Poincar e Duality for L p Cohomology and Characteristic Classes

DIMACS Technical Report 98-48 October 1998 Poincare Duality for Lp Cohomology and Characteristic Classes by Oliver Attie Dept. of Computer Science Rutgers University New Brunswick, New Jersey 08903 Jonathan Block Department of Mathematics University of Pennsylvania Philadelphia, Pennsylvania 19104 DIMACS is a partnership of Rutgers University, Princeton University, AT&T Labs-Research, Bell Labs, Bellcore and NEC Research Institute. DIMACS is an NSF Science and Technology Center, funded under contract STC{91{19999; and also receives support from the New Jersey Commission on Science and Technology. ABSTRACT We prove Poincare Duality for Lp cohomology, 1  p  1 We study the pairings between Lp and L1 and construct characteristic classes. 1 Introduction The purpose of this article is to prove Poincare duality for Lp cohomology. This relates the amenability criteria of Gromov (see [J]) and Block-Weinberger [BW], and of [ABW] to the criterion described in [A1]. Finally, we will give a complete proof of the statement in [1] relating the structure set of M  R to the Pontrjagin classes in uff cohomology. In the following we introduce a ne version of this coarse homology theory and prove that it is Poincare dual to bounded de Rham cohomology described by Gromov [G], Januszkiewicz [J] and Roe [R1]. This theory has also been studied by S. Gersten, who showed that a group is hyperbolic if and only if its second cohomology vanishes [Ge]. In a later paper the rst author will use the Poincare duality theorem to prove a Lefschetz xed point theorem, similar to the theorem of Heitsch and Lazarov [HL]. We will prove here, however, an analogue of the Hopf index theorem, which characterizes the Poincare duals of Januszkiewicz' classes as elements of homology. Jurgen Eichhorn [E] has previously de ned characteristic classes for Lp-cohomology, and J.Dodziuk [D] has proven the de Rham theorem for L2-cohomology, but the general Poincare duality theorem which we prove here does not appear to be in the literature, and it has some interesting consequences. 2 Uniformly nite homology We rst recall the de nition of the coarse theory given in [BW1], where only the case p = 1 was considered and was called uniformly nite homology. De nition 1 A normed abelian group is an abelian group G equipped with a \norm" function j
j: G ! R+ which satis es the triangle inequality. De nition 2 Let X be a metric space and X i+1 the i + 1-fold cartesian product with the metric d((x0; :::; xi); (y0; :::; yi)) = 0max d(xj ; yj ): j i Also, write
for the multidiagonal in X i+1 . Let CXi(p)(X ; G; j
j), (G; j
j) a normed group, denote the group of in nite formal sums c = axx x 2 X i+1, ax 2 G satisfying i. Given r > 0 there exists Kr so that for all y 2 X i+1 #fx 2 B (y; r) j ax 6= 0g j Kr ii. There exists R > 0 depending on c so that ax = 0 if d(x;
) > R. iii. j ax jp . X P {2{ Let U?1 (M) be the algebra of uniformly smoothing operators introduced by Roe in [R] (we will assume the operators in U?1(M) have bounded propagation speed). These are de ned to be operators whose propagation speed is bounded and whose kernels are uniformly bounded in C 1 norm. The following theorem of [BW2],[Yu] along with the results in [CM] generalizes index theorem in J.Roe's Ph.D. thesis: Theorem 1 There is a character map  : HC(U?1 (X )) ! HX1 (X ; R) There is also a \ ne" homology corresponding to the \coarse" theory. It is de ned as follows: De nition 3 A simplicial complex X is said to be a simplicial complex of bounded geometry, if there is a nite uniform bound on the number of simplices in the link of each vertex of X . This bound is called the complexity of X . Here a locally nite simplicial complex is considered to have the natural metric derived from barycentric coordinates. A simplicial map f : X ! Y between simplicial complexes of bounded geometry is said to have bounded geometry if there is a nite uniform bound on the number of simplices in the inverse image of a simplex under f. P P De nition 4 De ne the group of chains Cqp (X ; G; j
j) to be the group of formal sums of simplices in X , c = a  so that j a jp < 1. The boundary is de ned to be the natural linear extension of the singular boundary. We write the resulting homology Hp (X ; G; j
j). ( ) ( ) A uniformly contractible space is a metric space for which the metric ball of radius r is contractible within the ball of radius f (r) for some function f : R+ ! R+ . The following is an easy corollary of a Weil style double complex argument. Proposition 1 Let X be a uniformly contractible space, (G; j
j) a normed group. Then HXp (X ; G j
j) ' Hp (X ; G j
j). ( ) ( ) We can in fact introduce a \dual" cohomology theory for which there are two versions, one which involves di erential forms, and the other which is simplicial. We rst introduce the theory in terms of di erential forms. De nition 5 Let i(p)(M ) denote the Banach space of di erential i-forms bounded in the norm k kpp= (j (x) jp + j d (x) jp)dvol Z Then the maps di : cohomology by i p ( ) (M ) ! M i+1 (M ) (p) de ne a complex. We de ne the bounded de Rham i HDR p (M ) = [Ker(di )]=[Im(di? )] ( ) 1 {3{ The simplicial theory is de ned in the following manner. De nition 6 Let H(ip)(X ), for X a simplicial complex of bounded geometry denote the cohomology group whose cochains are simplexwise the same as the cochains of ordinary simplicial cohomology, but satisfying conditions j c() jp < 1 P Theorem 2 De Rham Theorem Let M be a smooth n-dimensional manifold of bounded geometry. Then there is an isomorphism i H ip (M ; R) ' HDR p (M ) ( ) ( ) To prove this we need the following result proven in [CMS] section 7 and [A]. Theorem 3 Let M be a manifold of bounded geometry. Then M admits a triangulation as a simplicial complex of bounded geometry. The resulting simplicial complex is quasi-isometric to the original manifold. Remark 1 This theorem yields a simplicial proof of the niteness theorem of [C]. To see this, simply take the class of manifolds with curvature bounded in absolute value, volume bounded below and diameter above and take the connected sum of all of them. This can be done in such a way as to give rise to a manifold of bounded geometry, by gluing along the boundary sphere in at most a nite number of ways. The triangulation theorem then says that each ball of xed diameter in this manifold can be covered by a xed number of simplices. Hence any manifold in the class above can be covered by this same number of simplices. This then implies that there are a nite number of PL types in this class. In addition to this we need a notion of subdivision, which we call regular uniform subdivision. This is a re nement of the notion of uniform subdivision in [A]. We rst recall that simpler notion. De nition 7 A subdivision of a simplicial complex of bounded geometry is said to be uniform if i. Each simplex is subdivided a uniformly bounded number of times on its n-skeleton, where the n-skeleton is the union of n-dimensional sub-simplices of the simplex. ii. The distortion sup(length(e); length(e)?1) of each edge e of the subdivided complex is uniformly bounded in the metric given by barycentric coordinates of the original simplex. De nition 8 A regular uniform subdivision is de ned in the following manner. Let  = k [p ; :::; pm] be a simplex in R , k  m. The vertices of the standard subdivision of  are the 0 points pij = 21 (pi + pj ); i  j: {4{ De ne a partial ordering of the vertices S by setting pij  pkl ; if i  k, j  l. The simplices of S are the increasing sequences of vertices with respect to the above ordering. We de ne a regular uniform subdivision to be any uniform subdivision which is a sequence of standard subdivisions. Uniform subdivision induces a map s : C(p)(K ; G) ! C(p)(K 0; G) which has bounded norm. Proof 1 The idea is to use Whitney and de Rham maps, with the triangulation above serving to show that these maps are well-de ned. The proof here imitates the ones found in [D],[W]. De ne the de Rham map : i(p)(M ) ! C(ip)(M ; R) by (!)
 = !:  Integration commutes with uniform subdivision as can be shown in the following manner. To de ne the Whitney map, which is the chain homotopy inverse of the de Rham map, let c be the cochain which assigns the value 1 to  and the value 0 to every other simplex. For each point q in M write q in barycentric coordinates as q =  (q)q , where q runs over the vertices of the triangulation, and is the corresponding index. For each , let Q 1 ; Q0 is the set of all and Q0 be subsets of M so that Q is the set of all p with  (p)  n+1 1 q with  (q)  n+2 : Then Q  star(q ), Int(Q0 )  M ? star(q ). Let 0 (p) be a smooth non-negative real function in M which has all of its derivatives uniformly bounded over and so that each is positive in Q and zero in Q0 . Construct the partition of unity Z Z Z P P 0  (p) =  (0p()p) We will choose normalizations of the 0 below, so we will assume 0 normalized suitably for the moment. Take any p 2 M ; since p has at most n + 1 non-zero barycentric coordinates at least one 1 of these, say  (p) is  n+1 : Hence p 2 Q , 0 (p) > 0 and  (p) is de ned for all . De ne the Whitney map W on each c by Xr W (c ) = r! (?1)i i d 0 ^ ::: ^ d^ i ^ ::: ^ d r ; i=0 where  = q 0 :::q r and  is de ned above. We can then extend by linearity. Note that suppW (c )  : {5{ Furthermore, because the triangulation is uniform, the resulting form is bounded. In fact, the map W de nes a bounded map on chains in the norm de ned by taking the supremum of the coecients of the chain. We now choose normalizations so that the de Rham theorem will be true. Choose normalizations of the 0 so that Z W (c Z ) =   W (c ) =  ; since suppW (c )   . Using this normalization, we have inductively for  0 a face of , R Z W (c ) = Z W (c    d 0 Z )= @ Z W (c ) =  0 0 W (c0 ) = 1: This normalizes all of the lower dimensional cochains. i De ne  : HDR (M ) ! H(ip)(M ; R) to be the map sending ! to the cohomology class (p) de ned by the linear functional  7!  ! on each simplex. De ne W  : H(ip) (M ; R) ! i HDR (M ; R) to be the map induced by the Whitney map W de ned above. (p) We claim that  and W  are inverses of eachother. In fact, by Stokes' theorem we can observe that and W are chain maps. That W = Id is proved by observing that the normalization conditions prove the result for each c() and then extending by linearity. That W   = Id follows from a series of easy estimates. Let ! be a closed di erential form in DR(p)(M ). Suppose the cohomology class [ !]=0 in Hi(p) (M ; R): Since integration commutes with uniform subdivision, we have [ 0 !] = 0, where 0 denotes the integration map over a uniformly subdivided triangulation. Fix 1 > 0. We claim that there exists a uniform subdivision so that R R R R R R R Z k ! ? W
! k<  R 1 Z Note that we have the estimate @! j j !(x) ? W
! j C
diam
sup j x2 @x over each simplex  of the triangulation, where C can be chosen independently of !,  . One then obtains an estimate in terms of the mesh of the triangulation h Z Z Z X k!?W
! k  j !(x) ? W
!(x) j dV 0  2K   4C
hN m k ! k m denotes the multiplicity of the intesections of the bg coordinate charts of M . Since Rwhere ! is a boundary, we can nd a cochain f so that Zk ! ? f k  2 {6{ Then Z Z k ! ? dWf kk ! ? W
! k + k W k
k ! ? f k   + k W k
 1 2 Since 2 is chosen independently of 1 and both can be made arbitrarily small, we obtain that the cohomology class [!] = 0. This proves the de Rham theorem. We have the corollary: Theorem 4 There is an isomorphism p i HDR p (M ) ' Hn?i (M ; R) ( ) ( ) for any orientable n-manifold M of bounded geometry. Our proof will be based on that in [Mau]. The following is also a consequence of the proof of the previous theorem: Theorem 5 There is an isomorphism H np?i (M ; R) = Hi p (M ; R) ( ) ( ) between Lp homology and Lp cohomology 1  p  1. De nition 9 A bg block dissection of a uniform triangulation of a bg simplicial complex is de ned to be a decomposition of the triangulation into polyhedra (ei;
ei) so that i. Hr (ei ;
ei) = 0, for r 6= n and Hn (ei;
ei) = Z. ii. Every simplex of the triangulation is completely contained in some ei . iii. The boundary of ei consists entirely of simplices of dimension m, for m < n. iv. The polyhedra (ei;
ei ) fall into a nite number of isometry types up to a xed amount of distortion. Proof 2 We use the existence of a bg block dissection of the triangulation of M which is dual to the uniform triangulation, in the sense that there is a one-to-one correspondance between the r-blocks and the n ? r-simplices. Suppose K is a bg simplicial complex, K 0 a uniform subdivision of K . For each simplex  of K de ne subcomplexes e() = all simplices of K 0 of the form (^n; :::; ^ 0) with n > ::: > 0 >  and
e() = all simplices of e() not having ^ as vertex. Proposition 2 Let K be a uniform triangulation of a manifold of bounded geometry. Then for each r-simplex  of K , (e();
e()) is an (n ? r)-block and the set of all (e();
e()) forms a block dissection of K 0. {7{ De nition 10 We need the analogue of the cap product. Let = X[bi ; :::; bni] 0 i be a generator of Cn(1)(K ) Let i0 = [b0i ; :::; bni ?r ], and i00 = [bn?r ; :::bn]. We de ne the cap product by \  = (i00 )0 X i Note that this de nes a product Hn1 (K ) H1r (K ) ! Hn1?r (K ) ( ) De nition 11 To calculate uniformly nite homology from the blocks e() we must identify the corresponding block chain complex with a subcomplex of C(p)(K ): this is done by choosing generators of each group of cycles Zn?r (e();
e()). If n-simplices of K are identi ed with elements of Cn(p)(K ), then z 2 Cn(p)(K ), the sum of the n-simplices of K , is a representative cycle for the generator of Hn(p)(K ) = Z. We totally order the vertices of K 0 so that ^ < ^ if dim > dim ; let s : C(p)(K ) ! C(p)(K 0 ) be the subdivision chain map, h : K 0 ! K a bg simplicial approximation to the identity, and h the map induced by h on 1f cochains. Finally, for each r-simplex  of K , let  2 C(rp) (K ; Z) be the cochain sending  to 1 and all other r-simplices to 0. De nition 12 De ne z() = h( ) \ s(z) 2 Cn(p?)r (K 0) Proposition 3 z() is a generator of Zn?r (e();
e()). Proof 3 Each generator [^n; :::; ^ 0] of Cn1(K 0) occurs in s(z) with coecient 1. Now h( ) \ [^n; :::; ^ 0] = [^; :::; ^ r] h[^r; :::^0] and this is zero except for just one simplex (^r; :::^0) contained in , where it is [^n; :::; ^ r]: Hence z() 2 Cn?r (e()): Moreover @z() = (?1)n?r h( ) \ s(z) since @s(z ) = s@ (z) = 0, so @z() = (?1)n?r h( ) \ s(z) which is an element of Cn?r?1 (
e()). Lastly, z () is a generating cycle since each simplex in z () has coecient 1. {8{ De nition 13 We de ne the duality map D : C rp (K ) ! Cnp?r (K 0) ( ) ( ) by D( ) = z(). If d is a boundary homomorphism for C(p) then dD() = @D( ) = (?1)n?r h( ) \ s(z)  being the boundary homomorphism in C(p). Thus X dD( ) = (?1)n?r ( ) where the sum is taken over those r + 1-simplices  so that @ ( ) =  + ::: But the interior of e( ) is in K 0 if and only if  is in K , so dD( ) = (?1)n?r D( ) So D induces an isomorphism, which proves Poincare duality. 3 Products We now set out to prove the Lefschetz theorem. De nition 14 Let c be an element of HXi1 (X ; R). We can de ne the corresponding mean cohomology class to c as follows. De ne the mean cochains of X to be the dual space Ci1(X ; R) i (X ; R) = (C 1(X ; R)) CM i with the coboundary de ned by (c) = (@c). We denote the corresponding cohomology i (X ; R). Mean homology is de ned similarly in terms of uniformly nite groups by HM cochains. De nition 15 There are cap products: n (X ; R) H 1 (X ; R) ! H M (X ; R) HM r n?r n M H1(X ; R) Hr (X ; R) ! Hn1?r (X ; R) n (X ; R) be a mean cochain and let These cap products are de ned as follows. Let 2 CM 0 00 .  2 Cr1(X ; R) be a 00uniformly nite chain. Then we can write = and = i i i i Now let 0 = 0i , = 00i . As an L1 cochain,  can be paired naturally with 00 to yield a real number. Thus we have a mean homology class = ( 00 ) 0 The other cap product is similar to this. P P P {9{ Proposition 4 Let M be a smooth manifold of dimension n. Then there is a non-degenerate pairing n (X ; R) H 1 (X ; R) ! H 1 (X ; R) = R HM 0 n n (X ; R). Hence, given any class x 2 H01 (X ; R), one can associate a dual class  2 HM Proof 4 This will follow from the proof of a second Poincare duality theorem, in which we will use the cap product introduced above in place of the one used in the proof of the rst Poincare duality theorem. This second cap product will be called the global cap product. We triangulate M via a uniform triangulation and introduce the dual bg block dissection. n (K ; R) be the mean cochain evaluating to 1 on the 1f chain  . De ne Now let  2 CM z() = h( ) \ s(z), where we now use the global cap product. This gives the duality map and proves that the cap product pairing is non-degenerate. De nition 16 We de ne an intersection product H1 (M ; Z) Hn1? (M ; Z) ! H01 (M ; Z) by taking the transverse intersection of cycles. More precisely, let C and D be simplicial complexes of bounded geometry representing cycles [C ] 2 Hr1 (M ) and [D] 2 Hs1 (M ). The Poincare dual of [C ] is then an n ? r form C and the Poincare dual of [D] is an s form D . Take the wedge product of these two forms. Then we de ne [C ]
[D] to be the Poincare dual of C ^ D . Claim: The class of [C ]
[D] is the same as the class of the transverse intersection of C and D. Proof 5 We rst make note of the following fact. The Poincare duality theorem proved above associates to each simplex  the form (c ). This was achieved via the cap product de ned above in the proof of Poincare duality. We call this cap product the local cap product. The local cap product can be utilized to show that over a given compact subcomplex  Z  = #(
)  where  is the form constucted by applying the cap product to . Making use of local projections of neighborhoods from M  M to M , and applying the local intersection formula to the diagonal yields the result. Thus: Z  where  ^  = #(
)#(
 ) 1 2 is obtained by a local cap product with  and therefore, locally Z #(
 ) =  ^ : Adding this up on both sides as a cycle in H01 (M ) gives the result. { 10 { Theorem 6 There is an isomorphism Cn1(M  N ; Z) ' M Ci1(M ; Cj1(N ; Z)) i+j =n Proof 6 Let M = fk g ;k and N = f0kg ;k be simplicial complexes of bounded geometry. The Cartesian products of the simplices give a cell decomposition of the product space M  N , with boundary operator @ (k  0l) = @k  0l + (?1)k k  @0l: The Cartesian product X A  B = a b k  0l of two cycles A = a k and B = b 0l in M and N is a cycle. We thus have a map P P Cn1(M  N ; Z) ! M Ci1(M ; Cj1(N ; Z)) i+j =n which we will claim to be an isomorphism. This follows from an Eilenberg-Zilber argument, namely that the cycles in M  N are seen to be generated by the product of cycles in M with cycles in N . We have a double complex Cp;q (M  N ; Z) = Cp(M ; Cq (N ; Z)) which is used to compute the uniformly nite homology, after the introduction of L1 estimates. Note that l1(X  Y ) = l1(X ; l1 (Y )). This implies that C
1(X  Y ) = C
1(X ; C
1 (Y )). There is already a norm on uniformly nite chains given by the supremum of the coecients. 4 Chern-Weil Theory Next we discuss Chern-Weil theory for manifolds of bounded geometry. This is taken directly from [J]. Given a Riemannian metric g on a compact manifold M one may compute, using second derivatives of components of the tensor g the matrix of the curvature form, and after evaluating a suitable O(n)-invariant polynomial on it, obtains the characteristic form. These are closed, and using the homotopy between two Levi-Civita connections coming from metrics g; g0 one shows that the cohomology class depends only on the quasi-isometry type. We therefore have the following theorem of Januszkiewicz Theorem 7 Any C bounded metric on M de nes a Chern-Weil homomorphism from the 2 ring of polynomials on the dual Lie algebra of the group O(n) invariant under the Ad-action  (M ). It is depends only on the quasi-isometry type. into HDR 1 The following discussion is based on [GH]. { 11 { De nition 17 Let E ! M be a vector bundle of bounded geometry and  = ( ; :::; k) be k global C 1 sections of E , where the rank of E is k. Then the degeneracy set Di () is the set of points x 2 M where the  ; :::; i are linearly independent, i.e., Di () = fx :  (x) ^ ::: ^ i(x) = 0g: The collection of sections  is generic if for each i, i intersects the subspace of E spanned by  ; :::; i transversely and if integration over Di () ? Di () forms a closed current. Then Di represents a cycle in Hi1 (M ). 1 1 1 +1 1 +1 De nition 18 A vector bundle of bounded geometry is a vector bundle  equipped with a Hermitian connection r so r and its associated curvature form are bounded in the norm k
k on di erential forms de ned by k k= fj (x) j + j d (x) jg Chern-Weil theory for vector bundles of bounded geometry works the same way as for the tangent bundle, so we will leave the details out. 2r Theorem 8 Let cr () 2 HDR 1 (M ) be the r-th Chern class of a bounded geometry vector bundle . Then its Poincare dual in Hn1?r (M ) is equal to the class of the degeneracy cycle Dn?r . Proof 7 This follows by transversality and the corresponding fact for the universal bundle on BU . Suppose 1 ; :::; k are generic sections of E . Using a partition of unity, construct sections k+1 ; :::; n of E such that 1(x); :::; n(x) span the ber of E over each point x. These sections de ne a map into the Grassmannian. The image of this map then meets the Schubert cycle 1;:::;1 exactly in the degeneracy set. However, we have that the Chern class cr (!) of the universal bundle ! is Poincare dual to the Schubert cycle 1;:::;1. Finally, we can deduce the following corollary for bg structure sets Corollary 1 Let M and N be two PL manifolds of bounded geometry which are bg homotopy equivalent S 2  S 3  R Then M and N are PL bg homeomorphic if and only if they have  (M ; R). the same Pontrjagin classes in HDR 1 Proof 8 We use the computation of the bg structure set in [A1] to nd that an element of the structure set of S  S  R is bg PL homeomorphic to the union of an in nite number of copies of a compact manifold homotopy equivalent to S  S  I which are classi ed up to 3 2 3 2 ordinary PL homeomorphism by their Pontrjagin classes, hence taking the Poincare duals, by their degeneracy cycles. These degeneracy cycles, glued together along the boundary of S 3  S 2  I yields the result. { 12 { References  [1] L.V. Ahlfors, Zur Theorie der Uberlagerungs achen, Acta Math. (1935) [2] O.Attie, J.Block and S. Weinberger, Characteristic classes and distortion of di eomorphisms, Journal of the A.M.S. 5, 919-921 (1992) [3] O.Attie, Quasi-isometry classi cation of some manifolds of bounded geometry, Math.Zeit. 216 (1994) [4] J. Block and S. Weinberger, Aperiodic tilings, positive scalar curvature and amenability of spaces, Journal of the A.M.S. (1992) [5] J. Block and S. Weinberger, Large scale geometry, Proceedings of the 1993 Georgia Topology Conference [6] J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. J. of Math. 90, 62-71 (1970) [7] J. Cheeger, W. Muller and R. Schrader, On the curvature of piecewise at spaces, Commun. Math. Phys. 92, 405-484 (1984) [8] J.Dodziuk, Homotopy invariance of L2 Betti numbers, Topology (1976) [9] J. Eichhorn,Characteristic classes of noncompact manifolds, 1986 Conference on Differential Geometry and its Applications, Brno, Czechoslovakia (1987) [10] S. Gersten, Bounded cocycles and combings of groups (1992), preprint [11] P.Griths and J.Harris, Algebraic Geometry (1978) [12] M. Gromov, Kaehler hyperbolicity and L2 -Hodge theory, J. of Di . Geom. 33, 237-263 (1991) [13] J.Heitsch and C.Lazarov, A Lefschetz theorem for open manifolds, Contemp. Math. 105 (1990) [14] T. Januszkiewicz, Characteristic invariants of noncompact Riemannian manifolds, Topology 23, 289-302 (1984) [15] C.R.F. Maunder, Algebraic Topology (1970) [16] J. Roe, An index theorem for open manifolds I, II, J. of Di . Geom. 28, 87-135 (1988) [17] J. Roe, Index theory and coarse cohomology on complete Riemannian manifolds, Mem. A.M.S. (1993) [18] H. Whitney, Geometric Integration Theory (1957) View publication stats