The tangential Thom class of a Poincare duality group

Abstract

For each Poincare duality group there exists a class, which we call the tangential Thom class of Gamma, in the group cohomology of Gamma x Gamma with a right choice of the coefficient module. The class has the crucial properties, even if stated in a purely algebraic language, which correspond to those of Thom class of the tangent bundle of a closed manifold. In particular the Thom isomorphism has been proved to exist by observing that certain two sequences of homological functors, one being the homology of Gamma and the other that of Gamma x Gamma, being regarded as functors defined on the category of Z Gamma-modules are homological and effaceable. (C) 2015 Elsevier B.V. All rights reserved.