The Homotopy Type of Spaces of Flat Connections for Classical Lie Groups

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.

💡 Research Summary

The paper investigates the weak homotopy type of spaces of flat connections on principal bundles whose structure groups are the classical compact Lie groups (U(n)), (SU(n)), (O(n)), (SO(n)) and (\mathrm{Spin}(n)). The author builds on the work of Baird and Ramras, who studied the map \