On Finite domination and Poincar\'e duality

TL;DR

This paper demonstrates that there are many non-homotopy finite Poincaré duality spaces, constructed using algebraic K-theory and group properties, with arbitrarily large dimension.

Contribution

It introduces a method to construct finitely dominated Poincaré spaces that are not homotopy finite, based on properties of the reduced Grothendieck group.

Findings

Existence of non-homotopy finite Poincaré duality spaces for certain groups

Construction of such spaces with arbitrarily large dimension

Use of stable Poincaré duality thickening in the proof

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.