Exotic presentations of quaternion groups and Wall's D2 problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909698433744896 |
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| author | Hofmann, Tommy Nicholson, John |
| author_facet | Hofmann, Tommy Nicholson, John |
| contents | The D2 problem of C. T. C. Wall asks whether every finite cohomologically 2-dimensional CW-complex is homotopy equivalent to a finite 2-complex. Several potential counterexamples have been proposed, the longest standing of which is a CW-complex constructed by Cohen and Dyer whose fundamental group is a quaternion group of order 32. We show that this CW-complex is homotopy equivalent to the presentation 2-complex of a presentation constructed by Mannan-Popiel, thus showing it is not a counterexample to the D2 problem. We next introduce an infinite family of presentations for a quaternion group of order $4n$ and prove that they achieve homotopy types which are not achieved by the presentations of Mannan-Popiel. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_15999 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Exotic presentations of quaternion groups and Wall's D2 problem Hofmann, Tommy Nicholson, John Group Theory Algebraic Topology Geometric Topology 57K20, 20C05, 57M05, 57Q12 The D2 problem of C. T. C. Wall asks whether every finite cohomologically 2-dimensional CW-complex is homotopy equivalent to a finite 2-complex. Several potential counterexamples have been proposed, the longest standing of which is a CW-complex constructed by Cohen and Dyer whose fundamental group is a quaternion group of order 32. We show that this CW-complex is homotopy equivalent to the presentation 2-complex of a presentation constructed by Mannan-Popiel, thus showing it is not a counterexample to the D2 problem. We next introduce an infinite family of presentations for a quaternion group of order $4n$ and prove that they achieve homotopy types which are not achieved by the presentations of Mannan-Popiel. |
| title | Exotic presentations of quaternion groups and Wall's D2 problem |
| topic | Group Theory Algebraic Topology Geometric Topology 57K20, 20C05, 57M05, 57Q12 |
| url | https://arxiv.org/abs/2507.15999 |