Stably free modules and the unstable classification of 2-complexes
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866915551181275136 |
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| author | Nicholson, John |
| author_facet | Nicholson, John |
| contents | For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental group $G$ and with Euler characteristic exceeding the minimal value over $G$ by $k$. This resolves Problem D5 in the 1979 Problem List of C. T. C. Wall. We also explore a number of generalisations and present a potential application to the topology of closed smooth 4-manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_02220 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Stably free modules and the unstable classification of 2-complexes Nicholson, John Algebraic Topology Group Theory Geometric Topology Rings and Algebras For all $k \ge 2$, we show that there exists a group $G$ and a non-free stably free $\mathbb{Z} G$-module of rank $k$. We use this to show that, for all $k \ge 2$, there exist homotopically distinct finite $2$-complexes with fundamental group $G$ and with Euler characteristic exceeding the minimal value over $G$ by $k$. This resolves Problem D5 in the 1979 Problem List of C. T. C. Wall. We also explore a number of generalisations and present a potential application to the topology of closed smooth 4-manifolds. |
| title | Stably free modules and the unstable classification of 2-complexes |
| topic | Algebraic Topology Group Theory Geometric Topology Rings and Algebras |
| url | https://arxiv.org/abs/2108.02220 |