Swan modules and homotopy types after a single stabilisation

TL;DR

This paper investigates Swan modules and their impact on homotopy classification, demonstrating the existence of non-free Swan modules and their implications for stabilisation in homotopy theory.

Contribution

It proves the existence of a non-free stably free Swan module, resolving a longstanding problem, and explores homotopy equivalences after stabilisation in specific dimensions.

Findings

Existence of a non-free stably free Swan module

Homotopy equivalence after multiple stabilisations but not after a single one in certain dimensions

A new proof regarding groups with k-periodic cohomology without computing Swan finiteness obstruction

Abstract

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.