Homology, Homotopy and Applications

Volume 26 (2024)

Number 1

On finite domination and Poincaré duality

Pages: 29 – 35

DOI: https://dx.doi.org/10.4310/HHA.2024.v26.n1.a3


John R. Klein (Department of Mathematics, Wayne State University, Detroit, Michigan, U.S.A. )


The object of this paper is to show that non-homotopy finite Poincaré duality spaces are plentiful. Let $π$ be a finitely presented group. Assuming that the reduced Grothendieck group $\widetilde{K}_0 (\mathbb{Z} [\pi])$ has a non-trivial $2$-divisible element, we construct a finitely dominated Poincaré space $X$ with fundamental group $π$ 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 Poincaré duality thickening.


Poincaré duality space, finite domination, Wall finiteness obstruction

2010 Mathematics Subject Classification

Primary 19J05, 57P10. Secondary 16E20.

Received 12 November 2022

Received revised 15 January 2023

Accepted 17 January 2023

Published 24 January 2024