On finite domination and Poincaré duality

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.

Keywords

Poincaré duality space, finite domination, Wall finiteness obstruction

2010 Mathematics Subject Classification

Primary 19J05, 57P10. Secondary 16E20.

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Received 12 November 2022

Received revised 15 January 2023

Accepted 17 January 2023

Published 24 January 2024