Homotopy invariance of foliation Betti numbers

Invent. math. 104, 321-347 (1991)

InuentioHe$ matbematicae 9 Springer-Verlag 1991

Homotopy invariance of foliation Betti numbers James L. Heitsch 1 and Connor Lazarov 2 University of Illinois at Chicago, Department of Mathematics, Chicago, Ill. 60680, USA Herbert Lehman College, CUNY, Bronx, NY 10468, USA Oblatum 14-IX-1990 Introduction Foliation Betti numbers, introduced by Connes [C I], bear a striking formal similarity to the Betti numbers of a Galois covering space [A]. Both appear in a index theorem for D e R h a m complexes, and both can be interpreted as the trace on a v o n N e u m a n n algebra applied to projection operators. Dodziuk proved that the covering space Betti numbers are invariants of h o m o t o p y type [D]. The purpose of this paper is to prove the corresponding result for foliations: the leafwise Betti numbers of a foliation of a compact manifold which admits an invariant transverse measure are preserved by a measure preserving leafwise h o m o t o p y equivilence. Given two compact foliated manifolds with invariant transverse measures and a leaf-wise h o m o t o p y equivalence, we construct a v o n N e u m a n n algebra containing the yon N e u m a n n algebras of the individual foliations and we show that there is an involution in this algebra which intertwines the corresponding projections which give rise to the Betti numbers of the two foliations. The assumption that the h o m o t o p y equivalence is measure preserving implies that there is a trace on this new algebra which restricts to the traces on the von Neumann algebras of the two foliations. Intertwining implies that the projections have the same traces, and thus that the corresponding Betti numbers are the same. The main technical problems are the construction of a transversely measureable leaf-wise triangulation whose simplicies have volume and diameter bounded away from zero, and the construction of simplicial approximations to the leafwise h o m o t o p y equivilence and its inverse. The techniques of I-D] and [H-L I] are then applicable. 1 von Neumann algebra of a foliation, the trace, foliation Betti numbers, and the statement of the main theorem Let F be a smooth oriented foliation of a compact oriented manifold M. We recall briefly some facts about transverse measures (see [M-S] chapter IV and [H-L I] w 1 Partially supported by NSF Grant DMS 8803208. 2 Partially supported by a PSC-CUNY grant.

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J.L. Heitsch and C. Lazarov

A transversal to F is a Borel subset of M which intersects each leaf in a countable set. A smooth transversal is a proper embedded submanifold of M which is also a transversal. The set of transversals form a a-ring and a transverse measure v is a measure on this a-ring. The measure is called invariant if it is invariant under the holonomy pseudo-group acting on smooth transversals. A Riemannian metric on M gives rise to a volume form 2L on each leaf L of F. The family 2={2L} and an invariant transverse measure v give rise to a measure #=2dr on M ([C I], [M-S], chapter IV). Let E be a smooth vector bundle on M with smooth metric, ElL this bundle restricted to L, and L~ the leaf of F through the point x. Let Hx = L2 (Lx, E ILx), the Hilbert space of L2 sections of the bundle E l L , over L~. A section of {Hx} is a function s: M ~ UHx with s(x)EH~,. A measurable structure is a sequence {s,} of sections such that for each x, {s,(x)} generates H~ as a Hilbert space. See [-Dix] part II, p. 161 for a complete discussion. We describe the measurable structure for this family in the Appendix. Let (,)~ be the inner product on Hx. A section s of {Hx} is called measurable if (s(x), s,(x))x is a Borel function on M for each n. The family of measurable sections has an inner product

(s, t) = S (s(x), t(x)):, d,u(x) M

and a section is called square integrable if I]s H2 = ( S , S ) < O(]. L e t / 4 be the collection of square integrable sections with this inner product and with two sections identified if they are equal a.e. on M. The Hilbert s p a c e / 4 is called the direct integral of {Hx}. In [ H - L I], (2.3) we describe certain spaces C~ (U, V, Y) of sections of E [] E* over M • M which give rise to bounded operators on L 2 (El L) for each L. Each such section gives rise to a measurable family of bounded operators of {Hx} and a bounded operator on /4 (see [ H - L I ] , (2.3.1) and [Dix], p. 181). The von N e u m a n n algebra W(F; E) of the foliation and the bundle E is the von N e u m a n n algebra generated by the C~(U, V,, ~) for all U, V, 7- Vv(F; E) is also generated by C"o(U, V, 7), the set of leafwise smooth, uniformly bounded measurable sections compactly supported on U • 7 V ( [ H - L I], (2.3)). (For further details see [C I] and [MS], chapter VI.) Note also this is not the same von N e u m a n n algebra defined by Connes as the two algebras act on different spaces. A n invariant transverse measure v determines a trace tr, on W(F; E). ([C I], [M-S], chapter VI, [ H - L I]). We briefly describe this trace on generators of W(F; E). An element of C"o(U, V,, 7) is given by a kernel k=k(x, y)2 where k(x, y)eEx| (and k(x, y ) = 0 if x and y are not on the same leaf). Thus the pointwise trace tr k(x, x) makes sense and

try(k)= ~ tr k(x, x) 2dr. M

The objects we study do not depend on the choice of metrics on M and E. In fact we may replace the given family of metrics {gL} on the leaves by a family of {WL} of leafwise smooth metrices which vary measurably transversely, provided we assume that there are constants C 1 and C2 such that (1.1)

Cl gL ~--WL~C2 gL

Homotopy invariance of foliation Betti numbers

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for all L. In particular for Ek= A k T* F | C , ( T * F is the cotangent bundle along the leaves of F), the Hilbert spaces L2(L,, Ek[L~) are unchanged, and the square integrable sections/~ and the von N e u m a n n algebra are also unchanged. The smooth sections of Ek[ L = A k T* L | C are the smooth k-forms on L. The leafwise exterior derivative induces L. dk. Cooe (Ek[L) ~ C~ (Ek +1 IL).

Extend d~ to a densely defined operator on I~(EkIL). The resulting cohomotogy groups H~ = ker d~/Om-dkV~_~c~ dom d L) are called the reduced L2 cohomology groups. Using the leafwise Riemannian metrics to construct metrics on U(Ek[L) the leafwise Laplacian is given by Akr =d~'_ i d~'*~+d~* d~~. F r o m [H-L I] Corollary (2.1.2) we have that the natural map ker (A~) --* H / is an isomorphism. Denote by PkL the projection U ( E k l L ) ~ k e r ( A ~ ) , and let P~X=pkL for x e L . The family of projections {Pkx} gives rise to a projection Pk in W(F; E,). The k-th foliation Betti n u m b e r for F is defined to be flk = tr~(Pk). (These were first considered by Connes [C I, C II]. Although we use different von N e u m a n n algebras, the Betti numbers are the same). The fig are independent of the choice of metric on the leaves for any family of leafwise smooth, transversely measurable bounded family of metrics on the leaves (see [H-L II], Proposition 2.2). We remark that we will identify H / with ker(AkL) when it is convenient. Now let M' be a second compact oriented Riemannian manifold with oriented foliation F ' and invariant transverse measure v'. Let f : M - - , M ' be a continuous map which takes each leaf of F to a leaf of F'. We say that f is a leafwise homotopy equivalence if there is a continuous map f ' : M ' - - , M which takes each leaf of F' to a leaf of F with the property that f ' f and f f ' are homotopic to the identity by homotopies that take each leaf of F (respectively F') to itself. Let/~ and fl~,be the foliation Betti numbers for F and F'.

Theorem 1.2 Assume that for each transversal T of F and each transversal T' of F' for which the restriction f : T-* T' is one to one and onto, we have v(T)=v'(r'). Then [lk=ff~ for all k.

2 Leafwise triangulations and simplieial approximations We use the following notation regarding simplicial complexes. If a is a simplex in a simplicial complex K, then ]a] and IKI will denote the underlying spaces. We will use a to denote the element of K, the map a: A k ~ M and the image of this m a p in M. The context should make clear what a means. If g: K1 ~ K2 is a simplicial map, [g[ will be the m a p from [Kll to IK2[. If G: C , ( K 0 ~ C.(K2)

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is a chain map and G(z)=Zala, the support of [G(z)I is U I~rr.We also occaa~*O

sionally use [a I to denote the singular simplex on IKI defined by treK. We denote by A j the standard j simplex in R j,

Bounded triangulations Let M, F and v be as above, and let p = d i m Y , q = c o d i m F. We fix once and for all two covers { U~} and { V,} of M by foliation charts. We require thai

(1) ~cu~ (2) each U~ and E is diffeomorphic by maps of bounded distortion to I~ x I~ and I~ x II respectively, where 1i = ( - j , D2. implies that in any given chart, with coordinates (xj . . . . , x,, Yt . . . . , yq), the natural map between two placques given by (xl, ..., xp, Yl . . . . , y,)--* (xl .... , x e, z 1 . . . . . zq) is a m a p of bounded distortion in the metrics induced by the metric on M, the bounds being independent of the placques_ Note that there is then a lower b o u n d on the volume of any placque and an upper b o u n d on the diameter of any placque. Let/~ be a smooth triangulation of M chosen with the following properties.

2.1 The simplices are small enough so that each simplex cr is contained in some V~ and if ~rn V~#=0, then the closure of the double star of a is contained in U~. The closure of the double star of a is defined as follows. m

c(~) =(a*)* where for any collection of simplices S, S* = U star v V ~S

where v is a vertex of S. Let P be a placque of the cover {U,} or {V,} and J > 0 . The 3 normal neighborhood of P is the set of points x such that there is a geodesic from x to P, perpendicular to P, of length less than 6. Because of the bounded geometry of F, there is 6 > 0 such that for any placque P there is a neighborhood of the zero section of the normal bundle of P in M which maps diffeomorphicaliy under the exponential m a p to the 5 normal neighborhood of P. 2.2 We require the simplices a of/~ to be so small that the diameter of C(a)< 6. 2.3 We require the leaves of F to be in general position with respect to /~. (See [Th], p. 219.) 2.4 If a: A " ~ M is an n-simplex, then a extends to a diffeomorphism of a neighborhood U of A" and F induces a p dim foliation F, of U in general position with respect to A'. We require that each leaf of F, satisfy the incidence and curvature conditions needed to apply the classical triangulation arguments to that leaf and a. (See [W], Chap. IV B). We may insure that conditions 2.1 to 2.4 hold by taking a sufficiently fine standard subdivision (see 2.9) of a given triangulation of M satisfying 2.1 and 2.2 and applying Thurston's Jiggling Lemma [Th]. In particular note that given > 0 we may choose the subdivision to be so fine that any leaf of any foliation


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F~ is e close in the C 2 topology to a hyperplane in R". Note also that the Jiggling L e m m a applied to successively finer subdivisions of a given triangulation gives universal (i.e., independent of the subdivision) lower bounds on the angle at which a leaf may intersect a j simplex for j < q. This insures that the triangulation procedure of [W] may be applied to each leaf of F. W e now describe how such a triangulation of M induces a triangulation of each leaf of F. F o r more details see [W] or [M]. This construction proceeds inductively over the dimension of the simplices a of /~. In essence what we do is construct for each leaf L a p dimensional linear submanifold o f / ~ which approximates L closely enough so that the normal projection of this submanifold onto L gives a triangulation of L. The main point is that each placque may be treated as an embedded compact submanifold of R" transverse to A". We then can apply the classical triangulation arguments to the placques. This procedure does indeed produce a triangulation because of conditions 2.3 and 2.4 on/~,

Let ~r be a simplex of/~. If dim ~ < q, then by 2.3, any nontrivial intersection of a placque P of a leaf L with a is a single point and this defines a zero simplex in L. Let j > q and assume that for each L and each simplex z with d i m z < j , we have defined (in the manner given below) simplices in L associated to z. Let P be a plaque in U, of a leaf L. Let ~r be a j simplex o f / ~ with (int t~) n P * ~. The intersection of P with the q skeleton of a is a finite number of zero simplices, say Vo, ..., v~, of the partial triangulation of L. Let 1

k

eo,,,...,~=~i~ Z vj j=O

be the barycenter of the points Vo, ..., Vk in a. The fact that Pc~ a is as close as we like to a convex linear p + j - n dimensional subspace of a, which is guaranteed by 2.4, insures that O=~0.1.....k is in the interior of a. As ~r is contained in the 6 normal neighborhood of P, we may define the zero simplex v = Vo.....k in P to be g(~) where g is the projection of the normal bundle of P to P. Let 7: A k ~ M be a k simplex o f / ~ and let ~1: A ~ A k be a linear map. We call T~ =7o~a a linear (i) subsimplex of 7. Let ~o be a simplex o f / ~ in Oa with (int a o ) C ~ P . 0 . Each simplex z that cro defines in P is obtained from a linear subsimplex z~ of t~o (and so of ~) under composition with the map n. N o w the join of zl with ~, z~,~, defines a linear subsimplex of t~ whose image under ~ defines a smooth simplex in P denoted z . v . The collection of such ~ , v are the simplices of P determined by a. See Figures 1 and 2. In Fig. 1, note that Vo, vx, v2, v3 are not necessarily coplanar, but they are close to being coplanar (by 2.4). Also note that if 7 is an adjacent 3 simplex in /~ with i n t [ 7 1 n P + 0 , the linear subsimplices of ~ and y defined by ~c~ 7 are the same, i.e., the collection of all linear subsimplices associated to L defines a p dimensional linear submanifold L o f / ~ which is homeomorphic to L under (where the simplices determined by P and t~ are mapped to P). In Fig. 2, 31 is a linear 1 subsimplex of ~ro (the front 2 face of a), ~ is in the interior of a and z~,~7 is a linear 2 subsimplex of tT. T = ~ ( z l ) and z * v = ~ ( z l . v - ) = P n a . Both ~ and v . v are simplices in P, and so also in L.

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Fig. 1. Alt the linear subsimplices of a determined by cr and P

V1

inl Fig. 2

It is convenient to g r o u p the k simplices of the leafwise t r i a n g u l a t i o n into families. First, o r d e r the vertices of/~. This induces an ordering of the simplices o f / s Namely, if S = (Vo. . . . . Vk) a n d 27'= (wo, ..., wr) are simplices of / ( with vertices in order, we say Z < 2 ; ' if Vk k and Vk=Wr, Z ) k - l = W r - l , , - , , UO=Wr-k, then S < Z ' . Let Z o . . . . , ~ k be faces o f an n simplex 7 o f K, where Ni- a < Zi. T h e 27o, ..., Z k determine a family of leafwise k simplices A = A ( Z o , ..., Zk), where a k simplex z of the leafwise


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triangulation is in A if its associated linear subsimplex z~ has vertices v0 . . . . ,/)k where vi~int S i. Suppose that z and ~' are two leafwise simplices in the same family, and let z~ and z'~ be the linear subsimplices used to define z and z'. Let vo, ..., Vk be the vertices of za and v~, ..., v~ those of T't. There is a natural linear map from z~ to z'l namely the one defined by v~~ v~. This induces a canonical diffeomorphism denoted h~,, between z and z', which is a map of bounded distortion. The bounds however depend on z and z' since it is possible that in any given family there are simplices of arbitrarily small diameter and volume. If we exclude from any family those simplices whose diameter or volume is less than a fixed e, we obtain a family of simplices, all of which are canonically diffeomorphic by maps of bounded distortion, the bounds being independent of the members of the family. N o t e that the map n: z~-~ ~ is a map of bounded distortion where the bounds are independent of T. Here ~ has the flat metric induced from R* and z the leafwise metric. This follows from the uniform bound on the curvature of any leaf of any foliation F, and the fact that there are only finitely many n simplices oegT. If we endow z with the fiat metric g it inherits from ~ and we denote the distance g defines by d then there are positive constants C1 and Ca independent of X, Y, x, y and ~ such that

2.5 C 1 dL(x, y)O such that if u, veLc~V, but not on the same placque, then dL(U, v)>c. Thus for n large, x. and y. are on the same placque P of V and f ( x . ) , f ( y . ) are in U~, but not o n the same placque. This contradicts the fact that f ( P ) is a connected subset of the leaf containing f(L). Now, for any leaf L, the original metric and the metric induced from ~(L) are quasi isometric a n d the constants involved are uniform over all leaves. Similarly for M'. Hence leafwise uniform continuity holds for the new metrics. Corollary 2.13 (of the proof). Let h: M x [0, 1] ~ M ' be a continuous map with the property that for each leaf L of F, h(L x I-0, 1]) is contained in a leaf of F'. Then h is leafwise uniformly continuous. Lemma 2.14 (Leafwise simplicial approximation). Let f: M ~ M' be as above and K and Is triangulations of M and M' which induce bounded leafwise triangulations K, K' respectively. There is a positive integer r and a Borel map g: M ~ M' with the property that for each L, gL is a map from L to E, the leaf containing f(L), which, on Sr(KL), is a simplicial approximation to fL.

Proof. Choose r > 0 so that each simplex of Sr(K) has diameter less t h a n the 6~ of 2.12, where ~ is that of (2.11) for K'. Denote by p' the dimension of the foliation F'. Let v be any vertex of St(K) on L. f(v) lies in the interior of a simplex z on E and z e A ( Z o . . . . , S~) where S t are simplices of/~' in order. Let Wo, ..., Wk be the ordered set of vertices of z, where wi=int(Z~)c~E. Let gL(v) be the first


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of the Wo. . . . . Wk with the property that the barycentric co-ordinate gL(V)(f(v)) > 1 / p ' + 1 . This determines a simplicial map gL from S'(KL) to KL,. To show that gL is a simplicial approximation to fL it is enough to show fL (S t (V))~ S t(gL (V)) for all v. Let v be a vertex of aeS*(KL). For x e a , dL(X, v) 1/2(p'+ 1). This implies fL (a) ~ s t (gL(V)) and hence fL (S t (V))~ S t (gL(V))). From the construction, the measurability of g is clear.

3 Leafwise simplicial f2 cohomology We now introduce the E2 cohomology groups H*(KL) and show that for a leafwise homotopy equivalence f : M ~ M' there are suitable simplicial approximations yielding isomorphisms H* (KL) _--__H* (K~,). Let K be a bounded leafwise triangulation, and K L the resulting simplicial complex on L. As in [D] we introduce the f2 simplicial co-chains C~(KL) consisting of the co-chains g with the property that ~ [ g ( z ) [ 2 < ~ , where the sum is taken over all k-simplexes of K L. C~ (KL) becomes a Hilbert space under the inner product (g, h ) = ~ g(z)h(z). We note the following lemma which will be useful and whose proof is straightforward. Lemma 3.1 Let F: C . (K O-~ C.(K2) be a chain map between two simplicial complexes. Assume there are positive numbers N1, N2 and B such that (i) given z, there at most N1 a such that {F(a), z}#0. (for simplices p and z,

{p, ~} = ~,. (ii) given a, there are at most N 2 r such that {F(a), z} 4=0. (iii) if F (a) = Z a~, z, then ~ [a~[ =