On symmetries of spheres in univalent foundations
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916106909777920 |
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| author | Cagne, Pierre Buchholtz, Ulrik Kraus, Nicolai Bezem, Marc |
| author_facet | Cagne, Pierre Buchholtz, Ulrik Kraus, Nicolai Bezem, Marc |
| contents | Working in univalent foundations, we investigate the symmetries of spheres, i.e., the types of the form $\mathbb{S}^n = \mathbb{S}^n$. The case of the circle has a slick answer: the symmetries of the circle form two copies of the circle. For higher-dimensional spheres, the type of symmetries has again two connected components, namely the components of the maps of degree plus or minus one. Each of the two components has $\mathbb{Z}/2\mathbb{Z}$ as fundamental group. For the latter result, we develop an EHP long exact sequence. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_15037 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On symmetries of spheres in univalent foundations Cagne, Pierre Buchholtz, Ulrik Kraus, Nicolai Bezem, Marc Logic in Computer Science Algebraic Topology 55P10 (Primary) 55U40, 03B38 (Secondary) F.4.1 Working in univalent foundations, we investigate the symmetries of spheres, i.e., the types of the form $\mathbb{S}^n = \mathbb{S}^n$. The case of the circle has a slick answer: the symmetries of the circle form two copies of the circle. For higher-dimensional spheres, the type of symmetries has again two connected components, namely the components of the maps of degree plus or minus one. Each of the two components has $\mathbb{Z}/2\mathbb{Z}$ as fundamental group. For the latter result, we develop an EHP long exact sequence. |
| title | On symmetries of spheres in univalent foundations |
| topic | Logic in Computer Science Algebraic Topology 55P10 (Primary) 55U40, 03B38 (Secondary) F.4.1 |
| url | https://arxiv.org/abs/2401.15037 |