Projective modules and the homotopy classification of (G,n)–complexes

Nicholson, John

doi:10.2140/agt.2024.24.2245

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Projective modules and the homotopy classification of $(G,n)$–complexes

John Nicholson

Algebraic & Geometric Topology 24 (2024) 2245–2284

DOI: 10.2140/agt.2024.24.2245

Abstract

A $(G, n)$ –complex is an $n$ –dimensional CW–complex with fundamental group $G$ and whose universal cover is $(n - 1)$ –connected. If $G$ has periodic cohomology then, for appropriate $n$ , we show that there is a one-to-one correspondence between the homotopy types of finite $(G, n)$ –complexes and the orbits of the stable class of a certain projective $ℤ G$ –module under the action of $Aut (G)$ . We develop techniques to compute this action explicitly and use this to give an example where the action is nontrivial.

Keywords

2–complexes, CW–complexes, $\mathbb{Z}G$–modules, projective modules, finiteness obstruction

Mathematical Subject Classification

Primary: 55P15

Secondary: 20C05, 55U15, 57K20

References

Publication

Received: 19 July 2022

Revised: 4 April 2023

Accepted: 28 April 2023

Published: 16 July 2024

Authors

John Nicholson
School of Mathematics and Statistics University of Glasgow Glasgow United Kingdom

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Volume 24, issue 4 (2024)