On Finite domination and Poincar\'e duality

Abstract

The object of this paper is to show that non-homotopy finite Poincar\'e duality spaces are plentiful. Let $\pi$ be finitely presented group. Assuming that the reduced Grothendieck group $\tilde K_0(\Bbb Z[\pi])$ has a non-trivial 2-divisible element, we construct a finitely dominated Poincar\'e space $X$ with fundamental group $\pi$ such that $X$ is not homotopy finite. The dimension of $X$ can be made arbitrarily large. Our proof relies on a result which says that every finitely dominated space possesses a stable Poincar\'e duality thickening.