e m m a 2.3 The distributional Betti numbers depend only on T and the class of X inX{M,T)/X{T). Proof. Suppose that V e X{M, J7) defines the same class as X in

X{M,F)/X{F).

Then, for all t, (X+s(Y-X))t:(M,F)^(M,F),

sGl,

is an leafwise homotopy between Xt and Yt- So n o / x:-f(s)ds

= Uo [Y;-f(s)ds

(6)

by (4). Take another bundle-like metric on M so that Y is of norm one and orthogonal to the leaves, and let II' : Q(J-) —> Ti'(J-) be the corresponding orthogonal projection onto the corresponding leafwise harmonic forms. Then n ' : H{T) 5 n'{T)

(7)

by (2). For any / e CC°°(R), let B} = IT o f Ys* • f(s) ds o IT : fl(T) -> fi(.F) , JR

and let B? denote its restriction to ^(T). Then the distributional Betti numbers /%lis, determined by J-, Y and the new bundle-like metric, are given

by
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KORDYUKOV

By (6), (7), and since 11,11' are projections, it follows that Tr (A.f)

= Tr (u o f X* • f{s)ds

: SV{F) - •

= Tr (u o f Y* • f(s) ds : Sl^F) -»

&{?)) n\T)

= Tr (II' o f Y; • f(s) ds : ft*(.F) -> ft* (.F)

= Tr(s}i)) . Therefore /3^is = /%\s as desired. 3 3.1

•

The distributional Euler characteristic and functions of the leafwise Laplacian A family of smoothing operators

Let UB(M) be the space of uniformly bounded Borel functions on R. Since the operator Dy = dj? + 5? is essentially self-adjoint in L2S7(.F), the Spectral Theorem defines a "functional calculus map" UB(R) -> End(L 2 ft(J r )) ,

<> / ^ <j>{Djr) ,

2

where End(L ft(jF)) denotes the bounded linear endomorphisms of LP$l(T). Let A be the set of functions <£ : R —> C that extend to entire functions on C so that, for each compact subset X c l , the set of functions x >-* —• e~tx with t > 0. By [21], the above functional calculus map, given by the Spectral Theorem, restricts to a "functional calculus map" A —> End(fi(.F)), which is a continuous homomorphism of C[2;]-modules and of algebras. Let Xi,..., xp, j / i , . . . , yq be foliation coordinates on a foliation patch U; i.e., (xi,...,xp,2/i,...,yg):t/-^RpxR9 is a diffeomorphism so that the slices R p x {*} correspond to the plaques of T in U. Then the differential forms dxj = dxix A . . . A dxir , for multiindices / = ( i i , . . . ,ir) with 1 < i\ < ... < ir < p, form a base of fl^lu) as C°°([/)-module. An operator B in ^(.T7) is local when, for

DISTRIBUTIONAL B E T T I NUMBERS OF TRANSITIVE FOLIATIONS

169

any a e ^(.F) and any x e M, the value (Ba)(x) depends only on the germ of a at x, and thus B defines an operator B\u in Q,{T\u) for any open subset U C M. Recall that a leafwise differential operator B in £l(T) is a local operator in fi(.F) such that, for arbitrary foliation coordinates x\,..., xp, 2/1,..., yq on any foliation patch U, with respect to the C°°(U)base dxj of fl (J-\u), the restriction B\u is given by a matrix whose entries are linear combinations, with coefficients in C°°(U), of the leafwise partial derivatives Qk

Qk

dxK

kl

dx

... dxpp '

for multiindices K = (fci,..., kp) e N p , where k = k\ + ... + kp. Now, a family {Bt | t £ R} of leafwise differential operators is called smooth when, for any foliation patch U with foliation coordinates x i , . . . , xp, y i , . . . , yq, in the corresponding expression of Bt\u, the above coefficients of the partial derivatives dk/dxK depend smoothly on t (they are C°° functions on!7xR). The support of such a family is the closure in R of the set of points t with P r o p o s i t i o n 3.1 Let <j> € A and let {Bt \ t e R} be a smooth compactly supported family of leafwise differential operators in Q(J-). Then B=

(jx*toBtdt\o4>(DT)

is a smoothing operator in J1(F) whose smoothing kernel depends continuously on W'fi(.F) || < f (I- dl)N4>(0 e™ d£ . (8) J& The proof of (8) is omitted here because it is rather technical. It is given in our work [3] with more generality. Observe that Theorem 1.1 is a direct consequence of Proposition 3.1. In fact, by Proposition 3.1, for / G C^°(M) and t > 0, the operator Bt,/=

[

X*s-f(s)dsoe-tAr

on £l(J-) is smoothing and its smoothing kernel depends continuously on / . Hence Af = HoBtj oil satisfies the same properties. This also shows that

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KORDYUKOV

the operator A'f = f X* • f(t)dtoU = BtJ oII Ju is smoothing with smoothing kernel depending continuously on / , and we have Tr(A'f) = T r ( ^ ) since II is a projection. 3.2

The distributional Euler characteristic and the leafwise heat operator

Let Btfl denote the restriction of Btj

to Ql(J-) for each degree i.

L e m m a 3.2 For any f G C~(R), we have TrB(tJ -> T r ^ as t -> oo. Proof. Note that BtJ

-A'f=

f X*s • f(s) ds o (e~tAJu = [ X*t • f(t)dtoe~Ar

- II) o (e-(*-D^ -if)

.

v Ju ' t 1 A r On the other hand, by [3], the operator e - ( - ) J — n has a continuous extension Sl{T~)' -» Sl{T)', and converges to zero in C(Q(T)',Cl(F)') (equipped with the topology of bounded convergence) as t —• oo. Therefore Btj — A'f converges to zero in £(J7(^r)',fi(^-")) as t —> oo, and thus its smoothing kernel converges to zero by Lemma 2.1, and the result follows.

•

For the sake of simplicity, it is worthwhile to use the supertrace notation. Consider ^(J 7 ) as a Z2-graded space: where n+(T) = neven(T) and Sl~(f) = fiodd(^"). For any Z2-homogeneous operator P on fl(J-), let P± denote its restrictions to fi±(^r). If moreover P is of trace class and Z2-degree zero, its supertrace is defined as Tr s (P) = T r ( P + ) - T r ( p - ) . In particular, (Xdis(^),/)=Trs(A/) for all / G C~(R). Choose an even function in A, which can be written a s m Then, for t > 0 and / G C~(R), let C U / = / ^a* • / ( s ) <** ° t ^ A ^ ) 2 : fi(.F) - fi(-F) •

tp(x2).

171

DISTRIBUTIONAL B E T T I NUMBERS OF TRANSITIVE FOLIATIONS

In particular, Btj = Ct^j when tp(x2) = e~x / 2 . Lemma 3.3 TrsCt^j is independent oft. Proof. It is similar to the proof of the corresponding result in the heat equation proof of the usual Lefschetz trace formula [7, 13]. We have

±' -Tr C ^ dt s

t f

= 2Trs(fx*-

f(s) ds o A ^ o $\tA^)

= 2Tr ( I X* • f(s) dsod^oS^o - 2Tr ( f X* • f(s) dsod+°5fO + 2Tr ( / X; • f(s) ds°5r°d+o

or/>(tAj:)\

?// (tA%) o i/> (tA

>£

tf

(tA^) o xj) (fA~)

^' (tA+) o V (tA+)

- 2 Tr ( / X ; • / ( s ) ds o 5+ o d~ o V' (*A~) o V (tA") On the other hand, since the function x \—> ip'(x2) is in .4, we have Tr ( f X* • f(s) dsodf°5$°

V' (t&$) o ^ (
= Tr ( d£ o /" Xs* • f(s) ds o V' (tA±) o ^ (iA£) o <5± = Tr (V> (tA%) o5%od^o

f X* • f(s) ds o -0' (tA%)

= Tr ( f X*s • f(s) ds o V' ( i A | ) o ^ (iA£) o 5% o d* = Tr ( /" Xs* • / ( s ) ds o «&± o d£ o 1/;' (*A|) o V ( * A | ) j , where we have used the well known fact that, if A is a trace class operator and B is bounded, then AB and BA are trace class operators with the same trace. Therefore ^ T V C * ^ , / = 0 as desired. • The following result follows directly from Lemmas 3.2 and 3.3. Corollary 3.4 We have TrsBti/ = (Xdis(n/) for anyt>0 and f G C~(R). Like in [21, p. 463], choose a sequence of smooth even functions m £ A, which we write as <j>m{x) — ipm(x2), with m(0) = 1, and whose Fourier

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KORDYUKOV

transforms (f>m are compactly supported and tend to the function
J | (id -
(9)

as m ~> oo. Consider the operator Ct,m,f = Ctti>mJ = f X* • f(g) dg o Vm(*A^) 2 JG

on Q(jc-). L e m m a 3.5 For any t > 0 and f € C^°(K), we have Tr s C t , m ,/ - TrsBtJ

= (xdi.(^),/)

as m ^ oo. Proof. Combining (9) and (8), we get that Ct,mj — Btj

—» 0

in ^ ( ^ ( J 7 ) ' , Q(!F)) as t —> oo. By Lemma 2.1, it follows that the smoothing kernel of Cttmj converges uniformly to the smoothing kernel of Btj, and the result follows. • 3.3

Description of the smoothing kernels

According to the structure of Lie foliations [12, 19], the foliation T can be described as follows. There is a finitely generated subgroup T C M that acts on the right in some manifold L such that: • M is diffeomorphic to the orbit space L x r K of the right T-action on M = L x l given by (x, s) • 7 = (a; • 7, s + 7) , say M = L x r l . covering map.

(x, s) £ L x M ,

7 e T;

Thus the canonical projection 7r : M —> M is a

• The leaves of the lifting T of T to M are the fibres L x {£}, t £ K, of the second factor projection D : M —> R. • The flow of the lifting X of X to M is given by Xt(x, s) — (x,t + s), t e M.

DISTRIBUTIONAL BETTI NUMBERS OF TRANSITIVE FOLIATIONS

173

Let Q, Q denote the holonomy groupoids of T, T respectively. Since the leaves of J", T have trivial holonomy groups, we have Q = {(x, y) € M | x, y lie in the same leaf of J7} , G = <(x,y) e M

x,y lie in the same leaf of T (fiber of D) > .

Thus Q, G are^C00 submanifolds o f M x M and M x M, respectively. Moreover it x n : Q —> (7 is a covering map whose group of deck transformations is

Aut (g -> g) = Aut(Tr) = r , where a € Aut(7r) corresponds to a x a G Aut (g —» g). Let s,r : g —> M the source and the range projections, which are the restrictions of the factor projections M x M —> M. Recall the definition of the global action of the convolution algebra

in Q,(!F). For any

* e C™ (<3,r* f\TF*

® s* /\TT}

, a e n(.F) ,

the element k • a G fi(^ r ) is given by (k-a)(x)

=

k{x,y)a{y)u)jr{y)

,

x£M,

where Lx is the leaf of T through x € M. Consider the lifting of the fixed bundle-like metric on M to M and its restriction to the leaves of T. Let w^j denote the Riemannian volume element of M, and UJ^ the Riemannian volume element of T. We also have a global action of any

keC°°(g,r*

f\TF*®s*

f\TT)

,

supported in an /^-neighbourhood of the diagonal M = A C g for some R > 0, on the space USl{T) of uniformly bounded differential forms in Cl(T): For any ex £ UCl^J7), the element k • ex E U£l(^F) is given by

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ALVAREZ LOPEZ AND Y.A.

KORDYUKOV

By [21], if h is a bounded Borel function on R such that its Fourier transform h e C£°(R), then the operator h{Dp) on £1{T) is represented by some element of Cc°° (g, r* f\ TT* ® s* f\

TT

Moreover, it follows from the proof of Assertion 1 in [21, p.461] that, for any function h in the Schwartz space <S(R) with supp/i C [—R, R], the operator h(Djr) on £l(T) is represented by a leafwise smoothing kernel on Q supported in the ^-neighbourhood of the diagonal M = A C §._ _ The map 7r x 7r : Q —> Q^ restricts to a diffeomorphism TT X TX : L x L —> Z- x L for any leaf L of T (L = ir(L)). Hence, the lift of the leafwise smoothing kernel of h{Djr) to Q is supported in the .R-neighbourhood of the diagonal M = A C Q, and thus defines an operator h (-D^.) on UQ, [T). It is clear that the diagram

USI(T\

^ l u n

(7TXX)*

Q(T)

(*) (7TX7r)*

^^

n( •n

commutes. Since
.

The action of kmj on fi(jF) defines the operator ipm(tAjr)2 any a G 0(J r ), we have (ipm(tAjr)2a)

(x) = (fcm,t -a)(x) = /

kmtt{x,y)a(y)ujjr(y)

in fi(jF): For

,

xeM.

This operator is equal to the operator ipm(tAjr)2 in fi(^") defined by the Spectral Theorem. Let km 0. For /? e E/fi ( TI, we have (Vm ( ^ )

2

/?) (i) = / fcm,t(x, y) /3(j/) w^(ji) ,

x e M .

175

DISTRIBUTIONAL B E T T I NUMBERS OF TRANSITIVE FOLIATIONS

For any / € C£°(R), consider the operator Ct,m,f = f X* • f(s) ds o m(iA^)2 : USl (.?) -» UQ ( > ) . Obviously, Ct,mj

o 7r* = IT* O Ct,m,f on

^T).

Lemma 3.6 For any f e C^°(M), £ > 0 and m, the operator Ct,mj is a smoothing operator whose smoothing kernel Ct,mj is supported in an Rneighbourhood of the diagonal in M x M for some R > 0, and ct,mj({x,

u), {y, v)) = X*__u o fem,t((x, v), (y, v)) • f(v - u)

(10)

for (x, u), (y,v) G L x K = M. Proof. For a e Ufl (F), we have (Ct,mja) = (

(x, u) X* • f(s) ds o 4,m (tAf)2aj

= J X; (ipm (tA^f = / ^-u JU

= L K-u

/

(x, u)

a ) (x, u + s)- f(s) ds km,t{(x,v),(y,v))a(y,v)u>f(y,v))

\JLX{V}

okm>t({x,v),(y,v))a(y,v)

-f(v-u)dv J

• f(v - u)w^(y,v)

,

JM

by using the change of variable s = v — u. Hence Ct,mj is defined by the smoothing kernel given in (10), and the result follows. • For each (x, u) € M, let [x, u] = TT(X, V). It is easy to see that the kernels Ct,mj and ctimj are related by the formula ([x,u],[y,v]) = ^2ct,m,f({x,u),(y-'Y,v

+ j)) ,

(11)

where we use the identity

via the map n. The sum in (11) is finite because Ct,m,f is supported in an .R-neighbourhood of the diagonal i n M x M for some R > 0.

176

4

J.A. ALVAREZ LOPEZ AND Y.A. KORDYUKOV

Distributional Euler characteristic and Connes' Euler characteristic

T h e goal of this section is t o prove Theorem 1.2. L e m m a 4 . 1 Given R > 0, there is some neighbourhood that TT:{{V,V)

| (y, u) e Bf((x,

u),R)

U of 0 in R so

, v - u e U} - • M

is injective for any (x,u) € M, where Bp((x,u),R) radius R and centred at (x,u) in the leaf L x {u}.

denotes

the ball of

Proof. Since M is compact, there exists a compact subset K C M with n(K) = M. Note t h a t , if t h e statement holds for (x, u) G K, then it holds for all {x,u) e M. Assume t h e result is false. T h e n t h e r e exist sequences (xi, Ui), (y,, Vi) G M , and a sequence 7i G T with (xi,Ui) G K, 7 4 ^ 0, and such t h a t {y%,Vi) and [\)i • 7i, Uj + 7i) approach B~((xi,Ui), R) in the sense t h a t t h e distance between t h e t e r m s of this sequences t o this set converges t o zero. Since K is compact, we can assume t h a t there exists limj(a;;,Ui)i = {x,u) e M. Hence, (yi,Vi) and (y; • 7i,«i + 7») approach t h e relatively compact set B^.((x, u), R). It follows t h a t , for infinitely many i, t h e points (J/J, Vi) and (t/j • ji, Vi + 7;) lie in some compact neighbourhood Q of Bp((x, u), R); thus Q • 7 J (~) Q =£ 0 for infinitely many i. This implies t h a t there exists some 7 € T such t h a t 7i = 7 for infinitely many i. In particular, 7^0. On t h e other hand, since(yj, Uj)and(yi -ji, v, + 7i) approach B^x, u),R), which is relatively compact, we can assume t h a t there exist \im(yl,vl) %

in B^((x,u),

,

lim(yi • 7 ; , ^ + -yt) i

R). Therefore, if lim^j/j, v^) = {y,v),

{y, v) G L x {u} ,

then

(y • 7, v + 7) = lim(yj • 7 ^ «, + 7 i ) 6 i x {u} ,

yielding z; = u = v + 7, and thus 7 = 0. T h e result follows from this contradiction. • L e m m a 4 . 2 For each m, there is a neighbourhood U of 0 in K such that the map IT is injective on the support of cttmj({x,u), •) for all (x,u) G M if t is small enough and the support of f is contained in U. Proof. For any fixed R > 0, choose some neighbourhood U of 0 in M satisfying t h e statement of L e m m a 4.1. For any m , we have supp (F (X I-> ipm (tx2)))

= Vt • supp (F {x >-> ipm (x2)))

,

177

DISTRIBUTIONAL B E T T I NUMBERS OF TRANSITIVE FOLIATIONS

where F denotes the Fourier transform. So, since supp (F (x H-> (f>m (z 2 ))) is compact, it follows that supp (F (x i-> ipm (tx2)))

C

R R 2"'"2~

if t is small enough, for each m. Thus the leafwise smoothing kernel of rpm (*A^) is supported in the ^-neighbourhood of the diagonal A c Q, and the leafwise smoothing kernel of tpm (tA^) is supported in the Rneighbourhood of the diagonal A C Q- From Lemma 3.6, we get supp(c t i m i / ((x,u),-)) C {(y,v) | (y,u) € B^((x,u),R)

, v - u e U)

for any (x, u) € M if t is small enough, and the result follows by Lemma 4.1.

•

By noncommutative integration theory [8], the holonomy invariant transverse measure A defines a trace on the von Neumann algebra of T', which can be shortly described as follows. The twisted convolution algebra Cc°° (S, r* /\ TF

® s* / \ TT)

is contained in the (twisted) von Neumann algebra W* {T, /\T!F*), and, for any k e Cc°°(g,r* l\TF* the trace

TTA(A;)

® s* /\TT)

,

is finite and given by the formula Tr A (fc)= / trk([x,u},[x,u])coM([x,u}).

(12)

Now fix U as in Lemma 4.2, and let / € C£°(R) with supp/ C U. Proposition 4.3 For all t > 0, we have TrsCUmJ

= /(0) • TrliPm(tAr)2

.

s

(13)

Proof. Recall that Tr C t , m j / is independent of t by Lemma 3.3, and TrsAipm(tAjr)2 is also independent of t by [21]. So we need only prove this statement for a single t. If t is small enough, we have TrsCt,mi/= /

trs(cttmj([x,u},[x,u])ujM([x,u\)

JM

tr s (c t , m ,/((:r, u), (x, u))) CJ^ (X, U) J M

178

J.A. ALVAREZ LOPEZ AND Y.A.

KORDYUKOV

JM

=/(0) • /

tTs(km,t([x,u],[x,

U]))CJM{[X,U})

JM

= /(0) • TrAm {t&f)2 by (11), Lemma 4.2, Lemma 3.6 and (12) since km,t G Cc°° (g, S* f\ TF ®r* f\ TT)

.

• Now we recall some facts on Connes' Betti numbers. The family {Pi,L | L is a leaf of T} , where each Piti is the orthogonal projection onto the space of square integrable harmonic i-forms on L, defines a projection Pt in the twisted foliation von Neumann algebra W* {T, [\TT*). As in [8], one can define A-Betti numbers /3A(.F) as P\{F) = Tr A Pi . Then the A-Euler characteristic of T is i

Using the corresponding supertrace notion, this formula can be rewritten as XKif) = Tr s A P , where P = ^ t ^ . By [21], we have that Tr A (tpm (tAp) J is independent of t, and TrA(^m(tA^)2)^XA(^)

(14)

as m —>- oo for alH > 0 (independently of m!). Proof of Theorem 1.2. Fix a neighbourhood U of 0 in R as in Lemma 4.2. Let / 6 C£°(R) with supp/ s U. Combining Proposition 4.3, Lemma 3.3 and (14), we have TrsCt,mJ

= /(0) • TrsA ( > m (tA?)2)

as m —> oo for any t > 0.

- /(0) • XA(F)

(15)

DISTRIBUTIONAL B E T T I NUMBERS OF TRANSITIVE FOLIATIONS

179

Fix any e > 0. From Lemmas 3.5 and 3.3, it follows that |TrsCt,m,/-Tr8flt,/|<e for any t > 0 if m is large enough. But 1rs5t,/ = (Xdis(n/> for any t > 0 by Corollary 3.4. So \TrsCttrnj-(Xdis(F)J)\<e

(16)

for any t > 0 if m is large enough. From (15) and (16), it follows that |(Xdis(n/)-/(0)-XA(^)|<£ foranye > 0, yielding Xdis(^) = are arbitrary. 5

XA(F)-5O

on U sincee > 0 and / € C%°(U) D

Localization theorem

Theorem 5.1 The distribution Xdis(^") is supported in the set of all s G M such that Xs has a fixed point in M. Proof. Let V be an open subset in R such that Xs has no fixed points for all s G V. We have to prove that Xdis^) — 0 on V. Note that the fact that Xs has no fixed points for all s G V is equivalent to Xs(x,u)

^(x-j,u

+ -y)

for any s G V, 7 G V and (x, u) G M. One can prove an analogue of Lemma 4.2, asserting that there exists a neighbourhood U of 0 in K such that, for any SQ G K and for each m, if / G C^°(U + s 0 ), then 7r is injective on the support of ct,mj((x,u), •) for (x, u) G M if £ is small enough. i,From this, it follows that, for each m, if t is small enough, then, for any [x,u] G M, either Ct,mj{[x,u],[x,u\)

= 0

or Ct,m,f{[x,

u], [X, U\) = Ct,m,f{{x,

U), ( l • 7 , U + 7 ) )

for some 7 G rn(C/ + so)- In the latter case, such a 7 is uniquely determined by (x,u). Take s0 G V and some neighbourhood [/ of 0 as above, satisfying also U + SQ C V. Since so is an arbitrary point of V, it is enough to show

180

J.A.

ALVAREZ L O P E Z AND Y.A.

KORDYUKOV

that (xdis,/) = 0 for any / e C°°(U + s0)- Then, by Lemma 3.5 and Corollary 3.4, it is enough to show that Tr s Ct,m,/ — 0 for each m and t small enough (depending on m). We have TrsCt,mi/= /

tTs(ct,mj([x,u],[x,u]))u>M({x,u\)

JM

-L -L

tr s (ct,m,f((x, u), (x--y,u + 7)))

UJ^(X,

u)

JM

tr s [X* o km,t((x,u

+ 7), (a; • 7 , u + 7))) • /(•y) w^(x,u) ,

JM

for the appropriate choice of 7 € T, where we use the identity /\T(VIV)F*

= / \ T{y.7tV+7) F*

given by the diagonal action of 7 on M. Since supp/ C U + So, we can consider only those [x,u] e M with 7 6 U + so for some choice of 7 € T with (x • 7, u + 7) in the support of Ct,m,/. Thus 7 € V, yielding (x • 7, u + 7) ^ (x, u + 7) by assumption. It follows that km,t((x, U + J),(X-J,U

+ 7)) -^ 0

as t —» 0 uniformly on (x,u) € M. Since Tr s C t , m ] / is independent of t, we get Tr s C t i m ! / — 0 for each m, as desired. • 6

The Lefschetz trace formula

The goal of this section is to prove Theorem 1.3. By Theorem 5.1, in order to evaluate TrsCtim,/= /

trscttmJ([x,u},[x,u})ivM([x,u})

JM

asymptotically as t —> 0, it is enough to integrate over small neighbourhoods of closed orbits. As in the proof of Theorem 5.1, take a neighbourhood U of 0 in K such that, for any s o € K and for each m, if / e C£°(U + so), then IT is injective in the support of ct,m,f{(x,u), •) for all (x,u) € M if t is small enough. Let s 0 be the period of some closed orbit of X. There exist finitely many closed orbits with the period in U + SQ. Hence, the neighbourhood

DISTRIBUTIONAL B E T T I NUMBERS OF TRANSITIVE FOLIATIONS

181

U can be chosen so that so is the only period that belongs to so + U, and thus only this period may be in supp/. Take a closed orbit of period So, and let c be the corresponding primitive closed orbit with length I — 1(c); thus so = kl for some integer k > 0. We also get that I € T, and (x, u + I) = (x • I, u + I) ,

(x, u + kl) = (x • kl, u + kl)

if [x, u] is in c. So x is a fixed point of the action of I on L, and there are no other fixed points of elements of T n (SQ + U) in some open neighbourhood W of x in L because all X-orbits are simple. Note also that 7r({z} x [0,/]) = c and 7r : {x} x (0,1) —> c

is a C°° embedding. Moreover 7r(W x [0,1]) is an open neighbourhood of c where there are no other orbits of period in so + U, and TT-.W

x(0,l)^

M

is a C°° embedding. Denote by fct e C°° (<J, r* / \ TT* ®s* /\ TiF) the leafwise smoothing kernel of the leafwise heat operator e _ t A ^ . Then, since km oo, by Lemma 3.6, it is enough to compute the asymptotics as t J. 0 of tr s (x; o kt ((y,v + kl), (y -kl,v + kl))) •

/ JWx[0,l]

= f(kl)-

V

f J

f(kl)uj^(y,v)

'

tT*(xh°h{{v,v

+ kl),{y-kl,v

+

kl)))wWiV(y)dv,

where u>w,v is the restriction of LJ^ to W x {v} = W, and we use the identity

given by the diagonal action of kl on M. But, by [7, 13], the integral J^ tr s (X*kl o & ((y, v + kl), (y -kl,v + kl)))

uw,v(y)

converges as t J, 0 to signdet (id - Xku

: T{x,v)(W

x {v}) -» T(XtV)(W x {v})) ,

which is independent of v, and the proof is finished.

•

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KORDYUKOV

References 1. J.A. Alvarez Lopez and S. Hurder, Pure-point spectrum for foliation geometric operators, preprint, 1995. 2. J.A. Alvarez Lopez and G. Hector, The dimension of the leafwise reduced cohomology, preprint, 2000. 3. J.A. Alvarez Lopez and Y.A. Kordyukov, Long time behaviour of leafwise heat flow for Riemannian foliations, preprint dg-ga/9612010, 1996, to appear in Compositio Math. 4. J.A. Alvarez Lopez and Y.A. Kordyukov, Distributional Betti numbers for Lie foliations, preprint, 2000. 5. J.A. Alvarez Lopez and P. Tondeur, Hodge decomposition along the leaves of a Riemannian foliation, J.Funct. Anal., 99 (1991), 443-458. 6. M.F. Atiyah, Elliptic operators and compact groups, in Lecture Notes in Mathematics, 401, 1-93. Springer Verlag, Berlin-Heidelberg-New York, 1974. 7. M.F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes I, Ann. of Math., 86 (1967), 374-407. 8. A. Connes, Sur la theorie non-commutative de I 'integration, in Algebres d'operateurs, Lecture Notes in Math. 725, Springer Verlag, 1979, 19-143. 9. A. Connes, Noncommutative differential geometry, Publ. Math., 62 (1986), 41-144. 10. C. Deninger, Some analogies between number theory and dynamical systems on foliated spaces, Doc. Math. J. Extra Volume ICM98, 1 (1998), 163-186. 11. C. Deninger and W. Singhof, A note on dynamical trace formulas, manuscript, 2000. 12. E. Fedida, Sur les feuilletages de Lie, C. R. Acad. Sci. Paris. Ser. A-B, 272 (1971), A999-A1001. 13. P.B. Gilkey, Invariance Theory, the Heat Equation and the AtiyahSinger Index Theorem, Publish or Perish Press, Berkeley, 1984. 14. A. Haefliger, Some remarks on foliations with minimal leaves, J. Diff. Geom., 15 (1980), 269-384. 15. Y.A. Kordyukov, Transversally elliptic operators on G-manifolds of bounded geometry, Russ. J. Math. Ph., 2 (1994), 175-198. 16. Y.A. Kordyukov, Functional calculus for tangentially elliptic operators on foliated manifolds, in Analysis and Geometry in Foliated Manifolds, Proceedings of the VII International Colloquium on Differential Geometry, Santiago de Compostela, 1994, 113-136, World Scientific, Singapore, 1995.

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17. Y.A. Kordyukov, Noncommutative spectral geometry of Riemannian foliations, Manuscripta Math., 94 (1997), 45-73. 18. C. Lazarov, Transverse index and periodic orbits, Geom. and Funct. Anal, 10 (2000), 124-159. 19. P. Molino, Geometrie globale des feuilletages riemanniens, Proc. Nederl. Acad. Al, 85 (1982), 45-76. 20. A. Nestke and P. Zuckermann, The index of transversally elliptic complexes, Rend. Circ. Mat. Palermo, 34 Suppl. 9 (1985), 165-175. 21. J. Roe, Finite propagation speed and Connes' foliation algebra, Math. Proc. Cambridge Philos. Soc, 102 (1987), 459-466. 22. K.S. Sarkaria, A finiteness theorem for foliated manifolds, J. Math. Soc. Japan, 30 (1978), 687-696. 23. I.M. Singer, Recent applications of index theory for elliptic operators, in Proc. Symp. Pure Appl. Math. 23, 11-31. Amer. Math. Soc, Providence, R. I., 1973.

Received November 3, 2000.

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Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2002 pp. 185-195

CIRCLE AT INFINITY INFLUENCES ON THE S M O O T H N E S S OF SURFACE FLOWS

SAMUIL ARANSON Department

of Applied Mathematics, Nizhny State Technical University 24 Minina Str., Nizhny Novgorod 603600, e-mail: [email protected]

Novgorod Russia,

EVGENY ZHUZHOMA Department

of Applied Mathematics, Nizhny State Technical University 24 Minina Str., Nizhny Novgorod 603600, e-mail: zhuzhomaQfocus.nnov.ru

Novgorod Russia,

Let M be a closed hyperbolic surface of negative Euler characteristic and A the universal covering space of M. Let Soo be the circle at infinity of A. Then there exists a continual set U 6 Soo with the following property. Suppose a flow / ' on M has a semitrajectory ft such that a lift I of ft has an asymptotic direction defined by a point of U; then / ' is not analytic and has a continual set of fixed points. Moreover, / ' has neither nontrivially recurrent trajectories nor closed transversals nonhomotopic to zero. The set U is dense and has zero Lebesgue measure on Soo. Given any point a & U, there exists a C°°-flow on M and the corresponding covering flow / on A such that some semitrajectory of / has an asymptotic direction defined by the point a.

Introduction The purpose of this paper is to consider some aspects of Anosov-Weil's theory. Most generally by the theory of Anosov- Weil one understands the study of asymptotic properties of curves with no self-intersections lifted to the universal covering, and their 'deviation' from the lines of constant geodesic curvature that have the same asymptotic direction (see reviews [4] and [15], Chapter 10). Throughout the paper M is a closed surface of negative Euler charac185

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teristic x(M) < 0. Let A be a hyperbolic (or Lobachevsky) plane, which is a simply connected complete Riemannian 2-manifold of constant negative curvature — 1. For A, we use the Poincare disk model \z\ < 1 of the complex 2-plane endowed with the metric ds = 2\dz\/(l — \z\ ). The circle 5oo = dA — (\z\ = 1) is called the circle at infinity. The geodesies of A are Euclidean circular arcs orthogonal to SQQ . We shall suppose that endpoints of any geodesic are ideal points belonging to the circle at infinity. Any M can be thought of as the orbit space A/T, where T is a discrete group of isometries of A acting freely on A. The group T is isomorphic to the fundamental group of M. The natural projection ir : A —> A / r = M is a universal cover. Suppose /* is a flow on M and / is a covering flow on A. Let Z* be a positive (negative) semitrajectory of /* and Z a semitrajectory of / which covers l^. If 7 tends to a unique point of Sex,, say a, as t —•» ±oo, we shall say that I has an asymptotic direction defined by a. Sometimes we shall say that l^ has an asymptotic direction whenever some lift of Z* (and so every lift) has an asymptotic direction. The point a € Soo is called a point achieved by fl. Denote by Aji C Soo the set of points achieved by all flows on M. The main result of the paper is the following theorem. T h e o r e m 2.1 Let M be a closed surface of negative Euler characteristic and Soo be the circle at infinity of the universal covering space A of M. Then there exists a continual set U{IJol h{M)) C Aji with the following properties. Suppose a flow /* on M has a semitrajectory Z± such that a lift I has an asymptotic direction defined by a point of U(IJol h(M)); then fl is not analytic and has a continual set of fixed points. Furthermore, /* has neither nontrivially recurrent semitrajectories nor closed transversals nonhomotopic to zero. The set U(IJol ^(M)) is dense and has zero Lebesgue measure on S ^ . Note that due to the papers [2] and [10], given any point of Ug = U{IJolh{M)), there is a C^-flow /* which satisfies theorem 2.1 (see also [3], where a C°°-flow /* with a preassigned smooth invariant measure was constructed) . 1

Preliminaries

A foliation F on M is called irrational if the following conditions hold: - F has a finite number of singularities.

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- every one-dimensional leaf of F is dense on M. - F has no fake saddles. It follows from the definition that any singularity s of F is of saddle type i.e., some neighbourhood of s is a union of saddle sectors divided by separatrices. The index of s is equal to ind s = 1 — ^ p , where v(s) > 1 is a number of separatrices of s. Recall that s is called a fake saddle if v{s) = 2. Denote by If0i(M) the set of irrational foliations on M. The following lemmas are needed for the sequel. L e m m a 1.1 Let M2 be a closed orientable surface of genus g > 2. There exists an irrational foliation T on M2 such that any singularity of T has negative index and there are at least two singularities of half-integer index. Proof. According to the paper [8], there exists an irrational foliation .F3 i on the projective plane P2 with four singularities: three thorns and one tripod. Let p : S2 —> P2 be a double covering, where S2 is the 2-sphere. By the Path Lifting Theorem, there is a covering foliation T§^. on S2. It is easy to see that TQ^ has six thorns ri\, . . . ,n% and two tripods t\, £2Since F$t\ is irrational and p is a double covering, ^6,2 is irrational as well. Fix a natural number g > 2 and take a set £29-4 C S2 of 2g — 4 points (if g = 2, then £29-4 = 0) so that these points belong to pairwise disjoint leaves and no points on separatrices of the foliation .7-6,2 • Calling every point of E29_4 a singularity (a fake saddle which has exactly two saddle sectors), we get the new foliation denoted by !F'& 2- By the choice of E2 g -4, T'§ 2 is a highly transitive foliation. Let q : M2 —* S2 be a branched double cover having the branched set S2 S -4Uf =1 {n,} with every point of branched index 2. Since S2 s -4Uf =1 {ni} belongs to the set of singularities of T'& 2 , it follows that there is a foliation T' on M2 that is a lift of T'§2 under q. The singularities of T' are six fake saddles q~l(ni), ... ,q~1(ne), and four tripods belonging to <7_1(£i), q~l(t2), and 2g — 4 Morse's saddles each of the index —1. Note that the index of a tripod equals —0,5. Since q is a branched double cover, J7' is a highly transitive foliation. Deleting the fake saddles, we get the desired foliation T on M2. • Another way to obtain irrational foliations with singularities of halfinteger negative index is to apply Theorem 2 [13] where irrational foliations are invariant foliations of pseudo-Anosov homeomorphisms. Denote by IJol(M2) the set of irrational foliations on the surface M2 which have only singularities of negative index. Denote also by IJol h{M) C IJol(M) the subset of foliations having singularities of half-integer index. By Lemma 1.1

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We recall that a geodesic lamination on M is a non-empty closed subset of M which is a disjoint union of simple (without self-intersections) geodesies. Given an irrational foliation T € IJol(Mg), one can constructs a special geodesic lamination called a geodesic framework. Consider a leaf L of T € IJol(Mg) which is not a separatrix. Then both positive and negative semileaves of L have asymptotic directions [12]. Let L be any lift of L to A. Then the ideal endpoints a{L), w{L) of L belongs to Sao and a(L) ^ w(Z-) because the index of every singularity is negative. Let g(L) be the geodesic with the endpoints a(L), UJ(L). This geodesic is called a geodesic corresponding to L. The geodesic 7r(g(L)) = g(L) does not depend on a lift of L and is called a geodesic corresponding to the leaf L. Since L is a simple curve (i.e., with no self-intersections), g(L) is a simple geodesic. Hence the topological closure clos[g(L)] of g(L) is a geodesic lamination, Lemma 3.1 in [9]. This geodesic lamination is independent of the choice of L because any leaf of T is dense in Mg. The geodesic lamination clos[g(L)} d^f G(F) is called a geodesic framework of T. Lemma 1.2 Let T € IJol(Mg) and let G(J-) be the geodesic framework of the foliation J-. Then G(J-) is a minimal geodesic lamination consisting of nontrivially recurrent geodesies, each dense in G(J-). Moreover, any ideal endpoint of a lift of every geodesic g £ G(F) is irrational, i. e. not a fixed point of an element of the Fuchsian group T. Proof. Since any leaf of T is dense in Mg, it follows that any geodesic of G(T) is nontrivially recurrent and dense in G(F). Hence, G{T) is a minimal geodesic lamination i.e., it has no proper non-empty sublaminations. Due to [7], any ideal endpoint of a lift of every geodesic g e G{!F) is irrational. • Lemma 1.3 Let T e IJol(Mg) and let G{T) be the geodesic framework of the foliation T. Given any geodesic g e G(^ r ) and closed geodesic go, the intersection g n go is nonempty. Proof. Due to [12], there is either a leaf L or generalized leaf L (denoted by the same letter) such that g(L) — g. Further, any component of Mg — G{F) is simply connected because T is an irrational foliation (see also [9] Chapter 4). Combining this with the density of g in G{!F), we get the desired result. • Recall that any component of Mg — G(J-) is an ideal polygon (see, e.g. [9]), Figure 1.

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Figure 1. An idea! polygon.

Let 51, g2 be adjacent sides of an ideal polygon P c M | - G(JF). Take a cutting segment S through gi, g2 so that some subsegment (o, b) C S flint P and geodesies rays gt C gi, g} C g2 with the initial points _a € S D gu b € E n g2 respectively form an ideal triangle denoted by T(£,gi,g2), see Figure 1. This triangle is called a tail triangle of P formed by the adjacent sides gi, g2 and the cutting segment E. L e m m a 1.4 Let G(F) be the geodesic framework of a foliation T €tfoi (M£) and P an ideal polygon of G{T). Then every tail triangle of P is dense in G(F). Proof. Let T{Y,,gt,g2) be a tail triangle of P formed by adjacent sides glt g2 of P and a cutting segment E. Then T(T,,gi,g2) is bounded by geodesies rays g? C gi, g2 C g2• It follows from Lemma 1.2 that both gx and gt are dense in G{F). This concludes the proof. • 2

P r o o f of t h e m a i n t h e o r e m

Let I be an irrational foliation on M that satisfies the conditions of Lemma 1.1 i.e., any singularity of T has negative index and there are at least two singularities of half-integer index. Denote now by IJolJl(M) c J / o i ( M ) t h e subset of suchfoliations on the surface M. Let f be a lift of T on A. Then any sernileaf L* of T that is not a separatrix goes to infinity and has an asymptotic direction
= y ALF),

where T ranges over

IJolyh(M).

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Due to [12], [ / ( / ^ ( M ) ) ^ . T h e o r e m 2 . 1 Let M be a closed surface of negative Euler characteristic and SQO be the circle at infinity of the universal covering space A of M. Then there exists a continual set U{l7ol h{M)) C Afi with the following properties. Suppose a flow fl on M has a semitrajectory l^ such that a lift I has an asymptotic direction defined by a point of U{I7olh{M)); then / * is not analytic and has a continual set of fixed points. Furthermore, /' has neither nontrivially recurrent semitrajectories nor closed transversals nonhomotopic to zero. The set U(IJol h(M)) is dense and has zero Lebesgue measure on S^. Proof. W i t h o u t loss of generality one can assume t h a t M is a closed orientable surface M% of genus g > 2, otherwise we pass to a double (nonbranched) cover. To simplify m a t t e r s denote U{IJol h{M)) = Ug. For t h e reader's convenience, we divide t h e proof into steps. T h e end of proof of a step will be denoted by o. We keep t h e notation of Section 1. S t e p 2 . 1 Take T € IJol h{M^) and let G{T) be a geodesic framework of the foliation T. Then G(J-) has at least two ideal polygons with odd number of sides. Proof of Step 2.1 Since T has at least two singularities with a halfinteger index, it immediately follows t h a t there are a t least two saddles, say s\ a n d S2, with odd number of separatrices each. Let s i and S2 be lifts of s\ a n d S2 respectively. Then t h e corresponding lifts of separatrices define t h e ideal vertices of t h e ideal polygons with odd number of sides, o S t e p 2 . 2 Let G{!F) be the geodesic framework of a foliation T G IJol h{Mg) and £ a geodesic segment which is transversal to G f (^ r ), int E n G(JF) ^ 0. We consider £ to be a segment endowed with some normal orientation. Then given any geodesic g £ G{T) endowed with a natural parametrization 9 : R —> g, there is a sequence of parameters U € R such that ti —> oo as i —> oo, 0(ti) n S / 0 , and the index of intersection g D E at the point 9(ti) equals (-1)*, i £ N. Proof of Step 2.2 By Step 2.1, there is an ideal polygon P C G(.F) with o d d number of sides. By Lemma 1.2, any positive ray g+ of g is dense in G{T), and thus dP C clos g = G(^"). Taking into account t h a t number of sides of P is odd, it follows t h a t there is a cutting segment E12 intersecting two adjacent sides 51, 52 G dP of P a t t h e points m i e <7in£i2, 77i2 € 52 n £12 respectively such t h a t 1. int PC\ ( m i , m 2 ) = 0. 2. There is a sequence of parameters t\ € R with 0 ( ^ ) —• m\ U m.2 as

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i —> oo. 3. The index of intersection g n S 1 2 at the point 9^) equals (-1)*, % € N (see Figure 2).

Figure 2. The tail triangle T ( S 1 2 , g i " , f l J )

Let gt C 3i (i = 1,2) be the geodesic rays which form together with S i 2 the tail triangle T ( £ 1 2 , gt, gt) of P, Fig. 2. By Lemma 1.4, T(Ei 2 , gf,gt) is dense in G(.F). Hence, T(TZ12> gf, gt) intersects S. To conclude the proof, it remains to apply the theorem on the continuous dependence of geodesies of G(T) on initial conditions, o S t e p 2.3 Suppose a flow f on M% has a semitrajectory l+ such that a lift I + ofl+ has an asymptotic direction defined by a point a(l ) S Ug; then f has no closed transversals nonhomotopic to zero. Proof of Step 2.S Assume the contrary. Then there is a closed simple transversal T of / ' which is nonhomotopic to zero [5]. It is well known that there exists a closed simple geodesic 30 which is freely homotopic to T. ^ By the condition, there is a semileaf L+ of some irrational foliation r € IJot g(L). Let I be the lift of L containing L+ and g(L) the geodesic on A corresponding to X". Then a+(g(L)) = a. Note that due to Lemma 1.3, g(L)

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intersects go- By Step 2.2, there is a sequence of parameters U € M. such that ti —> oo as i —> oo, 9(ti)r\go ^ 0, and the index of intersection g(L)(lgo at the point 9{ti) equals (— l) 1 , i € N. Hence there is a sequence of ~gi of lifts go which satisfy the following properties (see Figure 3): 1. g(L) intersects 'gi at a unique point, say m*. 2. The topological limit of geodesies gt equals a. 3. The index of intersection g~(L) n ~gi at m, equals (—1)\ Since the geodesic go is freely homotopic to T, it follows that 'gi is a corresponding geodesic of some lift Ti of T for every z G N, i.e., 7^ and Ti have the same ideal endpoints on S^. By item 2,1 intersects Tj beginning with i > io for some io sufficiently large, see Figure 3.

Figure 3. The sequence of gi.

By item 3, the index of intersection I n Ti, i > io is equal to ( — l) 1 . Hence, / + must intersect T in opposite directions, which is impossible, o As a consequence of Step 2.3, /* has no nontrivially recurrent semitrajectories and trajectories, otherwise / ' would have a closed simple transversals nonhomotopic to zero (Lemma 1.2.3 in [5]), which is impossible. S t e p 2.4 Suppose a flow /* on Mg has a semitrajectory l+ such that a lift I ofl+ has an asymptotic direction defined by a point a(l ) € Ug; then /* has a continual set of fixed points. Moreover, the uj-limit set of l+ consists of fixed points. Proof of Step 2.4 Assume the contrary, i.e., the set of fixed points of / ' is not continual. The w-limit set w(Z+) of l+ is not a unique fixed point because I has an asymptotic direction. Hence, u>(l+) contains at least two

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points. It is well known (see, e.g., [5]) that an w-limit set of any positive semitrajectory is a connected set. Combining this with our assumption, we get that co(l+) contains a one-dimensional trajectory, say l\. Let S\ be a transversal segment through a point a € h, a £ int S\. Then l+ intersects S\ infinitely many times because o £ w ( I + ) . As a consequence, there is a closed simple transversal T intersecting Z+, Corollary 1.1.2 [5]. Since I has an asymptotic direction, T is not homotopic to zero. This contradiction with Step 2.3 concludes the proof, o Step 2.5 Suppose a flow / ' on M^ has a semitrajectory l+ such that a lift I of l+ has an asymptotic direction defined by a point ail ) £ Ug; then / ' is not analytic. Proof of Step 2.5 Assume the contrary. Denote by Fix{ft) the set of fixed points of / ' . According to [1], pp. 38-39, Fix(ft) contains a finite number of isolated points, a finite number of non-isolated points S, which are endpoints of curves formed by fixed points, and closed curves, and nonclosed curves with endpoints in S. Outside of S, these curves of Fix(ft) are analytic, pairwise disjoint, and have no self-intersections. As a consequence of this description, Fix(fl) is a compact set. By Step 2.4, the w-limit set of l+ is a compact subset of Fix(ft). The description above of Fix(fi) shows that l+ tends spirally to a closed contour nonhomotopic to zero because I has an asymptotic direction. Hence, a{l ) is a rational point. This contradicts Lemma 1.2. o To conclude the proof it remains to show that Ug has zero Lebesgue measure. Following [11], we shall call a geodesic g c A transitive if given any intervals U\ and Ui C 5 ^ , there is a map 7 6 T such that one ideal endpoint of 7(5) belongs to Ui and the other ideal endpoint of 7(5) belongs to 1/2- Denote by TR(T) C S^o the set of the points a € 5oo with the following property: given any other distinct point b € Sooi the geodesic joining a, b and directed from b to a is transitive. Myrberg [14] proved that the Lebesgue measure of TR(F) is equal to the Lebesgue measure of Soo. In particular, Soo — TR(T) has zero Lebesgue measure. Step 2.6 Ug C Soo —TR(T).

In particular, Ug has zero Lebesgue measure.

Proof. In fact, it was proved by Hedlund [11] that any geodesic with an endpoint in TR(T) is transitive. So, if we assume that there is a point a € Ug n TR(T), then a is an endpoint of a lift g~ of some geodesic g € G{T), where T £ 77o( h(Mg) is an irrational foliation. Obviously, g has self-intersections because 7j is transitive. This contradicts the fact that a geodesic lamination consists of geodesic with no self-intersections. •

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Acknowledgments The authors are grateful to D. Anosov, V. Grines, R. Plykin, A. Stepin and A. Zhirov for useful discussions. We thank a referee for useful remarks which helped us to correct many errors. The research was partially supported by the INTAS grant 97-1843 and RFBR grant 99-01-00230. References 1. D.V. Anosov, On the behaviour of trajectories, on the Euclidian and Lobachevsky plane, covering trajectories of flows on closed surfaces, I, Izvestia Acad. Nauk SSSR, Ser. Mat., 51 (1987), no 1, 16-43 (in Russian); Transl. in: Math. USSR, Izv., 30 (1988). 2. D.V. Anosov, On the behaviour of trajectories, on the Euclidian and Lobachevsky plane, covering the trajectories of flows on closed surfaces, II, Izvestia Acad. Nauk SSSR, Ser. Mat., 52 (1988), 451-478 (in Russian); Transl. in: Math. USSR, Izv., 32 (1989), no 3, 449-474. 3. D.V. Anosov, On the behaviour of trajectories, in the Euclidian and Lobachevsky plane, covering the trajectories of flows on closed surfaces, III, Izvestia Ross. Akad. Nauk, Ser. Mat., 59 (1995), no 2, 63-96 (in Russian). 4. D.V. Anosov, Flows on closed surfaces and behaviour of trajectories lifted to the universal covering plane. Jour, of Dyn. and Control Sys., 1 (1995), 125-138. 5. S. Aranson, G. Belitsky and E. Zhuzhoma, An Introduction to Qualitative Theory of Dynamical Systems on Surfaces, Amer. Math. Soc, Math. Monogr., Providence, 1996. 6. S. Aranson, I. Bronshtein, I. Nikolaev and E. Zhuzhoma, Qualitative theory of foliations on closed surfaces, J. Math. Sci., 90 (1998), no 3, 2111-2149. 7. S. Aranson and E. Zhuzhoma, Maier's theorems and geodesic laminations of surface flows, Jour, of Dyn. and Contr. Sys., 2 (1996), no 4, 557-582. 8. P. Arnoux and J.C. Yoccoz, Construction de diffeomorphismes pseudoAnosov, C. R. Acad. Sci., 292 (1981), 75-78. 9. A.J. Casson and S.A. Bleiler, Automorphisms of Surfaces after Nielsen and Thurston, London Math. Soc. Student Texts, Cambridge Univ. Press, 1988. 10. C. Gutierrez, Smoothing continuous flows on 2-manifolds and recurrences, Ergod. Th. and Dyn. Sys., 6 (1986), 17-44.

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11. G. Hedlund, Two-dimensional manifolds and transitivity, Ann. Math., 37 (1936), no 3, 534-542. 12. G. Levitt, Foliations and laminations on hyperbolic surfaces, Topology, 22 (1983), no 2, 119-135. 13. H. Masur and J. Smillie, Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms, Comment. Math. Helv., 68 (1993), 289-307. 14. P.J. Myrberg, Ein Approximationssatz fur die Fuchsschen Gruppen, Acta Math., 57 (1931), 389-409. 15. I. Nikolaev and E. Zhuzhoma, Flows on 2-dimensional manifolds, Lecture Notes in Math. 1705, Springer Verlag, 1999.

Received June 7, 2000.

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Proceedings of F O L I A T I O N S : G E O M E T R Y AND D Y N A M I C S held in Warsaw, May 2 9 - J u n e 9, 2000 ed. by Pawel W A L C Z A K et al. World Scientific, Singapore, 2002 pp. 197-211

E N T R O P I E S OF H Y P E R B O L I C G R O U P S A N D SOME FOLIATED SPACES ANDRZEJ BIS Wydzial Matematyki, Uniwersytet Lodzki, ul. Banacha 22, 90-238 Lodz, Poland, e-mail: [email protected] PAWEL G. WALCZAK Wydzial Matematyki, Uniwersytet Lodzki, ul. Banacha 22, 90-238 Lodz, Poland, e-mail: pawelwalSmath. uni. lodz. pi

1

Introduction

Hyperbolic groups in the Gromov's [9] sense play an important role in geometric group theory (see Grigorchuk and de la Harpe [8] and the references there). In particular, any non-elementary hyperbolic group has exponential growth and the compact boundary of positive finite Hausdorff dimension (see, Ghys and de la Harpe [6], pp. 126 and 157). Also, a hyperbolic group G (generated by a finite symmetric set S) acts on the boundary of its Cayley graph X = C(G, S) via Lipschitz quasi-conformal maps (ibidem, p. 127). Roughly speaking, if (xn) is a sequence of elements of G representing a point £ of dX and g £ G, then g(£) is a point of dX represented by the sequence (g • xn). The dynamics of this action is of great interest. For instance, it has been shown (see Coornaert and Papadopoulos [3]) that this action is finitely presented, i.e. it is semiconjugate to a subshift of finite type in such a way that the fibres of the conjugating map are finite of bounded length and the equivalence relation determined by this map (two points are related whenever their images are equal) is another subshift of finite type. Also, one can consider the topological entropy h(G, S) of this action in the sense of Ghys, Langevin and Walczak [7]. Let us recall at this 197

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point that the entropy of a group action, as well as the related geometric entropy of a foliation, generalizes the topological entropy of a continuous transformation in the sense of Bowen [1]. For readers' convenience, suitable definitions are given in Section 2. Given a finitely generated group G and a finite symmetric generating set S, one can consider its rate of growth gr(G, S) defined as the rate of growth of the cardinality of the families of these elements of G which can be expressed as words of given length in the alphabet S (see Section 3 for more details). It seems that the numbers h(G, S) and gr(G, S) should be closely related. In fact, we expect that they should be equal as it happens in the case of free groups (compare Section 5) when the Cayley graph is just a tree. The graph of an arbitrary hyperbolic group, even for a good generating set, is not a tree and has some "unwanted" links. These links provided technical difficulties which did not allow us to get the expected equality in general. However, their role becomes less and less significant when the scale is enlarging. This drove us to introduce a technically complicated notion of the rate of growth gr rel (G, 5; fi, r) relative to some bounds [i and T depending on the geometry of G. For free groups, this relative rate of growth coincides with the standard one and this is in fact the reason for which the equality mentioned above holds in this case (see Section 5 again). The use of these bounds allowed us to overcome difficulties mentioned above and to prove the following. Theorem 1 The topological entropy h(G, S) (with respect to a finite symmetric generating set S) of a hyperbolic group G acting on its ideal boundary lies between the exponential rate gr rel (G, S ; / / , T ) of growth of G relative to suitable bounds fi and T, and the exponential rate gr(G, S) of growth of G (with respect to the same generating set): gr rel (G, 5; M, T) < h(G, S; dG) < gr(G, 5).

(1)

The precise description of the bounds mentioned in the Theorem can be found in Section 5 which contains also the proof of the Theorem and some final remarks. As was mentioned before, the notion of the topological entropy of a group action is recalled in Section 2. The same Section contains also a definition of geometric entropy of foliations which is used in Section 6, where we apply the Theorem to some foliated spaces and prove the following. Corollary 1 Let M be a compact Riemannian manifold with a hyperbolic fundamental group TT\ (M) = G. Let S be a finite symmetric set generating G, M be the universal covering of M, X = (dG x M)/G and TQ he the foliation of X obtained by suspending the natural action of G on dG x M.

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Then, the entropy h{Tc) of TG (with respect to the Riemannian metric along the leaves obtained by lifting the Riemannian structure of M) satisfies the inequalities - • gr rel (G, S, fi, T) < h{TG) < I • gr(G, S), (2) a b where a (resp., b) is the maximal (resp., minimal) length of free homotopy classes of curves corresponding to generating elements ofiri(M), while fi and T are the same as in (1). Finally, we define the exponential rate of growth relative to given bounds in Section 3 and we provide a short review on hyperbolic groups and spaces in Section 4. 2

Entropy

Let G be a finitely generated group of homeomorphisms of a compact metric space (X, d) and 5 be a finite symmetric (e 6 S, 5 - 1 = S) set generating G. Equip G with the word metric d$ induced by S and let B(n), n £ N, denote the ball in G of radius n and centre e. Two points x and y of X are said to be (n, e)-separated (e > 0, n e N) whenever d{gx, gy) > e for some g 6 B(n). Since X is compact, the maximal number N(n,e) of pairwise (n, e)-separated points of X is finite. Also, there exist finite (n, e)spanning subsets of X: A subset A of X is (n, e)-spanning whenever for any y e X there exists x G A such that d(gx,gy) < e for all g £ B(n). Let N'(n,e) denote the minimal cardinality of an (n, e)spanning subset of X. Similarly to the case of classical dynamical systems (compare, Walters' book [12], p. 169), the families N(n,e) and N'(n,e) of functions have the same type of growth (see Egashira's paper [4]), more precisely, they have the same rate of exponential growth and the topological entropy h(G, S, X) of G (w.r.t. 5) can be defined by the formula h(G,S,X)

= lim limsup — log N(n, e) = lim limsup — log./V'(n,e). e

*^ n—>oo

^

e

*® n—>oo

n

Note that If G = Z and S = {±1}, then h(G,S,X) coincides with the double of the Bowen's [1] topological entropy of the map / : X —> X corresponding to the integer 1.

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If all the maps of G are Lipschitz and X has finite Hausdorff dimension, then h(G,S,X) is finite for any S (compare, Ghys et al. [7], Prop.2.7). Also, if h(G,S,X) = 0 for some S, then h(G,S',X) = 0 for any other generating set S'. Therefore, one can distinguish between groups of positive and vanishing entropy without referring to generating sets. The same construction can be applied to finitely generated pseudogroups of local transformations of a metric space X. Also, the entropy of a foliation T of any compact Riemannian manifold M can be defined. We shall follow Langevin et al. [11]. The definition there is slightly different than that of Ghys e al. [7] but provides us with the same notion of entropy (called geometric). Let us consider a good covering U of M by charts distinguished by T, and a corresponding complete transversal T (compare, for instance, Candel and Conlon's book [2]). Given e > 0 sufficiently small and R > 0, let us say that two points x and y of T are (R, e)-separated whenever there exists a leaf curve 7 : [0,1] —> Lx originated at one of them (here, x) of length '(7) < R a n d such that its orthogonal projection 7 : [0,1] —> Ly to the other leaf (this time, Ly, the leaf through y) satisfies the inequality d(7(l),7(l))>£ whenever its origin 7(0) belongs to the plaque through y of a chart of U which contains both x and y. (Here, d denotes the Riemannian distance on M.) Denote by N(R, e,J-) the maximal cardinality of a subset A of T such that any two its points are (R, e)-separated in the above sense. Let h(J-) = lim lim sup N(R, e, F) and call h(T) the geometric entropy of T. Note that h{T) depends only on the Riemannian structure on M. The covering U involved in the definition plays only an auxilliary role. The value of h{T) does not depend on the choice of U. The geometric entropy of a foliation T can be compared to entropies of its holonomy pseudogroups corresponding to all good coverings by distinguished charts. If U is such a covering, Tiu is the corresponding holonomy pseudogroup, and Su is the set of all elementary holonomy maps corresponding to overlapping charts of U, then Tin acts on a suitable (built of connected transversals T\j C U, U G U) complete transversal Tu equipped with the Riemannian structure inherited from M, Tu carries the Riemannian distance, and the entropy h(Hu, Su,Tu) can be considered. With this

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notation the equality /i(JF) =sup——— •

h(Hu,Su,Tu),

where U ranges over the family of all good coverings of M and A(U) denotes the maximal diameter of plaques of charts of U, holds (see Ghys et al. [7]). Since the notion of holonomy extends directly to arbitrary foliated spaces, Candel and Conlon [2] proposed to use the last equality as the definition of a geometric entropy of any foliated space. 3

Growth

Let us keep the notation of the previous section and recall that the exponential rate of growth of G (with respect to S) is defined as gr(G, S) = lim - l o g i V ( n ) = lim - log N0(n), n—>oo n

n—>oo n

where N(n) = jj=B{n) and No(n) = #S{n) is the cardinality of the sphere S(n) of radius n and centre e. Also, if e > 0 (note that e is hereafter rather large than small !) and iV~o(n;e) is the maximal cardinality of an e-separated subset A of S(n), then g r ( G , S ) = lim - log N0(n;e). n—>oo n

In fact, iV0(n;e) < iVo(n) and, if A is such a subset of S(n), then UxtzAB(x,e) D S{n) and, therefore, N0(n;e)N(e) > N0(n) for any n € N. Moreover, if m e N and Ak, k = 1,2,..., are maximal e-separated subsets of S(mk), then for any n e N and any x € An we can find a sequence (xo,xi,- • -xn) of elements of the group G for which Xk € Ak, xn = x, XQ = e and d(xk,Xk+i) < m + e for all k < n. To construct such a sequence one can begin with xn = x, join x to e by a geodesic segment 7X, find the point x'n_1 of intersection of 7X with the sphere S((n — l)m) and a point x n _ i G An-i D ^ ( x ^ j j e ) , and continue by the induction. If y is another point of An, (yo,yi,... ,yn) is a corresponding sequence and k is the maximal natural number such that Xk =yk, then d(xk+i,Vk+i) > eTherefore, No(mn, e) can be defined as the maximal cardinality of a subset A of the sphere S(mn) which satisfies the condition (*) If x and y belong to A, then there exist sequences (xo,x\,... ,xn) and (j/o, 2/i, • • •, 2/n) of elements of G such that Xfc, j/fc G S(krn), x 0 = j/o = e, x n ^ x, y n =I y, rf(^j.a;j+i)irf(l/j,l/j+i)

< m + e, j = 0, ...,n-

1,

(3)

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y S(nm)

S(km) 'S((k+l)m) Figure 1.

and d(xk+i, Vk+i) > e when k is the maximal index for which Xk = ykTo define our relative growth we shall modify (*) by introducing an additional factor A in (3): Let us fix m € N, e > 0 and A e]0,1[, and denote by No(n;m,e,A) t h e maximal cardinality of a subset A of S(mn) satisfying t h e following condition: (**) If x and y lie in A, then there exist sequences ( x o , x i , . . . i „ ) and (yo,J/i, • • • ,yn) of elements of G such t h a t Xk, yk £ S{km), XQ = j/o = e , xn :rr x, yn = y, d{xj,Xj+i),d{yj,yj+i) for all j and (/(xfc+i^fe+i) Xk = Vk (Figure 1). T h e number


= m + \e

> e when A: is the maximal index for which

g r r e l ( G , S ; m , e , A) = l i m s u p

logNo(n;m,e,

A)

n—>oo TYITI

will be called the exponential rate of growth of G relative to m, e and A. Finally, if LI :]0, l [ x K + —> N and T :]0,1[—> R+ are arbitrary functions, t h e n we define the rate of growth of G relative to fj, and r by g r r e l ( G , S; p, r ) = sup{gr r e l (G, S; m , e, A); m >

M (A,

e), e > r(A), A e ] 0 , 1 [ } .

Obviously, grrel(G,5;M,r)>grrel(G,5;m,€,A) for any A G]0,1[, e and m sufficiently large. Since iV 0 (n; m, e, A) < N0(mn, e) for all m, n, e and A, we have grrel(G,5;M,r)
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203

for all JJL and r as above. For the free group Fk generated by the set Sk of k free generators we have always gr rel (F fc ,S fc ;M,T)=gr(F fe ,5 fe ), where Sk = Sk U S^1 U {e}. This is because F/. has no "dead ends" (see Grigorchuk and de la Harpe [8] for the definition and some information about some related problems) and in this case one can arrange e-separated subsets An of the spheres S(mn) in such a way that dist(x,An) — m for any x G An+\. In general, one can expect that a relative rate of growth is strictly less than the "true" rate of growth. However, the above observation about free groups and inequality (1) imply that the boundary entropy of several hyperbolic groups is strictly positive (see Section 4). 4

Hyperbolic spaces and groups

Let (X, d) be a metric space. A curve 7 : [a, b] —> X is a geodesic segment when dh(t)n(s))

=

\t-s\

for all t, s £ [a, b]. The space X is geodesic when any two points of X can be joined by a geodesic segment. For any finitely generated group G and any finite symmetric set S generating G, the Cayley graph C(G, S) is geodesic. Given three points Xo,y and z of a metric space X, the (based at XQ) Gromov product of y and z is given by (y\z)x0 = 2 (d(x<>, y) + d{x0, z) - d(y, z)). The space X is said to be hyperbolic (more precisely, <5-hyperbolic with 5 > 0) whenever the inequality (x\z)Xo > mm{{x\y)Xo,(y\z)Xo}

- 5

holds for arbitrary points xo,x, y and z of X. Clearly, the Cayley graph of any free group Fk (k = 1, 2,...) generated by the set Sk of k free generators is a tree, so becomes 0-hyperbolic. A finitely generated group is said to be hyperbolic whenever its Cayley graph with respect to some (equiv., any) generating set is hyperbolic. Free groups and fundamental groups of compact Riemannian manifolds of negative sectional curvature are hyperbolic. Assume that X is geodesic, take three points x\,X2 and £3 of X and connecting them geodesic segments 71,72 and 73. The union A = 7i U 72 U 73

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is a geodesic triangle with vertices Xi. The triangle A is r\-thin (r\ > 0) when the canonical isometry / A mapping A onto a tripod (i.e. the union of three segments with common origin) TA satisfies the condition

d(x,y)


for all x and y of A. In the proof of the Theorem we shall use the following. Lemma 1 (Ghys and de la Harpe [6], p.41) Let X be a geodesic metric space. IfX is S-hyperbolic, then all the geodesic triangles of X are 45thin. Conversely, if all the geodesic triangles of X are if thin, then X is 2r/-hyperbolic. To construct the boundary dX of a hyperbolic space X let us fix a base point xo and say that a sequence (xn) diverges to infinity whenever lim {xm\xn)

= oo,

m^n—>oo

where (-|-) denotes the Gromov product based at XQ. TWO such sequences (x„) and (ym) are equivalent whenever lim (xm\yn)

= oo.

m,n—»oo

The boundary dX of X consists of all the equivalence classes of sequences diverging to infinity. Note that dX can be described also in terms of equivalence classes of geodesic rays (i.e., maps 7 : [0,oo) —> X such that 7|[0,6] is a geodesic segment for any b > 0) or in terms of equivalence classes of quasirays (i.e. quasi-isometric maps of [0,00) into X): Two such rays (or, quasirays) 7 and a are equivalent whenever their Hausdorff distance ^#(7>°') i s finite. The boundary point corresponding to the equivalence class of such 7 is that determined by the sequence xn = 7(71), n € N. The equivalence of these constructions follows from the following fact which will be used later. Lemma 2 (Ghys and de la Harpe [6], p.87) Let X be a S-hyperbolic geodesic metric space. For any a > 1 and any c > 0 there exists D = D(5,a,c) > 0 such that any (a,c)-quasigeodesic segment 7 (i.e. any map 7 : [a, b] —> X (resp., 7 : [a,b] C\ Z —> X) such that the inequality —c — a\s — t\ <
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205

The set dX can be equipped with the metric structure as follows. First, for any £ and £ of dX put (£|C) = sup liminf

(xm\yn),

m,n—>oo

where (xm) and (yn) run over the set of all diverging to infinity sequences representing, respectively, £ and £. Note that if X is <5-hyperbolic, then (£|C)-25< liminf(im|yn)<(^|0 m,n—>oo

for all sequences (xm) and (i/m) representing £ and £. Next, choose r? > 0 and put ^(?,C)=exp(-r ? -(£|C)). Finally, let A:

d„(£,C) = inf{^/}„(&,&+!);& € <9X,£0 = £ and £fc+1 = ( , * ; € » } . i=0

If 77 > 0 is small enough, then d^ is a distance function on dX and (dX, dv) becomes a compact metric space of finite Hausdorff dimension (Ghys and de la Harpe [6], pp. 122 - 126). Moreover, the inequalities (1 - 277>„(£,0 < ^ ( £ , C ) < P„(£,C),

£,C e dX,

hold with rj = exp(r;5) — 1. Finally, let us recall that a hyperbolic group G is called elementary if either it is finite or contains a cyclic group of finite index. In this case, the boundary dG is either empty or finite (consisting of exactly two elements) and the entropy h(G, S, dG) vanishes. If G is ^-hyperbolic and non-elementary, then dG is infinite and G itself contains a free subgroup H of two generators. If the inclusion t : H —> G is (a, c)-quasi-isometric, then L induces a map (denoted by t again) of dH into dG which is Holder with a constant a — a(5, a, c). It follows that the image u{A) of any (n, e)saturated subset A of dH becomes (kn, Ce)-separated in dG, where k is a natural number depending on the choices of generating sets SH of H and So of G, while C is a positive constant depending on S, a, c and rj, a constant involved in the definition of the distance functions on dH and dG. This implies the inequalities h(G,SG,dG)>

yh(H,SH,dH) > 0, k where the last one follows from our Theorem and the final observation of Section 3.

206

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Proof of the Theorem

Let again G be a finitely generated group, S - a finite symmetric set generating G and consider C(G, S), the Cayley graph of G equipped with the distance function d satisfying d(9i,92) = \9i192\,

0i,02 e G ,

where | • | is the length function on G determined by S, and making the edges of C(G, S) isometric to Euclidean segments of length 1. Assume that C(G, S) is 5-hyperbolic and let dG — dC(G, S) be its boundary equipped, as in Section 4, with the distance function dv, r\ > 0 being small enough. The group G acts on C(G, S) via isometries which, when restricted to G C C(G, S), reduce to left translations Lg, G 9 h •-> gh. Therefore, each Lg extends to a Lipschitz homeomorphism, denoted by Lg again, of the boundary dG. Given 9 > 0 denote by N(n, 9; dG) the maximal number of points of dG pairwise (n, 0)-separated by this action of G. Also, let N'(n, 9; dG) be the minimal cardinality of an (n, #)-spanning subset of dG and h(G, S; dG) - the corresponding entropy. Since G is hyperbolic, there exists a constant CQ > 1 such that for any g € G there exists g' e G such that d(g,g') < CQ and \g'\ = \g\ + 1 (see Grigorchuk and de la Harpe [8], p. 60). Let D = D(5,co,co) be the corresponding constant such that any (co,co)-quasigeodesic segment lies at most D-apart (in the Hausdorff distance) from a true geodesic (compare Lemma 2). Let r(A) = —

—

and

n(\,e)

=

-. Co — 1

1 — A

We begin with the proof of the second inequality in (1). Choose 9 > 0 and a natural number k for which the inequality exp(—r\ • k) < 9 holds. For any n € N and any g G S(n + k) choose, if only possible, a point £g e dG which can be connected to e by a geodesic ray, say j g , which passes through g. We claim that the set A = {Zg;g€S(n

+ k)}

is (n, e)-spanning in dG. Indeed, if ( G dG, a : [0, oo) —> X is a geodesic ray connecting e to (, h £ S(n) and g = a(n + k), then g e S(n + k), the corresponding ray 7 9 exists and satisfies the conditions d(xj,yi)


+ k)

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207

g -h

V

Y<*

S(n) Figure 2.

and 2(h~1yi\h~1xj)

>{i-n)

+ (j - n) - (i + j) + 2(n + k) = 2k,

where Xj = a(j) and t/i = -y(i) for all i and j sufficiently large (Figure 2). Therefore, dv{Lh-it,Lh-i£g)

< Pv{Lh-i(,Lh-i£g)

< e~k71 < 6.

This shows the inequalities N'(n, 6; dG) < #A < #S{n + k) < N{n + k) which imply immediately the required inequality in (1). The proof of the first inequality in (1) is a bit more complicated. Fix A e]0,1[, e > r(A) and m > /j,(\,e). Choose n G N and a maximal subset A of S(mn) satisfying condition (**) of Section 3. For any x G A set xn = x, choose a point xn-\ G S(m(n — 1)) such that d(x, x n - i ) < m + Xe, then a point x n _2 G S(m(n — 2)) for which d{xn--\., ^n-2) < m + Ae and so on. Finally, put XQ = e. The map {0, m,..., ran} 3 j >-• xj/m is c-quasi-isometric with c = (m + Xe)/m < CQ. Each map considered above can be extended to a Co-quasi-isometric map j x : N 3 j H-> x'j e G such that x'im = Xi for i = 0 , 1 , . . . , n and the sequence (XJ) converges to a point £x of dG. We are going to show that the set {£x;xeA} is (n, #)-separated under the action of G for some 9 independent of n. To this end, let us take arbitrary points x and y of A, x ^ y, choose sequences (XQ,X\,..., xn) and (yo, 1/1,..., yn) satisfying all the conditions of (**) and let k be the maximal element of { 0 , 1 , . . . , n) for which Xk = yk-

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Denote this common value of x^ and j/fc by g and consider the quasi-geodesic rays j x and *yy obtained from -yx and 7 y by restricting their domains to {mk,mk + 1,...}. Then, 7X and 7V originate at g and converge to £a and £„, respectively. By Lemma 2, there exist geodesic rays % and 7^ originated at e and within the Hausdorff distance D from L g -i o 7X and L s -i o 7^, respectively. Then, for any j € N there exist positive real numbers Sj and tj such that <7x(5j),3 _ 1 7x(j)) < D

and

d(%(tj),g _ a %(i)) < £>•

and

d(%(tm),g~lyk+i)

In particular, d{;yx{sm),g'~1Xk+i)


< D-

Clearly, < m 4 D + Ae,

| s m - t m | < 2D 4 Ae

and d(7x(s m ),%(*m)) >

t-ID.

Assume that {%{sj)\%(tj)) > m 4- D 4- Ae for some j € N. The isometry / A corresponding to the geodesic triangle A with vertices e,7 x (sj),7 J / (tj) (compare Lemma 1) maps the points %(sm) and %{tm) onto some points of the originated at e edge of the tripod / A ( A ) and, therefore, satisfies the condition (7i(Sm),7y(*m)) < d ( / A ( 7 z ( S m ) ) , / A ( % ( t m ) ) ) + 45

= | s m - tm| + 45 < 2D 4 Ae 4- 45 < e - 2D. Comparing the inequalities above we obtain a contradiction which shows that

(%(sj)\%itj)) <m + D + t\ for all j e N . This inequality proves that

> (1 - 27]') exp(-r}(m + D 4- Ae 4 25)), i.e. that the points £x and £y are (nrn, 6>)-separated with 6 = (1 - 2TJ') exp(-Tj(m 4- D + Ae 4- 25)). The above argument implies the inequality N(nm, 6>; 9G) > # 4 = 7V0(G, 5; m, e, A)

ENTROPIES OF HYPERBOLIC GROUPS AND SOME FOLIATED SPACES

209

which holds for all n. Passing to suitable limits when n —> oo yields the required inequality in (1). Friedland [5] denned the minimal entropy hm\n(G) of a finitely generated group G of homeomorphisms of a compact metric space X: hmin(G)=mih(G,S), where 5 ranges over all finite symmetric sets generating G. Similarly, the minimal rate of growth gr min (G) of any finitely generated group G can be defined as follows (compare Grigorchuk and de la Harpe [8]): gr m i n (G) = infgr(G,S). The reader can define the minimal relative rate of growth grJ^}n(G) appropriately. If G is hyperbolic, then idc induces a Holder homeomorphism of boundaries of G obtained from different generating sets (Ghys and de la Harpe [6], page 128). Therefore, the boundary entropy of such G (w.r.t. a given finite symmetric generating set S) does not depend on the choice of a generating set used in the construction of dG and our Theorem implies immediately the following. Corollary 2 For any hyperbolic group G the equalities gf±{G)
= gr min (F fc ) = gr™{n(Ffc) = log(2fc - 1).

In fact, if S is any finite symmetric set generating Fk, then the elements of S represent members of a set S' generating Zfc, the abelianization of Fk. S' contains a symmetric set R' such that # i ? ' = 2k and the subgroup of Zk generated by R' has finite index. The corresponding subset R of S consists also of 2fc elements and generates the free group isomorphic to Fk. Therefore, h(Fk, S; dFk) > h(Fk,R; 0Fk) = h(Fk,Sk; 8Fk) > gr rel (F fc , Sk) = gv(Fk, Sk) = log(2fc - 1). The opposite inequality is obvious.

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The other natural case, that of the fundamental group Tg of a closed oriented surface of genus g > 1 is more complicated: The minimal rate of growth of rg is still unknown even if some estimates exist: g r m i n ( r s ) > 4g - 3, g r m i n ( r 9 ) < g r ( r 9 , 5 s ) « Ag - 1 - eg, where Sg is the canonical set of generators of Fg and eg is a small constant found numerically. In particular, 5 < gr m i n (r2) < 6.9798. The calculation or estimation of the value of the minimal relative rate of growth of Tg is yet more difficult.

6

Suspensions

Let now G be a hyperbolic group and M a compact Riemannian manifold with the fundamental group ni(M) = G. Let S be a finite symmetric set generating set and a (resp., b) be the maximum (resp., minimum) of lengths of free homotopy classes of curves homotopic to members of S. Let also M be the universal covering of M and X = (dG x

M)/G.

X carries a natural structure of a foliated space (X, TG) (compare Candel and Conlon [2], Chapter 11) whose leaves arise as 7r({£} x M), where £ G dG and 7r : dG x M —* X is a canonical projection. Also, X fibres over M with fibres homeomorphic to dG. Moreover, the distance function dv on dG (Section 3) and the Riemannian metric
(4)

Comparing (1) and (4) yields (2) and proves our Corollary 1.

Acknowledgments The second author was supported by the KBN grants P03A 066 10 and P03A 033 18.

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References 1. R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc, 153 (1971), 401-414. 2. A. Candel and L. Conlon, Foliations I, Amer. Math. Soc, Providence, 2000. 3. M. Coornaert and A. Papadopoulos, Symbolic Dynamics and Hyperbolic Groups, Springer Verlag, Berlin-Heidelberg, 1993. 4. S. Egashira, Expansion growth of foliations, Ann. Fac. Sci. Univ. Toulouse, 2 (1993), 15-52. 5. S. Friedland, Entropy of graphs, semigroups and groups, in Ergodic theory of Zd actions, eds. M. Policott and K. Schmidt London Math. Soc, London, 1996, 319 - 343. 6. E. Ghys and P. de la Harpe, Sur les Groupes Hyperboliques d'apres Mikhael Gromov, Birkhauser, Boston-Basel-Berlin, 1990. 7. E. Ghys, R. Langevin and P. Walczak, Entropie geometrique des feuilletages, Acta Math., 160 (1988), 105-142. 8. R. Grigorchuk and P. de la Harpe, On problems related to growth, entropy and spectrum in group theory, J. Dyn. Control Sys., 3 (1997), 51-89. 9. M. Gromov, Hyperbolic groups, in Essays in group theory, ed. S. M. Gersten, Springer Verlag, Berlin-Heidelberg-New York, 1987, 75-263. 10. M. Gromov, J. Lafontaine and P. Pansu, Structures metriques pur les varietes riemanniennes, Cedic/F. Nathan, Paris, 1981. 11. R. Langevin and P. Walczak, Entropy, transverse entropy and partitions of unity, Ergodic Th. and Dynam. Sys., 14 (1994), 551-563. 12. P. Walters, An introduction to Ergodic Theory, Springer Verlag, New York-Heidelberg-Berlin, 1982.

Received June 7, 2000, revised November 22, 2000 and September 25, 2001.

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Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2002 pp. 213-224

TAUTLY FOLIATED 3-MANIFOLDS WITH NO R-COVERED FOLIATIONS

MARK BRITTENHAM Department of Mathematics and Statistics, University of Nebraska, Lincoln, NE 68588-0323, e-mail: [email protected] We provide a family of examples of graph manifolds which admit taut foliations, but no R-covered foliations. We also show that, with very few exceptions, R covered foliations are taut.

1

Introduction

A foliation T of ^closed 3-manifold M can be lifted to a foliation T of the universal cover M of M; if the foliation T has no Reeb components, the leaves of this lifted foliation are all planes, and Palmeira [19] has shown that M is homeomorphic to R3. Palmeira also showed that T isjiomeomorphic to (a foliation of R2 by lines) xR. The space of leaves of T, the quotient space obtained by crushing each leaf to a point, is homeomorphic to the space of leaves of the foliation of R2, and is a (typically non-Hausdorff) simply-connected 1-manifold. If the space of leaves is Hausdorff (and therefore homeomorphic to R), we say that the foliation T is R-covered. Examples of R-covered foliations abound, starting with surface bundles over the circle; the foliation by fibres is R-covered. Thurston's notion of 'slitherings' [20] also provide a large collection of examples. A great deal has been learned in recent years about R-covered foliations and the manifolds that support them (see, e.g., [1], [8], [11]), especially in the case when the underlying manifold M contains no incompressible tori. The purpose of this note is to provide examples of 3-manifolds which admit taut foliations, but which do not admit any R-covered foliations. All of our examples are drawn from graph manifolds, and so all contain 213

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MARK BRITTENHAM

an incompressible torus, and, in fact, a separating one. The technology does not exist at present to identify examples which are atoroidal, and it is perhaps not clear that such examples should be expected to exist. That we can find the examples we seek among graph manifolds relies on two facts: (a) we have a good understanding [3] of how taut and Reebless foliations can meet an incompressible torus, and (b) we understand [4], [9], [17] which Seifert-fibered spaces can contain taut or Reebless foliations. These two facts have been used previously [2] to find graph manifolds which admit foliations with various properties, but no "stronger" properties; the examples we provide here are in fact the exact same examples used in [2] to illustrate its results. We will simply look at them from a somewhat different perspective. This paper can, in fact, be thought of as a further illustration of this 'you can get this much, but no more' point of view towards foliating manifolds. Namely, you can get taut, but not R-covered. R-covered is, it turns out, almost (but not quite) always a stronger condition: in all but a very small handful of instances, an R-covered foliation must be taut, as we show in the next section. This result seems to have been implicit in much of the literature on R-covered foliations; a different proof of this result, along somewhat different lines, can be found in [14]. 2

R-covered almost implies taut

In this section we show that in all but a very few instances, an R-covered foliation must be taut. We divide the proof into two parts; first we show that a Reebless R-covered foliation is taut, and then describe the manifolds that admit R-covered foliations with Reeb components. Recall that a Reeb component is a solid torus whose interior is foliated by planes transverse to the core of the solid torus, each leaf limiting on the boundary torus, which is also a leaf. (There is a non-orientable version of a Reeb component, foliating a solid Klein bottle, which we will largely ignore in this discussion. It can be dealt with by taking a suitable double cover of our 3-manifold.) We follow standard practice and refer to both the solid torus and its foliation as a Reeb component. A foliation that has no Reeb components is called Reebless. A foliation is taut if for every leaf there is a loop transverse to the foliation which passes through that leaf. Taut foliations are Reebless. Lemma 1 If a closed, irreducible 3-manifold M admits a Reebless, Rcovered foliation T containing a compact leaf F, then every component of M\F, the manifold obtained by splitting M open along F, is an I-bundle

TAUTLY FOLIATED 3-MANIFOLDS WITH NO R - C O V E R E D FOLIATIONS

215

over a compact surface. In particular, M is either a surface bundle over the circle with fibre F, or the union of two twisted I-bundles glued along their common boundary F. Proof. Because T is Reebless, the surface F is 7Ti-injective [18], and so lifts to a collection of planes in M. Their image C in the space of leaves of T is a discrete set of points in R, since it is closed, and any sequence of distinct points in C with a limit point can be used to find a sequence of points in F limiting on F in the transverse direction, contradicting the compactness of F. The complementary regions of F in M lift to the complementary regions of the lifts of F in M; in the space of leaves they correspond to the intervals between successive points of C. Each is bounded by two points of C, and so every component X of the inverse image of a component X of M\F has boundary equal to two lifts of F. Because M is simply-connected, as are the 9-components of X, X is simply-connected, and so X is the universal cover of X. Because F 7ri-injects into M, it 7ri-injects into M\F, and hence into X. The index of iri(F) in iti{X) is equal to the number of connected components of the inverse image of F in the universal cover of X. To see this, choose a basepoint XQ for X lying in F, and suppose that 7 is a loop based at XQ which is not in the image of ir\(X). Then the lift 7 of 7 to X must have endpoints on distinct lifts of F, for otherwise the endpoints can be joined by an arc a in the lift of F, whose projection, since X is simply connected (so 7*5 is null-homotopic) is a null-homotopic loop 7 * a in TTI(X). This implies that [7] — [a]e ni(F), a contradiction. Choosing representatives from each coset of TTI(F) in -K\{X), and lifting each to arcs with initial points a fixed lift So of XQ, we find that their terminal points must therefore lie on distinct lifts of F. But since X has only two boundary components, this means that m (F) has index at most two in 7ri(X). Since X is irreducible (because F is incompressible and M is irreducible), ([15], Theorem 10.6) implies that X is an /-bundle over a closed surface. The resulting description of M follows.

• The foliation by fibres of a bundle over the circle is always taut. In the other case, when F separates, we understand [5] the structure of the foliation T on each of the two /-bundles, since their boundaries are leaves. If the induced foliations can be made transverse to the /-fibres of each bundle, then by taking a pair of /-fibres, one from each bundle, and deforming them so that they share endpoints on F, we can obtain a loop transverse to the leaves of J7, so T is taut. If the induced foliations cannot be made transverse to the /-fibres, then F is a torus, and (after possibly passing

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to a finite cover) the foliation contains a pair of parallel tori with a Reeb annulus in between. It is then straightforward to see that the resulting lifted foliation T cannot be R-covered, since this torus xl will lift to R2 x. I whose induced foliation has space of leaves R together with two points (the two boundary components) that are both the limit of the positive (say) ray of the line. In particular, the space of leaves of J- would not be Hausdorff. Therefore Corollary 2 A Reebless, R-covered foliation is taut. We now turn our attention to R-covered foliations with Reeb components. Such foliations do exist, for example, the foliation of S2 x S1 as a pair of Reeb components glued along their boundaries; the lift to S2 x R consists of a pair of solid cylinders, each having space of leaves a closed halfline. Gluing the solid cylinders together results in gluing the two half-lines together, giving space of leaves R. We show, however, that, in some sense, this is the only such example. Recall that the Poincare associate P(M) of M consists of the connected sum of the non-simply-connected components of the prime decomposition of M, i.e., M = P(M)# (a counterexample to the Poincare Conjecture). Lemma 3 If J7 is an R-covered foliation of the orientable 3-manifold M, which has a Reeb component, then P(M) = S2 x S1. Proof. The core loop 7 of the Reeb component must have infinite order in the fundamental group of M, otherwise the Reeb component lifts to a Reeb component of T\ but since the interior of a Reeb component has space of leaves S1, this would imply that S1 embeds in R, a contradiction. The Reeb solid torus therefore lifts to a family of infinite solid cylinders in M, foliated by planes. The induced foliation of each closed solid cylinder has space_of leaves a closed half-line properly embedded in the space of leaves of T. Each such half-line is disjoint from the others; but since R has only two ends, this implies that the Reeb component has at most two lifts to M. This means that the inverse image of the core loop 7 of the Reeb solid torus, in the universal cover M, consists of at most two lines, and so the (infinite) cyclic group generated by 7 has index at most 2 in TTI(M). Because M is orientable, it's fundamental group is torsion-free, and so by [15], Theorem 10.7, 7Ti(M) is free, hence isomorphic to Z, and so ([15], Exercise 5.3) P{M) is an S 2 -bundle over S1. Since M is orientable, this gives the conclusion. • Note that the space of leaves in the universal cover does not change by passing to finite covers (there is only one universal cover), and so we can lose the orientability hypothesis by weakening the conclusion slightly. Putting the lemmas together, we get

TAUTLY FOLIATED 3-MANIFOLDS WITH NO R - C O V E R E D FOLIATIONS

Corollary 4 If a 3-manifold M admits an H-covered foliation T, either T is taut or P(M) is double-covered by S2 x S1 .

3

217

then

Taut but not R-covered

In [2], Theorem D, the authors exhibit a family of 3-manifolds which admit C(°)-foliations with no compact leaves, but no C^-foliations without compact leaves. Each of the examples is obtained from two copies M\,M% of (a once-punctured torus) x S 1 , glued together along their boundary tori by a homeomorphism A. What we will show is that for essentially the same choices of A, the resulting manifolds admit taut foliations, but no R-covered ones. A is determined by its induced isomorphism on first homology I? of the boundary torus, and so we will think of it as a 2x2 integer matrix with determinant ± 1 . We choose as basis for the homology of each torus the pair(*xS1,dFx*), where F denotes the once-punctured torus. (Technically, we should orient these curves, but because all of the conditions we will encounter will be symmetric with respect to sign, the orientations will make no difference, and so we won't bother.) Each Mi is a Seifert-fibered space (fibered by *xS1); the manifold MA resulting from gluing via A is a Seifert-fibered space iff A glues fibre (1,0) to fibre (1,0), i.e., A is upper triangular. We will assume that this is not the case. Let T denote the incompressible torus dMi = 9M 2 in MA- By [10], any horizontal foliation of Mj, i.e., a foliation everywhere transverse to the Seifert fibering of Mj, must meet dMi in a foliation with slope in the interval (—1,1). [Note that this disagrees with the statement in [2], where the result was quoted incorrectly.] If T is an R-covered, hence Reebless, foliation of M, then by [3], we can isotope T so that either it is transverse to T, and the restrictions Ti of T to Mj, i — 1, 2, have no Reeb or half-Reeb components, or T contains a cylindrical component, and therefore a compact (toral) leaf. In the second case, the torus leaf must hit the torus T, and is split into a collection of non-9-parallel annuli; these (essential) annuli must be vertical in the Seifert-fibering of each Mj, since F x S1 contains no horizontal annuli. But this implies that the gluing map A glues fibre to fibre, a contradiction. Therefore, we may assume that T restricts to Reebless foliations on each of the manifolds Mj. Note that [3] requires that we allow a finite amount of splitting along leaves to reach this conclusion; but since a splitting of an R-covered foliation is still R-covered (it amounts, in the space of leaves, to

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replacing points with closed intervals), this will not affect our argument. By [6], each of the induced foliations Ti of Mj has either a vertical or horizontal sublamination. Every horizontal lamination in Mj can be extended to a foliation transverse to the fibres of Mj, and so meets <9Mj in curves whose slope lies in (—1,1). If Ti has a vertical sublamination, it either meets 9Mj in curves of slope oo (i.e., in curves homologous to (1,0)) or is disjoint from the boundary. It is this last possibility, a vertical sublamination disjoint from T, that we wish to require, and so we will now impose conditions on the gluing map A to rule out the other possibilities. If both T\ and Ti have either a horizontal sublamination or a vertical sublamination meeting T, then for both M\ and M2, the induced foliations meet T in curves with slope in (—1,1) U {00}. Therefore, the gluing map

-^ ~ (

A I'

c dj '

which as a function on slopes, is v

'

must have A ((-1,1)

A(x) v

'

cx + d

U {00}) D ((-1,1) U {00}) ^ 0. But since

and A is increasing if ad — be = 1 (take the derivative!), we can force A (( —1,1) U {00}) to be disjoint from (—1,1) U {00} by setting „ . . , a+b \a\ > \c\, \d\ > c , and < —1. 11

1 h 1 1

in

c +

d

For example, we may choose a = —n, b — —nm — 1, c = 1, and d — m, with n, m > 2, so that a+b 1 = -n <-1. c+d m+1 As the figure below shows, the conditions —a > c, d > c and ad — bc= 1 are also sufficient; what is needed, essentially, is that neither the graph of A nor either of its asymptotes pass through the square [—1,1] x [—1,1]. For such a gluing map A and resulting manifold MA, either T\ or T2 (without loss of generality, T\) must contain a vertical sublamination £ disjoint from T = dM\. C is the saturation, by circle fibres, of a 1-dimensional lamination A in the punctured torus F. This lamination A cannot contain a closed loop, since then Ti, and therefore JF, would contain a torus leaf L missing T. Lemma 1 would then imply that M\L is an /-bundle, a contradiction, since it contains M^. A is therefore a lamination by lines.

TAUTLY FOLIATED 3-MANIPOLDS WITH NO R-COVERED FOLIATIONS

219

Figure 1.

By Euler characteristic considerations, the complementary regions of A, thought as in a torus, are products, and so the complementary region of A in F which meets OF is topological!}' a (d-parallel) annulus, with a pair of points removed from the 'inner' boundary. Therefore, the component N of MA\C which contains T is homeomorphic to M2 with a pair of parallel loops removed from dMz = T . Now we will assume that f is R-covered, and argue as in the proof of Lemma 1, to arrive at a contradiction. C lifts to a lamination in R3 by planes, whose image in the space of leaves R of T is a closed set. The two boundary leaves of £ in ON are both annuli; they are the complements of the two parallel loops in dM^ in the above description of MA\C A lift of this complementary region to M has boundary consisting of lifts of the two annuli. Since the lift is a closed set in M its image in the space of leaves R is a connected, closed set, and therefore an interval. This implies that the lift of MA\£ has (at most) two boundary components, implying that the inverse image of each of the annulus leaves is a single (planar) leaf of T. This implies, as in the proof of Lemma 1, that the image in 7 T I ( M A | £ ) = it\{Mi) of the fundamental group of each annulus has index at most 1, and so 7ri(M2) = F 2 x Z is cyclic, a contradiction. Therefore, T is not R-covered. This implies T h e o r e m 5 With gluing map A given as above, MA admits no R- covered foliations. On the other hand, every manifold MA built out of the pieces we have used admits taut foliations and, in fact, foliations with no compact leaves. We simply choose a vertical lamination with no compact leaves in each of the Seifert-fibered pieces Mi, missing the gluing torus T. The complement of this lamination is homeomorphic to T x J, with a pair of parallel loops

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removed from each of the boundary components. Treating this as a sutured manifold, and thinking of this as ( 5 1 x / ) x S1, we can foliate it, transverse to the sutures, by parallel annuli. Then, as in [12], we can spin the annular leaves near the sutures, to extend our vertical laminations to a foliation of MA with no compact leaves. This give us P r o c l a i m 6 There exist graph manifolds admitting taut foliations, but no R-covered foliations. 4

R-covered finite covers

Work of Luecke and Wu [16] implies that (nearly) every connected graph manifold is finitely covered by a graph manifold that admits an R-covered foliation. In particular, for any graph manifold M whose Seifert fibered pieces all have base surfaces having negative (orbifold) Euler characteristic, they find a finite cover M' (which is also a graph manifold) admitting a foliation T transverse to the circle fibres of each Seifert fibered piece of M', and which restricts on each piece to a fibration over the circle. Note that this implies that every leaf of T meets every torus which splits M' into Seifert-fibered pieces. Even more, every leaf of the lift, to the universal cover of M, of !F meets every lift Pi, P?, of the splitting tori. This can be verified by induction on the number of lifts of the tori that we must pass through to get from a lift we know the leaf hits, to our chosen target lift. The initial step follows by picking a path 7 between two 'adjacent' lifts P\ and P2, whose interior misses every lift of the splitting tori, and projecting down to M; this gives a path 7 in a single Seifert fibered piece of M'. This path can be made piecewise vertical (in fibres) and horizontal (in leaves of J 7 ), missing, without loss of generality, the multiple fibres of M' (just do this locally, in a foliation chart for T; the Seifert fibering can be used as the vertical direction for the chart). Each vertical piece can then be dragged to the boundary tori, since the saturation by fibres of an edgemost horizontal piece of 7 is a (singular) annulus with induced foliation by horizontal line segments; see Figure 2. [This is where the fact that T is everywhere transverse to the fibers is really used.] The end result of this process is a loop 7', homotopic rel endpoints to 7, which consists of two paths each lying in a circle fibre in the boundary tori, with a single path in a leaf of T lying in between. This lifts to a path homotopic rel endpoints to 7, consisting of paths in the two lifted tori, and a path in some lifted leaf. This middle path demonstrates that some lifted leaf L hits both Pi and P2. By choosing a point where any other lifted leaf V hits a lift P of a

TAUTLY FOLIATED 3-MANIFOLDS WITH NO R-COVERED FOLIATIONS

221

Figure 2. splitting torus, and joining it by a p a t h a to a point where L hits P i or P2, we can apply the same straightening procedure as above (see Figure 3), to show t h a t a is homotopic rel endpoints to paths, one of which lies in a lifted fibre and then lies totally in L', and the other of which lies totally in L, and then in a lifted fibre. This in particular implies t h a t L' also hits P i and P2. Therefore, every lifted leaf hits b o t h P j and P2, as desired.

P'E-

L' ?•••••

•

Figure 3. T h e inductive step is nearly identical; assuming our two leaves L\ and L2 b o t h hit P i , . . . , P n - i , and P n can be reached from P „ _ i without passing t h r o u g h any other lift of a splitting torus, the above argument implies t h a t two leaves, in the lift of the relevant Seifert fibered piece, and contained in L\ and L2, hit b o t h L n _ i and Ln, implying t h a t L\ and Li also b o t h hit P n . B u t this in t u r n implies t h a t the lifted foliation T has space of leaves R . This is because t h e foliation induced by T on any lift P of a splitting torus is a foliation transverse to (either of the) foliations by lifts of circle

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M A R K BRITTENHAM

fibres, and so has space of leaves R, which can be identified with one of the lifts of a circle fibre. [This is probably most easily seen in stages: first pass to a cylindrical cover of the torus, for which the circle fibers lift homeomorpically. The induced foliation from T is by lines transverse to this fibering, and so has space of leaves one of the circle fibers. The universal cover P is a cyclic covering of this, whose induced foliation has space of leaves the universal cover of the circle fiber.] The argument above implies that every leaf of T hits P at least once. But no leaf of T can hit a lift of a circle fibre more than once; by standard arguments, using transverse orientability, a path in the leaf joining two such points could be used to build a (null-homotopic) loop transverse to T, contradicting tautness of T, via Novikov's Theorem [18]. We therefore have a one-to-one correspondence between the leaves of T and (any!) lift of a circle fibre in any of the Seifert-fibered pieces, giving our conclusion Proposition 7 Any foliation of a graph manifold M, which restricts to a foliation transverse to the fibres of every Seifert-fibered piece of M, is ~R-covered. Combining this with the result of Luecke and Wu, we obtain Corollary 8 Every graph manifold, whose Seifert-fibered pieces all have hyperbolic base orbifold, is finitely covered by a manifold admitting an Rcovered foliation. Combining the proposition with our main result, we obtain: Corollary 9 There exist graph manifolds, admitting no R-covered foliations, which are finitely covered by manifolds admitting R- covered foliations. 5

Concluding remarks

Being finitely covered by a manifold admitting an R-covered foliation is nearly as good as having an R-covered foliation yourself. Any property that could be verified in the presence of an R-covered foliation, which remains 'virtually' true (e.g., virtually Haken, or having residually finite fundamental group), would then be true of the original manifold. It would then be of interest to know Question 1 Does every 3-manifold with universal cover R3 have a finite cover admitting an R-covered foliation? Or, even stronger Question 2 Does every irreducible 3-manifold with infinite fundamental group have a finite cover admitting an R-covered foliation?

TAUTLY FOLIATED 3-MANIFOLDS WITH NO R - C O V E R E D FOLIATIONS

223

Weaker, but still interesting Question 3 Does every tautly foliated 3-manifold have a finite cover admitting an R-covered foliation? The first two questions could be broken down into Question 3 and Question 4 Does every 3-manifold (in the appropriate class) have a tautly foliated finite cover? Note that showing that every irreducible 3-manifold with infinite fundamental group has a tautly foliated finite cover would settle the Conjecture 1 Every irreducible 3-manifold with infinite fundamental group has universal cover R3. Questions 1 and 2 can be thought of as weaker versions of the (still unanswered) question, due to Thurston, of whether or not every hyperbolic 3-manifold is finitely covered by a bundle over the circle; the foliation by bundle fibres is R-covered. Gabai [13] has noted that there are Seifertfibered spaces for which the answer to Thurston's question is 'No', although an observation of Luecke and Wu [16] implies that, via the results [10], the answer to our Question 1 is 'Yes', for Seifert-fibered spaces, since [7] a transverse foliation of a Seifert-fibered space is R-covered. [Note that the arguments of the previous section can be modified to give a different proof of this; look at how lifted leaves meet lifts of a single regular fiber, instead of lifts of the splitting tori.] Question 4, with its conclusion replaced by 'have a taut foliation', has as answer 'No'; examples were first found among Seifert-fibered spaces [4], [9]; there are no known examples among hyperbolic manifolds. Question 5 Do there exist hyperbolic 3-manifolds admitting no taut foliations? Finally, the result we have established here for graph manifolds is still unknown for hyperbolic 3-manifolds: Question 6 Do there exist hyperbolic 3-manifolds which admit taut foliations, but no R-covered foliations? References 1. T. Barbot, Caracterisation des flots d'Anosov en dimension 3par leurs feuilletages faibles, Ergodic Th. & Dynam. Sys., 15 (1995), 247-270. 2. M. Brittenham, R. Naimi and R. Roberts, Graph manifolds and taut foliations, J. Diff. Geom., 45 (1997), 446-470. 3. M. Brittenham and R. Roberts, When incompressible tori meet essential laminations, Pacific J. Math., 190 (1999), 21-40.

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4. M. Brittenham, Essential laminations in Seifert-fibered spaces, Topology, 32 (1993), 61-85. 5. M. Brittenham, Essential laminations in I-bundles, Trans. Amer. Math. Soc„ 349 (1997), 1463-1485. 6. M. Brittenham, Essential laminations in Seifert-fibered spaces: boundary behaviour, Topology Appl., 95 (1999), 47-62. 7. M. Brittenham, Exceptional Seifert-fibered spaces and Dehn surgery on 2-bridge knots, Topology, 37 (1998), 665-672. 8. D. Calegari, Foliations and the geometry of 3-manifolds, Thesis, University of California at Berkeley, 2000. 9. W. Claus, Essential laminations in closed Seifert-fibered spaces, Thesis, Univ. of Texas at Austin, 1991. 10. D. Eisenbud, U. Hirsch and W. Neumann, Transverse Foliations on Seifert Bundles and Self-homeomorphisms of the Circle, Comment. Math. Helv., 56 (1981), 638-660. 11. S. Fenley, Anosov flows in 3-manifolds, Ann. of Math., 139 (1994), 79-115. 12. D. Gabai, Foliations and the topology of 3-manifolds, J. Diff. Geom., 18 (1983), 445-503. 13. D. Gabai, On 3-manifolds finitely covered by surface bundles, in Lowdimensional topology and Kleinian groups, (Coventry/Durham, 1984), Cambridge Univ. Press, 1986, 145-155. 14. S. Goodman and S. Shields, A condition for the stability ofH-covered on foliations of 3-manifolds, Trans. Amer. Math. Soc, 352 (2000), 4051-4065. 15. J. Hempel, 3-Manifolds, Ann. of Math. Studies No. 86, Princeton University Press, 1976. 16. J. Luecke and Y.-Q. Wu, Relative Euler number and finite covers of graph manifolds, in Geometric topology (Athens, GA, 1993), Amer. Math. Soc, 1997, 80-103. 17. R. Naimi, Foliations transverse to fibres of Seifert manifolds, Comment. Math. Helv., 69 (1994), 155-162. 18. S. Novikov, Topology of foliations, Trans. Moscow Math. Soc, 14 (1965), 268-305. 19. C. Palmeira, Open manifolds foliated by planes, Ann. of Math., 107 (1978), 109-121. 20. W. Thurston, Three-manifolds, Foliations and Circles I, preprint.

Received October 31, 2000.

Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2002 pp. 225-261

E N D S E T S OF E X C E P T I O N A L LEAVES; A T H E O R E M OF G. D U M I N Y JOHN CANTWELL Department

of Mathematics, St. Louis St. Louis, MO 63103, e-mail: [email protected]

University

LAWRENCE CONLON Department

of Mathematics, Washington St. Louis, MO 63130, e-mail: [email protected]

University,

In 1977 Gerard Duminy proved that, if F is a semiproper leaf of a C 2 codimensionone foliation J of a compact n-manifold M and X is an exceptional local minimal set of 3", then the set £ X ( F ) of ends of F asymptotic to X, if nonempty, is homeomorphic to a Cantor set. No proof of this remarkable result has ever appeared, even in preprint form. Here, we offer a proof of our own.

1

Dummy's theorems

Let (M, J ) be a transversely oriented, C 2 -foliated manifold of codimension one, with M compact and oriented. We assume each component of dM, if any, is a leaf of 3". Remark that each leaf of 3" is oriented. Let U C M be an open, 3"-saturated subset and let X C U be an exceptional minimal set of $\U. As in [2] and elsewhere, we say that X is an exceptional LMS (local minimal set) of 3". If F is a leaf of 5", let £(F) denote the space of ends of F and let Ex (F) denote the subspace of ends that are asymptotic to X. Theorem 1.1 (Duminy) If the leaf F of 9 is semiproper and if it accumulates on the exceptional LMS X, then Ex (F) is homeomorphic to the Cantor set. Here, semiproper is taken to mean that at least one side of F is proper, 225

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J. CANTWELL AND L. CONLON

so we include the case in which F is a proper leaf. Corollary 1.2 If X is an exceptional minimal set of 7 and if F C X is a semiproper leaf, then E(F) is homeomorphic to the Cantor set. Indeed, since X is minimal, £ X ( F ) — £(F). Theorem 1.1 is a corollary of another more directly useful result. For this, let L be any semiproper leaf of "3 that is approached by a possibly different semiproper leaf F. By the transverse orientation, L has a positive and a negative side, and we assume (reversing the transverse orientation, if necessary) that the positive side is a side on which F accumulates. Let 0 € L and let [0, e] be a parametrized, ^-transverse arc issuing from 0 on the positive side. If a is a loop on L based at 0, then ha will denote the holonomy in [0,e] defined by a. If H C L is a compact, connected, nonseparating, oriented submanifold of codimension one (a "handle"), then the homological intersection number of the loop a with H will be denoted by a — H. Let L' be L cut apart along the handle H. Assume that 0 G H and let 0 + G H+ and 0~ G H~ be the two copies of 0 thus obtained. Let a be a curve from 0 + to 0 _ and a the curve obtained from a by identifying 0 + to 0~. Assume the definition of ± has been chosen so that a -^ H = + 1 . In Section 7 we show that there exists an e > 0, a contraction / of [0, e) to 0, a handle H, and curve a as above with a ^ H = +1 and f = ha. Theorem 1.3 Ife > 0, / , H, and a are as above and (a, b) is a component of (0,e) \ F then there is an integer N > 0 such that if a is a loop on V based at 0+ then ha(fk{b)) = fk(b) all k>N. Remarks. The integer N depends on the gap [a, b] but e, / , H, and a do not (see Proposition 7.16). The final choice of e, f, H, and a is made just prior to Lemma 7.15 while the choice of the integer N is made in the proof of Lemma 7.15. The proof of Theorem 1.3 occurs just after the proof of Lemma 7.15. Note that the theorem remains true if we replace b with a in its statement. A careful proof of Theorem 1.3 will occupy Sections 3 through 7. In Section 2 we will deduce Theorem 1.1 from Theorem 1.3 and we will also deduce the following theorem that was the basis of Duminy's original proof of Theorem 1.1. Theorem 1.4 (Duminy) If e G £.X(F), then every neighbourhood of e in F contains a complete submanifold that spirals on L with juncture H. Consequently, e is a cluster point of £.x (F) and this set is a Cantor set. It is natural to conjecture that, if F C X, the requirement in Theorem 1.1 that F be semiproper can be dropped. This is true for Markov LMS [4] but is unknown in general. A weaker conjecture would be that

BNDSETS OF EXCEPTIONAL LEAVES

227

the generic leaf of an exceptional minimal set X have a Cantor set of ends. Here generic can be taken either in the sense of [5] or [7]. It is not hard to show that the generic leaf has one end or a Cantor set of ends. In Section 8 we ask some other questions. Section 9 gives a general construction of examples of Markov minimal sets. The construction is general in the sense that it provides examples of any subshift of finite type occurring as holonomy of a codimension-one foliation. Remark. Duminy deserves full credit for Theorem 1.1. The proof given here is essentially one that we worked out about 15 years ago and we take responsibility for whatever shortcomings it may have. 2

The proof of Theorem 1.1 and Theorem 1.4

The foliation JF of Mn is described by a finite, C2 atlas $ = {(Ui, x;, yi)}[ =1 of foliated charts. Here, yt:Ul-^R and, on overlaps UiHUj, the coordinate changes are of the form x%

=

Xi(Xj ,yj),

Vi = ViiVj)-

(*J (**)

We can and do assume that the images j/i([/i) are bounded intervals in R with disjoint closures. As is standard, the set of transverse coordinate changes (**) is interpreted as a symmetric generating set {hi,... , hm} of a pseudogroup T of local C 2 diffeomorphisms in R, called the "holonomy pseudogroup of T'. To relate this pseudogroup to the geometry of J , it is useful to fix imbeddings of the intervals yi(Ui) C Ui as disjoint transverse curves to 3\ The union W of these transversals is then a complete transversal and the holonomy pseudogroup maps open subsets of W onto open subsets of W simply by sliding along the leaves. One also requires a certain "regularity" property of the foliated atlas, namely that each {Ui,Xi,y{) is actually a subchart of a foliated chart (Vi,Xi,yi) and that the closure Ui is a compact subset of Vi. Again, it is useful to fix imbeddings of the intervals yi(Vi) C Vi as disjoint transverse curves to 1. The union T D W of these transversals is then a complete transversal and the holonomy pseudogroup maps open subsets of T onto open subsets of T simply by sliding along the leaves. Again, the set of transverse coordinate changes (**) is interpreted as a symmetric generating set {hi,... , hm} of a pseudogroup T, of local C2 diffeomorphisms in 1R.

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Thus, the domain of hi is an open interval D{hi) with compact closure D{hi) and hi is the restriction of a C 2 imbedding hi : D(hi) —> R, 1 < i <m. Thus, every element of T extends C 2 -smoothly to the closure of its domain. Abusing notation we denote hi by hi, 1 < i < m, and f by T. In this section we assume the truth of Theorem 1.3 and deduce Theorem 1.1 and Theorem 1.4. Let X C M be an exceptional LMS of ?, let F be a semiproper leaf that accumulates on X, La, semiproper leaf lying in X. We assume we have chosen our atlas $ = {(C/j,x,,2/i)}[=1 of foliated charts so that boundary plaques of U, do not lie in F, 1 < i < r. L e m m a 2.1 The leaf L lies at finite level and the closure F of F in M accumulates on F from at most one side. Proof. No leaf at infinite level can have a proper side [1, Lemma 8.3.23], so F and L each lie at finite level. This means that F lies in a unique LMS Q and Q must either reduce to F itself (and F is a proper leaf) or it must be exceptional. In either case, F is a compact, nowhere dense, ^-saturated set containing X. Further, by [1, Corollary 8.3.12], F is a finite union of LMS, each of which is either a proper leaf or an exceptional LMS and none of which, except for Q, can accumulate on F. If Q reduces to F itself, F does not accumulate on F. Alternatively, Q is exceptional and accumulates on F only from the nonproper side of that leaf. • By Lemma 2.1, each open, ^-transverse arc meets F, if at all, in a relatively closed, nowhere dense set and each point in which F meets the arc is the endpoint of a gap of this set. Since X is exceptional, the closed subset X of F meets each open, ^-transverse arc, if at all, in an open subset of a Cantor set. The set Y = F C\ W is a compact, nowhere dense, T-invariant set in R, as is Z = X n W. By the above observations, Z is a Cantor set, but Y may contain isolated points. The leaf F accumulates on the semiproper leaf L C X. Since L accumulates on itself, F accumulates on L from the nonproper side. If F
229

ENDSETS O F EXCEPTIONAL LEAVES

such t h a t 0 £ Po and Pj fl Pj+i ^ 0, 0 < j < m. This defines a holonomy chain hT at 0. While this has involved choices, another such holonomy chain hT will necessarily be germinally equivalent t o hT. Germinal independence of choices will not be enough for our purposes, b u t what we do need is given in t h e next lemma. Let / and N > 0 be as in Theorem 1.3, x> = fN+i{b) and zt = fN+l(a), + Mi > 0. If r is a p a t h from 0 t o y € V, t h e n [Q,xQ] C D ( / i T ) . In fact, by T h e o r e m 1.3, since a = T - 1 O r is a loop at 0 + , ha is defined on [0, XQ\. L e m m a 2 . 2 Let T\ and r 2 be paths in V from 0 + to the same point y. Then, for any corresponding holonomy chains at 0, chosen as above, hT1(xi) = K2(xi) and hT1(zi) = hT2(zi). Proof. If a is a loop on L', based at 0 + , then Theorem 1.3 implies t h a t b o t h ha{xi) — Xi a n d ha{zi) = Zj. If a is t h e loop on L' formed by following T\ by T^" 1 , t h e n ha = h~2 o hTl a n d t h e assertion follows. • If y E V a n d r in V is a p a t h from 0 + t o y, then hT(0) can be interpreted as an J - p l a q u e P' containing y and hT{xo) is defined and can be interpreted as another 3"-plaque Po C F. Let no : P' —> Po be t h e projection along plaques of 3r±. In particular, no(y) £ F. If t h e points of F H W are lower endpoints of gaps of t h e set Y, we replace hT(xo) with hT(zo) in t h e definition of -KQ. L e m m a 2 . 3 A finite covering map no : LI —» F, with image a complete, connected submanifold BQ C F, is well-defined by the above procedure. By replacing XQ with Xi, i > 0, we obtain finite covering maps 7r» of L' onto complete, connected submanifolds Bi. Proof. By L e m m a 2.2, m is well defined, Vi > 0. It is a covering m a p by definition. Since H+ is compact and ni\H+ : H+ —> Hi is a covering m a p , it follows t h a t ni is a finite covering m a p . T h e remaining assertions are elementary consequences of this. • Let Hi = ni(H),i > 0. It is clear t h a t Hi ^ Hj whenever i ^ j . I t is also clear t h a t dBi = Hi U Hi+i,i > 0. L e m m a 2 . 4 If i < j and S , D Bj ^ 0, t/ien either j — i or j = i + 1 and Bi n J3t + 1 = H i + i . Proof. It will be enough t o show t h a t , if i ^ j , then int Bi n int P , = 0 . It is an easy consequence of t h e definition of 7Tj and nj t h a t this intersection is relatively open and closed in each of int Bi and int Bj. By connectivity, either int Bi = int Bj or these sets are disjoint. In t h e first case, it follows easily t h a t Hi = Hj, hence t h a t i = j . • D e f i n i t i o n 2 . 5 For each integer j > 0, Vj = ( J ~ . P , . C o r o l l a r y 2 . 6 The family {Vj}Cj°=Q is a fundamental

neighbourhood

system

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J. CANTWELL AND L.

CONLON

of an end e* G £ x (F) and e* is a cluster point of £X(F). Proof. It is obvious that {Vj}^0 is a fundamental neighbourhood system of an end e* G £(F). Since every Vj is asymptotic to L and L is asymptotic to X, it follows that e* G EX(F). Finally, each Bj is asymptotic to everything to which L is asymptotic. Indeed, if s € M, if / is the leaf of 3r± through s, and if {si}^ C L f l / clusters at s, then, for fixed j , lim |[si,7r,-(sj)]| = 0 (by [1, p. 170] or Lemma 4.6), so {^dsi)}°^l also i—»oo

clusters at s. Thus, each Bj is asymptotic to X and must be a neighbourhood of at least one end e.,- G EX(F). By Lemma 2.4, ej ^ e^ for j ^ A;, since Vj is a neighbourhood of {ek}^L,-, it follows that e* is a cluster point of {ej}f=0. U Proof.[Proof of Theorem 1.1] Let e G EX(F). Any neighbourhood V of e in F is a connected component of the complement in F of a compact, connected subset E C F. The set £J is also a compact subset of M. Let C be a compact submanifold of L such that the handle H and the curve a with a -^ H = +1 lies in the interior of C. A suitable normal neighbourhood y{C) of C in M will not meet .E. In the above discussion, we choose e so that [0,£) is a fibre of v(C). We take b G V D (0,e) such that all positive iterates ak lift to paths on F that start at b and stay in v(C). Since these paths cannot meet E, they stay in V, so each x, G V. But iJj C v{C) cannot meet E and i i e ^ fl 1/, so each iJj C V. By Lemma 2.4 and the connectedness of E, Bi n .E = 0 for alii > 0. It follows that Vj C V, if j is sufficiently large. By Corollary 2.6, V is a neighbourhood of a cluster point e* of 8.x(F). This cluster point may or may not be e but, in any case, the fact that V is an arbitrary neighbourhood of e will imply that e is a cluster point of £x(F). Since e G £ X ( F ) is arbitrary, it follows that EX(F) is a Cantor set. • Definition 2.7 A complete, connected submanifold V c F i s said to spiral (see [1, §8.4]) onto L with juncture H if (1) for each x G V, there is a choice [p(x),x] of a compact subarc of the leaf of IT-1- through x such that p(x) = y G L and p : V —> L is a surjection that is locally a homeomorphism; (2) for each y = »(x) G L, p~l(y) n [y, a;] = { x i } ^ 0 is a sequence such that Xi I y and xo = x; (3) p~l(H) falls into connected components H, = iJ, i = 0,1, 2 , . . . , such that HQ = 9 £ , and these copies of H partition B into a sequence {i?i}°^0 of complete submanifolds, dBi = ff, U -f/j+i, such that each Bi is homeomorphic to V via p.

ENDSETS OF EXCEPTIONAL LEAVES

231

Remark. We have actually defined the notion of V spiralling onto L from above. With obvious changes, one obtains a similar notion of spiralling from below. The two kinds of spiralling are interchanged by a change of transverse orientation, so no generality is lost in restricting attention to one type. Proposition 2.8 For each i > 0, iti : L' —> Bi is a homeomorphisrn. Proof. We deal explicitly with the case i = 0, the general case being no different. It will be enough to show that TTQ is one-one. Since TTQ is a finite covering map, it is enough to show that TTQ\H+ is one-one. Suppose not. Then there exists t/o = M t/i £ H+ so that 7To(j/i) = no(yo) = xo- Let o be a path in H+ from yo to y\ and let a be the loop at XQ in HQ which is the projection of a under TTQ. By reversing the direction of cr if necessary and rechoosing y\, we can assume that ha(0) = y\ e [0,x 0 ]. Then the set of points yj = hJa € [0, x 0 ], j > 0, satisfies yo < Vi • • • in [0, xo] and thus is a set of distinct points in TTQ (XO). This contradiction the fact that TVQ is a finite cover. D Proof.[Proof of Theorem 1.4] Given the end e, find an end e* as in the proof of Theorem 1.1. It is obvious that Vj spirals on L for all j > 0. • 3

Unbounded holonomy

In this section we use some notation and Lemma 4.6 from Section 4. Lemma 4.6 is the "key lemma" of [1, p. 170]. Let L and F be semiproper leaves of 1 and assume that F accumulates on L. We emphasize that the important case in which L — F is allowed. In that case L is exceptional. Fix a choice of transverse orientation and of basepoint 0 e L so that the leaf of J x through 0 has compact subarcs Jo = [yo, 0] and IQ = [0, xo], with Jo contained in a gap of the set Z = Lf]W and with A • |/o|/|^o| < 1 (A as in [1, p. 170] or Lemma 4.6). Definition 3.1 The holonomy of L is unbounded in IQ (respectively, in Jo) if there does not exist a sequence {xn}'i^L1 C IQ (respectively, Jo) such that xn converges monotonically to 0 and, Vn > 1, TL(xn) is the singleton {xn}. Here TL is the holonomy pseudogroup on transverse, open arcs containing 0 consisting of g £ T such that gj(0) € L, 1 < j < p, gp(0) = 0. Our goal in this section is to prove the following easy but important step in the proof of Theorem 1.3. Theorem 3.2 The holonomy of L is unbounded on any side that is approached by F. Remark. By the semi-stability theorem of Dippolito [6, Theorem 3], un-

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J. CANTWELL AND L. CONLON

boundedness on a proper side of L will fail if and only if there is a collar C = [0,1] x L of L — {0} x L on that side in M and a sequence {t n }^Li C [0,1], clustering at 0, such that {Ln — {£„} x L}^=1 is a sequence of proper leaves of 3\ In this case, no leaf can accumulate on L on the side in question. The proof of Theorem 3.2, therefore, is reduced to the case in which L is exceptional and the side in question is nonproper. In this case, L can always play the role of F. We fix these hypotheses. We suppose that the sequence {xn}^L1 C IQ, as in the above definition exists. We take n sufficiently large, let Ln denote the leaf of 5" through xn, and proceed very much as in the previous section, to construct a local homeomorphism 7rn of L onto Ln such that [y,TTn(y)] is a subarc of a leaf of J 1 , Vy e L. Indeed, since TL fixes xn, the construction of 7r, : L' —> Bi in Section 2 can be mimicked exactly, there being no need to cut L along a handle H, to produce a local homeomorphism nn : L —> Ln which is also surjective. By discarding finitely many of the points xn and renumbering the sequence, we assume the construction works for all n > 1. Lemma 3.3 For each integer n > 1, the points of L n (y, nn(y)) range over all of L as y ranges over L. Proof. By the definition of nn, the set L n {y,Ttn{y)) is the union of plaques, hence is an open subset of L. This set is also complete (hence closed) in L, so it exhausts L. • Proof.[Proof of Theorem 3.2] By an application of Lemma 4.6, we can choose n so large that the interval [y, 7rn(y)] has length less than that of ^0; vy 6 L. In fact, appealing to the octopus decomposition (see Subsection 4.2), one sees that there are at most finitely many Jp with |Jo| < \JP\. If Jp is one of these, one might have to increase n to guarantee that, for V = Vvi |[j/,7rn(2/)]| < l^ol- If Jp is not one of these, then for y = yp, \[y,nn(y)}\ < \IP\ < \JP\ < | Jol- But Lemma 4.6 implies that J 0 C [y,nn(y)], some y G L. This contradiction establishes Theorem 3.2. • 4 4-1

Some derivative estimates Basic inequalities

Recall that {h\,... , hm} is a symmetric generating set for the holonomy group r . Definition 4.1 A reduced word g = hi o- • o/i^ is called a chain of length \g\ = p. We write Qj = htj o • • • o hh, 1 < j < p, and # 0 = id. Convention 1 Whenever g is a chain of length p and Y C D(g), we set Yj=9j(Y),

0<j
233

ENDSETS OF EXCEPTIONAL LEAVES

In particular if Y = {yo}, then one has yo,.•• ,yPDefinition 4.2 If yo 6 D(g), 6 is a subset of the T-orbit of yo, and 2/0) • • • ,2/p G S, the chain g is said to be a chain at yo in G. This is a simple chain at yo if 2/? ^ 2/fc, whenever j ^ k. If |y| = p, if gp-\ is a simple chain at yo, and if g(yp) — yo, then g is called a simple loop at yo- Finally, if g = h~l o I o h, where h is a simple chain at yo and / is a simple loop at h(yo), then g is called a basic loop at yoDefinition 4.3 If Y C D(g) is a nondegenerate interval, the chain g is said to be a c/iain at Y. This is a simple chain at V if int Yj- n int Y^ = 0, whenever j ^ k. If \g\ — p, if g p _i is a simple chain at Y, and if g(Y) = Y, then g is called a simple loop at Y. Finally, if g = h~l o I o h, where h is a simple chain at Y and Z is a simple loop at h(Y), then g is called a basic loop at Y. Choose constants A > 0, B > 0, A > 0, and # > 1 such that 1. h't > A, I < i < m, 2. \h'(\ < B, 1 < i < m, 3. 6 = A/B, 4. A = exp(6#| W|) (where |Y| denotes the Lebesgue measure of a set Y). Lemma 4.4 Let g € T 6e o c/iam o/ length p, and let

UQ,VO

€ D{g).

Then

For this fundamental estimate see [10] or [1, p. 168]. Here is an easy consequence. Lemma 4.5 Let g e T and J C -D(<7) 6e a compact, nondegenerate interval. Suppose g is a simple chain of length p at J. Then ^<exp(0|W|),

\/u,veJ.

However, we need the following more delicate version of Lemma 4.5. It is the "key lemma" of [1, p. 170]. Lemma 4.6 Let j £ T and let Jo and IQ be compact subintervals of D{g) such that • Jo is non-degenerate, IQ is possibly degenerate, • Jo n Jo = {yo} is o, single point of d Jo, • |/o|/|Jo| < 1/A,

234

J. CANTWELL AND L. CONLON

If g is either a simple chain or a basic loop at JQ then 1. | 5 ( / 0 ) | < | < ? W | / 2. ^rl <\,Vu,v€ 9 »

JoU/0.

This lemma will give us significant control of holonomy on the nonproper side of a semiproper leaf. 4-2

The octopus decomposition

Fix a choice of a smooth, one-dimensional foliation 3 rX that is everywhere transverse to J . We assume the atlas $ from Subsection 4.1 is biregular relative to JF and J"1- in the sense defined in [1, §5.1]. Each Ui = Pi x /;, 1 < i < m. In standard fashion, the arc /j is to be identified with a set of 7-plaques by t <-> Pi x {t} and with the set yi(Ui), 1 < i < m. In the theory of foliated manifolds of codimension one, the open, connected, saturated sets U play a fundamental role [6, 8, 2, 1]. One considers the "transverse completion" U, obtained by taking the metric completion of U relative to any Riemannian metric on U relativized from a Riemannian metric on M. If U is connected, this simply attaches to U finitely many boundary components, completing 7\U to a foliation J of U tangent to the boundary. We emphasize that U is generally noncompact. The inclusion map i : U c-> M induces a natural immersion i : U <—> M that may identify some pairs of boundary leaves to a single leaf of 3\ The leaves of i(dU) are called the "border leaves" of U. If U is connected, there is an "octopus decomposition" U = Kl) Ax U---U Ar, where K is a compact, connected manifold with boundary and corners and each Ai is a foliated interval bundle, also with corners. The corners separate the subset d^K of dK (respectively, the subset d^Ai of dAi) transverse to F from the subset dTK (respectively, dTAi) tangent to 3\ The manifold K is called a nucleus of U and each Ai is an arm. Each arm has d^Ai connected and attaches to K so that d^Ai is identified with one component of dfaK. The nucleus can be chosen as large as desired, generally at a cost of increasing the number of arms. For more details, see [1, Chapter 5]. If L is one of the boundary leaves of U (generally L is taken to be a boundary leaf on the positive side of [/), we let C = KnL be the core of L. For 1 < i < m, the components of UnUi will be of the form Pi x (a, b). If both Pi x {a} and P, x {6} are subsets of Ui, then Pi x (a, 6) will be called an

ENDSETS OF EXCEPTIONAL LEAVES

235

ordinary product. Otherwise Pj x (a, b) will be called a special products. By the definition of nucleus [1, Chapter 5] every special product is contained in the nucleus. An ordinary product that meets both the nucleus and an arm is called a boundary product. Let U = U n W. The components of U are the intervals (a, b) from the previous paragraph. Define U to be the disjoint union of the closure in T of these intervals. That is, one defines the components of U to be the intervals [a,b]. If J is a component of U, then J is a component of an arm, if the product Pi x J is contained in an arm for some 1 < i < m. It is a nuclear component if the product is contained in the nucleus. It is a boundary component if the product meets both an arm and the nucleus. We let 3 be the set of all components in an arm. Let \3\ be the sum of the lengths of components in an arm. Remark. By choosing the nucleus larger one can make \3\ as small as desired. We let L be one of the boundary leaves of U (without loss assume L is a boundary leaf on the positive side of U). Let S = L n T. Let A C S denote the set of upper endpoints of components in arms, C C S the set of upper endpoints of nuclear components whose upper endpoints lie in S, and dS the set of upper endpoints of boundary components. The set C will be called the core of S and the set 9C will be called the boundary of the core. Then 8 = A U 6 and <9C = A n C. If y0 e S, then S is the T-orbit of y0. 4-3

Holonomy in the arms

If X C M is a compact, J-saturated set with empty interior, it is possible to select the regular foliated atlas so that Z = X O W is a compact, Tinvariant set that meets each of the open intervals D(hi) in a compact set. Assume this and choose fj, > 0 so small that any point in W that is /i-close to a point of the compact set Z n D(hi) lies in D(hi), 1 < i < m. Also assume any component of an arm has length less than /j,. Let U be a component of the complement of X. Let K be a nucleus and let Jo = [^OiJ/o] be a component of an arm. Then xo,yo £ Z. Remark. If g = hl o • • • o h^ is a chain at j/o in .A, then g is a chain at JoIt will be a simple chain, simple loop or basic loop, respectively, at Jo if and only if it is such at j/oLet IQ C W be a compact interval meeting Jo in exactly one of the endpoints, say yo- In order to use Lemma 4.6, choose IQ SO that |/o|/|-^o| < 1/A.

236

J. CANTWELL AND L. CONLON

Lemma 4.7 If g is a simple chain or basic loop at yo in A, then IQU JQ C D(g) and (1) g'{u)/g'{v)

< exp(60|3|) ifu,veI0U

(2) ifg is a basic loop, then exp(-60\d\)

J0; < g'(u) < exp(6<9|5|), Vu € I0UJ0.

Proof. By the remark, Jo C D(g). For 0 < j < p, we prove that Ij is denned. For j = 0, there is nothing to prove. Suppose that, for some i < p and 0 < J < i, I jis defined. Using Lemma 4.6, we obtain the inequality

In particular, J% e 3, so | / j | < \Ji\ < \x. Therefore, J, C D(hi+i) and U+i is defined. By finite induction, Ip is defined and IQ U JO C D(g)Let u, v e IQ U JO. By Lemma 4.4,

^M<exp(..g K -,,|). Furthermore, \UJ — Vj\ < \Ij\ + \Jj\ < 2|Jj| and, as in the proof of Lemma 4.6 (see [1, p. 170]), it follows that p-i

This proves (1). Finally, if g is a basic loop, find v e Jo with g'(v) = 1 and apply (1) to prove (2). • Corollary 4.8 Let 0 < r < 1 < s. If the octopus decomposition of U has large enough nucleus, if IQ U JO is as in Lemma 4-7, and if g is a simple chain or basic loop at j/o in -A, then r < g'{u)/g'{v) < s, Vu, v £ IQU JO- If g is a basic loop, then r < g'(u) < s, Vu £ IQ U JOProof. By making K large enough, we guarantee that exp(6(9|3|) < min{s, 1/r}. An application of Lemma 4.7 completes the proof. D Fix a boundary orbit § C X on the positive side of U, as above, together with an exhaustion K = Ka C Kx c • • • C Kk C . . . of U by nucleii of octopus decompositions. For a fixed A; = 0 , 1 , 2 , . . . , define Ak, dQk, and C^ as in Subsection 4.2. In particular, each Cfc is T-connected.

237

ENDSETS OF EXCEPTIONAL LEAVES

Fix a basepoint in Co and, by recoordinatizing, assume that this basepoint is 0 G T C K. Let int Cfc = Cfc \ dGk. The orbit S corresponds to a semiproper leaf L C X and any choice of Jo, as above, lies on a proper side. If the leaf is not proper, then IQ lies on the nonproper side and § fl Jo clusters on 0. It is our delicate task to analyze the holonomy along paths in L on this nonproper side. 4-4

The set r # C

TL

The set T# will serve as a set of generators for the germinal holonomy of L at 0, there exists an £o > 0 such that every element of T# is defined on the interval [—£o,£o], and we have very delicate control of the maps 7 for every 7 6 T#. Definition 4.9 The set To C TL consists of basic loops at 0 in Co. If Ffc c TL has been defined, some k > 0, then TL D T^+I D Tk is constructed as follows. The complement Tk+i \ Tfc is to be the set of basic loops at 0 of the form h~1 ogoh, where g is a basic loop in Cfc+i \ int Cfc at h(0) G 9Cfc and where ft is a simple chain at 0 in Cfc of length (say) p such that for 1 < j < k, 31 < ii < ... ij <•••< ik = p, satisfying: /ij(0) G int Cj, i < ij] htj(0) G dGj\ /i»(0) £ int Gj,i>

ij

Finally T# = Ufelo^fe- ^ n e P r °duct 7 = h~x o g o h is called the krepresentation of 7. L e m m a 4.10 Let L be semiproper. If the holonomy of L is unbounded on a given side of L, then T# has no fixed points arbitrarily close to L on that side. Indeed, in a standard fashion, each plaque-loop a on L at 0 is "factored" into a product of loops, each of which corresponds to an element of T#. This requires that a be modified by inserting finitely many plaque-chains of the form T*T~X (where * denotes the adjunction of plaque chains). It follows that, if T# has fixed points xn j 0, then ha(xn) = xn, Vn. Details are left to the reader. The following is an application of Lemma 4.7. Lemma 4.11 For some s0 > 0, [—£o,£o] C ^ ( 7 ) , V7 G T#. Further, for any such eo, one can choose 0 < e\ < £0 such that 7[—£i,£i] C [—£o,£o]; V7GIV Proof. Choose 6 sufficiently small that [—5,8} C .0(7) for the finitely many basic loops 7 G To. We can also assume that the finitely many simple chains in Co from 0 to points of <9Co are defined on [—5,(5]. Every 7 G T# \ TQ is of the form 7 = h~l o g o h where g is a basic loop in

238

J. CANTWELL AND L. CONLON

S \ Co at a point yo € dGo and h is a simple chain in Co from 0 to yoHere yo = IQ n Jo where Jo and Jo are chosen as in Lemma 4.7 and, by that lemma, Jo U IQ C D(g). Thus, choosing 5 smaller, if necessary, we can assume that goh is defined on [—<5, <5], for all choices of g and h as above. We must choose 0 < £o < S so that h~l is defined on g o /i[—£o, £<)]• By the mean value theorem, there is a point £ € /i[—5, <$] such that \g[h(-5),h{5)]\

=

\[h(-5),h{6)]\-\g'(0\.

Again appealing to Lemma 4.7, we note that the derivative g'(£) is uniformly bounded away from 0, for all basic loops g as above. Noting that g~l varies over all such basic loops as g does and that h varies over finitely many simple chains, one easily chooses EQ so that h[—£o, £o] C g~1oh[—S, 5], for all such g and h. Further, the previous argument yields -y[—£o,£o] C [—5,(5], V7 e T#. Replacing 6 by £0 in the previous argument yields an £1 < £0 so that

7[-£i,£i] c [-£0,£o], V7 e r # .

n

Every g e r # is a basic loop at 0. Choose £0 as in Lemma 4.11. Let Jo C [—£0, 0] have upper endpoint 0. Then g is a basic loop at Jo. If one chooses IQ C [0, £0] with |7 0 |/|Jo| < 1 A a n d Jo H Jo = {0}, an application of Lemma 4.6 yields |Jj U J,| < 2| Jj|. Definition 4.12 Let £ = \I0 U J0\ and 0 = (69\W\/£) • exp(6#|VF|). The constant © is fixed throughout the rest of the paper. Lemma 4.13 Let g £ T# and IQ and JQ be as above. Then ^fp-

< exp(0|u - v\),

Vu,veI0U

Jo-

Proof. Set A = 70 U JQ. Then £ = |A|. As usual, set Kj = gj(A), J

j = 9j(Jo), 1 < 3 < P- Then \uj-vA

g'AQ

|Ail


\u|A|

< !__Jll.exp(60|W|) " |A|

by Lemma 4.6. Since I0 is chosen with |Jo|/| Jo I < 1/^ at Jo p— 1

anc

^ 9 *s a basic loop

p—1

|A,|<2^|Jj|<6|W|. j=0

j=0

Thus

E K- - vi\ ^ l\reMw\w\) Y, iAii < nrpexp^i^o. &\w\ 3=0

' '

3=0

' '

ENDSETS OF EXCEPTIONAL LEAVES

239

By Lemma 4.4, p-i

9'{u) < exp( 922 \uj ~ vj\ ) < exp(0|u - v\). g'(v) 3=0

a Definition 4.14 A pseudogroup element 7 = gm o gm_y o • • • o glt where gi E r # , 1 < i < m is called a r#-chain of length m. Let 7 be such a chain and let A C D(7) be a compact interval. For 1 < p < m, set 7 P =
- ^ - T < exp(e|W|), 7 '(v)

Vtt,«eA.

Proof. Let Uj = jj(u) and Vj = Jj(v), 0 < j < m — 1. Then 7'M l'(v)

=

9m("m-i) gi("o) v 9'm( m-l) '" 9i(v0)'

Applying Lemma 4.13 to each gj gives ^ ^ e x p ^ G ^ K - ^ f )

<exp(6|^|).

D In Lemma 4.15, it is sufficient that 7 m _i be a simple r#-chain

Remark. at A. The following will be useful in Section 7. Corollary 4.16 Let f G F# be a contraction of [0,e) c Jo U Jo to 0. Let xo € [0,e) and Xk = / fc (xo), Vfc > 0. Then, for all integers p, k > 1 and allu,ve[xp,x0}, (fky(u)/(fk)>(v)<exp(pG\W\). Proof. Choose u = uo,u\,... ,up = v in [xp,xo] so that, for suitable choices of z, e [xp,x0], {ui,Ui_i} C A; = [f(zi),Zi], 1 < i < P- This

240

J. CANTWELL AND L. CONLON

is clearly possible. Then 7 = fk and A = Aj satisfy the hypotheses of Lemma 4.15, 1 < i < p, so we see that i'{ui-i)/l'{ui) < exp(0]H / |), 1
In Theorem 6.1 we will show that there does exist a contraction

/er#. Lemma 4.17 Let A and 7 be as in Lemma A.15 and let g € T be such that 7(io U Jo) C D(g). Let Q = [a, b] C A. Choose r < 1 so close to 1 that l-r<exp(-Q\W\)-\Q\/\A\. / / there exists x € 7(A) such that (1) g(x) = x < 7(6) (respectively, g(x) = x > j(a)), (2) g(j(b)) < 7(a) (respectively, g(j(a)) > f(b)), then there exists t £ 7(A) such that g'(t) < r. Proof. We treat the case x < 7(6) and 5(7(6)) < 7(0). The alternative case is obtained by an entirely similar argument. By the mean value theorem and Lemma 4.15,

MM _ m . JAI < e x p ( e i W |). JAI h(Q)\~Y(v)

| Q , <e X p C Oi^|J

|Q|,

for suitable values u € A and v G Q. Thus, r < e x p ^ u\w\) |A| S ^ ( A ) | <7(6)-7(a)

~

7(6) — x

^7(b)-g(7(b)) ~~

7(6) — x

Therefore r > 1

l(b) - 9(l(b)) _ g(j(b)) -x _ g( 7 (6)) - g(x) 7(6) — x 7(6) — x 7(6) — x

By the mean value theorem, there exists t 6 7(A) such that g'(t) < r.

O

Remark. A simpler version of Lemma 4.17 can be given with 0 replaced by 0. In the proof of the simple version, the use of Lemma 4.15 is replaced by a use of Lemma 4.5. Otherwise the proof is identical. We need the stronger lemma.

241

ENDSETS OF EXCEPTIONAL LEAVES

4-5

Better estimates

Every 7 G F# is defined on IQ U JO. Choose £\ in Lemma 4.11 such that [—£1, £1] U 7[—£i, £1] C / o U Jo, V7 G T#. In the rest of the paper we work in an interval (—e,e) C [—£i,£i] and take e smaller as needed. Lemma 4.18 Let 0 < r < 1 < s. Then k > 0 can be chosen so large and e > 0 so small that for all 7 G T# \ Fk, the inequalities r <j'(u) < s hold for —£ 0 such that, V7 e r # \ T^ with ^-representation 7 = / i - 1 o 5 o h, we have •v/r < g'(h(u)) < y/s,

-e < u < £.

By elementary calculus,

and each of the finitely many functions h' is continuous. By making e small enough, we guarantee that r

h

'(u) /i'(7(u))

r

/

/

hence r < 7'(u) < s,

—s
There remain the elements 7, e Tfc, 1 < i < n. By the assumption on these, we see that 7t'(0) = 1, 1 < i < n. Thus, making e possibly smaller, we complete the proof. • Lemma 4.19 Let£\ be as above andO < P < E\. Then k > 0 can be chosen so large that for 7 6 T# \ Fk, the inequalities 1/(2A) < 7'(u) < 2A hold for -Ei < u < £1. Further for this choice of k, [0,/3/(2A)] C j[0,(3] C [0,2A/?],

v7Gr#\rfe. Proof. By Corollary 4.8, we choose k > 0 such that, V7 G T# \ Tfc with fc-representation 7 = ^, _1 o g o /i, we have 1/2 < g'(h(u)) < 2,

-£i
By elementary calculus, 7 (u) =

'

^H)'5'(/l(u))'

-£^«*ei-

242

J. CANTWELL AND L. CONLON

By L e m m a 4.6 and Lemma 4.11, 1/A<

\ \ < A, /i'(7(u))

-£i
hence 1/(2A) < 7'(w) < 2A,

-ex

< u < £i.

By the mean-value theorem, |[0,/?/(2A)]| < |-y[0, /?] | = |[0,/3]| • -y'(£) <

|[0,2A/?]| 5

n

I n t h e a b s e n c e of a c o n t r a c t i o n

Let L be a semiproper leaf of 5". Fix a choice of basepoint 0 G L and the arc [—£i,£i] of t h e leaf of 3" x t h r o u g h 0. Let r # be as in Definition 4.9. By L e m m a 4.11, every element of T # is defined on all of [—£i,£i]. Further, [—£i,£i] C IQ U JO with I0 and Jo as in Definition 4.12. This allows us to use L e m m a 4.15 and L e m m a 4.17 with 0 as defined in Definition 4.12. We fix the following assumptions. (i) No 7 e r # restricts t o a contraction t o 0 on any subarc [0,77) C [0,e]. (ii) T h e leaf L has unbounded holonomy on the positive side. In t h e next section, we will see t h a t these assumptions lead to a contradiction. Here, we simply produce some technical consequences of the assumptions t h a t will be used in Section 6. It will not m a t t e r whether t h e positive side is a proper side of L. By L e m m a 4.18, we can choose the value of 0 < e < £\ so small t h a t 10

Remark.

K

7

' ^

<

10'

V

^

G F

# '

- £ < £ < £

(#)

9 In the proofs t h a t follow, it is enough t h a t the constants — <

1 < — be sufficiently close to 1. 10 L e m m a 5.1 There is a sequence {2/1,^2,21,2^3, £2, ?/4i.. • ,yi+\,Xi,...} (0,£), converging monotonically to 0, and a sequence {<7i}^i C r # that (a) gi\[xi,yi]

is a contraction

(b) 9i{yi) =Xi-i,

2
to Xi, 1 < % < 00;

in such

ENDSETS OF EXCEPTIONAL LEAVES

243

Xl

E-

X3

;

E

1 —

-3

32(2/2)

2/2

'.

E

i 33(2/3)

.

1 94(2/4)

:

xi

2/i

3 2/3

a 2/4

Figure 1.

Proof. In the rather technical argument, the reader may find the visual schema in Figure 1 helpful. By (ii) and Lemma 4.10, there is an a > 0 so small that the elements of r # | ( 0 , a ) have no common fixed point. On the other hand, by (i), none of these are germinally contractions to 0. Thus, given z G (0, a) C (0,e), choose gz e T# and [xz,bz) C (0, a) such that z G (xz,bz) and gz\[xz,bz) is a contraction to xz. We consider two cases. Case 1. Assume that a > 0 can be chosen so that there is a set of points z £ [0,a), clustering at 0, for each of which one can choose bz = a. Choose a sequence {gi,[xi,a)}^2=1 with g^ = gZi, Xi — xZi, and Xj J, 0. Choose xi so that 2\x± < a. In general the choice of ajj+j is to be made so that 2Xxi+i < Xi. If i ^ j , then it is clear that gt ^ gj. If i is sufficiently large, Lemma 4.19 implies that g^l(x\) < a, so renumber gi as 52, %% as £2, and set yi = a and 2/2 = 32~1(xi)- Again, for j sufficiently large, gj1(x2) < xi, so renumber gj as 53, Xj as X3, and set 2/3 = ff3~1(a;2). Iteration of this procedure yields a sequence {gi, [3^,7^)}°^ with all the desired properties. Case 2. Alternatively, a > 0 cannot be chosen as in Case 1. Thus, for some (0,??) and every z S (0,77), gz and [xz,bz) can be chosen as above with the additional property that gz{bz) = bz < a. By the local compactness of (0,77/2], choose a sequence {9i = gZl,[xi,bi] = [z Z i ,6 z J}£i such that 0 = {(xi,bi)}°l1 covers (0,77/2] and Xi J, 0. We claim that lim bi = 0. Otherwise, {6;}°^ would have a cluster point 5 > 0 and we i—>oo

could rechoose a = 5/2, contradicting the hypothesis that no choice of a > 0 places us in Case 1. Therefore, 0 is a locally finite cover of (0,77/2]. Also, wlog, we can assume that every component of 0 meets (0,77/2]. In

244

J. CANTWELL AND L. CONLON

particular, each component of 0 meets at least one other. We seek a subcover 0* C 0 without triple intersections. Set Aj = [xi, bi], i > 1. If int Ai C Ui^i int Aj, let 0 1 = O N {int Ai}. Otherwise, set 0 1 = 0. Inductively, suppose that 0* C O i _ 1 C • • • C O1 C 0, where 0 J covers (0,77/2] and, for 1 < j < i, int Aj C | J { i n t A| int Aj ^ int A G 0 4 } Define O

i+1

==»

int Aj £ 0 \

as follows. If intAj+i C y { i n t A | i n t A i + i ^ int A e 0*},

i+1

then set 0 = 0 l N {int A i + i } . Otherwise set 0 i + 1 = 0 \ Let 0* = f")i>i 0*We claim that 0* covers (0,77/2]. Indeed, if z e (0,77/2], then the local finiteness of 0 implies that z $ int Aj for sufficiently large values of j . Since every 0 l is a cover, some int Aj containing z is never discarded in the above process. Suppose that intAj,intAj, and int Afc are distinct elements of 0* and that int Ajflint AjPiint Afc ^ 0. Since these are intervals, we can assume that int Ai C int Aj U int Afc, in which case int Aj ^ 0* 2 0*, a contradiction. It

X2

E— X3

E

3

*

bl

—

E

3 bl

I

3 63

. #»

3 64

Figure 2. follows that we can renumber 0* = {(xi,bi)}^=1 so that (xi,bi)C\(xi+\,bi+i) is nonempty, Vi > 1, and conclude that b\ > 62 > ^1 > &3 > x2 > &4 > X3 > . . . (see Figure 2). Set y\ = bi and, if i > 2, set j/j = g~l{xi~\). It follows that this choice of {x%,Vi,gi}^Zi satisfies the requirements. D L e m m a 5.2 If F is a leaf and F n (xi, y{) ^ 0, then we can assume that be F n ( x i , y i ) is such that [51(6),6] C (3/2,2/1)Proof. We must consider the two cases in the proof of Lemma 5.1. In Case 1, choose x\ so that 2Axi < 6/(2A).

245

ENDSETS OF EXCEPTIONAL LEAVES

The application of Lemma 4.19 can be made to ensure that g\ and gi satisfy ya = 52 _1 (zi) < 2Axi < 6/(2A) < gi(b)


In Case 2, iterated applications of g± J to a point of F n (xi, j/i) produces a point 6 as desired. • Definition 5.3 Afe = (xk,yk+i), V/c > 1. Fix a choice of A = [gi(b), b] c (t/2,2/1) as in Lemma 5.2. Let A' = [91(b), b). We will consider elements 7 6 F i of the form 7 = gip o g i p l o • • • o g^ where ti = 1 and either ij = i,_i or i,- = 1 + ij-i, 2 < j < p, and such that A C D(7). Set 7o = id Ij = 9ij °9ij-i Aj=lj(A),

°---°9h 0<j
Definition 5.4 An element 7 G TL as above is admissible if, whenever ij = 1 + ij-i then Aj-inAj,.., ^ 0 . Also, 7 = 70 = id will be called admissible. Lemma 5.5 Let x G (0,6). Then there exists an admissible 7 G TL such that x G 7(A'). Proof. If x G (xi,6), we take 7 to be a suitable iterate g\, k > 0. If x G (0,xi], there is a unique integer n > 2 such that x G (x„,x n _i]. A suitable iterate of g^1 moves x to gnk(x) G (xn-i, Vn] = A n _ i . If n = 2, A„_i c (xi,6), while, if n > 3, then A n _ i C (xn-i,xn-2\Finite iteration of this process leads to A _1 (x) G (x\, 6), hence to g^r o A _1 (x) G A'. Then 7 = A o g\ is admissible and x G 7 (A'). • Lemma 5.6 If "f = gi ° • • • ° g^ is admissible, then Ao, A i , . . . , Ap have disjoint interiors. Proof. For 0 < i,j < p write Ai < Aj, if x G int Ai and y G int Aj implies x < y. For 0 < n < p, we show inductively that A„ < A„_i. This will prove the lemma. For n = 1, A0 = [91(b), b] and Ai = 31 (A0) = [gf(b),gi(b)}, so Ai < A 0 . Inductively, assume that, for some 0 < n < p, An < A„_i. There are three cases to consider. Case 1. Suppose that in+i = in- This together with the inductive hypothesis, gives A n +i = s? (A n _i) = 9in(An) < gin(An-i)

= An.

246

J. CANTWELL AND L. CONLON

C a s e 2 . Suppose t h a t in+\ = 1 + in and A n C A i n . Then A„+i = gi+in{An)

C 3i+i„(Airi) < A i n ,

hence A n + i < A n . C a s e 3 . Suppose t h a t in+i = 1 + in a n d A n % Ain. Set in = k, An = [u,v], and consider the overlapping of Afc and A n as indicated in Figure 3. By the inductive hypothesis, [u,v] < g^fav], so v < g^ (u).

An

Vk+i

Xk

Figure 3. Thus, using ( # ) , we obtain 0

<

<

V Xk

~ < Vk+i ~ Xk 9kl{yk+\)

9k1(u)-9k1(xl') 2/fc+i - Xk

-gkl{xk)

_ , -iw>x

.

n

Gfc T(0 < w

Uk+i - Xk T h a t is, v — Xk < (11/10) • |Afc|, hence 0
< ^-!Afcl

(*)

Similarly u-xk Vk+\ ~ Xk

>

gk{v) - gk{xk) Vk+i ~ Xk

9k(yk+i)

~

9k(xk)

Uk+i - Xk

g'k(0 > Y^It follows t h a t u - Xk > (9/10) • |Afc|, hence t h a t 0 < Vk+i -u

1 .. . < — • |Afc|

**)

247

ENDSETS OF EXCEPTIONAL LEAVES

P u t t i n g together (*) and (**), we obtain 0 < v - u < - • |Afc|. 5

(* * *)

Also, n

„ 9k+i(v) ~xk v - Vk+i

_ 9k+i{v) -gk+i{yk+i) v - yk+i

_

,

, ,

11 10

+

so (*) implies t h a t 0 < gk+i(v)

-xk<

— -(v-

yk+i)

< —

• |A fc |

T h a t is, gk+i{v)

<xk

+ —

We want t o show t h a t u > gk+i{v).

If not, we have 1

and

u < X k

+

11

,A

• l^fc|.

1

. A

loo ' '

I

fc

'

so

v u>

v x

~

~ k~

Yoo ''

,

'

11

>yk+1

89 . . , 1 .. , • A t > - • A*. , 100 ' kl 5 ' ' contradicting (* * *)

,A

,

Xk

~ ~ Too

=

•

C o r o l l a r y 5.7 Given k > 1, i/iere exists an admissible 7(A') and 7(6) G (xk,yk).

7 s-uc/i that Xk £

Proof. By lemma 5.5, choose 7 so t h a t Xk 6 7(A'). FVom t h e proof of Lemma 5.5, it is evident t h a t 7 = gk+i ° gk0! and t h a t [u, v] = gk o 7(A) is as in Case 3 of the proof of Lemma 5.6. T h u s , 7(6) = gk+i(v)

> gk+i(yk+i)

= xk

and 7(6) = gk+i{v) T h e proof is complete.

<x

k

+ jQQ- \Ak\ < y/c+i < Vk•

248

6

J. CANTWELL AND L. CONLON

Existence of the contraction

We make the same hypotheses as in Section 3 and use the same notation and conventions as we used there. In particular, the semiproper leaf F is asymptotic to L on the positive side of L. Theorem 6.1 There is f £ T# and rj > 0 such that [0,77) C D(f) and f : [0,77) —> [0,77) is a contraction to 0. Proof. Assume that there is no such contraction. Then, by Theorem 3.2, (i) and (ii) of Section 5 hold. We work in [0,e) chosen as in Section 5. We assume, wlog, that F is proper on its negative side. By Lemma 5.5, Fn (xi,yi) 7^ 0, hence Lemma 5.2 allows us to choose 6 £ Fn (£1,1/1) such that A = [gi(b),b] c (y 2 ,yi)_As before, let A' = [c?i(6),6). If Q = [a,b] is the component of (0, e) \ F with upper endpoint b, then Q C A. Let J = [0, b] and 0 be as in Subsection 4.4. Choose r < 1 so close to 1 that l-r<exp(-e|W|)|Q|/|A|. By Lemma 4.18, all but finitely many g £ T# satisfy g'{x) > r, Vx £ [0,e). Case 1. In Lemma 5.1, assume that infinitely many distinct elements of r # occur in the list {gi}^Zi- Select gk such that g'k(x) > r, 0 < x < e. By Corollary 5.7, let 7 £ TL be admissible such that xk £ 7(A') and 7(6) G (xj^yk). Thus gk{xk) = xk and xk < gkh(b)) < l(b). Since [a,b] is a gap of F n W, it follows that gk{l{b)) < 7(a)- By Lemma 4.17, there is t £ [rr/fc,7(6)] C [0,e) such that r > g'k(t), a contradiction. Case 2. Assume that only finitely many distinct elements of T# occur in the list {gi}^Zi- Let g £ r # be an element that occurs infinitely often in the list. By assumption, there is no value rj £ (0,e) such that g\[0,rj) is a contraction to 0, so g'(0) = 1. It follows that there is an integer k > 1 such that g = gk and <J'|[0, yk) > r. It is now possible to argue exactly as in Case 1 so as to obtain the contradiction that r > g'(t), some t £ 7(A) C [0,yk)-

a 7

A compactly supported cohomology class

In this section, we use the the contraction / of Theorem 6.1 to define a compactly supported cohomology class on L. We will use Poincare duality to show that this class yields a handle on L with the properties in Theorem 1.3. Unless otherwise specified, we keep the notation and assumptions of the previous section. By Theorem 6.1, we can assume that / £ T# and [0,e) C D(f) are such that / : \0,e) —» [0,e) is a contraction to 0. Let

249

ENDSETS OF EXCEPTIONAL LEAVES

(a,b) be a component of (0,e) \ F and set ak = / f c (o), bk = fk(b), Finally, let 0 be as defined in Definition 4.12.

k > 0.

L e m m a 7.1 Let h € T^, and suppose that neither h nor h~l is germinally equivalent to a contraction to 0 in [0,e). Then there exists an N > 0 such that for all k > N, h(bk) = bk- Furthermore, all but finitely many elements ° / r # fix every bk, k > N. Proof. Choose 0 < r < 1 such t h a t

l - r <

e X

p ( - e | W | ) - ^ ^ . 60 - o i

By assumption, h'(0) = 1. T h u s , we can choose an integer N >0 such t h a t h'(t) > r, 0 < t < hpf- If there is an integer n > N for which h(bn) ^ bn, t h e n such n can be found so t h a t h has a fixed point x € [ 6 n + i , 6 n ) . We fix t h e choice of integer n and assume wlog t h a t h(bn) < bn. It follows t h a t h(bn) < an. We are set up to apply Lemma 4.17 with A = [6i,&o], Q = [oo,6o], and 7 = / " • This gives the contradiction t h a t , for some to € fn[bi,b0] C [0,bN], h'(t0) r, 0 < t < e, for all but finitely many 7 e T # . Therefore, by t h e result of t h e previous paragraph, for all but these finitely m a n y 7 £ T#, j(bk) = bk, Vfc > N. D Since all but finitely many 7 € !*# fix bk, Vfc > iV, there can be at most finitely many elements in T # t h a t are germinally equivalent to contractions to 0 in [0,e). Let {30,91, • • • ,<7g} C r # b e t h e subset of these elements. By choosing e > 0 smaller, we can assume t h a t { f f o , 9 i , - - ,9q} are all contractions t o 0 in [0,e). Rechoose (a, b) to be a component of (0,£) \ F. Remark. L e m m a 7.1 holds with t h e contraction / equal any g e {go, gq}, (a, b) any component of (0,e) \ F, and bk = gk(b).

g\,..n

Let D be t h e smallest of t h e numbers exp(-e|Wl).

b °~"° bo ~ 9i{bo)

0
For notational simplicity, let g € { 3 0 , 9 i , - - - ,
h € T L and /i(6fc) < bk, some k > 0, i/ien

bk ~ 9(bk)

~

bk =

250

J. CANTWELL AND L. CONLON

Proof. Since h(bk) < bk, we have h(bk) < ak, so bk - h(bk) bk -ak _ gk(b0) - gk(a0) h - g{bk) ~ bk - bk+i gk(bo) - gk(bi) _ (/)'(0

fb0-a0\^

(b0-a0*

d^-PHW)-^)**

(9»)'K)

where the second last inequality follows from

(see Lemma 4.15). • Definition 7.3 If h € FL, the germ of /i|[0,e) at 0 will be denoted [h]. The group of all such germs will be denoted lK+(L). Theorem 7.4 There is a homomorphism v : !H+(L) —> R with infinite cyclic image. Furthermore, if x '• TTI(-^,0) —> "K+{L) is the canonical surjection, then the cohomolgy class [i/o^] 6 if 1 (L;]R) is compactly supported. The proof will be achieved in a series of lemmas. There are two cases to treat. In the harder case, g'AQ) = 1, 0 < j < q. We treat the easier case first. Case 1. There is g E {go,gi: • • • ,gq\ such that £f'(0) < 1. As in the proof of Lemma 7.2, bk = gk(bo), ak = gk(ao). The following lemma will be needed later in this section but it is convenient to prove it here. Lemma 7.5 Let h ETL be such that h(bk) ^ bk, for infinitely many values ofk. Then, h'(0) ± 1. Proof. Suppose that /i'(0) = 1 and, wlog, assume that h(bk) < bk for infinitely many values of k. Write g(t) = (p + r(t))t h(t) = (1 + s(t))t, where p ^ 1 and r(0) = 0 = s(0). Then t - h(t) t - g{t)

-s(t) l - p - r(t) •

It follows that t - hit) lira H- = 0, t—o+ t - g(t) contradicting Lemma 7.2.

•

251

ENDSETS OF EXCEPTIONAL LEAVES

Lemma 7.6 There is T £ (0,1) such that, whenever h € T L and h'(0) < 1, then h'{0) < 1 - r . Proof. If this were false, then, Vr € (0,1), there exists h € TL with 1 — T < /i'(0) = p < 1. Set a = g'(O). Choose such r and /i with 4r

< D.

Express g(t) = (a + s(t))t h{t) = (p + r(t))t, where s(0) = 0 and r(0) = 0. Then, for k sufficiently large, \ - a - s(bk) >

1 -a 2

l-p-r(bk)<2{l-p). Consequently, h-h(bk) = l-p-r(bk) ,4(l-p) ^ bfc - ff(bfc) 1 -
4T \-a

< g

which contradicts Lemma 7.2. Lemma 7.7 completes the proof of Case 1 of Theorem 7.4.

•

Lemma 7.7 Let the homomorphism v : {K+{L) —> R be defined by the formula v{[h\) = — log(/i'(0)). Then v has infinite cyclic image and the cohomology class [is o \] € lf 1 (L;R) is compactly supported. Proof. Since g'(0) ^ 1, v is nontrivial. By Lemma 7.6, the image of v does not cluster at 0. Furthermore, T# corresponds to a set of loops on L that generate 7Ti(L,0), so Lemma 7.1 implies that [v o ^] is compactly supported. • Case 2. We assume that <^(0) = 1, 0 < i < q, and we fix choices of elements g,h € {30, 0. Lemma 7.8 There is a strictly increasing sequence {nfc}^^ of positive integers on which I is bounded.

252

J. CANTWELL AND L. CONLON

Proof. (1) Suppose the assertion false. Then, liirin^oo l(n) = oo. Consider bn_- h(bn) bn - &„+/(„)_! K - g{bn) bn - bn+1 bn - bn+l

bn+i

— =

- bn+2

^n+/(n)-2

_|_

_

^n+J(")-l

_|_ . . . -|_

bn ~ bn+i bn — bn+i , 9(b ) ~ g(b x n n+1) |

{

9l{n)-2(bn)

bn — bn+i

-

bn — bn+i 9l^-2(bn+1)

bn — bn+\

2

= l + ff'(6) + (5 )'(C2) + • • • +

lin) 2

(g - y(tHn)-2),

where all & e (6„+i,MGiven r e (0,1), take n so large that r, V£ G [0,b„]. This is possible since <7'(0) = 1. Then ( ^ ) ' ( O = <7V' _ 1 (0) • g'(9j-2(0) • • -g'(0 > rj. Therefore, bn-h(bn) .,„, l-r'f")"1 ^-^ > 1 + r + r22 + • • • + rl{n)2~2 bn ~ g(bn) 1

—

Since l(n) —> oo, we can choose n so large that bn-g{bn)

- 2(1-r)

But the number r can be taken as close to 1 as desired, forcing n to be chosen sufficiently large. Therefore, given an arbitrarily large positive number R, we can choose an integer N > 0 such that

±^M>R, Vn>N. bn - g{bn)

(2) let R > 0 be given and choose N as in (1). Let n > N and let x £ [fen+ii ^n]- Let y e [&i, bo] be the point with x = gn(y) and let M

=

,ma£.

(y-9(y))-

bi
Then x - g ( x ) = gn(y) - gn(g(y)) _ (g")'(fl y - g ( y ) ^ - g(M 9n(bo) - gn(bi) (gn)'(ri) ' 6 0 - 6 1 ' for suitable £, 77 6 [62, bo]- By Lemma 4.16, x-g(x) exp(29|W|) • M bn - g(bn) b0 - 61

253

ENDSETS O F EXCEPTIONAL LEAVES

It follows that, for arbitrary x G [6 n + i,6„], x - h(x) > bn+i - h(bn) = bn+i -bn + bn- h(bn) > (R - l)(bn - g(bn)) > («-l)exp(-2e|W|)(6o-M . {x _

g{x)y

As N becomes arbitrarily large, so does R, hence (x — g(x))/(x — h(x)) becomes arbitrarily small, uniformly for x G (0,6^)- This contradicts Lemma 7.2. • Lemma 7.9 The sequence {l{n)}^Li

is bounded.

Proof. By Lemma 7.8, there is a strictly increasing sequence {nk}^Li of integers and a positive integer A such that l(nk) < A, Vfc > 1. Thus, g~x o h(bnk) > bnk. Suppose that l(ri) is not bounded. Then there are arbitrarily large values of n for which g~x o h(bn) < bn. Let 7 = g~x o h. By the above remarks, there exist arbitrarily large integers m such that 7(6 m ) < bm l(bm+l) > bm+lThus we can choose a 7-fixed point z G [6 m +i, bm]. Recall that g = git 6/- = gk(bo), and set a^ = gh(ao). In Lemma 4.15, take A = [6j, 60] and Q = [ao, 60]. Choose m as above so large that 1 - i{x) < e x p ( - 0 | W | ) • |Q|/|A|,

Vx G (0,bm].

Then, for suitable £ G [.z.6m], r1

frcs 1 - 7 (?) = 1

l(bm) - l{z) bm - j{bm) r = —r

- |
(5™)'(TJ)

bm-am > -r—-—

|A|-eXp(

°{Wl)

a contradiction.

|A|' •

Lemma 7.10 There is a positive number p such that 1

-<

x — q-j(x)

^H

0
0
p x- gi{x) Proof. Set g = gi and h = gj. Let b^ = gk(bo) as above. Let x G [bn+i,bn}, x = gn(y), y € [&i,&oj- Let \i = min6l<j,<60(y - d{y))- By

254

J. CANTWELL AND L. CONLON

Lemma 7.9, there exists an integer A > 0, independent of n, such that x - h(x)

gx+1(bn)


_ gn(b0) -

gn(bx+l)

x-g(x) gn(y) - gn(g(y)) (9 Y(0 b0 - 6A+1> ^ e x p ( ( A + i ) (gn)'(ri) y - g(y) '

g{x)

n

& m

(b0 - bx+1 \ v

C,r? e [&A+I,&O],

the final inequality being by Lemma 4.16. Thus, we take p to be the largest of the many bounds obtained for the finitely many choices of g = gt and h = gj,0 R having the properities in Theorem, 7.4Proof. Since every element of T# \ {g^1, • • • ,9^} fixes bk, for all k sufficiently large, and since T# defines a generating set G# C 3{+(L), we see that /i'(0) = 1, V [h] £ 5f + (L). We will use the method of Thurston's stability theorem to obtain the homomorphism v. By Lemma 7.10, 1 x — qAx)

- < p

^ V 4
0
0<x<e.

go{x)

Let bk — g^(bo). There is a subsequence {bnk}'kLisuch

that

>>nk-9t(bnk) lim fe^oo bnk - go(bnk) exists, 0 < i < q. Remark that Ao = 1. But, V7i € r # \ {g$ , . . . , g^ 1 }, h(bnk) = bnk so we also have -h(bnk) 0 = lim Kk fc^oo bnk - go(bnk) Given
fe—oo bnk -

Kk B = lim fc-oo bnk

go(bnk)

-6(bnk) -g0(bnk)

exist. Let r? = o9. For h e TL, write h(x) = x + h(x), where, as remarked above, h'(0) = 0. Then, r)(x) = x + fj(x) = (j)(9(x)) =x + 9(x) + 4>(x + 9(x)) = x + 9(x) + 4>'(£x)9(x) + 4>(x)

ENDSETS OF EXCEPTIONAL LEAVES

for some £x lying between x and x+6(x). 4>(x). Therefore

255

T h a t is, fj{x) = 9(x)+4>'(t;x)8(x)

+

lim . ^ " 7 ? ( S ) = lim ^ ( 6 " J

fc—oo bnk - go(bnk)

fc—oo

go(bnk)

(M =k-*°°\9o(bn i-(^4+^,.)-^4+) "" 9o(b„ ) g (b, k

k

0

= B+A Given e TL, suppose t h a t it has been shown t h a t A = lim

*nk ~ (bnk)

= lim

fc—oo bnk - go(bnk)

kKk)

fc-*oo

g0(bnJ

Set -r = 4>~1. T h e n x = -y(4>{x)) =x + 4>(x) + 7 ( 1 + 0(x)) = x + (x). T h u s , 7(0:) = —(1 + y'(£x))(x), hence fc—00 g0{bnJ

fc—00

ffol&nj

T h u s for each [h] G 5 { + ( L ) , we obtain a well-defined real number

^]

b = lim

fc—00 6„fc -

^-h{^\

go{bnk)

and 1/ : 9 { + ( £ ) —»1R is a group homomorphism, nontrivial since i/[go] = 1By L e m m a 7.2, if /i e T L , then either h(bnk) = bnk, V large values of fc, or t/[/i] > Z3 > 0, or ^[/i _ 1 ] > D > 0. T h a t is, t h e image of t h e homomorphism v must be infinite cyclic. It is clear t h a t v[h\ = 0, V?i G T # x { g ^ 1 , •. • , S * 1 } , hence the cohomology class \u o x] is compactly supported in L. • T h e proof of Theorem 7.4 is complete. We need a good holonomy interpretation of the cohomology class [z^°x]By Theorem 7.4, there is a smallest positive number r G Im(^). Normalize v so t h a t 7- = 1. T h e n $ = [1/ o ^] is a compactly supported, integral cohomology class in L. It is clear t h a t $ takes t h e value 1 on a suitable loop a in L, hence it is nondivisible. By s t a n d a r d theory, $ is t h e Poincare dual of a homology class t h a t is represented by a closed, oriented, connected, nonseparating submanifold H C L of codimension 1. We may suppose t h a t our basepoint 0 lies on H. For any loop a on L, based a t 0, the homological intersection number a -—^ H is t h e value of $ on a. Let a be a simple loop

256

J. CANTWELL AND L. CONLON

that meets the handle H once at 0 chosen so that a — ' - H = +1 and ha the holonomy around the loop a (a contraction). Let / = ha, g a contraction in {go,gi, • • • ,gq}, and as usual bk = gk{b). Remark. Since we do not know that / = ha is an element of T#, in order to apply Lemma 7.1 in the proof of the next lemma, we can not take the contraction to be / instead of g. Lemma 7.12 If h €TL and v[h] = 0, then h fixes bk, for all k sufficiently large. Proof. We consider separately the two cases of the proof of Theorem 7.4. In the first case, Lemma 7.5 and the definition of v in Lemma 7.7 implies that neither h nor / i _ 1 is a contraction to 0. In the second case, Lemma 7.2 and the definition of v in Lemma 7.11 implies that neither h nor h~r is a contraction to 0. Thus Lemma 7.1 implies h fixes bk, for all k sufficiently large. • For 0 < % < q, there exists a positive integer m.i so that v[gi] = rrii. Lemma 7.13 gi(bk) = fmi(bk), 0
,gq}.

ENDSETS OF EXCEPTIONAL LEAVES

257

Lemma 7.15 Ifj G r # and neither 7 nor 7 l is germinally a contraction to zero, then 7 fixes b^, for all k > N, for a possibly larger value of N. Proof. In fact, the last statement of Lemma 7.1, gives this result for all but finitely many elements of T#, while the first statement of the Lemma applied to each of these finitely many 7 G T# \ {go,gi,... ,gq} in turn gives the result for the remaining 7 G r # such that neither 7 nor 7 is germinally a contraction to zero, for a possibly larger N. • Remark. This is the N in Theorem 1.3. Proof. [Proof of Theorem 1.3] Any h„ as in Theorem 1.3 is a product of 7 G r # such that neither 7 nor 7 - 1 is germinally a contraction to 0 and the Theorem follows. • By Section 2, we have also established Theorem 1.1 and Theorem 1.4. Proposition 7.16 The handle H, curve a, and contraction f = ha do not depend on choice of gap (a,b). Proof. Starting with a different gap (a*,b*) of (0,e) \ F, we have the same set of contractions {go,gi, • • • ,gq} C T#. Take f = go- Repeat the argument of this section to obtain a compactly supported cohomology class [u* o x] and a contraction / * so that v*[/*] = 1. For 0 < i < q, there exists a positive integer m* so that v*[gi] = m*. Therefore v*[f] = mj so by Lemma 7.13 f{bk) = (f*)m°(°k), for all k sufficiently large, where °k = fk(b)Therefore v\f*\ = l/m^. Since 1 is the smallest positive number in lm(v), it follows that m j = 1 or v*[f\ — 1- Therefore, since fmi(bk) = gi{bk), for all k sufficiently large, it follows that v*[gi\ = mi — v[9i\, 1 < i < q. Further, since ^[7] = 0 = v*[y] for all 7 G T# with neither 7 nor 7 _ 1 germinally a contraction to 0 and since T# generates TL, it follows that v* = v. Therefore H* =H,a* = a, and / * = / . • 8

Problems

Besides the problems mentioned at the end of Section 1, a number of problems related to exceptional minimal sets have remained very stubborn. As with the problems at the end of Section 1, these are theorems for Markov LMS's [3]. If X is a LMS, an arc transverse to the foliation can be chosen to meet X in a Cantor set C. Fix x G C and let L be the leaf of 9" containing x. The holonomy group of L relative to X is the group HX(L, X) of germs at x of all holonomy maps (restricted to C) that fix x.. The following question was asked by Dippolito [6, §9]. Q u e s t i o n . Let X be an exceptional LMS and L c l a leaf. Is HX(L, X)

258

J. CANTWELL AND L. CONLON

either trivial or infinite cyclic and generated by the germ of a contraction that is unique in a suitable neighborhood of x in C? Do exactly a countable infinity of leaves in X have HX(L, X) ~ Z and are all the semi-proper leaves among these? Remark. In this paper we have shown that a semiproper leave can not have trivial holonomy but in a subtle sense have not proven that HX(L, X) = Z. For x not on a semi-proper leaf there are no results. Hector has proposed the following question [11]. Question. If X is an exceptional LMS does X have finitely many semiproper leaves? Finally, Question. |X| = 0? 9

If X is an exceptional LMS, does it have Lebesgue measure

Examples of Markov minimal sets

In this section we provide a construction of examples of arbitrary Markov minimal sets. One method of doing this, that of "branched staircases", is due to Takamura [12] and Inaba [9]. We present a different method. 9.1

Constructing the plug

The first step in our construction is to construct a plug (P, 3") where dP consists of two tranverse boundary tori a and p and 7 meets a and p in circles. Further a = <7iU
ENDSETS OF EXCEPTIONAL LEAVES

259

Figure 4. Constructing the plug

with the product foliation. Obtain a foliated manifold {N^^s) by pasting H x {0} to the once punctured dY and H x {1} to the once punctured dZ. Obtain a foliated manifold (iV^S^) by pasting the (now) twice punctured T x {0} to the (now) twice punctured T x {2} so that one end of W is matched to one end of U and one end of X is matched to the other end of X. The foliated manifold (N&, $4) has boundary consisting of two transverse tori a and r foliated by circles. The portion of (N4, $4) coming from the (now) twice punctured T x [1,2] is foliated by the product foliation. Denote the portions of a and r meeting this part of (N4, J4) by o\ and T\. Denote the portion of a meeting H x [0,1] by (73. Then a = a\ U ai U (73 U a4 where dW = ai U Pi = CT3) a n f i Pi = 04- The foliated manifold (P, J ) satisfies the properties of the first paragraph of this subsection.

260

9.2

J. CANTWELL AND L. CONLON

The construction

Suppose a Markov pseudogroup is given as on page 167 of [3] and use the m

notation of [3]. Since we can assume W — \\(R(hi)UD(hi))

is contained in

a compact subset of K, we may assume it is contained in the circle Sl. For 1 < i < m, let Ai - S1 \ D(hi) and Bi = S1 x R(hi). Let S be the surface obtained by removing 2m discs from the two-sphere S2. Then dS consists of the 2m circles Cu... ,Cm, C[,... ,C'm. Let ( P i . S i ) , . . . ,{Pm,7m) be m copies of the foliated plug constructed in Subsection 9.1. Let S2 x S1 be foliated by the product foliation. Construct a foliated manifold (M, 5") with a Markov minimal set X as in [3] as follows. For 1 < i < m, paste the plug (Pi, fi) into S x S1 by pasting
Ai,

XZD

\J

Ij,

Ulp«=i}

and

At D

|J

/j

{j|p«=o}

and p\ is pasted to C,' x Bi and

it follows that pasting in the plug Pi creates no additional holonomy on ZQ than hi and h^1. Thus r : ZQ —> Zo generates the holonomy of T|Zo and X is a Markov minimal set. References 1. A. Candel and L. Conlon, Foliations I, Amer. Math. Soc, Providence, RI, 1999. 2. J. Cantwell and L. Conlon, Poincare-Bendixson theory for leaves of codimension one, Trans. Amer. Math. Soc, 265 (1981), 181-209. 3. J. Cantwell and L. Conlon, Foliations and subshifts, Tohoku Math. Jour., 40 (1988), 165-187. 4. J. Cantwell and L. Conlon, Leaves of Markov local minimal sets in foliations of codimension one, Publicacions Matematiques, 33 (1989), 461-484. 5. J. Cantwell and L. Conlon, Generic Leaves, Comment. Math. Helv., 73 (1998), 306-336.

261

ENDSETS OF EXCEPTIONAL LEAVES

6. P. Dippolito, Codimension one foliations of closed manifolds, Ann. of Math., 107 (1978), 403-453. 7. E. Ghys, Topologie des feuilles generiques, Ann. of Math., 141 (1995), 387-422. 8. G. Hector and U. Hirsch, Introduction to the Geometry of Foliations, Part B, Vieweg and Sohn, Braunschweig, 1983. 9. T. Inaba, Examples of Exceptional Minimal Sets, in A Fete of Topology; Papers Dedicated to I. Tamura, ed. Y. Matsumoto and T. Mizutani and S. Morita, Academic Press, 1988, 95-100. 10. R. Sacksteder, Foliations and pseudogroups, Amer. J. Math., 87 (1965), 79-102. 11. P. Schweitzer, Some problems in foliation theory and related areas, in Lecture Notes in Math., 652, Springer-Verlag, 1978, 240-252. 12. M. Takamura,, Reeb stability for leaves with uncountable endset (in Japanese), Master Thesis, Hokkaido Univ. 1984.

Received

October 24, 2000, revised December

29,

2000.

This page is intentionally left blank

pTOCQCdiiTlQS

of

FOLIATIONS: G E O M E T R Y AND DYNAMICS held in Warsaw, May 2 9 - J u n e 9, 2000 ed. by Pawet W A L C Z A K et al. World Scientific, Singapore, 2002 pp. 263-273

S O M E R E M A R K S O N PARTIALLY HOLOMORPHIC FOLIATIONS MARIUSZ FRYDRYCH Wydzial Matematyki Uniwersytetu Lodzkiego, ul. Banacha 22, 90-238 Lodz, Poland, e-mail: [email protected] JERZY KALINA Instytut Matematyki Politechniki Lodzkiej, Al. Politechniki 11, 93-590 Lodz, Poland, e-mail: [email protected] The aim of this paper is to introduce the concept of Nijenhuis tensor on the submodule of sections of the normal bundle to the foliation endowed with the complex structure and its relationship to the formal integrability of canonically associated subbundle of complexified tangent bundle of the given foliated manifold.

1

Linear algebraic background

Linear algebraic background introduced in this section will be applied to the the tangent spaces of manifolds in the next parts of this paper. A complex structure on a real vector space V is a linear endomorphism J such that J 2 = — 1. The real space V may be considered as a complex vector space by the formula (a + ib)v = av + bJv, a, b G K, v G V. Proposition 1.1 Let V be a real vector space. There exist canonical correspondences between the following objects: (i) complex structures J on V (ii) idempotents r : V®^IRC (i.e. r 2 = T) such that or + ra = a, where a : V®]&C —> V
V®IRC

such that 263

V<8>IRC

= W © W.

264

M. FRYDRYCH AND J. KALINA

Proof. Assume (i). Let r = | ( 1 - iJc), where Jc = J ®R 1 denotes the complexification of the real operator J. Immediately we get that r 2 — r, and 1 1 1 1 which proves (ii). We have used the fact that endomorphism acting on V<8)RC is complexification of an endomorphism on V if and only if it commutes with a. Assuming (ii) put W = imager Then W = crW. We check that W C\W = image r n image(crr) = image r n image(er — ra) = imager n image(l — r)a = imager n image(l — r) = imager n kerr = 0. From the assumption it follows the identity r + OTO = 1 which together with the above gives desired decomposition W © W = V(8>RC which proves (iii). Assume (ii). Let J = i(2r — 1). We easily check that J 2 = i 2 (4r 2 - 4r + 1) = - ( 4 r - 4r + 1) = - 1 . Moreover, aJ = cri(2r - 1) = -iR 1 for a unique complex structure J : V —» V which proves (i). Assume (iii). Let r be a projection onto W along W. From this we get that 1 — r is a projection onto W along W so 1 — r = OTO which ends the proof. • Note that the projection r restricted to the V ® R 1 determines a C-linear isomorphism from complex space (V, J) to W. Similarly, the projection (1 — T)\V <8>R 1 is a C-isomorphism from (V, —J) to W. 2

Bundle structures and Bott connection

Throughout the paper M denotes an n dimensional manifold (real or complex) and T a foliation of M by submanifolds (real or complex) of dimension p and codimension q = n- p. If M is a complex manifold foliated by complex submanifolds, such a foliation will be called tangentially holomorphic to emphasis the holomorphic nature of its leaves. All objects in this paper are assumed to be of class C°°, except where explicitly stated otherwise. In a local computations the little Greek letters range between 1 and p and

S O M E REMARKS ON PARTIALLY HOLOMORPHIC FOLIATIONS

265

Roman letters between 1 and q. Except where explicitly stated otherwise the Einstein summation conventions will be in force. For a given foliation T of M, let TT denote the tangent bundle along the leaves. Then we have the short exact sequence 0-^TT->TM

-> TM/TT

-* 0.

(1)

For shortness we will denote the normal bundle TM/TT by vT and the projection by TM -^-> vT. The exactness of the sequence 1 is equivalent to the exactness of the sequence of the dual vector bundles 0 -> v*T -> T*M -> T*T -* 0.

(2)

Note the vector bundle v*T can be naturally identified with the annihilator of the the tangent bundle of T. Definition 2.1 We call a bundle mapping h: vT —> TM the splitting of the exact sequence 1 if irh = 1. The bundle % = image(fo) which is complementary to the vertical bundle TT in TM, is called horizontal. Given a splitting of the sequence (1), any vector Y € TM decomposes into Y = Yh +YV where Yh € image(/i) and Yv e TT. Let (U, ip) be a distinguished chart for T with local coordinates (xa,yi). Consider the horizontal parts Yj = {-^j)h of vectors -^-j. The vectors Yi,..., Yq form a basis of the horizontal subspace H. Since g§? form a local basis of TT, thus {-^JJ)V = A"(a;, y)gf^ for some smooth functions A"(a;, y), therefore we have Yj = ^-j — A"(x,y)gfs-. The local basis {-^s,Yj) will be called Tl-adapted, associated to the distinguished chart (U,ip). It is straightforward that (£ a , dyj) is the dual basis when £a = dxa+X'j(x, y)dyj. Now, if we demand, that the horizontal bundle Ti is integrable (involutive), we obtain equations d£,a e ideal ( £ \ . . . , £ p ) which are equivalent to the system of first order P.D.E. on functions A"(x, y) Yj\%-Yk\<*=0,

j,k = l,...,q;

where j,k

= l,...,q;

a = l,...,p.

a=l,...,p

266

M. FRYDRYCH AND J. KALINA

Definition 2.2 Let T be a foliation on M. A vector field X on M is called a foliated vector field (an infinitesimal automorphism of J-) if the local flow associated to X preserves the foliation. We have the well known fact Proposition 2.3 Let Y be a vector field on the foliated manifold (M,^). The following conditions are equivalent: (i) Y is foliated (ii) [Y,X] € T(TF) for all X £

T(TT)

(Hi) In every distinguished chart with the coordinates (x0,?/-7), Y has the form,

Let us recall [4] that a linear connection V in a vector bundle E —> M is a linear homomorphism V :T(E)

^r(T*M®£)

such that V / u = d/u + / V u ,

for

/ e C°°(M),

u e T(E).

As usual we define V x : T(E) -> T(E) for X e T(TM) by V x u = (V«)(X). The curvature R of a connection V is the skew-symmetric map R: Y{TM) x Y{TM) -^

T(End(jE;))

such that R-XY = [ V x , Vy] - V[x,y] for any X,Y eT(TM). In the case of the foliated manifold there a is natural partial connection Definition 2.4 ([1]) For any foliation J7 on M there exists a linear partial flat connection V: r > J O

-^>T(T*F®vT)

in a normal bundle vT given by S7xu = 7r[X,Y] for X e r(7\F), u e T{vF) and any Y € T{TM), such that TT(Y) = u.

S O M E REMARKS ON PARTIALLY HOLOMORPHIC FOLIATIONS

267

From the Jacobi's identity it follows that this partial connection is flat i.e. RXlX2 = 0, for XUX2 e T(TF). Let C^? be the sheaf of germs of functions which are locally constant along the leaves of T. o

In the usual way we can define the sheaf Y{yT} of germs of local sections of the normal bundle vT which are covariantly constant with respect to the o

partial Bott connection. It is evident that V(yT~) forms a sheaf of modules over the sheaf C^p of rings. o

Proposition 2.5 Every lift in T(TM) vector field .

of any section Y{vT) is a foliated

o

Proof. Take a local section u e ^{vT) in a neighbourhood of m € M. Let (U, ip) be a distinguished chart around m. Let Y be any lift of u over U. Thus in that chart Y = aa(x, y)-^s + W{x, y)-£jr- Therefore we have, d , dx^'

0 = V.

dbj{x,y) dxP

''

d dyj

Since 'K(-^T), • •., TT(g^-) form a local basis of u!F\U, the coefficients Q^P = 0 which proves that the vector field Y is foliated. We say that the normal bundle vT is complex if it is endowed with a bundle endomorphism J : v —> v which is a complex structure over the fibres. Suppose that V J = 0 i.e. V x J u = J ( V x w ) for all X £ TT and u e T(v!F). Such a connection will be called C-linear. Assuming vT is complex, we introduce the concept of the Nijenhuis tensor J\f on the o

sections of the sheaf Y(yT) as a skew-symmetric C^ linear mapping TV : A 2 f (vF) —* T(vT). Let u, v be arbitrary germs of sections of r(^jF) at a point. Choose any lifts u, v, Ju, Jv of u, v, Ju, Jv respectively. Define Af(u, v) = Jn ( Ju, v

u, Jv ) — i" ( Ju,Jv\

- [u,v\J

Since we assumed C-linearity of the partial Bott connection (V J = 0) o

o

we see that for every u € I ^ I A F ) it follows that Ju e T^J7). Using the above proposition, it is easy to see that the definition of TV doesn't depend of the choice of the lifts.

268

M. FRYDRYCH AND J. KALINA

Definition 2.6 We say that a complex structure J on the normal bundle o

vT is integrable if V J = 0 and N = 0. Let (yT, J) be a complex normal bundle as above. Consider the complexification of exact sequence 1 0 -» TJC-(g)RC -> TMKC ^ i VT®WL
According to Proposition 1.1 in Section 1 we get the following decomposition of the bundle: vF®w!C = W © W. Denote by V the subbundle of TM® R C defined by the formula V=

(7r®l) _ 1 (W).

It is easy to check that V = ker 7r where fr: TM®RC

—> vT

is given by 7r(u 1 + v 1 + v ® i by u + iv. Suppose that u + iv € V. We get (7r (8) l)(u + iv) = 7r(u) + in(v) e W — ker(l -I- iJ). This means that u + iv £ V if and only if 7r(u) = — J(n(u)). Let us now consider to local sections Yi, Yb of the subbundle V. From the above we obtain the following equations n(vk) = -J(n(uk)),

k = l,2,

(3)

where Yk = uk + ivk, forfc= 1,2. Calculating the complexified Lie bracket of the fields Y\, Y?, we get [Yi,Y2] = ([ui,u2]

- [vi,v2]) +i([ui,v2]

+

[vi,u2]).

269

S O M E REMARKS ON PARTIALLY HOLOMORPHIC FOLIATIONS

T h u s [Yi,y 2 ] € T(V) n([ui,v2] If u\ e T(TJ-),

means t h a t + [vi,u2])

= - J ( 7 r ( [ u i , u 2 ] - [vi,v2]))

•

(4)

we can choose Y\ = u\ (v\ = 0) and we get 7T ({UI,V2})

= - J (7r {[UI,U2]))

,

which means t h a t V Ul 7r(r; 2 ) = V U l ( - J ( 7 r ( « 2 ) ) ) = - J ( v u i 7 r ( u 2 ) ) . o

This proves t h a t V J = 0. o

.

o

Let H , t e r(i/.F) and u\ = J u , u 2 = Jv, v\ = u, v2 = u. Since V J = 0, we see t h a t u i + ivi,u2 + iv2 G r ( V ) , so substituting t h e m into (4) we get Af(u, v) = 0 which proves the necessity. To prove the sufficiency, let (U, ) be a distinguished chart for T with local coordinates (xa,yi). T h e local sections ^(-^jj),j — 1,...,
ponents of J with respect to the above basis. By the assumption, V J = 0 we infer t h a t gfs-•/£(£, y) = 0, a = l , . . . , p , which means t h a t Jlk(x,y), k,l = 1 , . . .,q don't depend on leaf coordinates. It is easy to see t h a t t h e vector fields gf^, gf? ~ iJj(y)w^> a = 1,. . .,p, j = I,.. .,q, form a set of generators of V(V\U). To prove the formal integrability of V it is enough to check t h a t space spanned over the last q fields is closed under the Lie bracket, which is equivalent to the vanishing of the Nijenhuis tensor. This ends t h e proof. • From t h a t proof we can see t h a t our foliation is given by the local submersions on pieces of complex manifolds in such a manner t h a t the transition diffeomorphisms are biholomorphisms, which means t h a t our foliation is transversely holomorphic. Conversely, if a foliation is transversely holomorphic then the normal bundle is canonically endowed with the integrable complex structure and, by Proposition 2.7, t h e bundle V is formally integrable. We summarize t h e above conclusion as C o r o l l a r y 2.8 The subbundle V is formally integrable if and only if T is transversely holomorphic. Remark.

In t h e dual description we consider the annihilator Q = V±

C T*M®RC

270

M. FRYDRYCH AND J. KALINA

of the " antiholomorphic" bundle V, which satisfies the conditions Q © Q = T ^ X ® R C C T*M®RC,

dT{Q) c ideal (T (Q ®Q)). It is straightforward, that formal integrability of V is equivalent to dT(Q) c ideal(r(Q)).

3

Formal integrability and main results

Using Proposition 2.7 from the previous section we are able to prove the following theorems. Theorem A Let M be a complex manifold of dimension m and T a smooth foliation of M by complex submanifolds of dimension p. Let further T M ® K C = U ®V

and TT®n.C = F © F , F c U be the decompositions in

the sense of Proposition 1.1. For the subbundle V = U © F the following conditions are equivalent:

(i) [r(v),r(v)]cr(n (ii) T is holomorphic, (iii) Bott connection is C linear. Proof. Let i/T®w>C = W ®W be the decomposition of the complexified normal bundle of (yT', J ) , where J denotes the induced complex structure on the quotient bundle vT. Now we prove that V = (ir®l)-1(W)

= U®F

so that we can apply Proposition 2.7. Let Z = X + iY e (TT <8> 1 ) - 1 ( W ) . where X,Y e TM, Z e TM® R C. Therefore n(X) + iir(Y) £ W which means that 7r(Y) = - J T T ( X ) .

(*)

Decompose Z in the following manner: (X + JY

.TX + JY\

, fX-JY

]

.TX-JY\

_

7

We observe that Z\ G U and Zi G U but, by (*), ir(Z2) = 0 and we conclude that Z2 G F which implies that Z G U © F. Thus we get the inclusion (TT<8>1)- 1 (W)

CU®F.

SOME REMARKS ON PARTIALLY HOLOMORPHIC FOLIATIONS

271

To prove the opposite inclusion, note that (ir<S>l)(F) = {0} C W and (TT 1)(U) C W since (TT 1)(X - UX) = ir(X) - iJw(X) e W, where X e T M . By Proposition 2.7 and the above considerations we get the equivalence between (i) and (iii). The equivalence (i) with (ii) is given by the following lemma. • Lemma 3.1 A tangentially holomorphic foliation T on a complex manifold M (i.e. such that leaves of T are complex submanifolds of M) is holomorphic if and only if it is transversally holomorphic. Proof If a foliation T on the complex manifold is holomorphic then it is in particular tangentially and transversally holomorphic. Now suppose that T is tangentially and transversally holomorphic. Let m 6 M be an arbitrary point in M, and ([/,
4>=(w,z),

w=(w\...,wp),

(z1,...,z"),

z =

be a complex chart on M around a point m, such that complex submanifolds {(w, z) £ U; w = const} are transverse to the given foliation T\U. Because the foliation T is tangentially holomorphic, we can define complex Informs of type (1,0) (i =dzi+\(w,zyadwa,

j =

l,...,q

such that

g=F ± =span{C 1 ,...,C 9 } for some complex functions \i€C°°(U,C),

a = l,...,p; j =

l,...,q

(see the Theorem A and Remark ). It is evident that (dw\...

,dwP, C\...,C9,

dw\...,duP,

C\... , f )

is a frame of T*M(g)RC|J/. Now, the fact that foliation T is transversely holomorphic implies that d< J 'e ideal ( C V . - . C ) , A simple computation shows that

j =

l,...,q.

272

M. FRYDRYCH AND J. KALINA

ice1,-

so dC,3 S ideal

,(q) implies dzl

= 0,

and Ti

dK

dK dw0

which means that functions A (z, w)i,

j = l,-

• •

, q ;

= 1 , . . . ,p,

are holomorphic. It follows that the annihilator T(Q) is generated by holomorphic 1-forms £•?', j = 1 , . . . ,q. Now the classical complex version of Frobenius theorem ends the proof. • Finally, we are in a position to prove the following. Theorem B Let M be a real manifold and V a subbundle of T M 0 i C such that V + V and V n V are formally integrable subbundles. Then there exists on M a flag of foliations (W, J-) and a canonical complex structure on THjTT, where TH®RC = V + V, Tf®RC = V r\V. Moreover, this complex structure is integrable if and only if V is formally integrable, which means that the foliation T is transversely holomorphic in Ji. Proof. Since V + V and V n V are invariant under complex conjugation, there exist distributions H and F in TM such that V + V = H®^ and VtlV = F®uC Standard algebraic observations give a sequence of natural isomorphisms H

F "

„

#<8>RC

V + V

F®RC

vnv

V

Vnv

V

Vnv

Proposition 1.1 in Section 1 gives a complex structure J on the quotient bundle H/F. Because of the formal integrability of V + V and V (~) V the distributions H and F are involutive, so actually determine the flag of foliations (H,!F). Theorem A gives us the last part of the statement. • We can regard Proposition 2.7 as a geometrical version of the classical Nirenberg theorem ([2], [3]). References 1. R. Bott, Lectures on characteristic classes and foliations, in Lecture Notes in Math. 279, Springer Verlag, 1972, 1-94.

S O M E REMARKS ON PARTIALLY HOLOMORPHIC FOLIATIONS

273

2. T. Duchamp and M. Kalka, Invariance of tangentially holomorphic foliations and Monge-Ampere equation, Mich. Math. J., 35, (1988), 91-115. 3. L. Nirenberg, A complex Frobenius theorem, in Seminars on analytic functions, I, Princeton, 1957, 172-189. 4. R.O. Wells, Differential analysis on complex manifolds, Springer Verlag, 1980.

Received December 18, 2000.

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Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawet WALCZAK et al. World Scientific, Singapore, 2002 pp. 275-295

FOLIATIONS A N D COMPACTLY G E N E R A T E D PSEUDOGROUPS ANDRE HAEFLIGER Section

de mathematiques, 2-4 Rue du Lievre, Geneve, e-mail: [email protected]. ch

Switzerland,

The holonomy pseudogroup of a foliation on a compact manifold satisfies a condition called "compact generation". Most properties enjoyed by the holonomy pseudogroup of a foliation on compact manifolds are also valid for compactly generated pseudogroups. In this talk we illustrate this principle by proving that a compactly generated pseudogroup of holomorphic transformations of a one-dimensional complex manifold has a finite number of closed orbits, unless all the orbits are closed, a property proved for transversely holomorphic foliations of codimension one on compact manifolds by M. Brunella and M. Nicolau [3].

It is well known that the dynamic properties of a foliation T on a manifold M are encoded in (the equivalence class) of its holonomy pseudogroup. Therefore it is natural to prove let say a theorem A in foliation theory, whose hypothesis and conclusion can be read on the holonomy pseudogroup, by proving first a theorem A' about pseudogroups satisfying suitable conditions reflecting the hypothesis of Theorem A, and then deduce Theorem A from Theorem A' (this is usually easy). As an example, Epstein, Millet and Tischler [5] proved that for a foliation T on a manifold M with a countable basis, the subset of M which is the union of leaves with trivial holonomy is a G& dense set (Theorem A). The hypothesis that M has a countable basis implies easily that the holonomy pseudogroup TioiT with respect to a complete transversal T contains an open set To with countable basis meeting all the orbits and such that the restriction of Ti to T> is countably generated. For such a pseudogroup Ji it is immediate that the set of orbits with trivial isotropy groups is a G$ dense set (Theorem A'). In turn, this implies immediately Theorem A. Sometimes the hypothesis of Theorem A involve conditions on the fundamental group or on the cohomology of M. This implies conditions on 275

276

ANDRE HAEFLIGER

the fundamental group or the cohomology of the holonomy pseudogroup H. of T. The notions of homotopy or cohomology for pseudogroups, or more generally for etale groupoids, have been introduced in Haefliger [11] and [12]; it is imperative that those notions depend only on the equivalence class of the pseudogroup. The holonomy pseudogroup of a foliation on a compact manifold satisfies a certain finiteness condition, called compact generation, which should be the hypothesis corresponding to the compactness of M in Theorem A'. In this talk we first recall the precise definitions of equivalence classes of pseudogroups and of compactly generated pseudogroups. In the second part we illustrate the above considerations in a particular case. Namely we first state a recent result of Brunella and Nicolau [3] about transversely holomorphic foliations of codimension one on compact manifolds and give the proof of this Theorem A. Then we state a corresponding Theorem A' concerning compactly generated pseudogroups of holomorphic transformations of a one-dimensional complex manifold. The translation of the proof of theorem A uses the notion of Cech cohomology for etale groupoids and we prove a finiteness theorem in 2.5 which is also at the heart of the proof of Theorem A. In the third section, we consider another etale groupoid attached to a foliation on a manifold M, its fundamental groupoid [1] (whose elements are the homotopy classes of paths in the leaves joining two points of a totally transversal submanifold). We exhibit a presentation for this groupoid and introduce the notion of compact presentation, reflecting conditions imposed on the fundamental groupoid by the compactness of the manifold M. The author thanks the referee for a constructive remark concerning the growth type of orbits. 1

Compactly generated pseudogroups

1.1

Pseudogroups and their equivalences

A pseudogroup of transformations {H, T) (or simply H) of a topological space T is a collection TC of homeomorphisms from open subsets of T to open subsets of T such that: 1) if h,h' £ 7i, then the composition0, hh' and the inverse h~l belong to 7i, a

T h e composition hh' of the homeomorphism h : U —> V with the homeomorphism h' :U' -> V is the homeomorphism h'~1(Ur\V) -> h(Ur\V) denned by x <-> h(h'(x)).

FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

277

2) t h e identity m a p of T belongs to H, 3) if a homeomorphism from an open set of T to an open set of T is locally in H, then it is in H. If T is a differentiable or a Riemannian manifold, then H is called a pseudogroup of differentiable or isometric transformations if all its elements are differentiable or are Riemannian isometries. T h e restriction of H to an open subset To of T is the set of elements of H with source (or domain) and target (or range) in To. T h e orbit H.x of a point x 6 T is the set of t h e images of x under the elements of H. T h e space of orbits, with the quotient topology, is noted H\T. For x e T , the isotropy group Hx is the group of germs at x of the elements h e H such t h a t h{x) = x. Two pseudogroups (Ho, To) and (Hi,T\) are equivalent if there is a pseudogroup (H,T) and homeomorphisms fi from T, onto open sets T[ of T , i = 0 , 1 , such t h a t 1) T'i meets all the orbits of W, 2) fi induces an isomorphism from Hi onto the restriction H\T' of H to T[. If Hi and H are pseudogroups of differentiable transformations, then Ho is differentiably equivalent to H\ if /o and f\ are diffeomorphisms. Remarks. Let (Ho, To) and (Hi,T\) be two equivalent pseudogroups. Then the spaces Ho\To and H\\Ti are homeomorphic. If T 0 carries an 'Ho-invariant measure finite on compact subsets, t h e n there is a corresponding Hi-invariant measure on T\. Examples. 1) Let To be an open subset of T meeting all the orbits of H. T h e n the restriction of H to T 0 is equivalent to H. 2) Let T 0 be t h e real line R and Ti be the circle R / Z . T h e pseudogroup of transformations Ho of R generated by the translation x >—> x+1 is equivalent to t h e pseudogroup of transformations Hi of R / Z generated by the identity m a p . To see this, consider t h e pseudogroup H of transformations of the union T of T 0 = R and Ti = R / Z generated by the integral translations of R and the restrictions of the projection R —> R / Z t o small open sets of R. T h e n a t u r a l inclusions fi of Ti in T induce isomorphisms from Hi to H\T{ • 1.2

The holonomy

pseudogroup

of a

foliation

A foliation T of codimension q on a manifold M can be given by a foliated cocycle over an open cover U = (Ui)iEi, namely submersions pi from Ui

278

ANDRE HAEFLIGER

onto open subsets Tj of Rq, with connected fibers, satisfying the following Compatibility condition: for every x £ C/j n Uj, there is an open neighbourhood U*j C Ui D Uj of x and a homeomorphism hfj from Pj(U^) to p»(C/y) such that pi = pj o /i?. on C/g. Note that hfj is uniquely denned and that for y £ t/£, we have h^ = /i^(see the remark below). The leaves of T are the connected components of M endowed with the topology having as a basis of open subsets the intersection of the fibers of the pi with the open subsets of M. A foliated cocycle {p'k}keK over another open cover W — {U'k}keK defines the same foliation T if the above compatibility condition is satisfied for the union of the two foliated cocycles. Let T be the disjoint union of the open sets T,; to simplify the notations we identify Tt with the corresponding open set of T. The holonomy pseudogroup of J- (with respect to the foliated cocycle {pi} ) is the pseudogroup of transformations (H, T) of T generated by the elements h*j. The holonomy pseudogroup of T associated to another compatible cocycle is equivalent to (TC,T). Therefore the holonomy pseudogroup of T is well defined only up to equivalence. By "the" holonomy pseudogroup of J7, we mean any pseudogroup equivalent to the holonomy pseudogroup associated to a foliated cocycle defining T. Note that if M is a differentiable manifold of class Cr and if the submersions Pi are of class Cr, the elements of the holonomy pseudogroup of T are differentiable of class Cr. Remark. It is well known that one can construct a foliated cocycle defining T such that the fibers of the submersions p, (called the plaques) are connected, that one plaque in Ui intersects at most one plaque in Uj and that the intersection of any two plaques is connected if it is non empty. In that case there is a unique homeomorphism hij : Pj(Ui C\Uj) —> Pi(Ui D Uj) such that pi = h^ opj on Ui n Uj. The holonomy pseudogroup of T is generated by the h^. There is a bijective natural correspondence between leaves of T and orbits of H, holonomy group of a leaf and isotropy subgroup of the corresponding orbit, closed minimal subsets of M saturated by leaves and closed minimal sets of T invariant by H, etc... There is a natural homeomorphism from the space of leaves to the space of orbits H\T. As the holonomy pseudogroup of a foliation is defined only up to equivalence, we shall respect the following rule. General Principle. We shall consider only notions or properties of a pseudogroup (H,T) which depend only on its equivalence class.

FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

1.3

279

Compact generation

Definition. A pseudogroup (H, T) is compactly generated if a) T is locally compact and contains a relatively compact open subset To meeting all the orbits of H, b) there is a finite set S — {si,... , Sk} of elements of Ji generating TL\Ta\ moreover each si : Vi —> W* is the restriction of an element s, e H whose domain Vi contains the closure of V,. Remarks. 1) By adding to S a finite number of elements, we can assume that S is closed under inverses, contains the identity of To and that the union T of the open sets s,(V; f~l To) contains To. It follows that the restrictions of the gi to T generate H,f. 2) In the definition above, it is understood that T is Hausdorff. If T is locally compact but not Hausdorff, one can find an equivalent pseudogroup (H',T'), where T" is Hausdorff, for instance the disjoint union of open Hausdorff subsets of T covering T. According to the above general principle, we have to check that compact generation depends only on the equivalence class of (H,T). This follows from the following more precise property. Lemma. Let (Ti, T) be a compactly generated pseudogroup as in the definition above. Let TQ C T be a relatively compact open subset meeting all the orbits. Then TL\T' satisfies the condition b) above. Proof. We can assume that the set 5 of generators of ~H\T0 is closed under inverses and contains the identity map of To. As TQ is relatively compact and TQ meets each orbit, one can find a finite set K of elements of H whose domains form an open cover of To, whose ranges are contained in TQ, each element k : V —> W of K being the restriction of an element k e TL defined on a neighbourhood of V. Interchanging the role of To and TQ, one find a finite set K' of elements of H verifying similar conditions. • It is easy to see that the finite set S' of homeomorphisms of the form ksk', (fcsfc')-1, k2sk^1, where s 6 S,k,ki,k2 € K,k' € K', generates Hyr^Examples. 1) The holonomy pseudogroup of a foliation on a closed manifold M is compactly generated. To see this, choose a foliated cycle {pi : Ui —> Tj}i e / defined over a finite cover of M and choose for each i £ / an open set U® whose closure is contained in Ui , such that the fibers of the restriction of Pi to U° are connected, and such that the U° form an open cover of M. Then the disjoint union To of the Pi{U°) is a relatively compact open set

280

A N D R E HAEFLIGER

in the disjoint union T of the Pi(Ui) = Ti meeting every orbit of H; the elements h^ restricted to To generate H\T0 and one can extract from this set a finite generating set. The same argument holds if M is compact and if T is either transverse or tangent to the boundary of M. 2) The pseudogroup of transformations TC of T — R generated by si : x >—> x/2,Vx, and by S2 : x H-> —X,VX ^ 0, is compactly generated. Indeed choose T0 =] - 1,+1[; then the restrictions of si and S2 to T0 generate 3) The pseudogroup of transformations of T = [-1, +1] generated b y m —x, Vrr ^ 0 is not compactly generated. 4) Let T be a subgroup of a simply connected Lie group G, and let H be the pseudogroup of transformations of G generated by the elements of T acting by left translations on G. This pseudogroup will be noted V tx G. If G is nilpotent, then T K G is compactly generated if and only if T is finitely generated and if T is cocompact, i.e. the quotient of G by the closure of T is compact. In that case r tx G is equivalent to the holonomy pseudogroup of a foliation on a closed manifold. When G is solvable, then r K G is compactly generated if T satisfies some arithmetic conditions. For instance, let G = Affi be the group of affine transformations of M preserving the orientation, and let fi : Aff i —> M* be the homomorphism associating to x >—> ax + b the number a. Then Gael Meigniez [17] has proved the remarkable theorem that r x G is compactly generated if and only if T is finitely generated and cocompact, and 1 is an integral combination of elements of /u(F)n ]0,1[ and also an integral combination of elements of /J,(T)C] ]1, oo[. As shown by Meigniez, many of those pseudogroups are holonomy pseudogroups of foliations on closed manifolds, but it is likely that it is not the case for all of them. In general it is very difficult to decide if T x G is compactly generated or not when G is non-compact and semi-simple. For example the case G = SL(2,R) is very mysterious, although many interesting compactly generated examples with T dense in G are described in Ghys, Gomez-Mont, Saludes [8] using arithmetic constructions. 5) Cavalier [4] has given a complete classification of equivalence classes of compactly generated pseudogroups of holomorphic transformations of a complex curve, preserving a non trivial meromorphic vector field. Growth type of orbits. Let (H, T) be a pseudogroup generated by a finite set S of elements. In general the type of the growth function of the orbit of a point x 6 T (associating to a positive integer n the number of images of

FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

281

x under composition of a t most n elements of S or their inverses) is not a n invariant of t h e equivalence class of Ji (see Example 2) in 1.1). Nevertheless for compactly generated pseudogroup (H,T), one can define the growth t y p e of orbits as the t y p e of the growth function of the corresponding orbit of H\T0 with respect to the finite generating set S satisfying the conditons of the R e m a r k 1) following the definition of compact generation. It is easy to see t h a t this definition does not depend on the choice of such generators and, by the lemma, is invariant under equivalences. T h e argument of Plante [19] shows t h a t , for a compactly generated pseudogroup (7i,T), the existence of an orbit with subexponential growth implies the existence of a measure on T invariant by Ti and finite on compact subsets. 1.4

Examples

of foliations

theorems versus

pseudogroups

1) T h e analogue of the Reeb stability theorem for compactly generated pseudogroups is the following. Let (7i, T) be a compactly generated pseudogroup. Let x € T be such that the orbit TC.x is closed and the isotropy group Tix is finite. Then there is a neighbourhood U of T and a finite group T of homeomorphisms of U such that the restriction of TL to U is generated by r . Moreover the map associating to an element of T its germ at x is an isomorphism. We prove this theorem as an illustration of the use of the condition of compact generation. We first observe t h a t the orbit of x is discrete (this is true for countably generated pseudogroups). We can t h e n choose an open relatively compact neighbourhood To of x meeting all the orbits of H such t h a t To n Ti.x — {x}. By the usual argument, we can find an arbitrarily small open neighbourhood U of x and a subgroup V of t h e group of homeomorphisms of U whose elements belong to Tt and such t h a t the m a p associating t o an element of T its germ at x is an isomorphism. Using the lemma in 1.3, we can choose a finite generating set S of HTO, stable under inverses, satisfying the condition b) in the definition of compact generation and U small enough such t h a t , for g € S, either U n domain g = 0 or U C domain g. In this last case, g has the same germ at x as an element of r . We can assume t h a t U is small enough so t h a t those two elements coincide on U. Let h e Ti.\u- We claim T. Indeed let y G domain in 5 such t h a t , on a small t h e conditions above, each 2) It has been proved by

t h a t h is locally the restriction of an element of h. By hypothesis there are elements gi, • • • ,gk neighbourhood of y, we have h = g\.. .gk- By gi is locally the restriction of an element of T. Edwards, Millet and Sullivan [6], generalizing

282

ANDRE HABFLIGER

arguments of Epstein, that for a foliation T of codimension 2 on a compact manifold M such that all the leaves are compact, then the leaves of T are the fibers of a generalized Seifert bundle. The corresponding statement for a compactly generated pseudogroup (T~t,T) of transformations of a 2-manifold T should be the following: if all orbits are closed, then (H, T) is the pseudogroup of change of charts of an orbifold structure on 7i\T. (I have not checked this statement.) 3) The following theorem is proved in Haefliger [10]. Let T be a transversely oriented foliation of codimension one on a connected paracompact manifold M. a) If the rank of the group Hom(7ri (M), Z) is finite, then the set of closed leaves is closed. b) If M is closed and T is real analytic, then either there is a finite number of compact leaves, or all the leaves are compact. The corresponding result for pseudogroups is the following. Let (7i, T) be a countably generated pseudogroup of orientation preserving transformations of a one -dimensional manifold T such that H\T is connected. a') If the rank o/Hom(7Ti(7Y),Z) is finite (where ni(7i) is defined in [11], see also [20] and [2]), then the set of closed orbits is closed. b') If (7i, T) is compactly generated and real analytic, either there is a finite number of closed orbits, or all orbits are closed. This statement implies immediately the corresponding statement for foliations because the fundamental group of M surjects onto the fundamental group of Tt. 2

Pseudogroups of holomorphic transformations of a complex curve

2.1 A theorem of Brunella and Nicolau The proof of the following theorem of Brunella and Nicolau [3] is an adaptation to the case of transversely holomorphic foliations of the proof of a theorem of Jouanoulou and Ghys on holomorphic foliations of codimension one (with possible singularities) on a compact complex manifold. For the real analytic case, see 3) in 1.4. Recall that a transversely holomorphic foliation of codimension one on a manifold M is given by a foliated cocycle {pi : [ / , — > € } such that the transition functions h^ are holomorphic.

283

FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

Theorem A. Let T be a transversely holomorphic foliation of codimension one on a compact connected manifold M. Then either there is finite number of compact leaves or all leaves are compact and there is a (non constant) transversely holomorphic meromorphic function constant on the leaves. Proof. [3] Consider the following short exact sequence of sheaves 0 -> O ^ -> f^log -> Vjr C -» 0. Here Qjr (resp. Q^-, ) is the sheaf of germs of transversely holomorphic 1-forms (resp. meromorphic 1-forms with polar singularities of order < 1) locally constant on the leaves. Restricted to J7j, such a form is locally the pull back by Pi of a 1-form f(z)dz where / is holomorphic (resp. meromorphic with poles of order < 1). The third sheaf is the sheaf of germs of functions locally constant on the leaves with support a finite number of plaques. The last map associates to a pole of the germ of a meromorphic 1-form its residue. Consider the associated long exact cohomology sequence . . . - H°(M,nJr i l o g ) - H°(M,Vr

® C) - • H\M,Q^)

-» . . .

The space H°(M, fi^r iog) of sections of fl^ j is the space of transversely meromorphic 1-forms locally constant on the leaves with polar singularities of order at most one. The space H°(M,V^ ® C) is a vector space with basis the compact leaves of J7. The main point of the proof is the fact that H1(M, fi^-) is finite dimensional (see Gomez-Mont [9]). Therefore if there are at least dim H1(M, Sl^) +2 compact leaves, one can find two linearly independent transversely meromorphic 1-forms constant on the leaves. Their quotient is a transversely meromorphic non-constant function which is constant on the leaves. Therefore all the leaves are compact. • The corresponding theorem for pseudogroups is the following. It is clear that it implies Theorem A. Theorem A ' . Let (W, T) be a compactly generated pseudogroup of holomorphic transformations of a one-dimensional complex manifold T such that 7i\T is connected. Then either there is a finite number of closed orbits, or all orbits are closed and there is a non-constant H-invariant meromorphic function onT. The rest of the section is devoted to the proof of this theorem which is just a translation in terms of pseudogroups of the above proof of Theorem A. This involves the definition of appropriate notions of cohomology with value in H-sheaves on T. It is more natural to define those notions in the framework of etale groupoids, and so we begin by recalling this notion.

284

2.2

ANDRE HAEFLIGER

Etale groupoids {G,T)

An etale groupoid (G,T) is a topological category Q with space of objects T, such that all the arrows (morphisms of the category) are invertible and such that the maps a : Q —> T and /3 : Q —> T associating to each arrow in G its source (initial object) and its target (terminal object) are etale maps, i.e. are locally homeomorphisms (see [12] and [2]). We shall often identify T with the space of units of GFor instance, a topological space T can be considered as an etale groupoid with space of units T and such that every arrow is a unit. A discrete group T can be considered as an etale groupoid with a single unit. To an etale groupoid (G,T) is associated a pseudogroup (H,T) whose elements are precisely the homeomorphisms from open sets U of T to open sets of T obtained by composing a section of a above U with /3. Conversely to each pseudogroup (7i, T) is associated an etale groupoid with space of units T, called an etale groupoid of germs; its set of arrows is the set of germs of elements of Ti with the usual germ topology, the projection a and p associating to a germ its source and target. The pseudogroup associated to it is again (7i,T). For this reason the notion of pseudogroups and the notion of etale groupoids of germs are equivalent notions, and we shall often use the same notation (H, T) for a pseudogroup or for its associated etale groupoid of germs. We now extend to etale groupoids the notion of compact generation. Definition. An etale groupoid (G,T) is compactly generated if a) T is Hausdorff locally compact and contains a relatively compact open set To meeting all the (/-orbits, b) there is an open set S c G generating the restriction G\T0 or" G t o ^o and contained in a finite number of compact subsets 6 . It is easy to check that a pseudogroup (H,T) is compactly generated if and only if its associated groupoid of germs is compactly generated. Remark. If G is Hausdorff (which is in general not the case, even if the associated groupoid of germs is Hausdorff), then S is relatively compact in Q. Localization. Let U = {Ui}iei be an open cover of T. Let Tu be the disjoint union of the L/j, i.e. the set of pairs (i,x) with x € C/j. The We use compact in the Bourbaki sense, i.e. for us a compact space is always assumed to be Hausdorff. In general the union of two compact subsets is compact if and only if this union is Hausdorff.

FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

285

localization (Gu,Tu) of (G,T) over U is the etale groupoid whose elements are the triple (J,g,i) with g € G,a-{g) G Ui,f3{g) € Uj. The projections a and /5 map (j,g,i) to (i,a(g)) and (j,(3(g)) respectively. The composition (k,g',j)(j,g,i) whenever defined is equal to (k,gg',i). Equivalence. Two etale groupoids (G,T) and (G',T') are equivalent if there exist open covers U of T and W of X" such that (Gu,Tu) is isomorphic to ( & „ ! £ , ) . Again it is an easy exercise to check that two pseudogroups are equivalent if and only if their associated etale groupoids are equivalent. Also an etale groupoid {G,T), with T Hausdorff, equivalent to a compactly generated groupoid, is also compactly generated. The analogue of the lemma in 1.3 is valid for compactly generated etale groupoids.

2.3

G sheaves

A C/-sheaf A is a sheaf of abelian groups on T with a continuous action of G'- for each g e G an isomorphism a — i > g.a from the stalk of A over a(g) to the stalk over /3(g), depending continuously on g and a. The ^-sheaves form naturally an abelian category (see Haefliger [12], Kumjian [16]). As before we are interested only in notions invariant by equivalence. If (G', T") is equivalent to (G, T), there is an equivalence between the category of (/-sheaves and the category of ^'-sheaves. For instance, if U = {J7j}ie/ is an open cover of T, the corresponding Gu-she&f is the sheaf Au whose stalk above (i,x), x € f/j, is (i,Ax). The action of (J,g,i) on (i,a) where a € •A.a(g-) is equal to (j,g.a). H°(G, A) will be the group of ^-invariant sections of A. Example. Let (G, T) be an etale groupoid of germs of analytic local automorphisms of a complex curve T. We can consider the short exact sequence of 5-sheaves

o -> n 1 -> n l g -> v c -• o where Q,1 ( resp. $ljog) is the <5-sheaf of germs of holomorphic 1-forms ( resp. meromorphic 1-forms with poles of order < 1) on T, and V ® C the sheaf of germs of complex valued functions with discrete support. The third homomorphism in the exact sequence associates to a pole of a meromorphic 1-form its residue. The action of G is the natural action by pull-back.

286

2.4

ANDRE HAEFLIGER

Cech Cohomology

For a CJ-sheaf A, we define the cochain complex [12] 0 - c ° ( g , A ) £ c \ g , A ) ^c2(g,A)^

...

Here C°(Q,A) is the group of sections of A and Ck{Q,A) is the group of continuous functions c associating to a sequence g 1,... , gk of composable elements of Q an element c(gi,... , <%) e Aprgi). The homomorphism 5° associates to a section c of A the 1-cochain S°c(g) = g.c(g) - c((3(g)). The homomorphism 51 : Cx{g,A) -» C2(G,A) is defined by &1c{gi,g2)) =9i-c(92) -c(gig2)

+c(gi).

Hk(Q, A) denote the k-th cohomology group of the above cochain complex. It is not invariant under equivalence. To remedy this, we pass to a limit. If U — {Ui\i^i is an open cover of T and V = {Vj}j C*(Qv,Av)- As usual, one checks that the homomorphism induced on cohomology is independent of the choice of r. So the next definition makes sense and is clearly invariant under equivalence. Definition. The Cech cohomology Hk(Q,A) Hk(g,A)

=

is defined as

\imHk(gu,Au)

where the limit is taken over the open covers U of T. In particular H°(G,A) = H°(g,A) is the group H°(g,A) of invariant sections of A. Note that, when the etale groupoid is the trivial groupoid equal to its space of units T, then our definition agrees with the usual definition of the Cech cohomology Hk(T,A). The proof of the following facts is like their proof in the particular case of the classical Cech cohomology. Facts. 1) A short exact sequence 0—> A ^> B —1 C —* 0 induces an exact sequence 0 -> H\g,A)

- • H°(G,B) -» H°(G,C) -> 1

H\g,A)

1

^H (G,B)^H (G,Q l

2) The natural map H (G,A) —> Hl(G,A) jff^T,^) = 0, it is bijective (Leray).

is always injective.

If

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FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

We prove only the last part of 2). We want to prove that, for every open cover U — {Ui}ieI of T, the homomorphism Hl{G,A) —> H1(Gu,Au) is surjective. Let z G Cl{Qu,Au) be a 1-cocycle; the value of z on {i,g,j) is of the form (i,a(i,g,j)), where a(i,g,j) e Ap(gy The cocycle condition means that, for composable elements (i,g,j),(j,g',k) € Qy, we have a(i, gg', k) = g.a(j, g', k) + a(i, g,j). We can consider a(i,lx,j) as a Cech 1-cocycle in Cl(U,A). By hypothesis it is a coboundary (because Hl{U,A) —> H1(T,A) = 0 is infective). Therefore for each i € / , there is a section li of A above Ui such that, for x £ Ui n Uj, we have a(i,lx,j) — U(x) — lj{x). The 1-cocycle z i(h9d)) = (ha{i,g,j)) - h{P(g)) + lj(a{g))) is cohomologous to z. For g = lx we have Z((i,lx,j)) = 0. We claim that Z((i,g,j)) depends only on g; this will imply that Z is the image of a 1-cocycle in Cl(Q,A). To check the claim, we note that, if a{g) G Uj n Uj> and [3(g) £ Ui, the cocycle condition implies

Z((i',g,j'))

= Z((i',l0{g),i))

+ Z((i,g,j)) + g.Z((j,la{g),j'))

=

Z((i,g,j)).

a Remark. There is a natural homomorphism from H*(G,A) to H*(Q,A), the cohomology groups defined in Haefliger [12] (isomorphic to the groups defined in Kumjian [16] and called Grothendieck cohomology groups). Like in the case of usual Cech cohomology for topological spaces, this homomorphism is bijective for * = 0,1 and injective for * = 2 2.5

A finiteness theorem

The proof of Theorem A' follows the proof of Theorem A. Using the short exact sequence given in 2.3 and the associated cohomology exact sequence (see 2.4), it remains to prove the following finiteness theorem applied to the 5-sheaf A = Q1. T h e o r e m . Let [G,T) be a holomorphic compactly generated etale groupoid and let A be the Q-sheaf of germs of holomorphic sections of a holomorphic vector bundle overT with a (holomorphic) action ofQ. Then dim H*(g, A) < oo for * = 0,1. Proof. Using the facts mentioned above, the proof is the same as the classical proof of the finiteness of the first cohomology group of a compact

288

ANDRE HAEFLIGER

complex manifold with coefficient in a locally free sheaf (see for instance paragraph 29 and Appendix B in the book of Foster [7] and the references therein). After localization, we can assume that T is the disjoint union of balls in C™. This implies that Hl{T, A) = 0. We can also assume that the vector bundle over T is trivial. Let To be an open relatively compact subset meeting all the orbits of Q, and choose an open set S in Q contained in the union of a finite number of compact subsets and generating G\T0 ( s e e the definition in 2.2). On the vector spaces C*(Q, A) or C*(G\To, A\T0), we consider the topology of the convergence on compact sets (they are Frechet spaces). The subspace of cocycles Z*(Q,A) is closed. Note that a 1-cocycle z € Z1(G^To, A\T0) is determined by its restriction to S, because z(9i92) = gi.z{g2) + z{gx). We have the following commutative diagram where the horizontal arrows are induced by the inclusion:

c°(g,A)^c°(glTo,AlTo) i i Z\g,A) - ^ Z\glTo,AlTo)

i

I

I

I

HHG,A)^H1(glT0,AlTo) 0 0 The second horizontal arrow p is a compact operator due to Montel theorem and the remark above. The last one is an isomorphism by the fact 2) proved in 2.4. Therefore

p®d0:c0(glTo,AlTo)®z1(g,A)^z1(glTo,AlT0) is surjective. A theorem of Schwartz implies that 6° = {p + 6°) - p has finite codimension, i.e H1(g,A) = Hl{{g\T0, -4|T 0 ) i s finite dimensional. • Remarks. 1) Let T be a transversely holomorphic foliation on a manifold M and let {H,T) be the groupoid of germs of its holonomy pseudogroup. Consider on M the sheaf Of of germs of transversely holomorphic vector fields on M and let 0 be the 7Y-sheaf of germs of holomorphic vector fields on T. It follows from 3.2 below that the group Hl(Ji.,Q) is naturally isomorphic to a subgroup of HX{M, 0 ^ ) . 2) Let (g, T) be a compactly generated holomorphic etale groupoid; this means that T is a complex manifold and that the associated pseudogroup

FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

289

(see 2.2) is made up of holomorphic transformations of T. Let 0 be the £-sheaf of germs of holomorphic vector fields on T. One should prove the existence of a germ of a complex space, whose Zariski tangent space is isomorphic to the finite dimensional vector space H1(G, 6 ) , which is versal for the germs of deformations of (G,T) in an appropriate sense [1] among holomorphic groupoids. 2.6

Holomorphic dynamic in dimension one

Let T be a transversely holomorphic foliation on a closed manifold M. E.Ghys, X. Gomez-Mont and J. Saludes [8] have constructed a decomposition of M in dynamically defined components analogous to the decomposition into Fatou and Julia sets for iteration of rational functions. They noticed that this decomposition comes from a decomposition of the holonomy pseudogroup of T and that their construction works for compactly generated pseudogroups of holomorphic transformations of a complex curve. We briefly describe part of their results in this framework. Let (H,T) be a compactly generated pseudogroup of local holomorphic automorphisms of a complex manifold T of dimension one. The complexification T ® C of the tangent bundle T of T splits as the direct sum T 1 ' 0 © T 0 , 1 of two complex line bundles on which H acts through the differential. The Beltrami differentials are the sections of the bundle T 1 ' 0 © T 0 ' 1 . Let C(T1'0) be the sheaf of germs of local sections a of r 1 , 0 with distributional derivatives locally in L2 such that da is an essentially bounded measurable Beltrami differential. The ^-invariant sections of C(T1,0) correspond to H-invariant vector fields a on T and the hypothesis of compact generation implies that the local flow <j)t generated by a is H-complete. This means that, for any x € T and s > 0, there is a partition 0 = to < ij < . . . < ifc = s, a sequence x = Xo, • • • , Xk of points of T and elements ho,... , /ife_i of H such that, for i = 0 , . . . , k — 1, the local flow (f>ti+1-ti(hi(xi)) is defined and is equal to Xi+\. The set of points x of T where the section a does not vanish is an open 7i-invariant subset U and Ti. preserves on U the parallelism x — i > (a(x),y/^la(x)). The ^-completeness of a implies that the restriction of Ti to U is complete in the sense of Salem [20] and one can apply the analogue of Molino theory to the complete pseudogroup 7i\u [20]. The Julia set of Ti is by definition the closed set of points x s T such that any ?i-invariant section of C(r 1,0 ) vanishes at x. Its complement is the Fatou set of Ti. The Fatou set is an H-invariant open set of T and its connected components are called Fatou components. For a Fatou component U, there are three exclusive cases for the restriction Ti^u to U :

290

ANDRE HAEFLIGER

1) H\u has closed orbits; it is equivalent to the pseudogroup generated by the identity map of the space of orbits, a compact Riemann surface minus a finite number of points. 2) H\u is equivalent to a pseudogroup generated by a subgroup T of the group G = C or Aff i acting by complex automorphisms on some strip {z e C | a < 9(z) < /?}, where —oo < a < (5 < +oo and the closure of T is one dimensional. 3) 7i\u is equivalent to the pseudogroup generated by a dense subgroup of the Lie group G — C or Affi acting by left translations on itself (the group Aff i has a left invariant complex structure isomorphic to the upper halfspace). 3 3.1

The fundamental groupoid and compact presentation The fundamental (or monodromy) groupoid of a foliation

Let T be a foliation of codimension q on a manifold M and let T be a total transversal to JF, i.e. a submanifold of M of dimension q transverse to the leaves and cutting every leaf. The fundamental groupoid (Q, T) of T relative to T is the etale groupoid defined as follows [1]. The set of elements of Q with source a point x € T and target a point y e T is non empty if and only if x and y are in a same leaf L; it is the set of homotopy classes in L of continuous paths c : [0,1] —> L joining y to x. Composition and inverse in Q are induced by the corresponding operations for paths. A basis of open sets for the topology on Q is obtained as follows. Let U be an open subset of T and let c : U x [0,1] —> M be a continuous map such that, for each u G U, the map cu : [0,1] —> M sending t to c(u, t) is a path in a leaf joining u to a point of T; then the set of homotopy classes in the leaves of the paths cu is an open set of the basis. Let (TC,T) be the associated etale groupoid of germs (cf. 2.2). It is equivalent to the holonomy groupoid of T and is called the holonomy groupoid of T associated to the transversal T. The kernel of the natural projection from Q to TC is a sheaf of groups on T. Its stalk above x € T is the fundamental group of the holonomy covering of the leaf through x. (The fundamental groupoid as defined above is the restriction to T of the Lie groupoid called by several people the monodromy groupoid of J-.) The fundamental groupoid plays the leading role in the study of germs of deformations of foliations (cf. Bonatti-Haefliger [1]). We now describe a presentation of the fundamental groupoid (G, T) of T. We can always choose a foliated cocycle {pi : Ui —> Ti}i6.r for T satisfying the properties mentionned in the remark in 1.2. Moreover we can assume

FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

291

that the fibers (plaques) of each Pi are simply connected and that there is an embedding Qj : Tj —> Ui such that piqi is the identity of Tj and such that the qi(Ti) are disjoint. Recall that there is a unique homeomorphism hij from Pj(U n Uj) to Pi(Ui n C/j) such that Pi(x) = hij(pj(x)) for each x G UiPiUj. To simplify the notations, we identify Tj to its image <7i(Tj) so that Pi can be considered as a retraction from Ui to Tj and the union T of the Tj as a total transversal to T. Let (G, T) be the fundamental groupoid of J- with respect to the transversal T. We now describe a set S of generators gij(y) G Q for (5,T) , where i,j G 7 and y G pj-(C/s D Uj). The element ^(2/) is the homotopy class of a path in the simply connected subset p~1(y) U p"1(hji(y)) of the leaf through y from hij(y) to y. Each element of Q can be expressed as a composition of elements of S. For x G UitlUj C\Uk, we have the relation 9ik(Pk{x)) =

gij{pj(x))gjk{pk{x)).

The set i? of all such relations is complete, i.e. two words in the elements of S represent the same element of Q if and only one can pass from one to the other by using the relations in R (see Higgins [15]). This follows from the fact that the nerve of the open cover of a leaf L by the plaques contained in it has the same 1-homotopy type as L. For a general discussion on presentations of etale groupoids, see 3.3. 3.2

Relations with the cohomology of M

As proved in Haefiiger [13], the fundamental group of (Q, T) is isomorphic to the fundamental group of M. In this section, we indicate a similar relation between first cohomology groups. Let T be a foliation on a manifold M given by a foliated cocycle Pi : Ui —> T over an open cover U = {[/jjjg/ of M like in the preceding section and let {Q,T) be the associated fundamental groupoid. Consider M as a trivial etale groupoid (Ai, M) with space of units M, where Ai is the space of germs of the identity map of M. Let {Mu,Mu) be its localization over U. There is a continuous homomorphism p : (Mu,Mu) —> (Q,T) mapping (i,lx,j) G Mu, where x e UiHUj, to 9ij(pj(x)) G Mu- Let A be a Q-sheai over T; its pull back p* A is a .M^-sheaf over the disjoint union Mu of the Ui. The stalk of p*A over (i, cr) G Mu is the group of triples (i, x, a), where x G Ui, a e APi(x). The action of (j, lx,i) € M u on (i,x,a) is given by (j, Is, «)•(*) x i ° ) =

(i,x,gji(j>i(x)).a).

292

ANDRE HAEFLIGER

In fact this sheaf is obtained by localization over U of a sheaf Aj= on M. The homomorphism p induces a natural homomorphism of chain complexes p*

:C*(g,A)-+C*(M,p*A)

mapping a cochain c to the cochain p*c defined by (p*c)((io> lx, h), (h, U, h), •••) = (io, x, c(gioil (ph (x)),gili2 (pl2 (x)), . . . ) ) • Note that H*(Mu,P*A) = H*(M,A?), the usual Cech cohomology. Proposition. The homomorphisms H*{g,A)-^H*(M,Ar) induced by p are bijective for * = 0,1. This follows from the following lemma. Lemma. In the commutative diagram below the horizontal homomorphisms induced by p C°(g,A)-^C°(Mu,P*A)

I

i Z\g,A)

^Z\Mu,p*A)

are isomorphisms. Proof. The first horizontal homomorphism is an isomorphism because the restriction of p*A to a plaque is a constant sheaf and each plaque is connected. To check the injectivity of the second homomorphism, let z € Z1(Q, A) be such that p*(z) = 0. This implies that 0 = p*(z)({i,lx,j))

=

(hzigijipjix))),

hence z = 0, because a 1-cocycle which vanish on the generators is zero. To prove the surjectivity, we note that a cochain c G Cl(Mu,P*A) associates to (z, lx, j) an element of p*A of the form (i, x, Q>ij(pj{x))), where dij is a section of A above Pj(Ui n Uj), because the intersection of two plaques is connected. If c is a cocycle, then 0 = Sc((i,lx,j),(j,lx,k))

= {i,lx,j).c((j,lx,k))-c((i,lx,k)

+

c((i,lx,j)),

i.e aikiPi{x)) = glj(pj(x)).ajk(pk(x))

+

aij(pj(x)).

1

A cocycle z 6 Z (g,A) such that p*(z) = c is defined as follows. For any g £ Q, we can find a sequence of indices io, • • • ,ik a n d a sequence of points xi,... ,Xk with Xj € Uij_1 n C/j. such that 9 = fftoii (Pii (xi))...

gi^lik

{pik

(xk))-

FOLIATIONS AND COMPACTLY GENERATED PSEUDOGROUPS

293

We define z(g) =

ai 0 ii(Pu(zi)) + . . . + • • • 9ik-2ik-i

(Pik-! (xk)).aik_ltk

gloil(Pii(xi)) (pik ( x f c ) ) .

This definition is independent of the choice of generators in the expression of g thanks to the relations. 3.3

Compactly presented etale groupoids

We first extend to etale groupoids the construction of groupoids given in Higgins [15] using a presentation by generators and relations. Given a topological space T, we consider a topological space 5, called an alphabet over T, endowed with two etale maps a : S —> T and (3 : S —> T. We also consider a topological space S' disjoint from S with a homeomorphism s 1—> (s)' from 5 to S' and two etale projections a, 0 : S' -4 T defined by a((s)') = p(s) and /3((s)') = a{s). For (s)' e 5 ' we also define ((s)')' = s. Let WW be the union of S and S', endowed with the union of the projections a and (3 to T. For a positive integer n, we consider the topological space W(") = W^1) Xj- . . . Xj- W^ of sequences ( s i , . . . , s„) (also noted s i . . . s n ) of n elements of W^ such that /?(st+i) = a(si). The maps a : W^ —> T (resp. (3 : W( n ) —> T) associating to such a sequence the point a(sn) (resp. /3(si)) are etale maps. A point in W(") is called a word of length n in the alphabet S. We also consider the space W^ of empty words homeomorphic to T, the point x E T corresponding to the empty word { } x at x, the projections a and (3 from W^ mapping the empty word at x to x. The space of words in the alphabet S is the union Ws of the W(n\ n > 0, endowed with the two etale projections a and (3 to T which are the union of the corresponding projections from the W^n>. We have a continuous etale map WW xTW(m) -> W ( n + m ) associating to (wi,w 2 ) with a(wi) = P{vj2) the word W1W2 obtained by juxtaposition. W^ acts by the identity. With this partial multiplication Ws is a topological category. If <j> is an etale map from S to an etale groupoid (Q,T) such that a

5 which induces the identity on T. Let i? C Ws be an open subset such that for each r € Ws we have a(r) = /3(r) = xr. If for each r £ R we have <3>(r) = lXr, we say that the relations R hold in 5 under . Given a word w = si... sn 6 W("), where n > 2 and such that for some fc < n we have Sfc+i = (sjt)', we get a word in W^n~2^ by suppressing in w the elements Sk and s/c+i. This operation generates in Ws an equivalence relation which is open. The quotient of Ws by this equivalence relation is

294

ANDRE HAEFLIGER

an etale groupoid (Qs,T), called the free etale groupoid over S. Any etale map from S to an etale groupoid (G,T) such that a(f> = a and P4> = P extends uniquely to a continuous homomorphism from (Gs,T) to (G,T) projecting on the identity of T. If $(S) generates Q and if the kernel of $ is the normal subgroupoid N generated by R, then Q is isomorphic to the quotient Gs/N and (S,4>,R) is called a presentation of G- Note that the quotient Gs/N is naturally an etale groupoid because N is open in G; it will also be noted {S;R}. We indicate a tentative definition of the notion of compact presentation for an etale groupoid. Compact presentation. An etale groupoid (G, T) is said to be compactly presented if: a) T is Hausdorff locally compact and contains a relatively compact open set To meeting all the (/-orbits. b) There is an open subset S C G generating the restriction G\T0 °f G t ° T0 and contained in the union of a finite number of compact subsets. Let S be an open set containing this union. c) In the space W$ of words in the alphabet S, identified to an open subset of Wg, there is an open set R which is contained in the union of a finite number of compact sets in Wg, and such that (G,T) is isomorphic to (S;R). One checks as in 1.3 that the fundamental groupoid of a foliation on a compact manifold is compactly presented, and that the notion of compact presentation depends only on the equivalence class. A test for this definition would be to prove the analogue of the global stability theorem of Reeb for compactly presented groupoid of dimension one. References 1. C. Bonatti and A. Haefliger, Deformations de feuilletages, Topology 29 (1990), 205-229. 2. M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften 319, Springer Verlag, 1999. 3. M. Brunella and M. Nicolau, Sur les hypersurf aces solutions des equations de Pfaff, C. R. Acad. Sc. Paris, 329 (1999), 793-795. 4. V. Cavalier, Pseudogroupes complexes quasi parallelises de dimension un, Ann. Inst. Fourier (Grenoble) 44 (1994), 1539-1565.

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5. D.B.A. Epstein, K.C. Millet and D. Tischler, Leaves without holonomy, J. London Math. Soc. 16 (1977), 548-552. 6. R. Edwards, K. Millett and D. Sullivan, Foliations with all leaves compact, Topology 16 (1997), 13-32. 7. 0 . Forster, Riemannsche Flachen, Heidelberger Taschenbiicher Band 184, Springer Verlag, 1977. 8. E. Ghys, X. Gomez-Mont and J. Saludes, Fatou and Julia components of transversely holomorphic foliations, preprint ENS-Lyon (2000). 9. X. Gomez-Mont, Transversal holomorphic structures, J. Diff. Geom. 15 (1980), 161-185. 10. A. Haefliger, Varietes feuilletees, Ann. Scuola Norm. Sup. Pisa 16 (1962), 367-397. 11. A. Haefliger, Homotopy and Integrability, in Manifolds, Amsterdam 1970, Lecture Notes in Math. 197, Springer Verlag, 1971, 133-163. 12. A. Haefliger, Differentiable Cohomology, in Differential Topology (Varenna 76), Liguori, Naples, 1979, 19-70. 13. A. Haefliger, Groupoides d'holonomie et classifiants, in Structure transverse des feuilletages, Toulouse 1982, Asterisque 116 (1984), 70-97. 14. A. Haefliger, Pseudogroups of local isometries, in Differential Geometry, Santiago de Compostela, Pitman Res. Notes in Math. 131, 1994, 174-197. 15. P.J. Higgins, Presentations of groupoids and applications to groups, Proc. Camb. Phil. Soc. 60 (1964), 7-20. 16. A. Kumjian, On equivariant sheaf cohomology and elementary C*bundles, J. of Operator Algebra 20 (1998), 207-240. 17. G. Meigniez, Holonomy groups of solvable foliations, in Integrable systems and foliations, Progress in Mathematics 145, Birkhauser 1997, 107-146. 18. P. Molino, Riemannian Foliations, Progress in Mathematics 73, Birkkhauser 1988. 19. J.P. Plante, Foliations with measure preserving holonomy, Ann. of Math. 102 (1975), 327-361. 20. E. Salem, Riemannian foliations and pseudogroups, in P. Molino, Riemannian foliations, Progress in Mathematics 73, Birkkhauser 1988, 265-296.

Received

November

16, 2000, revised December

28,

2000.

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Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2002 pp. 297-314

TRACES A N D INVARIANTS FOR NON-COMPACT MANIFOLDS

J A M E S L. H E I T S C H Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan Street (M/C 249), Chicago, Illinois 60607-7045, USA, e-mail: [email protected] We define a general trace for complete Riemannian manifolds of bounded geometry. This trace takes values in the Haefiiger functions of the manifold and its existence allows the extension of the classical theory of invariants for compact manifolds to complete Riemannian manifolds of bounded geometry.

1

Introduction

T h e heat equation method for invariants of manifolds is now quite well developed and classical results (index theory, Lefschetz theory, Betti numbers, Novikov-Shubin invariants, torsion, . . . ) can be extended to new situations (foliations, non-compact manifolds) provided t h a t an appropriate trace function can be defined, and t h a t the asymptotics of the heat operator as t —» oo (which is in general a global problem) can be handled. T h e asymptotics of t h e heat operator as t —> 0 present no problem as these are local. T h e L2 Index Theorem of Atiyah [1] is a classical example of this sort of extension. T h e case considered there is a covering of a compact manifold and an operator lifted to the cover from the compact base. T h e trace is given by integration over a fundamental domain of the covering. In [18], J o h n Roe provided an extension t o non compact manifolds which admit a regular exhaustion. T h e exhaustion allows one t o define a trace by averaging over the manifold, much as we do in the final section of this paper. Alain Connes extended the heat equation method to foliations which admit a transverse invariant measure. See [10] and the references therein. Here the trace is given by integration over the plaques of a cover by foliation 297

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JAMES L. HEITSCH

charts with the resulting function being integrated against t h e transverse measure t o obtain a real number. In [12] and [14] we showed how to remove the requirement t h a t the foliation admit an invariant transverse measure and obtained a general families index theorem. T h e trace in this case is given by integrating over the plaques and taking the Haefliger cohomology class of the resulting differential form on the transverse space. In this paper we define a general trace (the Haefliger trace) for any complete Riemannian manifold M of bounded geometry. This trace takes values in Ch(M) the space of Haefliger functions of M, which is a quotient space of t h e space of functions on a maximal uniformly separated discrete subset. T h e basic idea is to abstract the theory for foliations worked out in [13] to t h e case of a single leaf. T h e main innovation in this paper is the introduction of the Haefliger trace taking values in the space of Haefliger functions. As an example of the use of this trace, we outline a proof of a Lefschetz theorem for Dirac operators defined on any complete Riemannian manifold of bounded geometry. We note here t h a t Ch(M) agrees with the uniformly finite coarse homology group HQ ( M ) defined by Block and Weinberger, [7], and t h a t the application of uniformly finite coarse homology to index theory was anticipated in section 9 of [8]. T h e main idea of this paper is t h a t there is a naturally defined geometric trace taking values in Ch(M), and given this, the geometric arguments of [13] and [14] are readily translated to this setting, giving index theorems, Lefschetz theorems, etc. This paper may be seen as alternate approach to invariants for non-compact manifolds which provides immediate access to results not readily obtainable from the BlockWeinberger view: e.g. Lefschetz Theory.

2

Haefliger functions and the trace

In this section, we define the space of Haefliger functions for a complete Riemannian manifold of bounded geometry and a trace which takes values in this space of functions.

2.1

Haefliger

functions

Denote by M a complete Riemannian manifold of bounded geometry. This class of manifolds includes all compact manifolds and their covers as well as any leaf (and its covers) of any foliation of a compact manifold. There are also examples of such manifolds which cannot be a leaf of a foliation of any compact manifold, [6], [16].

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299

Let r > 0. A lattice T for M is cover by open balls of radius r whose centres are at least a distance r apart. The bounded geometry of M then implies that there is an integer k so that any element of T intersects at most k other elements of T non-trivially. To construct a lattice, let A C M be a maximal set of distinct points which are separated by a distance of at least r. Zorn's lemma implies that such a set must exist. Then let V be the set of balls of radius r whose centres are the elements of A. For any lattice V set C°(r) = {/ : T -> R | / is bounded}. The space of Haefliger functions on M is a quotient space of this space. Denote by d(-, •) the distance function on M. A permutation n of F is bounded if sup d(U,ir(U)) < oo. t/er The group G of all bounded permutations acts on C°(T) by composition, that is

(**/)(E0 = /(^(CO). Set Hr = linear span of {/ - it* f \ f € C°(T), n £ G}. The space of Haefliger functions Ch(M) for M is the space Ch(M)

= C°(r)/Wr,

where Tir is the sup norm closure of Hr- Denote the class of / in Ch{M) by [/]. Then [/] — 0 provided that for all e > 0 there is a representative / i G [/] so that sup \fi(U)\ < e. uer Note that if f(U) = ~f{rr{U)) for all U € f C T, and f n i r ( f ) = 0, then we may set f(U) = 0 for all U e T U 7r(r) without changing the Haefliger class of the function / . To see this, define g(U) = f(U) for U £ T and #(£/) = 0 otherwise. Then [g — n*g] = 0, and / i = / — (g — it*g) satisfies fi(U) = 0 for all U e f U 7r(f), and /i(C7) = /([/) otherwise. A priori, Ch{M) depends on the choice of lattice T. However, this is not the case. T h e o r e m 1 Ch{M) does not depend on Y.

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JAMES L. HEITSCH

Proof. Let Ti and T2 be two lattices for M and set T = Ti U T 2 . Now r may not be a lattice as the centres of distinct balls may be arbitrarily close, but there is still a k so that each element of T meets at most k other elements of T. It addition, C°(T)/Tir still makes sense. With these observations, it is an easy exercise to show C°(F1)/Hri

=* C°(T)/nr

* C0(r2)/Wr2.

• Similarly, one can show that Ch(M) does not depend on r. 2.2

The trace

Given a bounded measurable function g on M and any partition of unity * = {^u} subordinate to the cover of M given by the elements of the lattice r , define J g € C°(T) to be the function

J d){U)=

j^u(x)g(x)dx

where dx is the volume form on M. It follows easily from the definitions that the Haefliger function [J g] does not depend on * . For any bundle E, denote by C(E) the bounded measurable sections of E, by C°°(E) the bounded smooth sections, and by Co°(.E) the smooth sections with compact support. Denote by E E* the bundle n*(E) TT^E*) over M x M, where 7Ti,7r2 : M x M —> M are the two projections. For any k € C(E E*), k(x,x) is a linear map on Ex and so has a well defined trace tr(fc(x, x)) which is a bounded measurable function on M. We define the Haefliger trace Tr{k) of k to be the Haefliger class of the function /

tr(fc), that is Tr(fc)

|tr(/c)

(If M is compact, any k e C(E ® E*) defines an operator on C(E) by (ks)(x) = /

k(xJy)s{y)dy,

M

and the Haefliger trace of k is the usual trace of k as in this case, Ch(M)= H) Denote by Cf>{E E*) the set of k € C°°(E E*) so that there is s > 0 with k(x, y) = 0 for all (x, y) with d(x, y) > s. In addition we require that k and all its derivatives be bounded on M x M. Any k e Cf(E®E*)

T R A C E S AND INVARIANTS FOR NON-COMPACT MANIFOLDS

301

defines a bounded smoothing operator on L2(E) with finite propagation speed and ti(k(x,x)) is a bounded smooth function on M. An element k G C°°(E ® E*) is basic if there are f C T and n e G so that for all Ui ^ U2 G f, Ui n U2 = 0 and TT(L/I) n TT(C/2) = 0 and sup(fc) C Up U x TT(C/).

Note that if k is basic, then sup(/c) n (U x 7r(t/)) is a compact subset of the open set £/ x 7r([/). P r o p o s i t i o n 2 ylm/ fc G Cf>(E ® £*) may 6e written as a finite sum k = k\ + • • • + kg where each ki is basic. Proof. Choose s > 0 so that k(x, y) = 0 for all x and y with d(x, y) > s. Because of the bounded geometry of M and the fact that the centres of the elements of Y are at least a distance r apart, F can be partitioned into a finite number of sets, T i , . . . T n , so that for all distinct U\,U2 € 1^, £ = l,...n, d(Ui,U2) > 4r + 2s. This of course implies that Ui n U2 = 0. If ([/i x U3) n sup(fc) 7^ 0, then d{U\, U3) < s so it further guarantees that if (Vi x 1/3) n sup(fc) ^ 0 and (U2 x U4) n sup(fc) ^ 0 then U3 n C/4 = 0. This condition guarantees that the elements irtj £ G chosen below actually exist. Again by the bounded geometry of M and the fact that the centres of the elements of Y are at least a distance r apart, there is an integer p so that for each U € Y there are at most p elements of Y whose product with U intersects sup(fc) non-trivially. For a given U, denote these sets by Ui,... ,Up. (If there are fewer than p such sets, we complete the collection to a set of p elements by adding the empty set the required number jDf times). For i = 1 , . . . , n, j = 1 , . . . ,p, choose Ttij € G so that TT1J(U) = Uj for all U € IV Let {i/^} be a partition of unity subordinate to the cover given by Y and set

hi(x,y)

= Y^ V'l/(a;)V'7ri,j(t/)(y)fc(a;.2/)asf ;

Then each kij is basic and k = J2i, ^i,j • Let fci(x,y) G C{E £*), and k2(x,y) ki°k2£ C{E ® E*) by fci ok2(x,y)

= /

G Cf{E

^ E*), and define

ki(x,z)k2(z,y)dz,

M

and similarly for k2o k\. We have immediately P r o p o s i t i o n 3 ki o k2 and k2 o ki are elements of C(E ® E*).

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JAMES L. HEITSCH

We now come to the central result of the paper, namely the fact that TV satisfies the following trace property. It is this property which allows us to extend classical results from compact manifolds to complete manifolds of bounded geometry. Theorem 4 Suppose h(x,y) G C(E ® E*), and k2{x,y) G C™(E
Tr(k2oki).

Proof. By Proposition 2 we may assume that k2 is basic with corresponding 7r,r. Choose a partition of unity {ipv} subordinate to F so that for all U £ f, W x W E / ) I sup(fc2)n(t/x7r(C/)) = 1- Then the Haefliger function Tr(ki o k2) is represented by the function / € C°(T) whose value at C/GTis /([/) = / tpu(x)tr( / ki{x,y)k2(y,x)dy) u M = / U

dx

tv(ki(x,y)k2(y,x))dydx M

which, because k2 is basic, equals /

/ tr (ki(x,y)k2{y,x))

dy dx =

U TV(U)

I

/ tr (k2(y,x)k1(x,y))

dxdy

7r(C/) U

=

/ TT(U)

tv(k2(y,x)ki(x,y))dxdy M

again because k2 is basic, and this equals ipn(U)(y) / •n(U)

ti{k2(y,x)ki(x,y))dxdy.

M

This last integral is the value at TT(U) of the function g € C°(T) which represents Tr(k2oki), i.e. it is (Tr~1)*(g)(U). But g and (7r_1)*(g) represent the same Haefliger function. Thus Tr(fci o k2) = Tr(A;2 o k\). D 3

Dirac complexes and Lefschetz functions

The Haefliger trace adapts extremely well to any situation where the heat equation method is used. As an example of this, we will outline a proof of

T R A C E S AND INVARIANTS FOR NON-COMPACT MANIFOLDS

303

a Lefschetz theorem for complete Riemannian manifolds of bounded geometry. This of course includes various index theorems as special cases. Let M be a complete Riemannian manifold of bounded geometry. Fix a lattice T for M. We assume that each bundle over M comes equipped with a metric, and if the bundle is a complex bundle, that the metric is Hermitian. Unless otherwise stated, all objects considered here (e.g. metrics, connections, bundle maps, . . . ) are assumed to be bounded in the sense that there are global bounds for the local expressions of the object and all its derivatives is any local orthonormal framing of the relevant bundle. 3.1

Dirac complexes

A Dirac complex (E, d) on M consists of the following 1. E — (EQ, EI, ... ,Ei), a family of smooth finite dimensional complex vector bundles over M. 2. d = (do, d\,... , de-i), a family of differential operators where d% : C?(Ei)

-+ C™(El+1),

and di+\ di = 0. 3. Set E = ®iEi and denote the adjoint of di by d*. We assume that E is a Clifford bundle over the Clifford algebra of T*M, and that there is an Hermitian connection V on E, compatible with Clifford multiplication, so that the operator D = © ( ^ + dU)

• C^{E)

->

C^{E)

is given by the composition C^(E)

^ C^{T*M

®E)™

C?(E)

where m is Clifford multiplication. All the classical complexes (de Rham, Signature, Dolbeault, and Spin) give rise to Dirac complexes on M provided M supports the necessary geometric structures for these complexes to be defined. Denote the space of L2 sections of Ei by L2(Ei). The operators di and d*_t extend to densely defined unbounded operators dk : L2(Ei) - L2(Ei+l)

and

d*_x : L2(Ei) - •

As usual with possibly non-compact manifolds, we define H\M,d)

= ker(d l )/(imd i _i ndomdj).

L2(E%.X).

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J A M E S L. HEITSCH

It is necessary to mod out by imdj_i D dom di, the closure of the image of di-i in the domain of di, as imdj_i n domdj is in general not a closed subspace. Define A, = di.! d*_, + d* di : CF(Ei) -

C?(Ei).

The operator D = ©(d, © d*_x) is a Dirac operator and the fact that M is complete implies that D is essentially self adjoint [9]. Thus D2 = ©Aj is also essentially self adjoint, and we can apply the spectral mapping theorem to each of the A*. Hodge Theory extends to complete manifolds and we have the following facts: • L2(Ei) = ker A» © imaged»_i © imaged*. • The natural map ker A, —> Hl(E,d) 3.2

Geometric

is an isomorphism.

endomorphisms

An endomorphism T = (To, • • • , Tg) of a Dirac complex (E, d) is a collection of complex linear maps T% : C°°(£,) -> C°°(Ei) so that diTi = Tj + idj. The most interesting of these are the so-called geometric endomorphisms [2] defined as follows. Let / : M —> M be a smooth diffeomorphism of bounded distortion. For i = 0 , . . . ,£, let Ai : f*(Ei) —> Ei be a smooth bounded bundle map. For s € C°°(Ei) define (TiS)(x) =

Ai,x(s(f(x))).

We assume that the Ai are chosen so that T = (To,... ,Ti) defines an endomorphism of (E, d). Example. Let (E, d) be the de Rham complex of M and / : M —> M a diffeomorphism of bounded distortion. For Ai take the ith exterior power of the dual df* of the differential df. Then for each i, Ti is just the map / * on the differential i forms on M. As diTi = Ti+1du T induces a well defined map T* : Hl(E,d) -> l H (E,d). We would like our Lefschetz theorem to relate ^ ( - P ) t r T* to indices defined on the fixed point sets of / . In general, Hl(E,d) is not finite dimensional, so the classical trace does not make sense. To get our theorem, we must recast T* in terms of an operator on L2(Ei) in order to apply the more general Haefliger trace.

T R A C E S AND INVARIANTS FOR NON-COMPACT MANIFOLDS

305

Denote by Pi : L2(Ei) —> ker(Aj) the projection, and define T* = PiTiPi : L2{Ei) -> ker(A0 C L 2 ( ^ ) We also denote by T* the map PjTjPj o j : ker(Aj) —> ker(Ai) where j : ker(A;) —> L2{Ei) is the injection. It follows from results of [13] that the Haefliger trace Tr (T*) exists and is finite. Definition 5 The Lefschetz function L(T) of the geometric endomorphism T is the Haefliger function for M given by e

L(T) = £(-irn(ir). i=0

That this is a reasonable definition is a result of the fact that the diagram ker Aj Hl{E,d)

*

-

ker A*

-

H^E^d)

commutes. 4

Fixed point indices and the Lefschetz theorem

In this section, we describe the restrictions we place on the fixed point set N oi f and state our Lefschetz theorem. We assume that AT is a disjoint union of closed submanifolds ./V = UiVj of M. As they are disjoint and closed, each Ni admits a positive function ti{x) so that K(Ni)

= {x£M\

d(x, Nt) < ei(x)}

is an embedded normal disc bundle in M and we may assume that these disc bundles are pairwise disjoint. We further assume that / is non-degenerate on N. This means that /» restricted to the normal bundle of N in M does not have 1 as an eigen value at any point in TV, i.e. / has no transverse invariant directions. Note that the identity map IM of M satisfies this condition. Suppose that a is a smooth volume form on TV, and denote its restriction to TV; by di. We define the Haefliger function JN a as follows. Let {Vv} be a partition of unity subordinate to the cover by the elements of the lattice

306

JAMES L. HEITSCH

r . Then JN a = [g] where g is the function on T

i

NiHU

The fact that only a finite number of Ni n U ^ 0 follows from the facts that U is compact, /|AT = IN and that the Ni are disjoint and closed. Theorem 6 (Lefschetz Theorem) Let M, (E, d),f,A andT be as above. Associated to N is a smooth volume form a which depends only on f, A, the symbols of the Aj, the metrics and their derivatives to a finite order only on N, so that in Ch.(M)

L(T) = J a. N

For the classical complexes and T = f* acting on forms, a is given by the classical local integrand of the Atiyah-Singer G Index Theorem [4]. So, if M is a compact manifold, special cases of this theorem recover the Atiyah-Singer Index Theorem, [5], the Atiyah-Singer G Index Theorem [4], and the Atiyah-Bott Lefschetz Theorem for geometric endomorphisms, [2]. If (E, d) is an arbitrary Dirac complex and Ni = {x} is a single point, then

J2(-V)tvAitX i=0

| det(/ x - dfx For the classical complexes with T = f* and Ni = {x}, the a^ may be further identified. For this see [2] and [3]. If / = IM, then N = M and we define 1(D), the index of D by 1(D) = L(IM). AS an example of the theorems available in this case, we state only the following. Theorem 7 Let M be an even dimensional complete Riemannian Spin manifold of bounded geometry. Let p^, be the generalized Atiyah-Singer operator, [15], associated to the complex vector bundle E. Then in Ch(M) I(p+E)=

j

ch(E)A(M)

where ch(E) is the Chern class of E, and A(M) is the A genus of M. There are similar theorems for the other classical operators.

T R A C E S AND INVARIANTS FOR NON-COMPACT MANIFOLDS

5

307

An example

In this section, we give an example of a non-compact complete Riemannian manifold M which is a covering of S4, the surface of genus 4, and a diffeomorphism / : M —> M of bounded distortion which has two fixed points in the interior of each fundamental domain of the covering M —» £4. This example is a single generic leaf of the foliation defined in Section 4 of [13], to which the reader is referred for the proofs of the claims made here. Let II C SL2R be the subgroup generated by the elements a.j = #~ J a 0-?, j = 0 , . . . , 7 where 8 is rotation by 7r/16 and

a

d 0 Od-1

where d is chosen so that U\SL2R/S02 — £4. Let III C II be generated by {aflajl\i,j = 0, 3, 4, 7} U {af lafx \ i, j = 1,2,5,6}. Then M = Ui\SL2R/S02 and il/IIi ~ Z2 * Z2 ^ £>oo so M is non compact. Note that D^ is solvable so also amenable. Let / be the diffeomorphism of M determined by p, rotation of the Poincare disc by 7r/2. Take for the fundamental domains of £4 a regular 16-gon centred at zero in the Poincare disc and all its translates under II. This determines a set of fundamental domains in M. A careful reading of Section 4 of [13] shows that / fixes each fundamental domain of M, (i.e. for any fundamental domain D in the Poincare disc, there is an element 7 € III so that p{D) = 7(.D)). / acts on each fundamental domain of M by rotation by TT/2. In each such fundamental domain, the only fixed points are the centre and the 16 vertices. But the vertices are identified in groups of 8 by Fix so in each of these fundamental domains there are exactly three fixed points and at each fixed point, df acts by rotation by TT/2. By slightly altering the fundamental domains we have chosen (namely adding a small neighbourhood of every other vertex and deleting a corresponding neighbourhood of the other vertices), we obtain new fundamental domains so that each has exactly two fixed points contained in its interior. M has all four classical complexes defined on it, and the local indices at the fixed points for T = f* are all non trivial. For the de Rham complex they are all —2, for the Signature complex — 2i, for the Dolbeault complex with k = 0, they are 1/(1 — i), and for k = 1, they are i / ( l — i), and for the Spin complex they are ± i / \ / 2 depending on which lifting of df one uses.

308

6

JAMES L. HEITSCH

Proof of Lefschetz theorem

As noted above, each A, is an essentially self adjoint operator on L2(Ei). In addition, it is non-negative, so any bounded Borel function g on [0, oo) applied to Aj yields a well defined bounded operator p(Aj) on L2(Ei) whose Schwartz kernel we denote by kg(x, y). Denote by S(R+) the Schwartz class functions on R restricted to [0,oo). If g is a Schwartz class function on R, we denote its Fourier transform by g~. We now recall three theorems from [13]. The proofs for the case considered here are essentially the same as those given in [13] and so are omitted. Theorem 8 If g G S{R+), then kg{x,y) G C°°(E E*), (so Tr{g(Ai)) exists and is finite), and the map g —> kg is continuous. Ifg G CQ°(R), the compactly supported functions, then kg G Cf°(E <8> E*). Theorem 9 Suppose that g G S(R+), B is a differential operator on Ei (whose coefficients, with respect to orthonormal bases, and all their derivatives are uniformly bounded), and T = (To,... ,T() is a geometric endomorphism of(E,d). Then Tr(BTig(Ai)) and Tr(TiBg(Ai)) exist and are finite. Corollary 10 For i = 0 , . . . ,£ and t > 0, Tr(Tie~tAi) exists and is finite. Theorem 11 Tr{T*) = l i m ^ ^ Tr{Tte-tAi). Note that the convergence in Theorem 11 is pointwise and not uniform, so when applying a linear functional to Ch{M) as in Section 7 below, one needs to check that the functional commutes with taking limits. This does not happen in general. For an explicit counter example see [19], and for a systematic discussion, [11]. As the proof of the following theorem is where the trace property of Tr is most central, we will outline that proof here. For details see [13], Section 5. Theorem 12 For all t > 0, i

Y^i-lYTriTie-^) i=0

is independent oft. Proof. We use Quillen's formalism of the supertrace [17]. Let E = ®Ei and E+ = ®E2i and E~ = @E2l+l. An operator A on E which preserves this splitting is called even and one that reverses it is called odd. The super trace applied to an even operator A is defined to be Trs(A)=Tr(A|E+)-Tr(^|E-).

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T R A C E S AND INVARIANTS FOR NON-COMPACT MANIFOLDS

Note that if A and B are both even operators, then Tr s (AB) = while if both are odd, Tr a (AB) = -Trs(BA). Set T = ®U0T„

d = ®todi,

d*=@tld*,

Trs(BA),

A = ©t0Al.

T and A are even while d and d* are odd and these operators satisfy A = dd* + d*d,

dA = Ad,

d*A = Ad*, and Td = dT.

It is not difficult to show that if
d*
For s > t > 0 set ip(x) = e~tx - e~sx

and

ip(x) =
and note that both restricted to R+ are in S(R+).

As

e ^ ( - l ) l T r ( T ; e - ' A i ) =Trs(Te-tA), we must only show that Tr s (TV(A)) = 0. By Theorem 9, we have that both Trs(Tdd*ip{A)) exist and are finite, so we have

and

Txs(Tdd*xp{A)) + Tr,(Td*dV(A)) = Trs{Tdd*^(A)

Trs(Td*dip(A))

+ Td*#(A)).

But from the definitions, this last equals Tr s (TAy>(A))=Tr s (7V(A)), which is what we want to show is zero. As this will follow if we show that Trs{dTd*ip(A))

=

Trs(Tdd*ip(A))=Trs(dTd*ip(A)),

-Tvs{Td*dip{A)).

Theorem 8 implies that if ipn € S(R+)

converges to ip, then

lim Tr(dTd>„(A)) = Tr(dTd*V(A)) n—*oo

and lim Tr(Tn(A)) = T r ( T d * # ( A ) ) . n—>oo

310

J A M E S L.

HEITSCH

Set ViO) = {\/2)4>{x/2) and ip2{x) = e~txl2 + e~sxl2. Then V i > 2 restricted to R+ are in S(R+), and ip = ipifa- Choose sequences ipi,n £ S(R+) converging to tpi with xpi^n E CQ°(R). Then Tr s (dTcTV(A)) = lim Tr,(drd*(V>i,„tf 2 .n)(A)) = lim Tr s (dTd*Vi,„(A)^2, n (A)) = lim T r , ( ^ , n ( A ) d T d > i , n ( A ) ) by Theorem 4 (so it is here that we use the trace property of Tr). But lim Tr.(^ 2 ,n(A)dTd , ^i, n (A)) = - lim Tr s (Tcf Vi,„(A)V>2,n(A)d), n—»oo

n—»oo

since both V2,n(A)d and Td*tp\%n(&) are odd operators. This last term equals - lim Tr s (Td*# 1 , r i (A)V 2 ,n(A)) = - lim Tr s (Td*d(Vi,„V2,n)(A)) n~*oo

n—>oo

= -Tr s (Td*d^(A)) which completes the proof of Theorem 12.

• 4

Corollary 13 L{T) = lim t ^ 0 ^ ( - I J ^ f f ' e - ' ' ) To finish the proof of our Lefschetz theorem, we have Theorem 14 Associated to N is a smooth volume form a which depends only on / , A, the symbols of the Aj, the metrics and their derivatives to a finite order only on N so that in Ch(M) \hnJ2(--LYTr(T;e-^)=

/ a.

The proof of this theorem is a highly technical local computation and is identical to that given in [13] and so is omitted. 7

Functionals on Ch(M)

The space Ch(M) is somewhat difficult to understand. In the special case that M is compact, Ch(M) = R. One of the main results of [7], Theorem 3.1, is that Ch(M) ^ 0 if and only if M satisfies Condition (15) below. However, even in the simple case of M = JR, it is difficult to give a good description of this space. To obtain usable invariants, we need to exhibit interesting linear maps from Ch(M) to more tractable spaces, especially,

311

T R A C E S AND INVARIANTS FOR NON-COMPACT MANIFOLDS

maps to -R or C. Given the result of Block and Weinberger quoted above, it should not be surprising that Ch(M) is particularly susceptible to averaging operations, one of which we give below. For more in this vein, see [18] and [1]. Suppose that M is a complete Riemannian manifold of bounded geometry. Given xo e M and s > 0, denote by B{XQ, s) the ball in M of radius s and centre xo. Assume that for some XQ € M and any fixed t,

^ffo'*))

lim

= 1.

(15)

s-.oo V0l(-B(X(), s + tj)

Let T be a lattice in M and denote the centre of U € T by xu. Set

rs =

{uer\d(x0,xu)<s}.

For any [/] G Ch(M), define A : Ch{M) -* R as follows. A([f]) = lim

„°,

(

\..

P r o p o s i t i o n 16 A : Ck(M) —> R is well defined. Proof. We first show that F(s)

vol(B(x0,s))

converges as s —> oo. Since the centres of the Us are uniformly separated and M has bounded geometry, there is a constant C so that for any difference of two concentric balls, B = B(xo,s + t) - B(x0, t), the number of elements of T with centres in B is bounded by Cvol(B). If | / | is bounded by Cj, it follows immediately that < CfC vol(B). xuEB

In particular, this holds if t = 0, so F(s) is a bounded function and we need only show that for bounded t > 0, lim F(s + t) - F(s) = 0. s—>oo

Note that t may vary with s, but it is a bounded function of s. Given any set B C M, denote by f(B) the sum

f(B) = J2 f(U) xu&B

312

JAMES L. HEITSCH

and write B(s) for

B(XQ,S)

and vB(s) for vol(B(xo,s)).

Then

f(B(s + t)) f(B(s)) vB(s + t) vB(s) f{B(s)) f(B(s)) f(B(s + t)-B(s)) < + vB(s + t) vB{s) vB(s + t) vB(s + t) -vB(s) 1 1 + CfC < CfCvB(s) vB{s+t) vB(s vB(s + t) vB{s) ,vB{s f t ) - u B ( s ) 2CfC(l 2C/CvB(s + t) vB(s + t) which by our assumption goes to zero as s —> oo. To show that ^4([/]) is independent of the choice of / , we need only show that for any / e C°(T) and 7r e G, F(s +

t)-F(s)\

lim

Y,T.[W)-**f{u)]

0. vB{s) Let Cf be a bound for / (and so also for n*/), and let t be a bound on the distance that n moves elements of I\ i.e. for all U 6 T, d(xu,xn^)) < t. Then

Er.[/(^)-*V(tO]

/(B(s))-7r*/(B(s)) u5(s) u£(s) Since 7r moves elements of V at most a distance t, each element in f(B(s — t)) occurs in w*f(B(s)). Thus f(B(s)) and ir*f(B(s)) differ on at most the number of points in B(s) — B(s — t). Thus [vB(s) - vB(s - t)} < 2CfC vB(s) vB(s) As above, this last goes to zero as s —> oo. • Recall that the manifold M of our example above supports the four classical complexes. It is not difficult to see that M also satisfies Condition 15. For the classical complexes, the operators Aj commute with the action of the covering group of M over E4. The diffeomorphism / also commutes with this action, so we are in the situation considered by Atiyah in [1]. It follows easily that for the operators in Theorem 11, the linear functional A is the Atiyah trace (given by integration over a single fundamental domain, which is a compact set) divided by the volume of the fundamental domain. As the Schwartz kernel of the Tie~tAi is uniformly bounded and converges pointwise to the Schwartz kernel of T* and A is integration over a compact set, A does commute with the limit in Theorem 11. Thus we have the following. /(B(S))-7r*/(B(3))

T R A C E S AND INVARIANTS FOR NON-COMPACT MANIFOLDS

313

T h e o r e m 17 Let M and f be as in the example above and denote by V the volume of a fundamental domain for £4 in the Poincare disc. Then • For the de Rham complex, A(L(f*))

-4 = -r—

• for the Signature complex A(L(f*))

—Ai — -—-

• for the Dolbeault complex with k = 0, A(L(f*))

k = 1, A(L(f*))

2 = — —-, and for (l-i)V

2%

l-i)V

• for the Spin complex A(L(f*)) df one uses.

— ——— depending on which lifting of

References 1. M.F. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Asterisque, 3 2 / 3 3 (1976), 43-72. 2. M.F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes I, Ann. of Math., 86 (1967), 374-407. 3. M.F. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes II, Ann. of Math., 88 (1968), 451-491. 4. M.F. Atiyah and G.B. Segal, The index of elliptic operators: II, Ann. of Math., 87 (1968), 531-545. 5. M.F. Atiyah and I.M. Singer, The index of elliptic operators: I, Ann. of Math., 87 (1968), 484-530. 6. O. Attie and S. Hurder, Manifolds which cannot be leaves of foliations, Topology, 35 (1996), 335-353. 7. J. Block and S. Weinberger, Aperiodic tilings, positive scalar curvature, and amenability, Journal AMS, 5 (1992), 907-918. 8. J. Block and S. Weinberger, Large scale homology theories and geometry, in Geometric Topology, ed. W.H. Kazez, Amer. Math. Soc, Providence, R.I., 1997. 9. P. Chernoff, Essential self adjointness of powers of generators of hyperbolic equations, J. Func. Anal, 12 (1973), 401-404. 10. A. Connes, Geometrie Non-Commutative, InterEditions, Paris, 1990. 11. D. Guido and T. Isola, Noncommutative Riemann integration and Novikov-Shubin invariants for open manifolds, J. Func. Anal., 176 (2000), 115-152.

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JAMES L. HEITSCH

12. J.L. Heitsch, Bismut Superconnections and the Chem Character for Dirac Operators on Foliated Manifolds, K-Theory, 9 (1995), 507-528. 13. J.L. Heitsch and C. Lazarov, A Lefschetz theorem for foliated manifolds, Topology, 29 (1990), 127-162. 14. J.L. Heitsch and C. Lazarov, A general families index theorem, KTheory, 18 (1999), 181-202. 15. H.B. Lawson and M.-L. Michelsohn, Spin Geometry Princeton University Press, Princeton, N.J., 1989. 16. A. Phillips and D. Sullivan, Geometry of leaves, Topology, 20 (1981), 209-218. 17. D. Quillen, Superconnections and the Chem Character, Topology, 24 (1985), 89-95. 18. J. Roe, An index theorem on open manifolds I, J. Diff. Geo., 27 (1988), 87-113. 19. J. Roe, An index theorem on open manifolds II, J. Diff. Geo., 27 (1988), 115-136.

Received July 31, 2000, revised November 17, 2000 and February 14, 2001.

Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2002 pp. 315-332

HILBERT MODULES OF FOLIATED MANIFOLDS W I T H BOUNDARY MICHEL HILSUM Institut de Mathematiques, C.N.R.S., 175, rue du Chevaleret, 75 013 Paris, e-mail: [email protected] We define in this article the notion of boundary Hilbert module over a C*-algebra B and show that the class in K*(B) defined by such a module is null. This generalizes the bordism invariance of the index of elliptic operators. Some consequences on foliated manifolds are derived, such as obstructions on the existence of a riemannian metric with longitudinal positive scalar curvature

1

Introduction

The bordism invariance of the index of the Dirac operator on smooth spin manifolds has been established by M.F. Atiyah and I.M. Singer, and play an important role in the first proof of the index theorem for elliptic operators [17]. It has been generalized in various geometric situations: for smooth families of Dirac operators by W. Shih [20], for coverings of a manifold with principal countable discrete groups by J. Rosenberg [19]; for correspondences between smooth manifolds, using bivariant K-theory, by A. Connes and G. Skandalis [4]; alternative analytical proofs where given of [20] by R. Melrose and P. Piazza [14] and by L. Nicolaescu [16], and of [19] by E. Leichtnam and P. Piazza [12]. In each of these situations, the corresponding index is a class in the K-theory group of a C*-algebra. Here we wish to settle an general result in the context of Hilbert modules over C*-algebras. We introduce a notion of Hilbert module with boundary over some C*-algebra B. We also define partitioned Hilbert module over B. These objects behave well with respect to processes of pasting and cutting, similarly as cutting a partitioned manifold along the separating manifold gives two manifolds with boundary. 315

316

MICHEL HILSUM

The main result of this article states that the index class in K*(B) defined by the boundary of such an Hilbert module is null (Theorem 4.3). There are in fact two cases, the even-dimensional case, in which the boundary module is Z/Z 2 -graded, and the index has value in K0(B), and the odd-dimensional, with no grading, where K\{B) is concerned. We have treated here only the odd case. For proving this, we develop an idea due to N. Higson which relates this problem to Callias or Dirac-Shrodinger type operator on partitioned manifolds [6], [18]. These operators have been first studied by C. Callias, and have been since the subject of several works by N. Anghel, J. Bruning, H. Moscovici, U. Bunke and M. Lesch (cf ref. in monograph [13]). Given an even partitioned module, Theorem 4.2 states the equality between the index class in K\(B) of the Callias type operator and the index class of the operator on the partitioning module. These notions are quite general and it appears that index bordism invariance is a rather analytical property. We specialize then these results to Hilbert modules arising from foliated manifolds. Let (W, F) be a spin foliated closed manifold of class C 0 , o °. We use the longitudinal Dirac operator A acting on some Hilbert module £ over the C*algebra of the foliation C*(W, F), defined by A. Connes [3] (cf. minicourse in these proceedings by H. Moriyoshi [15]). We obtain then a bordism invariance property of the longitudinal index class [£, A] e K*(C*(W,F)) (Theorem 5.2). Let (Woo,g) be a smooth Riemannian manifold, and K : W^ —> R the scalar curvature of g, which is a smooth function. One important problem in Riemannian geometry is to determine topological obstructions to the existence on Woo of a Riemannian structure with positive scalar curvature (cf. [11] for a survey). Recall A. Lichnerowicz's Theorem stating that if Woo is closed and spin with K > 0, then (AiWoo), [Woo]} = 0,

(1.1)

where A(W) is the A-genus of the tangent bundle of the manifold. Then Theorems 4.2 and 4.3 gives as corollaries obstructions for a manifold (W, F) to be equivalent by a transverse bordism to a foliated manifold with strictly positive scalar curvature (Proposition 6.2), which straightens an earlier result of J. Rosenberg [19]. Similar conditions occur when (W, F) separates a foliation with strictly positive scalar curvature and Proposition 6.3 generalizes a theorem of J. Roe [18]. When the foliation is of class C°°'°°, using the fundamental class of A. Connes sect. III.7 [3], we derive, in Section 7, cohomological obstructions, among them A. Lichnerowicz's (1.1).

H l L B E R T MODULES O F FOLIATED MANIFOLDS W I T H BOUNDARY

2

317

Preliminaries

Let B a C*-algebra and £ a Banach right B-raodule ; then £ is said to be a Hilbert module over B if it comes with a sesquilinear map (.,.) : £ x £ —> B such that for £,77 G £, a G B, one has (£,77)* = (»7, £)\ (£>0 ^ 0,{^, 077) = (£,77)0, and the map £ —> ||(£,£)ll 3 i s the norm on £. As usual we denote by £(£) the C*-algebra of bounded adjointable operators of the B-module £, and by K.(£) the closed ideal of compact operators off, generated by the rank one operators of the form ( —> £(77, £), for £,77 G £. Let D a densely defined closed operator on £, commuting with the action of B. The adjoint of D is the operator with domain the set of £ G £ such that there exists 77 G £ satisfying for every ( G £ the equality (£>(,£) = {C,,rj) and then D*£ = 77. The operator is said to be symmetric if (D^,rj) = (£,,Dr)) for £, 77 G dom-D, and selfadjoint if D = D*. For a densely defined operator D of B-module, we define C{D) the subalgebra of C{£) of operators a G £•{£) such that a(dom D) C dom Z? and such that the graded commutator [D, a) extends to an element of £(£). Lemma 2.1 Let D be a closed densely defined B-operator on £ such that D* is densely defined. For any a G £(D), one has a* G C(D*), and [a,D]* is the closure of[D*,a*]. Proof. For £ e domD, 77 edomD*, the equality (D£,,a*rj) = ([a, D]£,,ri}+ (a£, D*ri) shows that a*n € dom D* and (D*a* - a*D*)r) = [a, D}*n. D Given a closed linear operator D on £, we shall denote by W(D) the Hilbert B-module domD equipped with the B-product (£,77) + {£>£, D77). Recall that a core or an essential domain for a closed operator D is a dense subspace of W(D). An operator D is said to be regular or affiliated if 1 + D*D and 1 + DD* are surjective operators of £, and in that case the inverse operators (1 + D*D)~1 and (1 + DD*)-1 belongs to C{£) [10]. If £> = D*, then these conditions are equivalent to the surjectivity of im(i -I- T). The following is elementary Lemma 2.2 An closed densely defined B-operator D on £ is regular if and only if the natural continuous injection W(-D) —> £ is adjointable We precise now some terms which will be used in the sequel, i) An unbounded module over B is the data of a Hilbert module £ over B with an unbounded closed densely defined regular operator T commuting with B. We shall call (£, T, r) an even unbounded module if r is a unitary involution such that TT + TT = 0. ii) A symmetric module over B is an unbounded module (£,T) with T c T*.

318

MICHEL HILSUM

iii) A closed module over B is an unbounded module (£ , T) with T = T* and (i + T)-1 e K(£). Recall that if (£,T) is a closed module, we can associate an element [£,T] G Ki{B) as follows [21]: as the operator F = T(l + T2)~i satisfies F* — F G fC(£) and F2 — 1 G /C(£) and its image in the quotient C*-algebra C{£)/K,{£) is a projection, it gives a class in Ko{C(£)/K.(£)). Then [£,T] is the image of this class by the index map KQ{C{£)/K,{£)) >-> K\{K.{£)), followed by the natural inclusion K\{K{£)) —* K\{B). Analogously, to a closed even module (£,T,T) corresponds a class in K0(B). We end this section with a lemma which will be useful later. Let £\, and £2 two Hilbert B-modules, T\ a selfadjoint regular operator on £\ and T2 a selfadjoint operator on £2 possibly singular. Let / € £(£i,£ 2 ) such that /(domTi) C domT 2 , and such that the operator T 2 / - fT\ defined on domTi extends to a bounded adjointable map which we shall denote by a. By the lemma above, the adjoint of a is the closure of the densely denned /*T 2 — Tif*. The map ti/(£) = / £ for £ £ domTi defines a continuous map from W(Ti) to W(T2), commuting with B and of norm less than

VlNP + ll/ll2L e m m a 2.3 The map Uf is adjointable and we have, for 77 € domT 2 u*fV = f*ri + (i + Ti)" 1 ^?? + (1 + T i 2 ) - V ( z + T2)VProof. Let denote by Wf the operator on the right hand side above. For £ G domTi and r\ G domT 2 , we have WfT] G domTi. Put £ = (i + T2)?7 and 0 = (i + Ti)£, we have then

= {t, (i+Ti)-x(r + m - t)-v)cr2+o^?> + (Tie ) T 1 (i + T i ) " ^ / * + (Ti - t) _1 o*)(T2 + »)>7> = ((Ti + i)" 1 *, (Ti + i ) " 1 ^ * + (Ti iTla*)Q + ((Ti + i)-1^, (Ti + i ) " 1 ^ / ' + (Ti - i)"1a*)C>

= (e,(r + (Ti-»)-V)c> = ((/ + a(Ti+t)-1)fl,0 = {(T2+i)/(Ti+i)-^,C) = (uf£,ri) + (T2UfS,T27i).

= ((T2 + i)ft,(T2

+ i)ri)

a Let us finally recall the definition of the external tensor product of two regular operators Tj acting on £;, for i = 1,2, this is an operator S acting

HlLBERT MODULES OF FOLIATED MANIFOLDS WITH BOUNDARY

319

on the external tensor product of Hilbert modules £\ ® £2 with domain defined as follows: let W(Ti) W(T2) be the tensor product ohf Hilbert module, and Ij : W(T\) —> £ — j the canonical injection. Then I = I\ (&I2 • W(Ti) W(T 2 ) -» £1 £ 2 is adjointable [8, 10]. Then let domS = im J and for ^ e domT,-, one set S(£ <8> £2) = Tifi £2 + £1 ® T2£2- It is essentially defined on the algebraic tensor product V\ 2?2 for any essential domains T>i of Ti: and moreover, if (1 + T*Ti)~l 6 )C(£i), for i = 1, 2, then

3

Partitioned and boundary modules

We define now an operator useful for the sequel. Let (£, T) be an unbounded module over B, with T = T*. Let Tj, i = 0,1, be the unitary on C 2 , given by

Let ^(^(T) acting on the tensor product of Hilbert modules £(,.L2(R)
+

l®d1®Ti,

where d\f{u) = ^ . Then ^^(T) is essentially selfadjoint on the algebraic tensor productf d o m T ® C£°(R) C 2 , and for £1,^2 in that subspace, one has the more familiar form

+

+

3l

•^-(*v*. 0(iM-& £)- <> We denote by *(T) acting on £ L 2 (R) C 2 be the closure of the restriction \&ooCT) on the algebraic tensor product domTC£°(]0,1[)®C2. Given
the operator in £(£ L ([0,1])) C 2 ) obtained by multiplication by ip on the second term of the tensor product. Lemma 3.1 The operators *(T) and^^T) are regular. For0eC°°([0,1]) one has m(6) e £(*(T)*) and moreover i/ 0(0) = 0(1) = 0, t/ien m(0)dom*(T)* C d o m ^ ( T ) . / / ( i + T ) " 1 e /C((T) is an external tensor product, the first and last assertions follow. As m(0) G £( , I'(T)), Lemma 2.1 gives the third assertion. It suffices then to prove that 0£ € dom^'oo(T') for support 0 c]0,1[. Let

320

MICHEL HILSUM

£ G dom*(T)* and ip e Cc°°(]0,1[) such that
D We introduce now the notion of partitioned module. Let - (£, £), r) be an even unbounded module with D = D*. - (£b,T) a closed module over B. Suppose that there exist projections q~, p and q+ in C{£) such that 1 = q~ + p + q+ and imp~£ 6 ®L 2 ([0,l])C 2 . Assume that the involution T commutes with q± and restricts on imp to 1 1 ® T 0 TI. For any ip £ C([0,1]), let a(ip) G C(£) be the operator given by the formula a(ip) = ip(0)q~ + m(ip) + ip(l)q+• Definition 3.2 With these hypothesis, the unbounded module (£, D, T) is partitioned by (£b,T) if the following properties hold a. For every

L2(R) C 2 with D = ^' 0 0 (T), with obvious formulae for p and 9*. Lemma 3.4 For any v? G C°°([0,1]) , we have a(y?) G £(£>). 7 / X ° ) = + i)~l G £ ( £ ) .

HlLBERT MODULES OF FOLIATED MANIFOLDS WITH BOUNDARY

321

Proof. Suppose first support(y) C]0,1[, and let ip G C£°(]0,1[) such t h a t tpip = ip. T h e n by properties b, c of t h e definition, one has [D, a( = [$>(T),a( tp'k <E> Ti)p£ + lim <£>&£>£ = (1 ®.D£. As D is closed, Zfy£ = lim D, y ' <8> Ti)p is bounded. In general every

2 where support ipi C [0, A [ and (/?2(0) = 2(l)- Suppose then t h a t s u p p o r t ^ C [ 0 , | [ and let ) of £ 6 C 2 is isomorphic W ( ^ , ( T ) ) , a n d we have a factorization of a(ip)(i + D)~l by £

i ! ± ^

W{D)

J^U

Wo

• £•

By Lemma 2.3, t h e m a p £ —> a()£ from W ( D ) to Wo is adjointable, and t h e injection Wo —> £ is compact. • We come now to t h e main notion of this article. Let - (£, S,T) be an even symmetric module over B. - (£b,T) a closed module over B. Suppose that there exist projections q, p in £(£) such that 1 = q + p and imp~£ b C 2 . Assume that the involution r commutes with q and restricts on imp to 1 <8> 1 ® TQT\. For any ip G C([0,1]), let b(ip) G £(£) be the operator given by the formula 6(V>) = m(ip) + ip(l)q. Definition 3.5 With these notations, the closed module (£a,T) is the boundary of (£, S, r) if the following properties hold a. For every ip G C~([0,1]), with G C c °°(]0,1[), and if £ G d o m S or £ G d o m ¥ ( T ) , then S
322

MICHEL HILSUM

c. For every ip G C°°([0,1]) with supportip f~l support C 2 with S the closure on Cc(]0, +oo[) of the matrix given by the formula (3.1), in which case the boundary is given by (£b,T). Remark 3.7 Suppose (£, S, r) and (£;,, T) satisfies the preceding conditions, but with b(ip) replaced by b(ip) = m(ip)+ip(l)q and the condition a replaced by the analogous condition a with L2(R, C) implemented by the map

MU)J 2

2

V-&(i-«).

2

from L ([0,1]) C to L ([0,1]) ® C 2 , the operator * ( - T ) becomes

/

0

U**(-T)U = fa."

\ « 3u i -

--£- £a + T N

" o

We see then that if (£b, —T) is the boundary of {£, S, r) in the sens of the definition 3.8, then replacing b(ip) by b(ip) = (q + pU)b(tp)(q + pU), the closed module (£b,T) becomes the right boundary of (£,S,T) (and inversely). Lemma 3.8 Let {£, S, r) be an unbounded module with boundary as in the definition above. Then for any (f G C°°([0,1]) ; one has b( such that a(<^)£ = 0 for some
H l L B E R T MODULES O F FOLIATED MANIFOLDS WITH BOUNDARY

f € domD, j] £ d o m * ( T ) , one has a(l - 9)£ £ domS,

6(0)T?

323

G d o m S and

S(a(l - 0)£ + 6(0)77) = Da(l - 0)£ + *(T)6(6»)r/. From Lemma 2.3, we see that that the map (£,77) —> (o(l — 0)£, 6(0)77) from W(D)ffiW(*(T)) to W(5) is a surjective and adjointable and the formally same map to £ is adjointable too. By a classical result of Mishchenko, there exists a right inverse to this map, and thus the canonical injection W(5) —> £ which factors through a product of adjointable maps, is in £(W(S),£). For the second assertion, let W C W(D) be the closed subspace generated by the elements £ G domZ) such that q+£ = 0. The hypothesis implies that the natural injection W —> £ is compact but as W = W(S), this is equivalent to suppose that the natural injection W —> £ is compact. • Inversely, let (£,S,T) a symmetric module with boundary (£b,T) and projectors p,q satisfying to the definition 3.5. Let (£i,Si) another symmetric module with right boundary (£{,,— T), pi,Qi the projectors, as in the remark above. We can form the Hilbert module £ = imqi © £b £S> L 2 ([0,1],C 2 ) ffiimq. Let 0 £ C°°([0,1]) such that 0 EE 1 in a neighbourhood of 1, and 0 = 0 near 0, and set D the operator with domO = 61 (9) dom Si + 6(1 - 9) dom S and for f <E dom S and C e dom Si D(6i(<9)£ + 6(1 - 0 ) 0 = (1 ® U)Slbl{9)i

+ S6(l - 0)C

(3.2)

In order to ensure that D is well defined, we have to show that 61 (0)£ 46(1 - 0)( = 0 implies Si&i(0)£ + S6(l - 0)C = 0. But there exists ip £ C£°(]0,1[) such that ip = 1 on support(0(l - 0) and thus Si&i(0)£ + S6(l 0)C = *(T)V»(6i(0)C + 6(1 - 0 ) 0 = 0. Analogously, Lemma 3.1 shows that the operator D is symmetric and does not depend of the choice of 0. P r o p o s i t i o n 3.10 The triple (£,D,f) D* and is partitioned by (£b,T).

is an unbounded module with D =

Proof. Let us show that D is selfadjoint, and let £ £ domD*. We have identically (D£, () = (£, D*Q for £ = 6i(0)£ + 6(l -9)n, and then by letting 77 = 0, Lemma 3.8 implies 61 (0)£ G domSi, and thus 6i(0)C G dom Si. Analogously, one has b(9)( G domS, which shows that domD = domD*. It follows then from Lemma 2.1 that the map (£, rf) —> (61(0)£, 6(1 — 0)77) from W(Si) © W(S) to W(D) is a surjective and adjointable map. There exists a right inverse to this map, and thus the canonical injection W(D) —> £ is adjointable. •

324

MICHEL HILSUM

4

Callias type operators

In this section we introduce Callias type operator for partitioned Hilbert modules (cf. mem. [13]). Usually, such operators are defined for odd dimensional manifolds. The present definition concerns the even dimensional case, and agrees with U. Bunke's work [2]. Let now an even unbounded module (£, D, r) over B, with D selfadjoint. For a G £(£), the operator D + ra is a bounded perturbation of a regular operator, and is also regular, and let

F(a) =

(D2 + 1)-HD

+ TCL).

As (D + T ) 2 = D2 + 1, one has (D + T)'1 G C(£) and F{\) = (D2 + l)-i{D + r) is a unitary. The equality F{a) = F ( l ) +T{D2 + l)~i(a1), shows then F(a) G £{£)• Lemma 4.1 Let a = a* G £>(D), satisfying [D,a](D 4- i ) _ 1 G fc(£) and (a2 -l)(D + i)~l G K(£). Then the following relations hold F'(a)- F'(a)* £ K{£) andF(a)2 - 1 eK{£). Proof. We have D(D2 + l)~\ = (D2 + l)~\D and (D2 + l)~ir = T(D + 1)~2. Thus to prove the first relation, it remains to show that [a,(D2 + 1)-*] G K(£). It suffices to prove that [a, (D2 + l)" 1 ] £ K(£) and as (i + D)~1(—i + D)~x = (1 + D 2 ) - 1 , we are reduced to show that [a,(D ± i)-1} G K(£). For £ G £, one has (i + £>) _1 £ e domD and [a, (i + D)-1}^ = (i + D) _1 [Z?,o](i + £>) _1 £, and by hypothesis this last operator is compact. For the second relation, we have that F(a)2 = F(a)F(a)* modulo K.(£) and thus the following operator is compact 2

F(a)F(a)* - 1 = (D2 + l)~i(D2 + a2 + T[D,a])(D2 + 1)"* - 1 = (D2 + l)-i(r[D,a}+a2-l)(D2 + l)-i. D This lemma shows that under these conditions the couple (£, F(a)) determines a class in K\(B) [21]. Recall that to the unbounded module (£b,T) with T = T* and (i + T ) " 1 G K.(£) is also associated a class in Ki(B) as follows: the operator F = T(l + T 2 ) " 1 satisfies F - F* G K(£b) and F2 — 1 G IC(£b), and the class of (£b,T) is by definition the one of the couple (£b, F) in KX(B). For any V € C°°([0,1]) such that ip(l) = - 1 and V>(0) = 1, y/(°) = V'(l) = 0. w e set ify = F(a(rp)). From the Lemma 4.1, the couple (£, F^) determines also a class in K\(B).

H l L B E R T MODULES O F FOLIATED MANIFOLDS W I T H BOUNDARY

325

Theorem 4.2 Let (£b,T) be an unbounded module over B, which partitions the unbounded module (£,D,T). For any ip as above, the classes in Ki{B) of (£b,T) and of (£,F^,) are equal. Proof. We prove first the assertion in the particular case where £ = £b L 2 ( R , C 2 ) and D = *oo(T). We know that Ki(B) is canonically isomorphic with the bivariant group KKl(C, B) of G. Kasparov. Let a € KK\C,C0(R)) and (3 e KK1 {C0{H), C) be the dual Dirac and Dirac elements as introduced in [9]. They are mutually inverse, in the sense that a ®c0(R) 0 — Id and (3 <8>c a = Id . Let z — [£b,T], and form the external cup-product (3 c0(R) (P ®c z) = (a <S>Co(R) /3)®cz = z. For the general case, one can suppose that £ = £b ® £ 2 (R) <8> C 2 . Indeed,let £\ = £ © £o where £0 = B L 2 (R) <8> C 2 , n = T ® T0 and Di = D ® 1 ® L where L = -^ ® r i . By the first step the module (5o, 1 <8>L) gives the null class in Ki(B), and the classes defined by D and D\ in K\(B) are then equal. By the stabilisation theorem of G. Kasparov ([21], Theorem 6.2), the modules ker(ri ± 1) are stabilized, and isomorphic to £b <8> L 2 (R). Let now for t e]0,1] the Hilbert module given by

£t = imq~

®(£b®L2(l-r\t-l},C2))®imq+,

and £0 = £b ® L 2 (R, C 2 ). By using the Propositions 3.10 and 3.9, one gets an evident selfadjoint operator Dt on £t- Let Do = ^oo{T). Then with obvious notations the family Ft = (Dt+at('ip))(D2 + l)~2 gives a homotopy between F0 and Fi in KKl{C, B), and thus [Fi] = [F0] = [£b, T}. D T h e o r e m 4.3 If the closed module (£b,T) is the boundary of a symmetric module (£,S,T) such that (1 + S*S)~1 £ K.(£), then one has [£b,T] = 0 in the group K\(B). Proof. Let £ be the partitioned Hilbert module obtained by adding £b ® L2(] - oo,0]) ® C 2 and let D the corresponding operator. For any ip G C°°[0,1] with V(0) = 1, V(l) = - 1 and V'(°) = V>'(1) = 0, we have a ( » - 1 G /C(D), and thus F(a(V>)) - F ( l ) £ /C(5), where F(o) = (D 2 +1)~ 2 (D + ra) is defined as before. As F ( l ) is invertible, by Theorem 4.2, we have [£b,T] = [£,F(1)\ = 0 in K^B). D

326

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MICHEL HILSUM

B o r d i s m of foliations

Let (W, F) be a closed (i.e. compact boundaryless) foliated manifold of class C 0 ' 0 0 . We suppose for simplicity that d i m F is odd. We need the longitudinal index class of the foliation (W, F) [3], which is the subject of the minicourse by H. Moriyoshi in these proceedings [15]. We recall briefly its definition. Let G be the holonomy groupoid of the foliation (W, F) with source and target maps r, s : G —> V, and C*{G) the reduced C*-algebra of G, also denoted C*(W,F). We assume that the tangent bundle to the foliation F is spin and let S be a spinor bundle, and A\ the longitudinal Dirac operator acting C°'0O{W, S). Then by taking the pull back s*(S) of S on G, one can form a Hilbert module £ over B, and the pull back of A\ on G is a closable selfadjoint regular operator on £. Letting A its closure, then the couple {£,A) is a closed module. The class of (£,A) in Ki(C*(W,F)) is the longitudinal index class of the spin structure on F. Let (V, E) be a connected compact foliated manifold such that W is a union of connected component of the boundary V, and that the foliation E is transverse to W and intersects W along F. By [5], Lemma 3.5, one has a tubular neighbourhood U of W with a C0'°°-diffeomorphism 9 : U -> W x [0,1], such that E ~ F © (U x R) on U w.r.t. 9. We may choose a riemannian structure g along the leaves on V satisfying g = dt2 © /i, where /i is a riemannian structure along the leaves of F, and we fix a spinor bundle SE for E compatible with S, in the sens that the restriction of SE to U is equal to the product of S by the standard spin bundle of T[0,1] = [0,1] x R. Now let D/ be the longitudinal Dirac operator acting on C°'°°(V, SE)- Then one has an isomorphism C°'°°(U, SE) ^ C°'°°(W, S) ® Cc°°([0,1]) ® C 2 , and one has [13]

Let iJ the holonomy groupoid of the foliation (V, E) and let H^ = {l'j'r(l)->s(l) ^ W} C H. Then G is a closed and open subgroupoid of H$, in general non trivial, and H$ is a closed subgroupoid of H. The restriction of the source map s : H —> V to i/$f is a covering of the foliation F. The pullback by this map of S and of Ai gives rise to a closed

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H l L B E R T MODULES OP FOLIATED MANIFOLDS WITH BOUNDARY

module {£b,A) over C*{H$). As G is open and closed in H$, C*(H%), and a m a p $ : K»(C*(W,F)) tion, one has $([£,A))

one has a morphism C*(W, F) —» -> K „ ( C * ( # $ 0 ) , and by construc-

= [Sb,A].

(5.2)

Let H.y = { 7 ; s ( 7 ) e V,r(7) e W} C H. T h e n similarly, there is a Hilbert C*(H$)-module J ' associated to S*(SE), and the longitudinal Dirac operator on E determines an operator D on J. Let £+ C J the submodule generated by section of S*(SE) whith support in the complementary of r*{U). Recall t h a t a submanifold W of V transverse to E is said to be faithful if it intersects all the leaves of E. L e m m a 5.1 Using 9 : U —> [0,1] x W, one has an isomorphism J = (£b <8> L 2 ( [ 0 , 1 ] , C 2 ) ) © £+. With respect to this isomorphism, the closed module (£b,A) is the boundary of (J,D,T). Moreover, ifW is a faithful transverse to the foliation on V, then (1 + D*D)~l is a compact operator of J. Proof. T h e restriction groupoid H^ gives rise to a C*{V, E) C*(H^) Hilbert bimodule. Let Dy be the longitudinal Dirac operator on C*(V, E). Then D = Dv

<2>C-(V,B)

Id.

On t h e complete manifold V = (] - oo, 0] x W) U V one has a foliation E of class C°'°° which extends trivially E. W i t h evident notations, by [Proposition 5.2] [7], the longitudinal operator Dy is regular on E. T h u s , by Proposition 3.9, Dy is regular and (1 + DvDv)~x is compact. This implies t h a t D is regular. If W is faithful, there the bimodule gives a Morita equivalence between C*(V,E) and C*(H%). Then as (1 + DvDv)~l is compact, the operator l (1 + D*D)~ is compact. • This last lemma and Theorem 4.3 give now immediately T h e o r e m 5.2 Suppose that dV = W and that W is a faithful (V,E). Then one has $({£,A\) = 0 in K*(C*(H%)).

tranverse

to

Remark 5.3 Last theorem is not true if the hypothesis W faithful is dropped. Here is a counter-example communicated to us by the referee whom we t h a n k s gratefully. Let T 2 be the two-torus and F C T(T2) a Kronecker foliation of irrationnal slope. T h e n t h e T h o m isomorphism in K-theory for R-actions [3] shows t h a t t h e longitudinal index class from the spin structure is a generator in K!(C*{T2,F)) = Z2.

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MICHEL HILSUM

Let Vx = T2 x [±, 1] and let W = T2 x {1} and Wx = T2 x {£}. There exists a two dimensional foliation on V\ transverse to W and inducing F, and for which W\ is a leaf: this foliation can be obtained by extending the product foliation near W and tabulating leaves by rotating along Wi (leaves wind around W\). Let E the foliation on the solid torus V — D2 x S1 obtained by extending this foliation by a Reeb component on the complementary of V\ — W\. Then W is not faithful, C*(H$) is Morita equivalent to C*(T2, F), and 3>([£,J4]) 7^ 0. Therefore, the conclusion of the theorem is not satisfied. Now suppose that dV = W = W1l)W2 where (Wi, Fi), (W2,F2) be are closed manifold with spin foliation of class C 0 ' 0 0 , and that there exist a spin transverse foliated bordism (V, E) between them. Let Bj be the reduced C*-algebra of (Wj,Fj) and Bj the reduced C*-algebra of the groupoid Gw\, and $j : K*(Bj) —> K*(Bj). under these conditions

Again , we assume that W is faithful; the

Corollary 5.4 / / ^ ( [ ^ A ] ) ^ 0, then [£2,A2] ^ 0. Proof. There is a Hilbert module £j over C*{H^) built from H$ = s-1(Wj)r\r-1(W), with a representation C*(Bj) -> £(£,•). This bimodule gives a class in the bivariant K-theory group [Sj] € KK(Bj,C*(H^)), such that the associated homomorphism Ki(Bj) —> K\{C*{H^)) is injective. Keeping the above notations, one has C*(W, F) = B\® B2 and $ = [£\] o $1 — {£2] ° ^2- By the last proposition and using injectivity, one gets the conclusion. D Remark 5.5 As mentioned in the introduction, there are analytical proofs of bordism invariance in various geometric situations [4, 1, 14, 16, 12]. It relies essentially on the exact sequence in bivariant K-theory for B a C*algebra of G. Kasparov [9] -> KK°(C0(V

- W), B) -> KK\C(W),

B) -> KK\C(V),

B) ->,

where dV = W and B is a separable C*-algebra. It is less elementary as it uses pseudodifferential calculus. However, one can guess that its works for B = C*(V,E) and this would give in that situation an alternative proof of Theorem 5.2. 6

Foliation with longitudinal positive scalar curvature

Let (W, F) be a foliated manifold of class C 0 - 00 and g a C 0 ' 0 0 -riemanman structure along the leaves. The longitudinal scalar curvature is the smooth

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329

function KI : V —> R the value of which at x E V is equal to the scalar curvature of the restriction of g to leaf which contains x. One may asks under which condition m should be positive. J. Rosenberg [19] and A. Connes [3] obtained the following Proposition 6.1 Let (W, g) be a smooth spin foliated manifold and g a longitudinal riemannian metric with strictly positive scalar curvature. Then the longitudinal index class of the spin structure in K*(C*(W,F)) is nul. Using the result of the last section , one then obtains Corollary 6.2 Let (W\, F{), {W2,F2) be closed manifold with spin foliations of class C 0 ' 0 0 , and let (V,F) a transverse faithful bordism between (Wi,Fi) and (W2,F2). If ®i([£i, Ai}) ^ 0, then (W2,F2) cannot carry a riemannian structure with strictly positive longitudinal scalar curvature. Now we focus on the following situation: (V, E) is foliated manifold of class C 0 ' 0 0 (V is not assumed to be compact), and (W, F) a codimension one closed foliated submanifold, faithfully transverse to E. We choose a tubular neighboorhood U ~ [—1,1] x W and the foliation E and F are assumed to have compatible spin structure. We define as above (£,A) (resp. (£b,A), (J,D,T)) the longitudinal index classes oiC*{W,F) (resp. C*(G%)). Suppose now that Vis separated or partitioned by W, which means that V - W = V+ U V- where V_, V+ are open and V-DV+0. Then as in Lemma 5.1, the unbounded module (J,D,T) is partitioned

by(£,i). In this situation, we obtain a generalization of a previous results of M. Gromov-B. Lawson [11] and of J. Roe [18]. Proposition 6.3 Let (W, F) be a closed spin foliated manifold of class C 0 ' 0 0 which separates faithfully (V, E) as in conditions above. If there exists c > 0 such that the longitudinal scalar curvature satisfies K > c onV, then $([£,A])=0inK.(C*{H%)). Proof. The Bochner-Weitzenbock formula tells us that D2 = V*V + R, where k = K O r and V are the pullbacks of the Clifford connection and of the scalar curvature on J. Let tp be a smooth function on V which modulo CC(V) is equal to 1 on V_ and to -1 on V- and such that \\dtp\\ < %. Let then F^ = (1 + D2)?(D + Tip) is invertible, with i> is the pullback of xjj. One | | F ^ , i ^ - l | | = ||(1 + D2V)?{$2 - 1 + [ D V l $ ) ( l + £>v)*ll < !- w h i c h s h o w s t h a t Fiinvertible. By Theorem 4.2, it follows that [£, A] = 0 in Ki(C*(H%)).

is

D

330

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MICHEL HILSUM

Example: the smooth case

We illustrate this by several geometrical examples. There are spin foliated manifolds for which we know that [£,A] ^ 0 , for example all foliations which comes from a free action of a simply connected solvable Lie group II.C [3]. Here are another examples. Suppose that (W, F) is a foliation of classe C 0 0 ' 0 0 . Let 11 C H* (W, R) be the subring generated by Pontryagin classes of the quotient vector bundle TW/F, Chern classes of holonomy invariant complex vector bundles on W, and the image of WOq in H*(W,R) by the Weil homomorphism (q = dim W — dim F). Then A. Connes has proved that every w e K determines a homomorphism u>„ : K*(B) —> R such that III.7 [3]

u,.([£,A]) = {A(F)/\w,[W]),

(7.1)

where A(F) is the A-genus of the tangent bundle of the manifold, a universal polynomial in the Pontryagyn classes of F. Proposition 7.1 Let (W,F) be a foliated manifold of classe C00'00 which is a faithful tranverse boundary component of a compact foliated manifold of class C0'00. If there exists u e 11 such that (A(F) A w, [W]) ^ 0, then QifrAVjLOinKWiHft)). Proof. Form the construction of the fundamental class [3], Section III.7, as Hyy is itself a groupoid of local tranverse diffeomorphisms of (W, F), each u> G 1Z defines similarly a map C such uJ*([£b,A\) = (A(F) Aw, [W]). The assertion follows now from relation (5.2). • Thus if (W, F) satisfies conditions of Theorem 5.2, or of Proposition 6.3, the longitudinal index class implies the annulation of all the caracteristic numbers in the r.h.s. of Equation (7.1). There always exists w G H*(W, R) such that A(F)Acu = A(W) ; in particular, if $([£, A]) = 0, then one recovers Lichnerowicz's condition (1.1). Remark 7.2 In the case where W is the boundary of V, the construction works if we only suppose that the foliations are oriented, by using the signature operator alongs the leaves instead of the Dirac operator. The condition in the last proposition becomes (L(W) A ui, [W]) =£ 0, where L(W) is the L-genus of the bundle F. Furthermore, if (V, E) is assumed to be a class C°°'°° then Stokes' Theorem gives directly (A(F) A u>, [W]) = 0 for every ui £ H.

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References 1. P. Baum and R.G. Douglas, Relative K homology and C*-algebras, K-Theory, 5 (1991), no. 1, 1-46. 2. U. Bunke, A K-theoretic relative index theorem and Callias-type Dirac operators, Math. Ann., 303 (1995), 241-279. 3. A. Connes, Non-Commutative Geometry, Academic Press, New-York, 1994. 4. A. Connes and G. Skandalis, The longitudinal index theorem, Tohoku J. Math., 317 (1994), 521-526. 5. C. Godbillon, Feuilletages: etudes geometriques, Progress in Mathematics, no. 98, Birkhauser Verlag, Leipzig, 1991. 6. N. Higson, A note on the cobordism invariance of the index, Topology, 30 (1991), 3. 7. M. Hilsum, Fonctorialite en K-theorie bivariante pour les varietes lipschitziennes, K-theory, 3 (1989), 401-440. 8. K. Jensen and K. Thomsen, Elements of Ki^-theory, Birkhauser, Boston, 1991. 9. G.G. Kasparov, The operator K-functor and extensions of C* -algebras, Math. U.S.S.R. Izv., 16 (1981), 513-572. 10. E.C. Lance, Hilbert C*-modules, L. N. Ser., no. 210, London Mathematical Society, Cambridge University Press, 1994. 11. B. Lawson and M.-L. Michelsohn, Spin Geometry, London Mathematical Society. L. N. Ser., no. 210, Cambridge University Press, Saint Louis, 1994. 12. E. Leichtnam and P. Piazza, Spectral sections and higher AtiyahPatodi-Singer index theory on Galois coverings, G.A.F.A., 8 (1996), 3-26. 13. M. Lesch, Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte ziir Mathematik, no. 136, B.G. Teubner, Stuttgart, 1997. 14. R.B. Melrose and P. Piazza, Families of Dirac operators, boundaries and the b-calculus, J. Diff. Geom., 46 (1997), 99-179. 15. H. Moriyoshi, Operators algebras and the index theorem for foliated manifolds, Foliations: Geometry and Dynamics (Warsaw), in these proceedings, World Scientific, 2001, 127-155. 16. L. Nicolaescu, Generalized symplectic geometries and the index of families of elliptic problems., Mem. Amer. Math. Soc, 609 (1997). 17. R.S. Palais, Seminar on the Atiyah-Singer index theorem, Annals of Math Studies, no. 21, Princeton University Press, Princeton, 1965. 18. J. Roe, Partinioning noncompact manifolds and the dual Toeplitz prob-

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lem, O p e r a t o r algebras and applications, no."?" London Math. Soc. Lecture Notes Ser., 135, Cambridge University Press, 1988. 19. J. Rosenberg, C*-algebras, positive scalar curvature, and the Novikov conjecture., Publ. Math., Inst. Hautes E t u d . Sci., 5 8 (1983), 409-424. 20. W. Shih, Fiber cobordism and the index of a family of elliptic differential operators, Bull. Amer. Math. S o c , 72 (1966), 984-991. 21. N.E. Wegge-Olsen, K-theory and C*-algebras, Oxford University Press, 1993.

Received November 3, 2000, revised July 3, 2001.

Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawet WALCZAK et al. World Scientific, Singapore, 2002 pp. 333-349

N O N - E U C L I D E A N A F F I N E LAMINATIONS VADIM A. KAIMANOVICH CNRS UMR-6625, IRMAR, Universite Rennes-1, France, e-mail: [email protected] The purpose of the present paper is to discuss examples of affine Riemann surface laminations which do not admit a leafwise Euclidean structure. The first example of such a lamination was constructed by Ghys [5]. Our discussion is based on the geometric methods developed by Lyubich, Minsky and the author [13], [12], which rely on the observation that any affine surface A gives rise in a natural way to a hyperbolic 3-manifold SjA with a distinguished point at infinity. In particular, we give a new interpretation and a generalization of the example of Ghys.

1

Affine and hyperbolic laminations

In this Section we recall the basic facts on the relationship between affine and hyperbolic laminations. Although our exposition is self-contained, more details on this relationship can be found in [12]. l.A

Affine and Euclidean surfaces

By endowing a Riemann surface S with an atlas of coordinate charts with transition maps from a given pseudo-group C (contained in the pseudogroup of all holomorphic maps) one can define finer geometric structures on S. Definition 1.1 We shall say that S is (i) an affine Riemann surface, if C is the group of all complex affine maps z H-> az + b, a , i ) 6 C , a / 0 ; (ii) a Euclidean surface, if C is the group of all maps z <—> az + b, a,b 6 C, \a\ = 1 (so that the transitions are Euclidean motions). 333

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VADIM A. KAIMANOVICH

If S is an affine surface, then its tangent and cotangent bundles (and hence all tensor bundles) are endowed with a natural flat connection. Being parallel with respect to this connection means to have constant coefficients in any affine coordinate chart (the reader is referred to [3] and [8] for general notions from the theory of affine manifolds). So, one can talk about parallel vector fields, forms, Riemannian metrics, etc. on S. In these terms a Euclidean surface is just an affine surface endowed with a parallel conformal metric. An affine Riemann surface structure is the same as a complex affine structure, or, in "real terms", a projective Euclidean (= similarity) structure. In particular, an affine plane is R 2 endowed with the class of all multiples of a given Euclidean structure. Any complete affine surface is a quotient of the affine plane by a freely acting discrete group of Euclidean motions. Therefore, for any such surface the affine structure can be refined to a Euclidean one, i.e. there exists a parallel conformal metric. However, this is no longer true if we pass from a single surface to a collection of surfaces assembled into a lamination. l.B

Hyperbolic space

Before describing a relationship between affine surfaces and 3-dimensional hyperbolic geometry, let us first recall the basic notions concerning the hyperbolic space. The sphere at infinity <9H3 is the boundary of the visibility compactification of the 3-dimensional hyperbolic space H 3 . The space H 3 with a distinguished boundary point q € <9H3 is called pointed at infinity. By Vq — 9 H 3 \ {q} we denote the punctured visibility sphere. The hyperbolic space H 3 is fibred over Vq by means of the projection pq which assigns to any h € H 3 the uniquely determined point pq{h) 6 Vg such that h lies on the geodesic joining q and pq(h). Definition 1.2 ([9]) Let (H 3 , q) be a pointed at infinity hyperbolic space. The choice of the point q £ 9 H 3 determines the Busemann cocycle (3q on H 3 x H 3 by the formula

Pq(hi,h2) = \im[d(hi,h) -d(h2,h)]

,

h

where h € H 3 converges to q in the visibility topology. Definition 1.3 The horosphere centred at q and passing through a point h £ H 3 is the level set of the Busemann cocycle: Hor q (/i) = {h' e H 3 : f3q(h, h') = 0} .

335

NON-EUCLIDEAN AFFINE LAMINATIONS

By Hor(H 3 ,g) we denote the space of' horospheres centred at q, and by Hor(H3)=

Hor(H 3 ,g)

(J 9S9H

3

the space of all horospheres in H 3 . The center of a horosphere T 6 Hor(H 3 ) is denoted T ^ e <9H3. Below we shall usually omit the subscript q when the point at infinity q is fixed. These notions are best illustrated by looking at the upper half-space model H ^ C x l

+

dp2 =

= {(z,t):zeC,(>0},

|rfz|

^2+ ^

,

(1.4)

where \dz\2 is the standard Euclidean metric on C. The geodesies in this model are either Euclidean half-circles orthogonal to the boundary plane or vertical lines, and the visibility sphere is the union of the distinguished point at infinity q — oo and the boundary plane V = {(z,t) :z€C,t

= 0} =

dU3\{q}.

The Busemann cocycle with respect to the point q = oo is MM

=**$$•

(1-5)

Thus, the horosphere Hor(/i) is the horizontal coordinate plane passing through h, and the map b is the coordinate projection h — i > z(h) (see Figure 1). l.C

Hyperbolization of affine surfaces

P r o p o s i t i o n 1.6 The punctured visibility sphere V = 9 H 3 \ {q} of a pointed at infinity hyperbolic space (H 3 ,^) is endowed with a natural structure of an affine plane. Proof. The projection p allows one to identify any horosphere Hor(/i) with the punctured visibility sphere V. Denote by Eh the Riemannian metric on V obtained by restricting the hyperbolic metric onto Hor(h) and then projecting it onto V. By formula (1.4)

K - ^

,

(1.7)

336

VADIM A. KAIMANOVICH

<7 =

Hor(/t)

CO

h

V^C

(»

Figure 1.

so that (V,Eh) is a Euclidean plane. Further, by (1.5) the structures £/> for different h are all proportional, and Eft! £/i 2

exp[/?(/n,/i 2 )]-

(1-8)

• Conversely, let us show that any affine surface gives rise to a pointed at infinity hyperbolic 3-manifold. Definition 1.9 Let 5 be a Riemann surface. The elements of the scaling bundle p:SjS-^S over S are conformal circles in the tangent spaces TZS, z £ S. Any circle h G SjS can be considered as the unit circle of the associated conformal Euclidean metric Sh on T p ^5. Below we shall often identify h and Eh, and consider SjS as the bundle of conformal Euclidean metrics on tangent spaces TZS, zeS. We can use formula (1.8) to define (3(h\, /12) for any two points hi, /12 € SjS from the same fibre. Therefore, any fibre of p is endowed with the metric d(hi,h2)

= \P(hi,h2)\

,

phi = ph2

(1.10)

NON-EUCLIDEAN AFFINE LAMINATIONS

337

Proposition 1.11 If A is an affine plane, then SjA is given a natural structure of a pointed at infinity hyperbolic space. Proof. We shall realize A as the boundary plane of the hyperbolic space in the upper half-space model. The fibres of the bundle p : SjA —> A are endowed with the metric (1.10), and the points of 9)A are themselves metrics on tangent spaces TZA. In order to combine them and produce a metric on SjA we shall need the affine connection over A. Since A is simply connected, the affine connection (being flat) gives natural bijections between all fibres of p which preserve the distance (1.10). These bijections are obtained by the parallel transport of metrics e^, h G SjA and provide us with a product structure on SjA. In other words, any metric s^ extends from the tangent space TPhA to a Euclidean metric (also denoted Eh) on A (note that because of this formula (1.8) for the Busemann cocycle now makes sense for all h\,h,2 € SjA, not just for those from the same fibre). Now taking the product of the metric Sh and the fibre metric (1.10) gives precisely the Riemannian metric (1.4). D We shall call the constructed correspondence Sj between affine planes and pointed at infinity hyperbolic spaces the hyperbolization functor. Remark 1.12 In the same way one can easily see that the hyperbolization functor Sj is bijective between the category of complete affine surfaces and the category of complete hyperbolic 3-manifolds obtained by factorizing a pointed at infinity hyperbolic space (H3,q) by a freely acting discrete horospheric group (i.e. the one which preserves the horospheres centred at q). l.D

Affine and Euclidean laminations

Let now £ be a Riemann surface lamination, i.e. the leaves of C are Riemann surfaces, and there is an atlas of local charts ("flow boxes") such that the transition maps are conformal and transversely continuous. Further restricting the class of leafwise transition maps one gets affine and Euclidean laminations (cf. Definition 1.1). Their leaves are affine and Euclidean surfaces, respectively. If the leaves of an affine lamination are isomorphic to the standard affine plane C, we also call it a C-lamination. Now one can ask whether, given an affine lamination A, its structure can be refined to that of a Euclidean lamination. In other words, whether one can choose a Euclidean {= parallel) metric on every leaf of A (in [5] this is called uniformization). As we have already mentioned, a single leaf of an affine lamination is

338

VADIM A. KAIMANOVICH

always uniformizable. However, the problem is that the leafwise Euclidean metrics have to be consistent with the lamination structure, i.e. have to be "well-behaved" when one passes from one leaf to another one. In the lamination setup (with no transverse smooth structure) one can deal either with the continuous or with the Borel category. Definition 1.13 An affine lamination is uniformizable in the continuous (resp. Borel) category if it admits a leafwise parallel metric which is transversely continuous (resp. Borel). Below for simplicity we shall only deal with the affine C-laminations, although all the considerations carry over to general affine laminations as well (cf. Remark 1.12 and Remark 1.17 below). Let A be a C-lamination. Then the application of the hyperbolization functor $j to the leaves L of A gives rise to the lamination H = f)A (the hyperbolization of A) whose leaves are pointed at infinity hyperbolic 3spaces f)L. The lamination Ti is a 1-dimensional fibre bundle p : Ti —> A over A. The leafwise Busemann cocycles piece together a global Busemann cocycle (3 on Ti, The elements of H can be considered as Euclidean conformal metrics on the tangent spaces to points of ,4 (cf. Definition 1.9). Therefore, sections a : A —> Ti of the fibre bundle p are in one-to-one correspondence with leafwise conformal Riemannian metrics on A- Sections corresponding to parallel (= Euclidean) leafwise metrics are precisely those which consist of leafwise horospheres, i.e. those for which (3{a{zi),a{z2)) = 0 for any zi,z2 from the same leaf in A. Theorem 1.14 A C-lamination A is uniformizable in the continuous (resp. Borel) category if and only if the Busemann cocycle on the lamination Ti is cohomologically trivial in the transversely continuous (resp. Borel) cohomology, i.e. if and only if there exists a transversely continuous (resp. Borel) function f : Tt —> R such that P(h1,h2)

= f(h2)-f(h1)

(1.15)

for any two points hi, hi from the same leaf in Tt. Proof. If a : A —> Ti is the section corresponding to a parallel leafwise metric on .4, then P(hi, h2) = P(hi,o- o p(hi)) + P(a o p(hi), a o p(/i 2 )) + /3(a o p(h2), h2) =

f(h2)-f(hi)

with f(h) = (3(<j o p(/i), h). Therefore, the graph of the section a coincides with the level set / _ 1 ( 0 ) .

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NON-EUCLIDEAN AFFINE LAMINATIONS

Conversely, if p satisfies (1.15), then t h e level set / _ 1 ( 0 ) is t h e graph of a uniquely defined parallel section a. Clearly, t h e function / and the section a are continuous (resp. Borel) simultaneously. • C o r o l l a r y 1.16 Let an affine lamination Abe a quotient of a C-lamination A with respect to a discrete group G of automorphisms. Then A is uniformizable in the continuous (resp. Borel) category if and only if the Busemann cocycle on A is cohomologically trivial by means of ascertain transversely continuous (resp. Borel) G-invariant function on fiA. Remark 1.17 In order t o make Theorem 1.14 valid for general affine laminations one has t o pass t o the leafwise de R h a m cohomology by replacing t h e B u s e m a n n cocycle with the appropriate differential 1-form (the differential of t h e Busemann cocycle with respect to the second argument). T h e n t h e above Corollary would follow from Theorem 1.14 directly applied to the quotient lamination A. 2

2. A

F o l i a t i o n s a n d l a m i n a t i o n s a s s o c i a t e d 'with t h e h y p e r b o l i c space Tautological

foliations

T h e simplest building blocks of an affine lamination are the s t a n d a r d affine planes. As we have seen, such a plane arises as the punctured visibility sphere Vq = 9 H 3 \ {q} of a pointed at infinity hyperbolic space ( H 3 , g ) . Conversely, the hyperbolization functor ^ allows one to recover the space ( H 3 , q) from Vq. Varying t h e b o u n d a r y points q € 9 H 3 we obtain a family of affine planes Vq, q G <9H3. D e f i n i t i o n 2.1 T h e tautological cally compact t o t a l space

C-foliation

«9 2 H 3 = <9H3 x d H 3 \ diag =

Ao is the foliation of the lo( J P , x {q} qedn3

(2.2)

with t h e leaves Vq. Its hyperbolization Ho = $)Ao is called the tautological pointed at infinity hyperbolic foliation. T h e total space of Ho is

H 3 x <9H3 =

( J H 3 x {q} qedH3

and the leaves are pointed at infinity hyperbolic spaces ( H 3 , g ) . T h e Busem a n n cocycle on Ho is P{(h1,q),(h2,q))=pq(h1,h2).

340

2.B

VADIM A. KAIMANOVICH

Parametrizations

of the unit tangent bundle

Denote the unit tangent bundle of the hyperbolic space by UH3 with the canonical projection p : f/H 3 -+ H 3 , and let 7 = {"fT}Tem be the geodesic flow on UH3. The endpoints of the geodesic determined by a tangent vector v 6 UH are denoted -y±00(v) G dH 3 . By Hor(w) = Hor 7 «. ( „ ) (p(u)) = {h £ H 3 : / ^ ( ^ ( p ^ K / i ) = 0} we denote the horosphere centred at the point 7°°(t;) and passing through the point p(v). Clearly, Hor(ui) = Hor(u 2 )

<=>•

Hor(7Ti>i) = Hor(7 T u 2 ) Vr e R ,

so that the formula 7 T Hor(u) = Hor(7 T u) ,

r e R

(2.3)

determines an action of the geodesic flow on the space Hor(H 3 ). There are

7°°(tO=<7+ = T,

7-°°(w)=9-

Figure 2.

two natural parametrizations of the space t / H 3 (see Figure 2).

341

NON-EUCLIDEAN AFFINE LAMINATIONS

Proposition 2.4 The map v^{p(v)n°°(v)) 3

3

(2-5)

3

from UH to the space H x 9 H is a dijfeomorphism. For any (h,q+) G H 3 x 9 H 3 the associated vector v G C/H3 is the directing vector of the geodesic ray issued from the point h in the direction q+. Proposition 2.6 The map v^(y-°°(v),Ror{v))

(2.7)

from UH.3 to the space 0H3xHor(H3)\{(g,T):g = Too}=

|J

7>Too x {T}

(2.8)

T<EHor(H3)

is a dijfeomorphism. For any g_ G <9H3, Y G Hor(H 3 ) with g_ ^ T ^ the associated vector v G C/H3 is the tangent vector to the geodesic joining qwith Too o-t the point of its intersection with the horosphere T. 2. C

Stable and strongly stable foliations

Recall the definitions of two natural foliations associated with the geodesic flow on UH3: Definition 2.9 Two vectors ui,i>2 G UH3 belong to the same leaf of the stable foliation Ws of the geodesic flow if lim sup dist(7*i>i, 7*^2) < 00 ,

(2-10)

t—>+oo

and to the same leaf of the strongly stable foliation Wss if lim dist(7'ui,7^ 2 ) = 0 ,

(2.11)

t—>+oo

where dist denotes the metric on UH3. Condition (2.10) means that 7°°(vi) = 7°°(i>2)- Therefore, Proposition 2.12 The identification (2.5) establishes an isomorphism between the foliations Ho and Ws. Condition (2.11) means that Hor(t;i) = H o r ^ ) (this is why W s s is also often called horosphere foliation). Therefore, Proposition 2.13 Under the identification (2.7) the foliation W s s is isomorphic to the foliation of the space (2.8) with the leaves Vy^ x {T}. Corollary 2.14 The foliation W s s is a Euclidean foliation.

342

VADIM A. KAIMANOVICH

Proof. Indeed, the leaves of W s s can be identified with horospheres in H , so that they are endowed with the natural Euclidean structures induced by the Riemannian metric on H 3 . • In terms of the coordinates (
7 T (g_,T) = (
(2.15) ss

so that it acts by affine laminar automorphisms of W mapping (not isometrically!) concentric horospheres one onto the other. The foliation .Ao is the result of factorization of Wss by the geodesic flow (so that the space of .4.0 has one dimension less than the space of Wss). This is the reason why the leaves of A) carry just a natural affine structure (the "scale" having been lost as a result of the action of the geodesic flow). On the hand, Ao is, of course, uniformizable. The total space of the hyperbolization $jWss is $)VToo x {T} = H 3 x Hor(H 3 ) .

|J 3

TeHor(H )

The Busemann cocycle on fyWs is f3((h1,T),(h2,T))=pTx(h1,h2).

(2.16)

The action of the geodesic flow on UH3 (2.15) induces its action on SjWss 7 T (/i,T) = (/ 1 , 7 T T)

(2.17)

by isometries between leaves. 2.D

Laminations associated with Kleinian groups

The roles of two factors <9H3 in the definition (2.2) of the total space 9 2 H 3 of the foliation Ao are quite different: the first one (the "leafwise direction" ) is indispensable if we want to have a C-lamination, whereas nothing prevents us from replacing the second one (the "transverse direction") with an arbitrary subset of 5 H 3 . Therefore, for any subset X C <9H3 the space

Ax = [J Vg x {q} qex is endowed with a lamination structure (this is not a foliation unless X is a submanifold of 9H 3 ). Definition 2.18 ([12]) Let G be a Kleinian group. The lamination AG — 3 -4A(G) > where A(G) C 9 H is the limit set of G, is called the affine lamination associated with the group G. The corresponding hyperbolic lamination

NON-EUCLIDEAN AFFINE LAMINATIONS

343

Ti-G — &-A.G associated with the group G is t h e product lamination of t h e total space H 3 x A(G). Since t h e limit set is G-invariant, the group G acts on AG by laminar affine maps and on Tic by laminar isometries. Moreover, the action of G on H 3 (and, therefore, on t / H 3 = H 3 x 9 H 3 ) is discontinuous, so t h a t it is also discontinuous on HG — H 3 x A(G). Denote by MG the corresponding quotient hyperbolic lamination. Remark 2 . 1 9 Although the quotient hyperbolic lamination MG always makes sense, the quotient of the affine lamination AG by the action of t h e group G is well-defined ( = the action of G on AG is discontinuous) only if G is an elementary group. 2.E

Non-triviality

of the Busemann

cocycle

If the group G is non-elementary, then the Busemann cocycle of the lamination MG is non-trivial for an obvious reason: it is non-trivial for t h e leaves in M G corresponding to fixed points of the hyperbolic elements in G. Denote by A 0 = Ao(G) C A t h e set of all such fixed points. It is not h a r d t o see t h a t even if we discard the set AQ (i.e. remove from MG all leaves with non-trivial Busemann cocycle), t h e n t h e Busemann cocycle of the lamination M'G = H'a/G

,

H'G = H 3 x (A(G) \ A„(G))

is non-trivial in the continuous cohomology. result:

We shall prove a stronger

T h e o r e m 2 . 2 0 For any non-elementary Kleinian group G the Busemann cocycle of the lamination M'G is non-trivial in the Borel cohomology. Proof. Fix for convenience a reference point o € H 3 (its choice is irrelevant for w h a t follows) and take a probability measure \x on the group G such t h a t t h e first moment ^Z„eG d(o, go) ^(g) is finite and fi(g) > 0 for all g £ G. Denote by /J,°° the product measure on the space G°° of sequences g = (01,52, ••• )• Every g € G°° gives rise to the (random) sequence hn = hn(g) = g\gi.. .gno € H 3 which /u°°-a.e. has t h e following properties (see [11]): (i) There exists a limit /loo = hoo(g) — lim hn £ dH3 , n—>oo

and t h e image v of the measure ^°° under the m a p g >-* h00(g) purely non-atomic.

is

344

VADIM A. KAIMANOVICH

(ii) If U denotes the Bernoulli shift (51,32, • • •) *-* (
.

(iii) There exists a number I = 1{JJ) > 0 (the same for a.e. g € G°°) such that -d(o,hn) 71

—> I . n—>oo

(iv) The distance between hn and the geodesic ray joining the points o £ H 3 and hoo e <9H3 is o(n). Combination of (iii) and (iv) implies that /i°°-a.e. -Phx(ho,hn)^l. (2.21) n Assuming that the Busemann cocycle on M.'G is trivial, i.e. there exists a G-invariant Borel function / on H.'G such that 0q(huh2)

= f(h2,q)-f(h1,q).

(2.22)

put F(g) =

f(o,h00).

As it follows from (i) above, i^(Ao) = 0, so that the function F is /z°°-a.e. well-defined. Then /3/joo (o, hn)

= f(hn,

/loo) - / ( O , /loo) = / ( 5 1 5 2 • • • 3 n » , /loo) - / ( O , /loo) 1

= f(o,g- ...

fljV^oo)

- /(o-^oo) = ^(f/ n 9) - ^ ( 9 ) •

Since U preserves the measure n°°, (2.21) would be then impossible by the Poincare recurrence theorem, which gives the sought for contradiction.

•

Corollary 2.23 There is no Borel G-equivariant map assigning to every point q G A(G) \ Ao(G) a horosphere centred at q. 3

An example of a non-Euclidean affine foliation

The first example of an affine foliation A which is not Euclidean was given by Ghys [5] (also see [6]) on the base of a construction of non-standard deformations, of Fuchsian groups due to Goldman [7] and Ghys [4] (note that a completely different example is given in [12]). Here we shall recast the example of Ghys by making more transparent its connection with the

NON-EUCLIDEAN AFFINE LAMINATIONS

345

foliations associated with t h e geodesic flow on H 3 . T h e dimension of our example is lower as instead of the group SL(2, C) of isometries of H 3 considered by Ghys in [5] we deal directly with the space C/H 3 (on the other hand, our example is real, whereas the foliation of Ghys is holomorphic). T h e argumentation is also different: it is based on the recent results [2], [10] on t h e ergodicity of the horocycle foliation on abelian covers of hyperbolic manifolds rather t h a n on the more restrictive ergodicity of the geodesic flow used by Ghys (which allows us to drop t h e compactness assumption, see Remark 3.13). For an arbitrary Kleinian group G the Busemann cocycle on t h e hyperbolic foliation A 4 G is cohomologically non-trivial (Theorem 2.20); however, M.G does not correspond to any affine foliation (see Remark 2.19). On t h e other hand, recall t h a t the tautological affine lamination *4o can be considered as a quotient of the strongly stable foliation W s s of the geodesic flow with respect t o the action of the flow (see formula (2.15) and the ensuing discussion). This factorization preserves t h e leafwise affine structure, but destroys the Euclidean structure. T h e example exploits t h e same idea, but, in order to have a discontinuous action, we replace the "whole" geodesic flow with a one-dimensional representation ( = character) of the group G. 3. A

Twisted

action

Take a compact hyperbolic manifold H and put G = iri(H). T h e actions of G and of t h e geodesic flow 7 on ITH3 commute (note t h a t only the first of these actions is isometric!). We shall now define a new "twisted" action of G on [ / H 3 by combining the original action of G with the geodesic flow. From now on we shall assume that the first Betti number of G is positive, i.e. the group Hom(G, R) = HX(H, R) of additive real-valued characters of G is non-trivial. Remark 3.1 Although H o m ( G , R ) is trivial for certain cocompact Kleinian groups, it is plausible (according to [15, p. 98]) t h a t if G is a lattice (in particular, if G is cocompact), t h e n it always has the so-called Millson property: there exists a finite index subgroup G ' c G such t h a t H o m ( G ' , R) is non-trivial. This property has been proved for several classes of lattices. D e f i n i t i o n 3 . 2 T h e twisted action Tx of the group G on UH3 determined by a character \ G Hom(G, R) is r»v =

9

o 7x<») (v) = 7 x ( 9 ) o g(y) ,

(3.3)

where in t h e right-hand side v 1—> gv is t h e s t a n d a r d action of G on C/H 3 (see Figure 3). We shall also use the same notation Tx for the action of the

346

VADIM A. KAIMANOVICH

group G on the space of horospheres Hor(H 3 ) defined by the formula T

ST

=

7 X(9)

0

ff(T)

=

g

0 7 X(9)( T ) ]

where 7 now stands for the action (2.3) of the geodesic flow on Hor(H 3 ) Then in the coordinates (q, T) (2.7) on C/H3 TS(q,T) = (gq.TST) .

Figure 3.

3.B

Admissible characters

Definition 3.4 A character \ S Hom(G,R) is admissible if the action Tx is free, proper and totally discontinuous. One can easily see that all characters sufficiently close to the identity in Hom(G,IR) are admissible. The following complete description due to Salein [14] was pointed out to us by Ghys. Note that although Salein deals with compact hyperbolic surfaces only, his proof based on a criterion of Benoist [1] almost verbatim carries over to compact hyperbolic manifolds of higher dimensions as well. Definition 3.5 The stable norm on Hom(G,R) is

11X11 = sup ^ gea Kg)

347

NON-EUCLIDEAN AFFINE LAMINATIONS

where 1(g) = min{d(h,gh)

: h € H3} ,

ge G,

denotes the length of the closed geodesic on H associated with the conjugacy class of g in the group G. Proposition 3.6 The action Tx is admissible iff \\x\\ < 1. Remark 3.7 As a motivation for this result note that if \(g) = ~K9) f° r a certain element g € G, then T£v = v for any vector v G C/H3 tangent to the axis of g. 3. C Absence of Euclidean structures The action Tx preserves the foliation Wss and its affine (but not Euclidean, unless x = 0!) structure. Definition 3.8 For an admissible character x £ Hom(G, R) denote by Bx the affine foliation of the quotient manifold UH.3/TX obtained by factorizing the strongly stable foliation W s s by the action Tx. For x = 0 the foliation BQ is just the strongly stable foliation of the geodesic flow on U"H3/G, so that it is endowed with a natural leafwise Euclidean structure. Theorem 3.9 If X ¥" ®> then the affine foliation Bx does not admit any Borel Euclidean structure. Proof. Since Bx is defined as the quotient of the strongly stable foliation Wss by the action Tx (3.3), by Corollary 1.16 we have to prove that the Busemann cocycle (2.16) on the hyperbolization f)W s s is not cohomological to zero by means of a certain Borel T x -invariant function: /3((hi, T), (h2, T)) = f(h2, T) - f(hlt

f(T°(h,?))=f(h,r)

T) ,

VgeG,

(h,T)ef)Wss-

We shall prove it under the only assumption that x ? ^ 0 (i.e. without requiring that the character x D e necessarily admissible). Indeed, if (3.10) were satisfied, then, as it follows from formula (2.16) for the Busemann cocycle on SjWss, the function / would be expressed as /(/>,T) = ^ ( T ) + / 3 T o o ( T , / i ) ,

(3.11) 3

where ip is a function on the space of horospheres Hor(H ), and fir^ (T, h) denotes the common value Pr^fi', h), h' e T. On the other hand, as it follows from (2.17), the action Tx on the hyperbolization SjWss has the form T9(h,T) = ( 5 / i , T | T ) .

348

VADIM A. KAIMANOVICH

Therefore, using (3.11) and the fact that (7TT)oo = Too for any r £ l and T € Hor(H 3 ), we obtain f(T°(h,

T)) = f(gh,T9T)

= tp{T°T) +

0groo(T°r,gh)

= cp(T9T) + /3 Too (7*<»>T, T) + /3 Too (T, h) =


The latter formula compared with (3.11) implies that / is T x -invariant iff the function ip on Hor(H 3 ) satisfies the relation V ^ T ) - x(9) =
V 5 e G, T € Hor(H 3 ) .

(3.12)

In particular, the function

349

NON-EUCLIDEAN AFFINE LAMINATIONS

5. E. Ghys, Sur I'uniformisation des laminations paraboliques, 73-91, in Intagrable systems and foliations, Progress in Math., 145. Birkhaiiser, Boston, 1997. 6. E. Ghys, Laminations par surfaces de Riemann, Panoramas et Syntheses, 8 (1999), 49-95. 7. W. Goldman, Non-standard Lorentz space forms, J. Diff. Geom., 21 (1985), 301-308. 8. W. Goldman, Projective geometry on manifolds, Preprint, 1988. 9. V.A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincare, Phys. Theor., 53 (1990), 361-393. 10. V.A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups, J. Dynam. Control Sys., 6 (2000), 21-56. 11. V.A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Math., 152 (2000), 659-692. 12. V.A. Kaimanovich and M. Lyubich, Conformal and harmonic measures on laminations associated with rational maps, Stony Brook, preprint 2001-5. 13. M. Lyubich and Y. Minsky, Laminations in holomorphic dynamics, J. Diff. Geom., 47 (1997), 17-94. 14. F. Salein, Varietes anti-de Sitter de dimension 3 possedant un champ de Killing non trivial, C. R. Acad. Sci. Paris, Ser. I 324 (1997), 525-530. 15. E.B. Vinberg, V.V. Gorbatsevich and O.V. Shvartsman, Discrete subgroups of Lie groups, 1-124, in Lie Groups and Lie Algebras II (A.L. Onishchik, E.B. Vinberg eds.), Springer, Berlin, 2000.

Received

November

7, 2000, revised September

20,

2001.

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Proceedings of F O L I A T I O N S : G E O M E T R Y AND D Y N A M I C S held in Warsaw, May 2 9 - J u n e 9, 2000 ed. by Pawel W A L C Z A K et al. World Scientific, Singapore, 2002 pp. 351-354

TRANSVERSE LUSTERNIK-SCHNIRELMANN CATEGORY AND NON-PROPER LEAVES REMI LANGEVIN Laboratoire de Topologie, Departement de Mathematiques, Universite de Bourgogne, B.P. 47 870, SI 078 Dijon, France, e-mail: [email protected] PAWEL G. WALCZAK Wydzial Matematyki, Uniwersytet Lodzki, ul. Banacha 22, 90-238 Lodz, Poland, e-mail: pawelwal@math. uni. lodz.pl T h e transverse Lusternik-Schnirelmann category for foliated manifolds (as well as its s a t u r a t e d version) has been defined in [2]. T h e article [1] contains t h e definitions, motivation and the survey of results concerning these notion obtained until now. In this note, we present some observations made recently by the authors and related to the presence of non-proper leaves in foliated manifolds. We would like to mention t h a t our results are valid in any codimension on arbitrary, a priori non-compact, manifolds. To begin with let us accept the following definition: A leaf L on a foliated manifold ( M , JF) has pathwise non-trivial holonomy when there exists a loop 7 : [0,1] —> L on L such t h a t the holonomy m a p /i 7 : V —> T, V being a neighbourhood of the origin 7(0) in a transversal T, admits no non-trivial p a t h s 5 : [0, e) —> T consisting of points fixed by /i 7 . More precisely, if S is such a p a t h , then there exists a sequence (sn) of positive reals such t h a t sn —> 0 as n —> 00 and h-y(5(sn)) / S(sn) for any n. Proposition neighbourhood for which HQ Ht(L) C L for

1 If a leaf L has pathwise non-trivial holonomy, U is an open of L in M and H : U x [0,1] —> M is a foliated homotopy = idu and H\{U) C L', V being a leaf, then V = L and any t 6 [0, 1].

Proof. Let A C [0,1] be the set of all t such t h a t HS(L) C L for any s
352

R. LANGEVIN AND P.G.

WALCZAK

Take a loop 70 : [0,1] —» L with pathwise non-trivial holonomy. Let o = 7(0), xt = Ht(x0) and j t = Ht ° 70 for t G [0,1]. Since -yto is a loop on L one can consider the holonomy map along it. This holonomy map is conjugated to that one along 70, so holonomy along 7t0 is also pathwise non-trivial. The curve [to, to + e) 3 t —> xt is non-trivial in the transverse direction, therefore there exists a sequence (s„) of reals such that sn > to, sn —* ^o as n —> 00 and x

h-,t0(xSn) ¥= xSn

(1)

for any n. Using a foliated chart around xto one can see that (1) implies that the leaf distances dyr(yn,xtri), where yn = /i 7t (xXn) satisfy the condition djr(yn,xsJ>A,neN,

(2)

where A > 0 is a constant which depends on the choice of the foliated chart only. (See Figure 1, where t — sn for a large n.)

T

L

U

Figure 1.

On the other hand, the leaf curves 7 t and j t , 7t(s) = hltQ|[0is](xt), have the same origin xj, stay in the same leaf and - for t sufficiently close to to project to curves in L. From the uniform continuity of H\jt0 ([0,1]) x [0,1] it

TRANSVERSE LUSTERNIK-SCHNIRELMANN CATEGORY

353

follows that these projections are uniformly close in L, therefore, 7t and j t are uniformly close in Lt, the leaf through xt, for t close to to- Consequently, dr(yn,Xs„.) —> O a s n ~ » oo.

(3)

Clearly, (3) contradicts (2). ^ P r o p o s i t i o n 2 If L is non-proper and has pathwise non-trivial holonomy, then L admits no open neighbourhoods which can be deformed into a leaf by a foliated homotopy. Proof. Assume that U is an open neighbourhood of L &nd H : Ux [0,1] -> M is a foliated homotopy with H0 = idy and HX{L) C L', U being a leaf. By Proposition 1, V = L and Ht(L) C L for any t. Fix xo € L and set j(t) = Ht(x) for t € [0,1]. 7 is path in L, so one can choose a chain V0,VU... ,Vk of foliated charts, members of a good foliated atlas on M, along 7. We can assume that Vi C V; C £/, We can also choose another good foliated chart W0 around x 0 and make it small enough to satisfy the condition (Vt € [0, l])(3i € { 0 , 1 , . . . , *}) Wt C V,

(4)

where Wt = Ht(W0). (If we like, we can add that Wi C 14.) Since WQ is connected, so is Wu and therefore Wx is contained in a single plaque of Vk. On the other hand, since L is non-proper, L n W0 consists of infinitely (but countable) many plaques. Therefore, there exist j € { 0 , 1 . . . , k} and arbitrarily close real numbers *i and t2 such that h < *2, Wtl U Wt2 C Vj, WH n L intersects at least two plaques while Wt2 is contained in a single plaque of Vj (see Figure 2), a contradiction. •

Figure 2.

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The above observation can be expressed in terms of transverse LusternikSchnirelmann categories as follows: Corollary 1 If a foliated manifold (M, T) contains a non-proper leaf with pathwise non-trivial holonomy, then the saturated transverse LusternikSchnirelmann category of T is infinite. • In the case of foliations of codimension-one we obtain also the following. Corollary 2 If a codimension-one C2'-foliation J7 of a compact manifold M contains an exceptional minimal set E, then the saturated transverse Lusternik-Schnirelmann category of J- is infinite. In fact, by Sacksteder Theorem, E contains a resilient leaf with linearly contracting holonomy, such a leaf is non-proper and has pathwise non-trivial holonomy. Comparing the above observations with those of [2] (and other papers listed in [1]) one can expect that any basic function on a foliated manifold (M, J7) has infinite number of critical leaves provided that J- has a nonproper leaf with pathwise non-trivial holonomy. Acknowledgments While preparing this article the second author was supported by the Universite de Bourgogne and by the Polish KBN grant P03A 033 18. References 1. H. Colman, LS-categories for foliated manifolds, in Foliations: Geometry and Dynamics, eds. P. Walczak et. al., World Sci. Publ., 2001, 17-28. 2. H. Colman, E. Macias-Virgos, The transverse Lusternik-Schnirelmann category of a foliated manifold, Topology, 40 (2000), 419-430.

Received June 18, 2001.

Procccdi/Tids of FOLIATIONS: G E O M E T R Y AND DYNAMICS held in Warsaw, May 2 9 - J u n e 9, 2000 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2002 pp. 355-370

S T R U C T U R A L L Y STABLE DIFFEOMORPHISMS HAVE N O C O D I M E N S I O N ONE PLYKIN A T T R A C T O R S O N 3-MANIFOLDS VLADISLAV

MEDVEDEV

Department

of Applied Mathematics, Nizhny State Technical University 24 Minina Str., Nizhny Novgorod 603600, e-mail: [email protected] EVGENY

Novgorod Russia,

ZHUZHOMA

Department

of Applied Mathematics, Nizhny State Technical University 24 Minina Str., Nizhny Novgorod 603600, e-mail: [email protected]

Novgorod Russia,

Let / be a structurally stable diffeomorphism of a closed 3-manifold M3. We show that the spectral decomposition of / has no non-orient able codimension one expanding attractors. This means that a structurally stable diffeomorphism of a closed 3-manifold has no codimension one Plykin attractors.

Introduction The most investigated structurally stable systems are codimension one Anosov diffeomorphisms and Morse-Smale systems (see reviews [1], [15], and [25]). For example, any codimension one Anosov diffeomorphism is conjugate to a hyperbolic torus automorphism ([4], [14]). The natural generalization of Anosov diffeomorphisms are diffeomorphisms satisfying the axiom A of Smale [25], so called yl-diffeomorphisms. A structurally stable A-diffeomorphism of the n-torus T " (n > 2) with an orientable codimension one expanding attractor fi can be obtained by Smale's surgery ([25], p.788-789) a codimension one Anosov diffeomorphism. They are called DAdiffeomorphisms, where the abbreviation DA means Derived-from-Anosov. 355

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Note that due to the orientability of ft, the accessible boundary of any component of T™ - ft from within T™ - ft consists of pairs of codimension one unstable manifolds, so-called 2-bunches (see definitions in Section 1). A first example of a codimension one non-orientable expanding attractor was constructed by Plykin [19] on a two-sphere S2. A feature of Plykin's structurally stable diffeomorphism / : S2 —» S2 is the existence of so-called 1-bunches. To be precise, if ft is a non-orientable codimension one (i.e., onedimensional) expanding attractor of / , then there exists a component of S2 - ft with an accessible boundary consisting of a unique one-dimensional unstable manifold. In [21], Plykin constructed a codimension one nonorientable attractors with 1-bunches on open n-manifolds, n > 3 (in the same paper, he classified, up to conjugacy, diffeomorphisms restricted on some neighbourhoods of attractors). Bearing in mind these examples, we call that a codimension one expanding attractor ft is a codimension one Plykin attractor whenever ft has 1-bunches. Let us give the sketch of Plykin's construction. Denote by J : T™ - • Tn the involution x —> —x (mod 1) which has 2™ fixed points v\,... , «2" • Take the DA-diffeomorphism g :Tn -» Tn with the codimension one orientable expanding attractor ft9 such that g commutes with J and has the fixed points vi, . . . ,i>2"- Due to Theorem 2.3 [21] (see the careful construction in [11], [23] for n = 2 and [5] for n = 3 as well), such g exists. Denote by q : Tn —> Tn IJ the natural projection which is a double branched covering with the branch points v\, . . . , v2^• It is not hard to see that the quotient space T2/J is a 2-sphere. Since g{—x) — —g(x), we obtain an induced diffeomorphism f : S2 -^ S2 with the one-dimensional Plykin attractor g(ft s ). As to n > 3, the branch points must be removed to get a codimension one Plykin attractor on a smooth manifold. Thus, M = q(Tn — U2=1i>i) is an open smooth n-manifold, n > 3, which admits an induced diffeomorphism with the codimension one Plykin attractor g(ftg). As to closed n-manifolds, n > 3, up to now, there are no examples of diffeomorphisms with codimension one Plykin attractors. Nevertheless, such diffeomorphisms were investigated. In [12], one proved that if a closed n-manifold Mn, n > 3, admits a codimension one expanding attractor (orientable or non-orientable), then Mn has a nontrivial fundamental group. In particular, there no diffeomorphisms of n-sphere Sn, n > 3, with codimension one Plykin attractors. The more precise result was obtained in [21], where one proved that ^ ( M " ) contains a subgroup isomorphic to the integer lattice Z n . In [17], Plante proved that an orientable codimension one expanding attractor defines a nontrivial element of homology group. In particular, a closed manifold Mn has a nontrivial homology group Hi ( M n ) .

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In this paper we prove that a structurally stable diffeomorphism of a closed 3-manifold has no non-orientable codimension one expanding attractors. Theorem 3.1 Let f : M3 -> M3 be a structurally stable diffeomorphism of a closed 3-manifold M3. Then the spectral decomposition of f does not contain non-orientable codimension one expanding attractors. This theorem and the following lemma Lemma 2.2 A codimension one expanding attractor fi of an A-diffeomorphism f : M3 —> M3 is nonorientable iff ft is a Plykin attractor. imply that a structurally stable diffeomorphism of a closed 3-manifold has no codimension one Plykin attractors. 1

Main definitions

We begin by recalling several definitions. Further details may be found in the books [11], [16], [23], and reviews [1], [25]. A-diffeomorphisms. Throughout the paper M " is a closed n-dimensional manifold endowed with some Riemannian metric d. Recall that a diffeomorphism / : Mn -> Mn is said to be an A-diffeomorphism if the nonwandering set NW(f) of / is hyperbolic and periodic points are dense in NW(f). The stable manifold Ws(x) of a point x 6 NW(f) is defined to be the set of points {y £ Mn such that d{f^x, fiy) —> 0 as j —> +oo}. A stable manifold is an injectively immersed Euclidean space for each x e NW(f). The unstable manifold Wu(x) of x is the stable manifold of x for the diffeomorphism f-1. By definition, let W*{x) C Ws{x) (resp. W?(x) C Wu{x)) be an eneighbourhood of x in the intrinsic topology of the manifold Ws(x) (resp. Wu(x)), where e > 0. Basic sets. Let / be an A-diffeomorphism of a n-dimensional manifold Mn. It was shown by Smale [25] that NW(f) is a finite union of pairwise disjoint /-invariant closed sets fii, . . . ,fifcsuch that every restriction f\ilt is topological^ transitive. These ft, are called the basic sets of / . A basic set ft is called an expanding attractor if there exists a closed neighbourhood U of fl such that f(U) C int U, C\j>of*{U) = Q, and the topological dimension dimO of fi is equal to the dimension dim(£Q) of the unstable splitting EQ. If dimfi = n — l, then Q is called an expanding attractor of codimension one. It is well known (see e.g., [18], [26]) that a codimension one expanding attractor consists of the (n — l)-dimensional unstable manifolds Wu(x),

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i £ ( l and is locally homeomorphic to the product of (n - l)-dimensional Euclidean space and a Cantor set. For dimE^ = 1, we shall denote by (x,y)s (resp. [a;,2/]s) an open (resp. closed) arc of Ws(z), z € ft, with endpoints x, y G W(z). By definition, let W*{x) C Ws{x) (resp. W?{x) C Wu(x)) be the eneighbourhood of x in the intrinsic topology of the manifold Ws(x) (resp. Wu(x)), where e > 0. Following [6], a basic set ft is called orientable if for any a > 0, /? > 0 the index of the intersection W£(x) n WJf(a;) does not depend on a point of this intersection (it is either + 1 or —1). Structurally stable diffeomorphisms. Let Diff1(Mn) be the space of C 1 diffeomorphisms of Mn endowed 1 with the uniform C topology. Two diffeomorphisms f,g£ Diff1(Mn) n n are called conjugate if there is a homeomorphism ip : M —> M such that

denoted by

Bunches of codimension one expanding attractors. Let G C M be an open set with a boundary dG. The subset 5(G) C dG is called an accessible boundary from G if for every point x E S(G) there exists an open arc a which is in G and x is an endpoint of a. Let ft be a codimension one expanding attractor of / . Due to [6], [20], the accessible boundary S(M —ft)is a finite union of unstable codimension

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one manifolds, each one is called a boundary unstable manifold through a boundary periodic point of ft. The boundary unstable manifolds of fi splits into a finite number of so called bunches in the following way. The family of pairwise disjoin unstable manifolds W u ( p i ) , . . . , Wu(pk) is said to be a k-bunch if there are points Xi £ Wu(pi) and arcs (xi,yi)s^, yi £ Wu(pi+\), l
Figure 1. (a) 1-bunch, (b) 2-bunch, and (c) 3-bunch.

Ply kin [21] proved that if n = dim M > 3, then any bunch of £} is either a 1-bunch or 2-bunch. It is not difficult to see that if Cl is orientable, then every bunch of 0 is a 2-bunch (see the beginning of the proof of Lemma 2.2). So the boundary periodic points of orientable fi split into disjoint pairs of associated points. A codimension one expanding attractor ft is called a codimension one Plykin attractor if fi has 1-bunches. 2

Preliminaries

In the next lemmas we keep the following notation. Let fi be an expanding attractor of codimension one and let B be a bunch of fi. According to [21], B is either a 1-bunch or 2-bunch. In the last case, B consists of two unstable manifolds Wu(p) and Wu(q), where p, q are boundary associated periodic points. If B is a 1-bunch, we will assume that Wu(p) = Wu(q) and p — q. The following lemma holds.

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Lemma 2.1 There is a homeomorphism V : (Wu(p) -p)U

(Wu(q) - q) -> (Wu(p) - p) U (Wu(q) - q)

with the following properties • Given any point x £ Wu(p) — p,
Ws(x)

and moreover, (x,
intersects

• If m is the period of p, then m is the period of q as well and moreover, V ° fnrn\(W(P)-P)U(W(q)-q)

= I""1

°
where n £ Z. In particular, the maps f , p)U(Wu(q)-q).

(1) u

ip commute on (W (p) —

• (p extends to the homeomorphism (denoted by the same letter) ip : Wu{p) U Wu(q) -^ Wu(p) U Wu(q), by letting ip(p) = q and ip(q) = p. • (f2 = id. Proof. This lemma is a consequence of Lemmas 2.1, 2.2 [6], and Theorem 2.1 [21], For the reader's convenience we give the sketch of the proof. Given any point x 6 Wu(p) there is a unique point y 6 Wu(q) such that (x,y)s = (x,2/)g, and vise versa (see Section 1). Let the map ip : {Wu{p) - p) U (Wu(q) -q)->

(Wu(p) - p) U {Wu{q) - q)

be given by (f(x) = y whenever (x, y)s = (x, y)^. By this definition, ip2 = id. Thus, ip(Wu(p) -p)

= (Wu(q) - q) and
According to the theorem on the continuous dependence of stable manifolds on initial conditions [25], ip is a homeomorphism. Due to the invariance of fi, the point q has the same period m as p. Since Ws(Cl) is invariant under / , we have that f"1 °
=V°

fm\(W(p)-p)U(W»(q)-q)-

It implies the relation (1). Because of the restriction fm\wu(p)-P has the only hyperbolic fixed point p, there is a closed disk Dp C Wu(p) bounded by the closed curve Sp = dDp homeomorphic to the circle such that p € int(Dp) and Dp C

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STRUCTURALLY STABLE DIFFEOMORPHISMS

int(fm(Dp)). Since tp is a homeomorphism, S^ =
C\fim(DP)=P,

f)fjm(Dq) = q.

j<0

j<0

Due to (1), V o fjm(Dp

-p)=

Pm o V{Dp -p)

= fjm(Dq

- q).

Similarly, tp o fJm(Dq — q) = fjm(Dp — p). Hence, ip extends to the homeomorphism Wu{p) U Wu{q) ->• Wu{p) U Wu{q) if we put
a

The following theorem is the crucial step to prove the main theorem. Theorem 2.1 Let f : M3 -> M 3 be an A-diffeomorphism of a closed 3manifold M3. If f has a codimension one Plykin attractor Ct, then M3 is non-orientable. Proof. Because of Q, is a Plykin attractor, fi has 1-bunches. Let Wu(mo) C Q be the unstable manifold forming a 1-bunch of ft, where mo is a boundary periodic point. Without loss of generality we can assume that mo is a fixed point of / . According to Lemma 2.1, there is the homeomorphism tp : Wu(m0) —> u W (m0) such that: 1. For x <= Wu(m0) - m0, tp(x) <E (Wu(m0)

- m 0 ) n Ws{x).

2. For x € Wu{m0) - m 0 ,

(x,V(x)y

=

(x,^(x))llcWs(x).

3-

• Do n ip(Do) = 0.

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• Dai c Da2 whenever a\ < a.^. • The boundary dDa depends continuously on a. • i.

Such a family of the disks Da exists because the unstable manifold Wu(mo) is homeomorphic to the Euclidean plane R 2 and XQ ^ ip(xo)-

9(D S ) Figure 2. The curve C = C 0 i U C\%.

Step 2.1 There exists the index a^ € (0; 1) such that

d(ip(Dao)).

Proof of Step 2.1. If ct\ < a.2, then tp(Dai) C ip(Da2) because

(xo) £ -^l- Since the boundary 9£) a depends continuously on a and ao with no self-intersections and such that C01 connects the points XQ, x\. Due to step 2.1, the interior of Dao and the image of the interior under

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363

ip have no intersections. Therefore, y(Coi) C tp(Dao) is a path with no self-intersections. Moreover, (C) = {a;0,
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Figure 3. The disk Dc and the strip P = Pi U ip(Pi) Ul0U
Let C\ be the side of the rectangle Pi C Dc which is opposite to the side de f

C = C0. Note that the other pair of the opposite sides /0 and (x))l, which is homeomorphic to an open Mobius band Mj,. Since stable manifolds are pairwise disjoint, we see that the sets Ma are pairwise disjoint as well. The theorem on the continuous dependence of invariant manifolds on initial conditions [25] implies that the sets Ma depend continuously on a. Since the strip P is homeomorphic to the product S1 x (0,1) (here S1 is a circle), it follows that the open subset C 3 = ^xeP{x,ip{x))l

= UQ(E(o,i)Ma

is homeomorphic to the product Mb x (0,1). o It is well known that a Mobius band is a non-orientable surface. Hence, C 3 is a non-orientable 3-manifold. As a consequence, M 3 is a non-orientable 3-manifold as well [9]. This concludes the proof. • Corollary 2.1 Suppose the condition of Theorem 2.1 holds and R is a component of M3 — Q, such that the accessible boundary S(R) of R contains

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a 1-bunch of fi; then the fundamental group m (R) is nontrivial. Proof. By the proof of Theorem 2.1, there is an open subset C3 C R n (Ws(Cl) - ft) homeomorphic to the product M& x (0,1), where Mb is an open Mobius band. Therefore, R is a non-orientable open manifold. It follows the nontriviality of 7r1(J?). • Recall that a codimension one foliation T on M 3 is called a Reeb foliation if every leaf of T is homeomorphic to a plane K2 (in the intrinsic topology) [24]. The following lemma says that the nonorientability of a codimension one expanding attractor fl is equivalent to fi being a Plykin attractor. L e m m a 2.2 A codimension one expanding attractor fl of an A-diffeomorphism f : M 3 —> M3 is nonorientable iff ft is a Plykin attractor. Proof. Let 0 is a codimension one Plykin attractor. Then VI has a 1bunch Wu(m0), where m 0 is a boundary periodic point. Hence there exists an arc (x,y)ll = (x,y)s

C\Vs(x),

(x,y) s f~l ft = 0,

x,y € Wu{m0) - m0.

Take a disk D C Wu{mQ) such that x,y £ D. Since D C fl, D has a neighbourhood U with a local product structure such that the component oiUr\Wu{m0) containing D divides U into two open domains, say Ui and U2, each one homeomorphic to a 3-ball. Note that the unstable manifold Wu(mo) can be endowed with a normal orientation because Wu(m0) is homeomorphic to Euclidean space M2. Therefore, is we suppose that the arc [x, y]J intersects Wu(mo) at x, y with the same index of intersection, then (x,y)^ has to intersect both Ui and U2- Since (x,y)s n 0 = 0 and U has a local product structure, [/; fl fi = 0, i = l , 2 . This contradicts the property of O being locally homeomorphic to the product of a Cantor set and a ball [18], [26]. As a consequence, arc [x,y]0 intersects Wu(mo) at x, y with opposite indexes of intersection, and thus fl is nonorientable. Let us assume now that Q is a nonorientable codimension one expanding attractor. We have to prove that ft is a Plykin one. Suppose not. Then all bunches of il are 2-bunches. Take a 2-bunch B C fl and denote by K the component of M3 — O such that the accessible boundary of K contains B, C SK. Since / is structurally stable, it follows that KUB is homeomorphic to the product M2 x [0; 1] (in the intrinsic topology) [8]. This fact and Plykin's Theorem on the embedding of Cl into a codimension one foliation (see the Theorem on invariant foliations [21], p.93) imply that Vl can be embedded into a Reeb foliation denoted by T. Moreover, since the unstable manifolds of ft form a C1 lamination [10] (see also [23], Ch. 12), we can

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assume that T is a C1 foliation as well. In [24], Rosenberg proved that if a closed three-manifold M3 admits a C2 Reeb foliation, then M 3 is homeomorphic to the three dimensional torus T 3 . The main technic in the paper [24] is a moving of an embedded 2-disk D2 C M 3 in a general position under the C 2 Reeb foliation T so that J- induces on D2 a foliation with Morse's singularities of the center and saddle types. Franks (see the proof of Lemma 5.1 [4]) showed that Rosenberg's technic works for C 1 foliation as well. So, M 3 — T3. Let us show that T is transversally orientable. Let TT : M3 —> T3 = 3 R / Z 3 be a universal cover, where Z 3 is the group of integer shifts of R 3 which is isomorphic to the fundamental group 7Ti(T3). Let T be the covering foliation for T under TT. Obviously, T is a Reeb foliation. Since R 3 is simply connected, T is transversally orientable. Due to [2], one can assume that T is defined by a Pfaff 1-form of the kind dz = P(x, y, z)dx + Q(x, y, z)dy, where (x,y,z) are Cartesian coordinates. It follows that T is transversally oriented because the group Z 3 of covering maps consists of Euclidean translations which preserve orientation of 1R3. Hence the lamination formed by the unstable manifolds of points of fi is transversally oriented as well. This implies that 0 is an orientable attractor. The contradiction concludes the proof. • 3

Proof of the main theorem

Theorem 3.1 Let f : M 3 -> M3 be a structurally stable diffeomorphism of a closed 3-manifold M 3 . Then the spectral decomposition of f does not contain non-orientable codimension one expanding attractors. Proof. Assume the converse. Then the spectral decomposition of / contains a codimension one Plykin attractor, say O. By definition of Plykin attractors, there is an open component R C M3 —fisuch that the accessible boundary of R contains the 1-bunch Wu(mo) C fi, where mo is a boundary periodic point. Without loss of generality we can assume that m 0 is a fixed point, passing if necessary to some degree of / . According to Theorem 2.1, M 3 is non-orientable. Let M3 be an orientable manifold such that p : M3 —> M 3 is an (nonbranched) double covering for M3. Then there exists a diffeomorphism / : M3 —> M3 which cover / i.e., f op = po f. Let us prove that / is an A-diffeomorphism. It is obvious that if x 6 M3 is a nonwandering point of / , then p(x) is a nonwandering point of / because p is a local homeomorphism. Therefore,

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367

NW(f) C p~1(NW(f)), where NW(g) means the nonwandering set of the map g. Because / is an A-diffeomorphism, the periodic points of / are dense in NW(f). Since p is a double covering, we see that a preimage of any periodic orbit of / is either one periodic orbit or two periodic orbits of / . As a consequence, the periodic orbits of / are dense in p~1(NW(f)). Hence, p-^NWif)) c NW(f). This proves the equality NW(f) = p-^NWif)). The hyperbolicity of NW{f) implies the hyperbolicity of NW(f) because p is a local diffeomorphism. Thus, / is an A-diffeomorphism. Because of / is a structurally stable diffeomorphism, / satisfies to the strong transversality condition [13]. Since p is a local diffeomorphism, we see that / satisfies to the strong transversality condition as well. By Robinson's theorem [22], / is a structurally stable diffeomorphism. Let us show that the preimage p _ 1 (fi) contains a codimension one expanding attractor. By definition of a basic set, there is an orbit O(xo) of some point xo £ fi which is dense in fi. The preimage p~1(xo) consists of two points xi, x2. Clearly, the union 0(xi) U 0(x2) is dense in p _ 1 ( 0 ) , where 0(xi) is the orbit of the point Xi (i = 1,2). Thus, clos O(xi) U clos 0{x2) = clos {0(xx) U 0(x2)) = P _ 1 ( ^ ) , where clos (N) means a topological closure of N. Obviously, each of the set clos 0{x\), clos 0(2:2) is closed, invariant, and contains an everywhere dense orbit. As a consequence, clos 0(xi) and clos 0(x2) are basic sets of / . Since a spectral decomposition is unique [25], it follows that either p _ 1 (ft) = clos 0(xi)

= clos

0(x2)

is one basic set or clos 0{xi), clos 0{x2) are different basic sets. In both cases, we denote clos 0(£i) by Cl. Because 0 is a codimension one expanding attractor, Cl is locally homeomorphic to the product of a Cantor set and the plane R 2 . Hence, 0 is locally homeomorphic to the product of a Cantor set and the plane R2 as well. It follows that the topological dimension of Cl is equal to the dimension of the unstable splitting restricted on Cl. Since p is a local diffeomorphism, dimE% = 2, i 6 fi. As a consequence, 0 is an expanding attractor of codimension one. Recall that the manifold M 3 is orientable. Due to Theorem 2.1, the expanding attractor fi has no 1-bunches. Plykin [21] proved that for the manifold dimensions dim M™ > 3 any codimension one expanding attractor can contain 1-bunches and 2-bunches. Hence, H contains only 2-bunches. Let fh\ G fi be a lift under p of the point mo. Because p is a double covering, rhi is a boundary periodic point of the codimension one attractor

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Cl. Since every bunch of fi is a 2-bunch, it follows that there is the boundary periodic point m 2 £ 0, m 2 ^ rhi, associated with rhi. Let W^(rhi) (1 = 1,2) be the component of Ws{fhi) —rrii such that flr\W^(rhi) = 0. In [8], it was shown that W^(rhi) and W^rh-z) belong to the unstable manifolds Wu{a.\) and Wu(ak+i) respectively of the repelling periodic points d i , dfc+i (note that the proof of this result uses the fact that M3 is closed), Figure 4. Moreover, there are repelling periodic points d i , . . . ,&k+i and saddle periodic points Pi, . . . , Pt of index 2 such that the following conditions hold: 1. The set 012 = {m 1 }UTy 0 s (mi)UdiUP^ s (Pi)UPiUd 2 U...UP fc Ud i;+ iUTy 0 s (m 2 )U{m2} is homeomorphic to an arc with no self-intersections whose endpoints are rhi and m 2 . 2. a i 2 — (mi Um 2 ) C M3 — 0 . 3. The repelling periodic points di alternate with saddle periodic points P, on the arc ai2-

Figure 4. The arc 012.

Let us prove that p(m 2 ) = mo- According to Lemma 2.1, if xn —» mi as n —> oo, then yn —>• m 2 , where x n 6 W u ( m i ) , y„ € Ws{fh,2), and {.xn,yn)s = {xn,yn)s0- Since (£„,y„) s Uf2 = 0 and p(fi) = ft, it follows that M ( * n > y n ) 0 ) = (P(^n),P(jfn))0-

The inclusionp(x n ) € Wu(m0) implies p(yn) £ Wu(rrio) because of Wu(rrio) is a 1-bunch. Since p is a continuous map, p(xn) —> p(rhi) = mo and P(Vn) -> p(m 2 ). From the inclusion p(yn) G Wu(mo) and Lemma 2.1 it follows that p(yn) —> mo. Hence, p(m 2 ) = m 0 . Because of the relation P°f = f°p, p maps the invariant manifolds of / into the invariant manifolds of / . Then

# | W )

=p(Wi(fh2)),p(a1)

=p(ak+1),

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STRUCTURALLY STABLE DIFFEOMORPHISMS

p(W s (Pi)) = p{Ws{Pk)),p{Pl)

= p(Pk),p(a2)

=

p(ak),....

Since the repelling periodic points en alternate with saddle periodic points Pi on the arc a\2, we see that the number of periodic points on a\2 is odd because the endpoints both are of saddle type. As a consequence, there is either a periodic point ctj with p(Ws(Pi-i)) = p(Ws{Pi)) or a periodic point Pi with p{W({Pi)) = p(W£(Pi)), where W?(Pi), Wj(Pi) are different components of Ws(Pi) — Pi. In both cases, there is a point (a; or Pj) at which p is not a local homeomorphism. This contradiction concludes the proof. • Acknowledgments The research was partially supported by the INTAS grant 97-1843 and RFBR grant 99-01-00230. The authors are grateful to D. Anosov, S. Aranson, V. Grines and M. Malkin for useful discussions. We thank a referee for useful remarks. References 1. D. Anosov. and V. Solodov, Hyperbolic sets, in Sbornik of ser. "Modern Problems of Math." ed. D. Anosov, Dynam. Systems-9, 66 (1991), 12-99. 2. S. Aranson and E. Zhuzhoma, On topological equivalence of codimension one Reeb foliations on a three dimensional torus, in Methods of Qual. Theor. of Diff. Eq., Gorky, (1978), 41-44 (in Russian). 3. R. Bowen, Periodic points and measures for axiom A diffeomorphisms, Trans. Amer. Math. Soc, 154 (1971), 337-397. 4. J. Franks, Anosov diffeomorphisms, in Global Analysis, Proc. Symp. in Pure Math., AMS 14 (1970), 61-94. 5. J. Franks and C. Robinson, A quasi-Anosov diffeomorphism that is not Anosov, Trans. Amer. Math. Soc, 223 (1976), 267-278. 6. V. Grines, On topological conjugacy of diffeomorphisms of two a dimensional manifold onto one-dimensional orientable basic sets I, Trans. Moscow Math. Soc, 32 (1975), 31-56. 7. V. Grines, On topological conjugacy of diffeomorphisms of two a dimensional manifold onto one-dimensional orientable basic sets II, Trans. Moscow Math. Soc, 34 (1977), 237-245. 8. V. Grines and E. Zhuzhoma, Structurally stable diffeomorphisms with codimension one basic sets, preprint, Universite de Bourgogne, Laboratoire de Topologie, Dijon, no 223 (2000).

370

V. MEDVBDEV AND E.

ZHUZHOMA

9. J. Hempel, 3-manifolds, Princeton, Annals Math. Studies, 86 (1976). 10. M. Hirch and C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis, Proc. Symp., Amer. Math. Soc, 14 (1970), 133-163. 11. A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems, Encyclopedia of Math, and its Appl., Cambridge Univ. Press. 12. H. Kollmer, On hyperbolic attractors of codimension one, in Geometry and Topology, Lect. Notes of Math., 597 (1976), 330-334. 13. R. Mane, A proof of C1 stability conjecture, Publ. Math. IHES, 66 (1988), 161-210. 14. S. E. Newhouse, On codimension one Anosov diffeomorphisms, Amer. J. Math., 92 no 3 (1970), 761-770. 15. I. Nikolaev and E. Zhuzhoma, Flows on 2-dimensional manifolds, Lect. Notes in Math.,1705, Springer Verlag, 1999. 16. Z. Nitecki, Differentiable dynamics, MIT Press, Cambridge, 1971. 17. J. Plante, The homology class of an expanded invariant manifolds, Lect. Notes Math., 468 (1975), 251-256. 18. R. V. Plykin, On the topology of basic sets of Smale diffeomorphisms, Math. Sbornik, 84 (1971), 301-312 (in Russian). 19. R. V. Plykin, Sources and sinks of A-diffeomorphisms of surfaces, Mat. Sbornik, 94 (1974), 223-253. 20. R. V. Plykin, On hyperbolic attractors of diffeomorphisms, Usp. Math. Nauk, 35 3 (1980), 94-104 (in Russian). 21. R. V. Plykin, On the geometry of hyperbolic attractors of smooth cascades, Usp. Math. Nauk, 39 (1984), 75-113 (in Russian). 22. C. Robinson, Structural stability of C 1 diffeomorphisms, J. Diff. Eq., 22 no 1 (1976), 28-73. 23. C. Robinson C, Dynamical Systems: stability, symbolic dynamics, and chaos, Studies in Adv. Math., Sec. edition, CRC Press. 24. H. Rosenberg H, Foliations by planes, Topology, 7 (1968), 131-138. 25. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc, 73 1 (1967), 741-817. 26. R. Williams, Expanding attractors, Publ. Math. I.H.E.S., 43 (1974), 169-203.

Received June 7, 2000, revised January 5, 2001

Proceedings of FOLIATIONS: GEOMETRY AND DYNAMICS held in Warsaw, May 29-June 9, 2000 ed. by Pawel WALCZAK et al. World Scientific, Singapore, 2002 pp. 371-386

ON EXACT POISSON MANIFOLDS OF DIMENSION 3

TADAYOSHI MIZUTANI Saitama

Department of Mathematics, Faculty of Science, University, 255 Shimo-Ookubo Urawa, Saitama 338-8570, e-mail: [email protected]

Japan,

A Poisson manifold is called " exact" if its Poisson bi-vector field represents a trivial element in the Poisson cohomology. We investigate some topological properties of the foliation which is associated with an exact Poisson manifold and construct explicit examples of exact Poisson structures on closed 3-manifolds which are different from homogeneous ones.

1

Introduction

A Poisson manifold is a pair (M, II) of a C°°-manifold and a 2-vector field on it, which satisfies [11,11] = 0 , where [• , •] denotes the Schouten bracket. We call the condition [II, II] = 0 for a 2-vector field II the Poisson condition and II the Poisson bi-vector field. The Poisson bracket {/, g} of f,ge C°°(M) is then defined by {f,g} = U(df,dg). It satisfies the following well-known properties. (1) (/,-> {/, g) f,g e C°°(M) gives a Lie algebra structure (over R) on C°°(M), that is, the pairing {/,g} is skew-symmetric bilinear and satisfies the Jacobi identity {/, {g, h}} + {g, {h, / } } + {h, {/, g}} — 0, (2) {/, gh} = {/,g}h + g{f, h} holds for f,g,he

C°°{M).

For any 2-vector field II on M, we define a homomorphism of bundles I = IU- T*M -> TM 371

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TADAYOSHI MIZUTANI

which is given by Ix{ax) = Ux(ax, •) = i Q x n ,

ax € T*M,

at each point x G M. Here, we used the notation of interior product to express a contraction of tensors. The rank of the linear map Ix is called the rank of H at x and it is denoted by rank Hx. If the rank Il x is constant on the whole manifold, (M, II) is called regular. In this paper, we are mainly concerned with regular Poisson manifolds. One of the geometric aspects of a Poisson manifold (M, II) is the fact that the distribution (plane field) given by Image/ X
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ON EXACT POISSON MANIFOLDS OF DIMENSION 3

the p-th exterior space bundle of the tangent bundle of M and T(APTM) denotes the set of smooth section of it, that is, the space of p-vector fields. 2

Generalized divergence and the Schouten bracket

Let M be a smooth manifold and P a p-vector field, that is, P £ ( p > 0 ) . Let c : T{{T*M))®T{AP(TM))

->

T(AP(TM))

T{A{p-l)(TM))

denote the contraction. Definition 1 Let V be a connection (covariant differentiation) on M and P a p-vector filed. Then the (p — l)-vector field DivyP given by D i v v P = c(VP) is called a generalized divergence of P associated with the connection V. It is shown that if V is the Levi-Civita connection of a Riemannian metric and P = X is a vector field, Divv-X coincides with the usual divergence divX with respect to the Riemannian volume Cl, i.e. L\Q = (divX)Q (Lx is the Lie derivation). Although the generalized divergence of a p-vector field depends on the choice of the connection V, we often omit V and write DivP for DivvP. It is not always true that Div2 = Div o Div = 0. It is proved, however, if one chooses a connection which preserves a volume form that Div2 = 0 holds. In fact, if V preserves a volume form Ct, one can see the following relation of Div and d(= exterior differential) holds. d(Q(P)) = ( - l f f t ( D i v P ) ,

(p = deg P).

One of the definition of the Schouten bracket [P, Q] is the following ([6]). Definition 2 Let Div be a generalized divergence associated with a torsion free connection of M. Let P, Q be p-vector field and q-vector field on M, respectively. The (p + q — l)-vector field [P,Q] defined by [P, Q] = Div(P A Q) - (DivP A Q + (-l)pP

A DivQ)

is called the Schouten bracket of P and Q. It is proved that [P, Q] is well-defined, namely, it is independent of the choice of the torsion free connection involved. The following is a list of some basic properties of the Schouten bracket ([9]). Here, f,g are smooth functions and P,Q,R are a p-vector field, a q-vector field and an r-vector field, respectively. Also, we use the interior product notation for the contraction.

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TADAYOSHI MIZUTANI

1. For f,geC°°{M),

{f,g} = 0,

2. [/, Q] = idfQ, more generally, [fP, Q) = ( - 1 ) " P A (idfQ) + f\P, Q], 3. [P,Q] = (-1)">[Q,P], 4. Let P = X be a vector field, then [-X", <J] = Xx<3 (the Lie derivative), 5. [P,QAR]

= [P,Q]AR + ( - 1 ) ( P - D « Q A [P,R],

6. Div[P,Q] = -[DivP,Q] - (-l)P[P,DivQ], when Div 2 = 0, 7. [P, [Q,P]] = ( - ^ [ [ P . Q l . P ] + ( - l j d ' - 1 " ' - 1 ) ^ , [P,J2]]'(generalized Jacobi identity). 2.1

Integrability of a plane field

Here, we briefly discuss the integrability of the plane field which is given as the image of a 2-vector field. Let II be a 2-vector field on M. Assume the rank of II is equal to 21 (0 < 21 < dimM) everywhere on M. Recall that II defines a distribution T\i which gives the following subspace of TXM; ^n,x = { n x K , •) G TxM\ax

£ T*M}.

We prove the following Theorem 1 The distribution JFn defined by a regular 2-vector field U whose rank is 21, is integrable if and only if [U, Ul] — [n, II A • • • A n] = 0 holds. We first prove following formula. Lemma 2 If rank II = 21, then

[n,ul] =-2Dwn AU1 , where Divn is defined by choosing any torsion free connection on TM. Proof. Since II A II' = 0, we have [n,n'] = - D i v I l A l I ; - I T A D i v I l ' .

(1)

Plugging the following

Divn' = Divn A n'- 1 + n A Divn'-1 + rn, n'-1]

(2)

into the above (1), we have

[n,n'] = -2DivnAn / -n 2 ADivn'- 1 -nA[n,n'- 1 ].

(3)

Again plugging (2) for I — 1 into (3), we obtain

-3Divn A n' - n 3 A Divn'-2 - n2 A [n,n;-2] - n A [n.n'-1].

(4)

ON EXACT POISSON MANIFOLDS OF DIMENSION 3

375

Repeating this we have 1-1

[n,n'] = -(z + ijDiviiAii'-J^ir A(n,n'-i]

(5)

»=i

Using [II, Uk] = k [II, n] A Uk~l for k > 1, we get

^

+1

)[n, n] A n'- 1 = -{i +1) Divn A n'.

From this, we obtain

[n,n(] = I[U,U]AU1-1 = -2DiviiAir\ D Proof of Theorem 1. Let Tu be of codimension q and assume it is defined by a local equation of 1-forms a.\

= • • • = aq

— 0.

Then we have n ( o , , - ) = 0,

j =

l,...,q.

Taking the covariant derivative, we have (Vn)(ai,.) + n(VaJ-)-)=0. By contraction, we obtain (DivU)(aj)+U(daj)

=0.

(6)

Thus, if { a i . . . a , } satisfies the Frobenius integrability condition, H(dotj) = 0 and we have DivII(aj) = 0. This shows that DivII is a vector field tangent to Tn and DivII A XI' = 0 , since rankll = 21. Conversely, if DivIlAlT' = 0, fromO = (c^,DivIlAll') = (Divn)(a.,)An' we can see (Divn)(a.,) = 0 and hence by (6), we get H(daj) = 0 for each j . This means that each dotj should be of the form Q

^2 ak A pkj fc=i

for some 1-forms {Pkj}- This shows {a\,... integrability condition. We get the following well-known fact

,aq} satisfies the Frobenius •

376

TADAYOSHI MIZUTANI

C o r o l l a r y 3 Let (M, II) be a regular Poisson teristic distribution is integrable.

manifold.

Then the charac-

Proof. Let rank U = I. Since [11,11] = 0 , [II, II'] = 2l[U,U] A I I ' - 1 = 0. T h u s , t h e result follows. • Remark. In a similar way, if dim M = 21 + 1 and r a n k l l = 21, we can show t h a t t h e characteristic distribution is a contact plane field if and only if [II,n ( ] is nowhere zero. 3

E x a c t P o i s s o n manifolds of special kind

If a Poisson manifold ( M , II) has a vector field Z which satisfies L z I I = [Z, n ] = —II, it is called exact. In this case Z is called a homothetic vector vector field of ( M , II) ([1]). Prom t h e view point of t h e Poisson cohomology, an exact Poisson manifold is a Poisson manifold whose Poisson bi-vector field II represents 0 in H^P(M). Recall t h a t t h e Poisson cohomology is a cohomology whose p - t h cochain group is T ( A P T M ) and t h e coboundary operator a : T ( A P T M ) -> T(AP+1TM) is given by cr(P) = - [ I I , P] ([9]). It is not difficult t o give examples of exact Poisson manifolds which are non-compact. T h e following examples are standard. E x a m p l e 1 ( C o t a n g e n t b u n d l e s ) Let (T*M,d\) be t h e canonical symplectic structure of t h e cotangent bundle of M , where A is t h e Liouville form. Let II be t h e 2-vector field on T*M such t h a t II = (dX)~l. Here we regard dX as an isomorphism T(T*M) —> T*{T*M). Note t h a t this means 7n(<^A) = —II. Using a general formula (du)(X,-)

=

Lxoj-d(uj(X,-))

for a 2-form u> a n d a vector field X, we see L r i ( d / ) ^ = 0 and II satisfies the Poisson condition [H(df, -),Ti] = 0. Let Z = n(A, •)• T h e n by t h e above, we have LzdX = d(dX(U(X))) = dX. From this 0 = LZ(ITI

° dX) = Lz(In)

° dX + In o LzdX

= Lz(In)

° dX + In o dX.

This means

[n(A,-),n] = -n. T h u s , (T*M, n , I I ( A , •)) is an exact Poisson manifold. E x a m p l e 2 ( L i e P o i s s o n s t r u c t u r e ) Let (fl*,n) be t h e Lie Poisson structure on t h e dual space of a Lie algebra g. T h e Poisson bi-vector field II is defined as follows. We have an identification T*g* « g* x g " = g* x g by

377

O N E X A C T POISSON MANIFOLDS OF DIMENSION 3

translations. If T*g* 9 p,q are represented as p = (a,x),q this identification, then II is given by

— (a, y), under

n Q (p,g) = a([x,y]). Let Z be the radial vector field on Q*. Namely, Za(a G fl*) is a tangent vector which corresponds to the curve e* • a in g*. We regard p,q as the constant 1-forms. In other words, we consider p as a 1-form on g*, given by a — i > (a, x), where i £ g = g** is independent of a. Then we have Lz{U{p, q)) = Lz(a({x, y])) = ^ | 4 =o(e'a([x, y})) = a([x, y]) = U(p, q). On the other hand, Lz(U(p, q)) = (LzU)(Pl q) + U(Lzp, q) + U(p, Lzq) = (LzU)(p,q). since p and q are constant fields. From these we have (LzU)(p,q)

= U(p,q).

This shows (g*,II, — Z) is an exact Poisson structure. Of course, the homothetic vector field in the above is not unique. In fact if Z, Z' are both homothetic vector field for II, then the Lie derivative LZ-Z'^- vanishes. Thus the set of homothetic vector field of a Poisson structure forms an affine subspace of the vector space of all the vector fields, whose associated vector space is the space of vector fields which preserve the Poisson bi-vector field. Recall that any Hamiltonian vector field I(df), f e C°°(M) preserves IX Now, we are interested in the following problem: Problem. On a closed manifold what kind of codimension one foliations do appear as underlying foliations of exact Poisson manifolds ? We will consider this problem in the case where M is a closed 3-dimensional manifold. First, we note that every orientable foliation of dimension 2 is an underlying foliation of a Poisson structure. In fact, let (M, J7) be a foliation whose leaves are 2-dimensional and let IT € T(A2J7) be a non-zero cross section. Then II is naturally considered as a 2-vector field on M. It is easily checked that the image of In coincides with T. The Poisson condition on II is satisfied since in this dimension, it is equivalent to the integrability of T (see Section 2).

378

3.1

TADAYOSHI MIZUTANI

Exact Poisson structure of special kind

In this subsection, we give two examples of exact Poisson manifolds which we call 'special'. The underlying manifolds are closed and quotients of 3-dimensional Lie groups. E x a m p l e 3 Let X\, X2, X3 be the right invariant vector field of G = SL(2, R) corresponding to - ( o i ) ' ( o o ) ' ( l o ) satisfy the following bracket relations; [Xi,X2} = -X2,

[Xi,X3] = X3,

res

[X2,X3] =

Pectively-

The

y

-2X\.

The 2-vector field II = X\ AX2 satisfies the Poisson condition [II, II] = 0 hence defines a Poisson structure. If we choose a uniform discrete subgroup r of G = SL(2, R), we obtain an induced Poisson structure on M = G/T = SL(2,i?)/r which is a closed manifold. The underlying foliation Tn is known as an Anosov foliation spanned by X\ and X2. It is also known that each leaf of this foliation is dense in M. Let Z = X\ + aX2, (a is a constant) then

Lzn=[Z,U}

= [X1,X1AX2] = Xi A {x1,x2} = -Xi AX2 = - n .

Thus (M, II, Z) is a closed exact Poisson manifold. Similarly, we have the following second example. E x a m p l e 4 Let G be a simply connected 3-dimensional solvable Lie group whose Lie algebra is generated by Xi, X2, X3 with the relations [Xi ,X2] = -X2,

[Xi, X3] = X3,

[X2, X3] = 0.

Like as in the case of Example 3, let II = Xi A X2, Z — X\ + aX2 be the right invariant fields on G. By the same computation, we see that II defines an exact Poisson structure and Z is a homothetic vector field. Also, if we choose a uniform discrete subgroup T, we obtain an exact Poisson structure on a closed 3-dimensional manifold. In this case, M = G/T is known to be diffeomorphic to a T 2 -bundle over S1 and the foliation is a suspension of a dense linear foliation of T2, hence the leaves of Fn are all dense again. Note that, in both cases of the above examples, the symplectic leaves of the characteristic foliations are generated by the vector fields X\, X2 with the relation [Xij-XVj = —X2, which generate the Lie algebra of 2dimensional affine group GA. From this, we can see that the leaves are the orbits of a locally free actions of GA.

O N E X A C T POISSON MANIFOLDS OF DIMENSION 3

379

We call the Poisson manifold which is obtained in the above examples a special exact Poisson manifold. It has a homogeneous structure which is induced from an invariant Poisson structure on a 3-dimensional Lie group. One of the property of the homothetic vector field of our special Poisson manifold is the fact that its divergence (with respect to the canonical volume form) vanishes everywhere. Indeed, let fi be the volume form on M, such that Q,(Xi A X2 A X3) = 1. Then by an easy computation we can see Lz^l — 0 which is, by definition, equal to (divZ)Q, and divZ = 0. In the following , we will show that this property characterizes the special exact Poisson manifolds. T h e o r e m 4 Let (M, H, Z) be a regular exact Poisson manifold, where M is a closed 3-dimensional. Suppose that the homothetic vector field Z is divergence free with respect to some volume form Q, on M. Then (M, II, Z) is diffeomorphic to a special exact Poisson manifold. Proof. Choose a Riemannian metric on M whose associated volume form is equal to O. We will use the generalized divergence with respect to the Riemannian connection of this metric. In the next section, we investigate some topological properties of the foliation associated with a codimension one exact Poisson structure. Especially, we prove the homothetic vector field on a closed manifold is necessarily tangent to leaves everywhere (see Theorem 9). If we admit this fact, we have Z A II = 0, DivZ = divZ = 0, hence, - I I = [Z, n] = DW(Z A n ) - DivZ A II + Z A DivII = Z A DivII. Since II is nowhere vanishing, this shows that Z and DivII are two vector fields tangent to the leaves of Txi, and are linearly independent at each point of M. Taking Div of both sides of [Z, II] = - I I (see Section 2), we have [Z, DivII] = -DivII. This shows that on M there exists an locally free action of the 2-dimensional affine group GA. Since divZ = 0 and div(DivII) = Div 2 II = 0, we have Lz$l — 1/Divn^ — 0. This means that the action of GA preserves the volume fi. Now a theorem of Ghys ([2]) concerning the rigidity of the action of GA on 3-manifolds says that this action is smoothly conjugate to one of the standard ones. That is, it is equivalent to a natural action of GA which is the action on one of the quotient manifolds G/T in the examples of this section. This means that there is a diffeomorphism tp : M —• G/T sending Z to Xi and DivII to X^.. •

380

4

TADAYOSHI MIZUTANI

Exact Poisson structures on closed manifolds

As before, we consider regular Poisson structures and find some topological properties of exact Poisson manifolds. The main result is Theorem 9 in which we show that the homothetic vector field is always tangent to leaves provided the manifold is closed and the associated foliation is of codimension one. We start with the following Lemma 5 Let (M, II, Z) be a regular exact Poisson manifold. Then the homothetic vector field Z preserves the foliation Tu • Proof. Let the foliation T\\ be defined locally by Pfafnan forms ai,...,aq which span Ker/ri- The relation n ( a j , •) = 0 leads to the equation {Lzn)(ai)

+ n(Lzai,-)

= 0.

Since (LzTV)(ai) = —II(a,, •) = 0, we have U(Lzai,-)

= 0,

(i =

l,...,q).

Thus each Lzcii is a functional linear combination of a i , . . . , aq. Now let X be a local vector field which is tangent to the leaves. Then we have ax{LzX)

= Lz{oi(X))

- (Lzai)(X)

= 0,

(t = 1 , . . . , q).

Thus LZX is also tangent to the leaves. This means Z preserves the foliation Tj\. • By the above lemma, the subset of M, where Z is transverse to !Fu, is an open saturated subset (the subset which is a union of leaves ) of M. Lemma 6 Let (M, II, Z) be a codimension one exact Poisson manifold. That is, M is an exact Poisson manifold such that $n is a codimension one foliation. Let U be an open saturated subset of M, where Z is transverse to the foliation 3n- Then the foliation !Fn\u restricted on U is defined by a closed 1-form. Proof. Take a 1-form a which satisfies In(a) = 0 and a(Z) = 1 on U. It is easy to see that da = Lza A a. By Lemma 5, Lza is a functional multiple of a, hence we have da = 0 on U. D Lemma 7 Let (M, II, Z) be a regular exact Poisson manifold. If L is a leaf of J-JI such that Z is tangent to L, then L is a non-compact leaf.

O N E X A C T POISSON MANIFOLDS OF DIMENSION 3

381

Proof. Take a leafwise symplectic 2-form u> on M such t h a t In{^>) — n . If r a n k l l = 2k, the 2fc-form wk = u> A • • • /\LJ

(/c-times)

restricts to a volume form on each leaf. Since the pairing (uik,H ) is a non-zero constant and LzU.k = — kUk, (Lzu)k,Ilk} is also a non-zero constant. T h u s t h e restriction (Lzi^k)\L of Lzwk to L is a non-zero multiple of tjk\L- Since Z is tangent to L, this shows diV( u fc|£)Z = C (the divergence with respect t o t h e volume w f e |i). for some non-zero constant C. This is impossible when L is compact. • L e m m a 8 Let (M, II, Z) 6e a codimension one exact Poisson manifold, where M is a closed manifold. Then the subset of M, which is the union of leaves where the homothetic vector field Z is tangent to each leaf is a non-empty closed saturated set. Proof. Closedness of the set is clear. If it is empty, Z is transverse to F n everywhere on M and Z A IIfc is nowhere zero on M (2k is the rank of II). Let f l b e a volume form on M dual to Z A LTfe ( i.e. ft satisfies Q(Z A II fc ) = 1). T h e n it is easily seen t h a t d i v n Z = k which is impossible on a closed manifold M. • Let ( M , II, Z) be an exact Poisson manifold of a closed manifold, which is of codimension one. We put rank II = 2k. As we have seen in Lemma 7, t h e vector field Z is not tangent to a compact leaf. Assume t h a t Z is transverse to a compact leaf L. Since the 1-parameter subgroup 4>t generated by Z preserves the foliation F*n, the union U 4£ jj(/>t(L) consists of compact leaves which are diffeomorphic t o L. If \J jf(j>t(L) is not whole M, t h e r e exists a leaf which is the limit leaf of a subset of U t e jcj0((L), which itself should be compact and Z must be transverse to it. This implies t h a t it has to be contained in U t e j ^ t ( L ) which should be whole M (we assume M is connected). This contradicts the above Lemma 8 T h u s , we can conclude t h a t (M, LT, Z) has no compact leaves as long as M is closed. Therefore, for example, there is no exact Poisson structures on S3 since every codimension one foliation of S3 has a compact leaf diffeomorphic to T 2 ([7]). Also, we saw Z is not everywhere transverse to F n - Moreover in the case of special exact Poisson manifold, the homothetic vector field is everywhere tangent to leaves on the whole manifold. Hence it is n a t u r a l to ask the following question: Q u e s t i o n . Are there any examples of ( M , II, Z) on which Z is tangent to the leaves of F n on one p a r t and transverse to t h e m on the other p a r t ? T h e following theorem shows there is no such example on a closed manifold provided (M, II, Z) is of codimension one.

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TADAYOSHI MIZUTANI

Theorem 9 Let (M, II, Z) be an exact Poisson structure of a closed manifold, which we assume regular and of codimension one. Then the homothetic vector field Z is tangent to the foliation Tn everywhere on M. Proof. To the contrary, we assume that there exists an open subset of M, where Z is transverse to Tj\- Let M — U U U' be the partition into two part; on U, Z is transverse to Tw and on U', Z is tangent to !Fn- By Lemma 5, both U and U' are saturated subsets. By Lemma 8, U' is a nonempty closed set. Since TXL has no compact leaves, the closed saturated subset U' ^ M has to contain an exceptional minimal set. Let E denote it. By a theorem of Sacksteder([8]), if a codimension one foliation is of class C 2 , the exceptional minimal set contains a leaf which has a contracting holonomy. Take such a leaf L contained in E. Choose a point x e L and a transverse small arc / « (—1,1) through x, where 0 corresponds to x. The contracting holonomy gives a germ of a map ip : (—e, e) —> ( — 1,1) at 0 (e > 0 is small). The intersection IDE is a Cantor set and ItlU is a union of open intervals. Let {
{v0,V!,...,Vfe-i, vfc - v0, Vnv+1

+ 0(i = o,...,k- i)}

which covers I. Each Vi is assumed to be diffeomorphic to Dn~l x Iit Ii = (-1,1) and Dn-1 x {t}, (t e U) is a plaque in Vi and Dn~l x {0} is the plaque where I f~l Vi lies (dim M = n). The diffeomorphism ipto for small to induces a germ of diffeomorphism of each Ii at 0, which we denote by 4>i- Cm t n e other hand, we have a local holonomy translation along t which gives a germ /ij+i,* : (it,0) —» (/i + i,0). Since the diffeomorphism (pto preserves the foliation, we have /it+i,i o (pi = (pi+i o hi+iti for each i = 0 , 1 . . . k — 1. From this relation we have ip °