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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.21975 |
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| _version_ | 1866908965354340352 |
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| author | Hofmann, Tommy Nicholson, John |
| author_facet | Hofmann, Tommy Nicholson, John |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21975 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Swan modules and homotopy types after a single stabilisation Hofmann, Tommy Nicholson, John Algebraic Topology Group Theory Geometric Topology K-Theory and Homology 55P15, 20C05, 57Q12 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. |
| title | Swan modules and homotopy types after a single stabilisation |
| topic | Algebraic Topology Group Theory Geometric Topology K-Theory and Homology 55P15, 20C05, 57Q12 |
| url | https://arxiv.org/abs/2507.21975 |