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 ${\pi }_1(M)$-representations, where $G$ is a compact Lie group. We then relate these families to the functorial map $$\text{Hom}({\pi }_1(M),G)\to {\text{Map}}_{\ast }(M,BG)$$ 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.