The tangential Thom class of a Poincaré duality group

Abstract

For each Poincaré duality group Γ there exists a class, which we call the tangential Thom class of Γ, in the group cohomology of Γ×Γ 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 Γ and the other that of Γ×Γ, being regarded as functors defined on the category of ZΓ-modules are homological and effaceable.