Poincaré duality and signature for topological manifolds (Q952609)

This paper describes how to define the signature of a topological \(4k\)-manifold without using a cell structure induced by a handle decomposition. Since the chain complexes associated to the homology theories such as singular homology are infinitely generated, the challenge is to get a finite dimensional complex on which an index can be defined. The authors achieve this by building a theory of infinitely generated modules over a \(C^{*}\)-algebra \(\mathcal A\). They show how one can define a signature for Hermitian forms in this case, by splitting the duality operator into a direct sum where the matrix for the operator has an almost hyperbolic form for which there is a finite dimensional subspace on which the signature can be defined. They show that this definition is independent of the splitting of the module.