On symmetries of spheres in univalent foundations
Abstract
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.
- Publication:
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arXiv e-prints
- Pub Date:
- January 2024
- DOI:
- 10.48550/arXiv.2401.15037
- arXiv:
- arXiv:2401.15037
- Bibcode:
- 2024arXiv240115037C
- Keywords:
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- Computer Science - Logic in Computer Science;
- Mathematics - Algebraic Topology;
- 55P10 (Primary) 55U40;
- 03B38 (Secondary);
- F.4.1
- E-Print:
- comments welcome