Swan modules and homotopy types after a single stabilisation

Tommy Hofmann; John Nicholson

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arXiv · 2507.21975

Swan modules and homotopy types after a single stabilisation

Tommy Hofmann·John Nicholson

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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.

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Tommy Hofmann, John Nicholson. 2025-07-29. Swan modules and homotopy types after a single stabilisation. https://arxiv.org/abs/2507.21975

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