The space of commuting elements in an exceptional Lie group and maps between classifying spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917789506207744 |
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| author | Takeda, Masahiro |
| author_facet | Takeda, Masahiro |
| contents | Let $π$ be a discrete group, and let $G$ be a compact connected Lie group. $\mathrm{Hom}(π,G)_0$ denotes the null-component of the space of homomorphisms from $π$ to $G$, and $\mathrm{map}_*(Bπ,BG)_0$ denotes the null-component of the space of maps from $Bπ$ to $BG$. Since the classifying space functor is continuous, there is a continuous map $Θ\colon\mathrm{Hom}(π,G)_0\to\mathrm{map}_*(Bπ,BG)_0$. Atiyah and Bott studied this map when $π$ is a surface group, and proved surjectivity in rational cohomology. In this paper, we obtain the condition that the map $Θ$ is surjective or not in rational cohomology when $π$ is $\mathbf{Z}^m$ for $m\geq 3$ and $G$ is a compact connected Lie group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_19500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The space of commuting elements in an exceptional Lie group and maps between classifying spaces Takeda, Masahiro Algebraic Topology 55R37, 57T10 Let $π$ be a discrete group, and let $G$ be a compact connected Lie group. $\mathrm{Hom}(π,G)_0$ denotes the null-component of the space of homomorphisms from $π$ to $G$, and $\mathrm{map}_*(Bπ,BG)_0$ denotes the null-component of the space of maps from $Bπ$ to $BG$. Since the classifying space functor is continuous, there is a continuous map $Θ\colon\mathrm{Hom}(π,G)_0\to\mathrm{map}_*(Bπ,BG)_0$. Atiyah and Bott studied this map when $π$ is a surface group, and proved surjectivity in rational cohomology. In this paper, we obtain the condition that the map $Θ$ is surjective or not in rational cohomology when $π$ is $\mathbf{Z}^m$ for $m\geq 3$ and $G$ is a compact connected Lie group. |
| title | The space of commuting elements in an exceptional Lie group and maps between classifying spaces |
| topic | Algebraic Topology 55R37, 57T10 |
| url | https://arxiv.org/abs/2409.19500 |