The Homotopy Type of Spaces of Flat Connections for Classical Lie Groups
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915681079918592 |
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| author | Davis, Andrew |
| author_facet | Davis, Andrew |
| contents | Let $M$ be a smooth manifold. We use Chern-Weil theory to study the characteristic classes of principal $G$-bundles built from continuous families of $π_{1}(M)$-representations, where $G$ is a compact Lie group. We then relate these families to the functorial map $$\text{Hom}(π_{1}(M), G)\rightarrow\text{Map}_{*}(M,BG)$$ and use this relationship to study the weak homotopy type of the space of flat connections for $U(n)$, $O(n)$, $SO(n)$, and $\text{Spin}(n)$ bundles. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_15004 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Homotopy Type of Spaces of Flat Connections for Classical Lie Groups Davis, Andrew Algebraic Topology Differential Geometry K-Theory and Homology Let $M$ be a smooth manifold. We use Chern-Weil theory to study the characteristic classes of principal $G$-bundles built from continuous families of $π_{1}(M)$-representations, where $G$ is a compact Lie group. We then relate these families to the functorial map $$\text{Hom}(π_{1}(M), G)\rightarrow\text{Map}_{*}(M,BG)$$ and use this relationship to study the weak homotopy type of the space of flat connections for $U(n)$, $O(n)$, $SO(n)$, and $\text{Spin}(n)$ bundles. |
| title | The Homotopy Type of Spaces of Flat Connections for Classical Lie Groups |
| topic | Algebraic Topology Differential Geometry K-Theory and Homology |
| url | https://arxiv.org/abs/2512.15004 |