Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866913385228009472 |
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| author | Nicholson, John |
| author_facet | Nicholson, John |
| contents | In previous work, we related homotopy types of finite $(G,n)$-complexes when $G$ has periodic cohomology to projective $\mathbb{Z} G$-modules representing the Swan finiteness obstruction. We use this to determine when $X \vee S^n \simeq Y \vee S^n$ implies $X \simeq Y$ for finite $(G,n)$-complexes $X$ and $Y$, and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective $\mathbb{Z} G$-modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case $n=2$, difficulties arise which lead to a new approach to finding a counterexample to Wall's D2 problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2005_01664 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction Nicholson, John Algebraic Topology Group Theory K-Theory and Homology Number Theory In previous work, we related homotopy types of finite $(G,n)$-complexes when $G$ has periodic cohomology to projective $\mathbb{Z} G$-modules representing the Swan finiteness obstruction. We use this to determine when $X \vee S^n \simeq Y \vee S^n$ implies $X \simeq Y$ for finite $(G,n)$-complexes $X$ and $Y$, and give lower bounds on the number of homotopically distinct pairs when this fails. The proof involves constructing projective $\mathbb{Z} G$-modules as lifts of locally free modules over orders in products of quaternion algebras, whose existence follows from the Eichler mass formula. In the case $n=2$, difficulties arise which lead to a new approach to finding a counterexample to Wall's D2 problem. |
| title | Cancellation for $(G,n)$-complexes and the Swan finiteness obstruction |
| topic | Algebraic Topology Group Theory K-Theory and Homology Number Theory |
| url | https://arxiv.org/abs/2005.01664 |