Some asymptotic formulae for torsion in homotopy groups
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866929597040295936 |
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| author | Boyde, Guy Huang, Ruizhi |
| author_facet | Boyde, Guy Huang, Ruizhi |
| contents | Inspired by a remarkable work of Félix, Halperin and Thomas on the asymptotic estimation of the ranks of rational homotopy groups, and more recent works of Wu and the authors on local hyperbolicity, we prove two asymptotic formulae for torsion rank of homotopy groups, one using ordinary homology and one using $K$-theory. We use these to obtain explicit quantitative asymptotic lower bounds on the torsion rank of the homotopy groups for many interesting spaces after suspension, including Moore spaces, Eilenberg-MacLane spaces, complex projective spaces, complex Grassmannians, Milnor hypersurfaces and unitary groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_02497 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Some asymptotic formulae for torsion in homotopy groups Boyde, Guy Huang, Ruizhi Algebraic Topology 55Q52, 55Q05 (Primary) 55Q15, 55P40 (Secondary) Inspired by a remarkable work of Félix, Halperin and Thomas on the asymptotic estimation of the ranks of rational homotopy groups, and more recent works of Wu and the authors on local hyperbolicity, we prove two asymptotic formulae for torsion rank of homotopy groups, one using ordinary homology and one using $K$-theory. We use these to obtain explicit quantitative asymptotic lower bounds on the torsion rank of the homotopy groups for many interesting spaces after suspension, including Moore spaces, Eilenberg-MacLane spaces, complex projective spaces, complex Grassmannians, Milnor hypersurfaces and unitary groups. |
| title | Some asymptotic formulae for torsion in homotopy groups |
| topic | Algebraic Topology 55Q52, 55Q05 (Primary) 55Q15, 55P40 (Secondary) |
| url | https://arxiv.org/abs/2301.02497 |