Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://higeom.math.msu.su/~asmish/Lichnaja-2010/Version2010-11-20/Trudy/Publications/2008/Mishchenk-Popov-Poincare_duality_and_signature_for_topological_manifolds.pdf Äàòà èçìåíåíèÿ: Thu Sep 25 20:33:52 2008 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 01:54:42 2016 Êîäèðîâêà:

Topology and its Applications 155 (2008) 2041­2047 www.elsevier.com/locate/topol

PoincarÈ duality and signature for topological manifolds
A.S. Mishchenko
, 1



, P.S. Popov



Department of Mechanics and Mathematics, Moscow State University, 119992 Moscow, Russia Received 2 December 2006; accepted 11 June 2007

Abstract The signature of the PoincarÈ duality of compact topological manifolds with local system of coefficients can be described as a natural invariant of nondegenerate symmetric quadratic forms defined on a category of infinite dimensional linear spaces. The objects of this category are linear spaces of the form W = V V where V is abstract linear space with countable base. The space W is considered with minimal natural topology. The symmetric quadratic form on the space W is generated by the PoincarÈ duality homomorphism on the abstract chain­cochain groups induced by singular simplices on the topological manifold. © 2008 Elsevier B.V. All rights reserved.
MSC: 19Lxx; 55N25; 57Nxx Keywords: Topological manifolds; Noncommutative signature; PoincarÈ duality

1. Introduction It is well known that each nondegenerated Hermitian form over complex numbers assigns an integer which is equal to the difference of dimensions of positive and negative subspaces of the form for proper splitting of the underlying space. This integer is called by the signature of Hermitian form and does not depend of reduction to diagonal matrix. In topology Hermitian forms naturally appear in cohomology groups of orientable manifolds. Given two cohomology classes in the middle dimension of an orientable closed manifold one can take the product of the classes and then take the integral along the fundamental class. In the case of dim M = 4k we obtain nondegenerated symmetric quadratic form for which the signature is defined. The signature has natural properties both with respect to disconnected union of manifolds, that is the signature of sum equals to the sum of signatures and with respect to Cartesian product, that is the signature of product equals to the product of signatures. More of that the signature is invariant of oriented bordisms, that is the signature of the boundary of oriented manifold vanishes. All these properties are useful for study of manifolds from the point of view of their classification.


Expanded version of a talk presented at the International Conference on Topology (Aegion, Greece, June 2006). E-mail address: asmish@mech.math.msu.su (A.S. Mishchenko).

* Corresponding authors. 1 Supported by the Russian Foundation of Basic Research, No. 05-01-00923-a.

0166-8641/$ ­ see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.topol.2007.06.021


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The natural aim is in building of the signature type invariants for nonsimply connected manifolds with nontrivial coefficient systems. The most natural is to consider a representation of fundamental group of the manifold in a C algebra. Such kind of the signature type invariants belongs to Hermitian K -theory of fundamental group. But the building of the signature type invariants in Hermitian K -theory faces with an essential obstruction, which is that the cohomology groups for arbitrary local system of coefficients may not be projective modules. This obstruction was overcome in the paper [2] for triangulated combinatorial manifolds. In this case the chain and cochain groups are free finite generated modules. The PoincarÈ duality can be realized by the intersection operator with fundamental cycle from the cochain module to the chain module and induces the isomorphism in homology groups. Using standard algebraic machinery one can construct a nondegenerated Hermitian form on a free finitely generated module that induces an element in Hermitian K -theory. So symmetric noncommutative signature can be build for triangulated topological orientable manifolds. Following the argument by M. Gromov [3] (see also [4]) the signature should be constructed for nonsimply connected topological manifolds. If the representation of the fundamental group is finite dimensional then the signature can be calculated without using of the PoincarÈ complexes of chain groups. In the case of the group algebra of the fundamental group the problem of direct construction of the signature was still open. The existence of triangulation for arbitrary topological manifolds is still unknown. One can use some cell structure of topological manifolds induced by the handle decomposition which is based on heavy and inefficient theory of microbundles. All natural constructions, such as Cech homology, singular homology, Alexander­Spanier homology lead to infinitely generated chain and cochain groups. Two problems appear. The first problem consists in building algebraic category which includes infinitely generated free modules with natural topology and which admits an extension of signature functor to Hermitian K -theory. The second problem consists in application the signature functor to topological manifolds. The first step in this direction was done in [5] where the construction was developed for simply connected manifolds. The main idea how to extend the construction of signature to infinitely generated modules over the C -algebra A was presented in the papers [6­8] where all modules were countably generated. We present here general case that includes uncountably generated modules for the chain groups of singular simplices. The main Theorem 5 says that one can construct the noncommutative signature of nonsimply connected manifold staring from the PoincarÈ complex of singular chains of the manifold. Here we present a proper category of infinitely generated modules over the C -algebra A. Consider direct limit h of the countable sequence (uncountable directed family) of finitely generated free A-modules of increasing dimensions and maps which send basis to basis. The module h is free with countable (uncountable) family of generators. Endow the module h with the topology of direct limit. Let H be the module of all continuous functionals on h, H = HomA (h, A). The module verse limit. The modules h and of compact manifold the chain homology the cochain module i Let
def

H is the inverse limit of free finitely generated H are mutually dual, that is HomA (H , A) = h. module is inverse limit and the cochain module s the inverse limit and the chain module is direct

modules and has the topology of in In particular for the Cech homology is direct limit whereas, for singular (uncountable) limit.

: H j (H j) h J be self-adjoint continuous linear bijective map. Here h and j are two direct limits, H and J are inverse limits of Amodules. It turns out, that for is possible to define the signature type invariant as an element of Hermitian K -theory which coincides with classical signature of Hermitian form in finite dimensional case. Namely, there is a splitting of the module H j into a direct sum such that the matrix of the operator has almost hyperbolic form = 0 0
def

0



,

where is finite dimensional. By definition the signature of equals to the signature of , sign = sign . This definition is independent of the choice of splitting.


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2. Signature and infinite generated modules Let A be a separable unital C -algebra. Let be an infinite abstract set, # be the set of all finite subsets in which has natural direction by inclusion. If # that is , then A denotes free finite dimensional A-module with free generators xi , i . Consider a direction of maps h : A A , , induced by natural inclusion of free generators. Then the module h is defined as the direct limit of the spectrum h = lim A ,h .


(1)

The elements of h can be described as families of elements of the algebra A with finite support, that is xi A, i , xi = 0 for i .The A-module h has the natural topology generated by the direct limit. / The family of conjugate modules (A ) = HomA (A , A) induces the inverse limit H = lim A , h


,

h



: A



A .



(2)

The elements of the module H can be described as arbitrary families of elements of the algebra A: xi A, i . The module H has the natural topology of inverse limit. Two modules, h and H are mutually dual as topological C -modules. Namely, Proposition 1. One has natural isomorphisms H = HomA (h, A), h = HomA (H , A). The pairing between H and h has the natural form: x h, x ={xi , i }, y H , y ={yi , i }, x, y =
i Top Top

(3) (4)

ai bi .

(5)

The sum is defined correctly as one of the vector (x ) has finite support. Consider the submodule of h. We shall call that is the submodule of finite type if there exists a finite subset such that x , i , / xi = 0 . (6)

In general case let be a minimal subset such that x , i , /
def

xi = 0.

(7)

Put c = #( ). Then we shall call that is the submodule of the cardinal type c. If the submodule has the family of generators of cardinality c then is of the cardinal type c. Inverse, if the projective module is the submodule of the cardinal type c then it has at least c generators. Theorem 1. If the module is not the submodule of finite type, then the dual module is not finite or countable generated over the algebra A. Theorem 1 is reduced to the case when is countable. The latter was proved in [8]. For arbitrary cardinal number c one can prove similar property. Assume that the C -algebra A is separable. Theorem 2. If the module is the submodule of the cardinal type c, then the dual module has at least 2c generators over the algebra A.


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Lemma 1. Any linear continuous operator Hom(H , h) is finite generated that is it is represented as a composition = pr, ~ H
pr

An

(8)

~

h where pr is the projection onto coordinates i , #( ) = n. For proof it is sufficient to take the neighborhood of zero in h, which does not contain any linear subspace. For example such a neighbourhood is the set B(0, 1) = x h: i , x xi = 0 , then x
-1 i

<1 .

(9)

With respect to the definition of topology on H there is a finite subset such that if x H satisfies the condition i , (10) (B (0, 1)). Hence (x ) = 0.

Theorem 3. Let : H j (H j) h J be linear continuous self-adjoint bijection. Then there is self-dual splitting H j = H1 l j1 , 0 0 0
(H j) = H1 l j 1

(11)

(12)

such that l is a finite generated projective A-module, the operator has the matrix form = , (13)

and , , are bijections. Define the signature of as the equivalence class of the pair (l , ) in Hermitian K -theory sign( ) = sign(l , ) L0 (A).
def

(14)

Theorem 4. The class sign(l , ) in L0 (A) does not depend of the choice of the constructions in the previous theorem. 3. Signature of topological manifolds Consider topological Hausdorff compact manifold X of dimension n. Let = 1 (X, x0 ) be fundamental group, let A = C [ ] be the group algebra and A = C [ ] be its completion to C -algebra. Denote be Ci (X, A) the chain group of singular simplices with local coefficient system A generated by the natural representation of fundamental group 1 (X, x0 ). Then denote Ci (X, A) = A A Ci (X, A). All of them are free left A modules. The cochain groups are defined as C i (X, A) = HomA Ci (X, A), A . The manifold X is oriented if there is an element [X ] Hn (X, R) such that for any point x0 X the restriction of [X ] to the homology group Hn (X, X - x, R) R is the generator. Then there is the intersection operator (see for example [1]) D
[X ] def def

: C i (X, A) C

n-i

(X, A),

D

[X ]

( ) =[X ] ,

(15)


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2045

which induces isomorphism in the homology groups D
[X ]

: H i (X, A) Hn-i (X, A).

(16)

Define a formal category of singular algebraic PoincarÈ complexes following [2]. The difference consists of that all modules are not finite generated. Therefore surgeries of singular algebraic PoincarÈ complexes differ from that in [2]. Consider a complex of free A-modules C ={Ci ,i , i = 0,..., }, and dual complex C ={C i , i }. We say that we have a singular PoincarÈ complex of dimension n if the diagram 0 ···

n+3

C

n+1

0 D
0

1

C

1 D
1

2

··· ···



n

C

n D
n

n+1

C

n+1



n+2

C

n+2



n+3

··· (17)

C

n+2



n+2

C

n+1

C

n



n

C

n-1



n-1



1

C

0

0

has the properties: (1) The image of Di lies in a finite dimensional coordinate submodule of Ci . (2) Commutativity and self-adjointness i Di = (-1)i D Di = (-1)
i -1



n-i +1

,

(18) (19)

(n-i)i

D

n-i

.

(3) Induced homomorphisms Di : H(C ) H(C ) are isomorphisms. Hence the system C (X ) ={C (X, A), , D[X] } is the PoincarÈ complex of singular chains for the manifold X . One can similar define a singular PoincarÈ pair {C ,B , ,D } of dimension n + 1 and the boundary {B , ,E } of dimension n as a singular PoincarÈ complex where E
i -1

= i Di + (-1)i

+1

D

i -1



n-i +2

.

(20)

The notion of singular PoincarÈ pair generates the bordism relation which is equivalent relation. Two singular PoincarÈ complexes c ={C ,d ,D } and c ={C ,d ,D } are called homotopy equivalent if there exists a homomorphism f : C C such that f d = d f , D = fD f , (21) and f induces isomorphism in homology. If two singular PoincarÈ complexes are homotopy equivalent then they are bordant. 3.1. Infinite dimensional surgeries Consider the singular PoincarÈ complex (22) and the left part 0 ···

n+3

C

n+1

0 D
0

1

C

1 D
1

2

··· (22) ···
0

C

n+2



n+2

C

n+1

C

n



n

C

n-1



n-1

The image of D0 lies in a finite dimensional free submodule B , Im D0 B C0 . Consider a representation of C as direct sum of free modules C0 = C0 B. One has 0
n+3 n+2 n+1

(23) B C
D 0 1 D
n 1

C

2

··· (24)

0

···



C

n+2



C

n+1



C

n

C

n-1



n-1

···


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where is surjective. Hence one can change the complex (24) to the homotopy equivalent complex 0
n+3 n+2 n+1

B
D
0



C

1 D
1

2

··· (25) ···

···



C

n+2



C

n+1



C

n



n

C

n-1



n-1

and using standard finite dimensional surgery to the homotopy equivalent complex 0
n+3 n+2 n+1

0
D
0 n

C

1 D
1

2

··· (26) ···

···



C

n+2



C

n+1



C

n



C

n-1



n-1

Finally, for n = 2k one can obtain a homotopy equivalent complex of the form 0 ···

k +2

C

k +1

k D
k

k +1

C

k +1



k +2

··· (27)

C

k +1

C

k

0

and for n = 2k + 1 a complex of the form 0 ···

k +3

C

k +2

k D
k

k +1

C

k +1 D



k +2

C

k +2



k +3

··· (28)

k +1

C

k +2

C

k +1



k +1

C
k +1

k



k

0
k +1

In the case n = 2k let : Ck ker Dk +
k +1

, : Ck C

such that

= Id .

The operator P = Dk is projector with Im( Dk ) = ker k +1 . Really, if x ker k +1 then Dk ( Dk x - x) = (Dk - Id)Dk x = k +1 x . This means that in the homology one has [Dk ( Dk x - x)]= 0. Since the operator Dk is isomorphic on the level of homology we have that Dk x - x = 0 or Dk x = x . Hence the diagram (27) is splitted into direct sum of diagrams 0
0
k +2 k +1

(Im P )
0



k +1

C

k +1 0



k +2

··· (29)

··· and



C 0

k +1

(Im P)

k

0 0
0

Im P

··· (30)

0

D

···

0

Im P

0

The last diagram defines the signature as the element of K0 (A). In the case of n = 2k + 1 we should consider the manifold M â S1 and the algebra A = C [1 (M â S1 )] = C(S1 , A). Then K0 (A ) K0 (A) K1 (A) and the signature of manifold M can be described as the second component of sign(M â S1 ) K0 (A ) K0 (A) K1 (A). Thus we have the following Theorem 5. The PoincarÈ complex C (X ) of singular chains on the manifold X generates the signature sign(C (X )) K (A) that is invariant with respect to both bordisms and homotopy equivalences.


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References
[1] E.H. Spanier, Algebraic Topology, McGraw­Hill, New York, 1966. [2] A.S. Mishchenko, Homotopy invariants of nonsimply connected manifolds 1. Rational invariants (in Russian); English translation: Izv. Math. 34 (3) (1970) 501­514. [3] M. Gromov, Positive curvature, macroscopic dimension, spectral gaps and higher signature, in: Functional Analysis on the Eve of the 21st Century, in: Progr. Math., vol. 132, 1996, pp. 1­213. [4] A.S. Mishchenko, The Hirzebruch formula: 45 years of history and contemporary art, Alg. Anal. (Saint-Petersburg) 12 (4) (2000) 16­35. [5] A.S. Mishchenko, P.S. Popov, On construction of signature of quadratic forms on infinite-dimensional abstract spaces, Georgian Math. J. (9) 9 (4) (2002) 775­785. [6] P.S. Popov, Signature of infinite dimensional maps, in: Proceedings of 25th Conference of Young Scientists of MSU, 2003, pp. 52­54 (in Russian). [7] P.S. Popov, Algebraic construction of signature for topological manifolds, in: Abstracts of International Conference "Topology, Analysis and Application in Mathematical Physics", Devoted to Memory of Yu.P. Solovjov, 2005, pp. 15­16. [8] P.S. Popov, Extension of functor of Hermitian K -theory and signature of topological manifolds, Mat. Sb. 198 (8) (2007) 83­102; English translation: Sb. Math. 198 (8) (2007) 1145­1163.