Topological components of spaces of commuting elements in connected nilpotent Lie groups
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911465458368512 |
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| author | Antolín-Camarena, Omar Villarreal, Bernardo |
| author_facet | Antolín-Camarena, Omar Villarreal, Bernardo |
| contents | We study the homotopy type of spaces of commuting elements in connected nilpotent Lie groups, via almost commuting elements in their Lie algebras. We give a necessary and sufficient condition on the fundamental group of such a Lie group $G$ to ensure $\mathrm{Hom}(\mathbb{Z}^k,G)$ is path-connected. In particular for the reduced upper unitriangular groups and the reduced generalized Heisenberg groups, $\mathrm{Hom}(\mathbb{Z}^k,G)$ is not path-connected, and we compute the homotopy type of its path-connected components in terms of Stiefel manifolds and the maximal torus of $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_09652 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological components of spaces of commuting elements in connected nilpotent Lie groups Antolín-Camarena, Omar Villarreal, Bernardo Algebraic Topology We study the homotopy type of spaces of commuting elements in connected nilpotent Lie groups, via almost commuting elements in their Lie algebras. We give a necessary and sufficient condition on the fundamental group of such a Lie group $G$ to ensure $\mathrm{Hom}(\mathbb{Z}^k,G)$ is path-connected. In particular for the reduced upper unitriangular groups and the reduced generalized Heisenberg groups, $\mathrm{Hom}(\mathbb{Z}^k,G)$ is not path-connected, and we compute the homotopy type of its path-connected components in terms of Stiefel manifolds and the maximal torus of $G$. |
| title | Topological components of spaces of commuting elements in connected nilpotent Lie groups |
| topic | Algebraic Topology |
| url | https://arxiv.org/abs/2405.09652 |