Paper

Swan modules and homotopy types after a single stabilisation

Hofmann, Tommy, Nicholson, John

2025

Overview

We study Swan modules, which are a special class of projective modules over integral group rings, and their consequences for the homotopy classification of CW-complexes. We show that there exists a non-free stably free Swan module, thus resolving Problem A4 in the 1979 Problem List of C. T. C. Wall. As an application we show that, in all dimensions \\(n \\equiv 3\\) mod \\(4\\), there exist finite \\(n\\)-complexes which are homotopy equivalent after stabilising with multiple copies of \\(S^n\\), but not after a single stabilisation. This answers a question of M. N. Dyer. We also resolve a question of S. Plotnick concerning Swan modules associated to group automorphisms and, as an application, obtain a short and direct proof that there exists a group with \\(k\\)-periodic cohomology which does not have free period \\(k\\). In contrast to the original proof our R. J. Milgram, our proof circumvents the need to compute the Swan finiteness obstruction.

• Publisher: Cornell University Library, arXiv.org
• Subject: Automorphisms / Homology / Modules / Stabilization