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

TL;DR

This paper investigates the homotopy types of spaces of flat connections on principal bundles over smooth manifolds, using Chern-Weil theory and characteristic classes to relate representation spaces to mapping spaces for classical Lie groups.

Contribution

It introduces a novel approach connecting characteristic classes to the homotopy types of flat connection spaces for classical Lie groups.

Findings

Characterizes the weak homotopy type of flat connection spaces for U(n), O(n), SO(n), and Spin(n).

Establishes a relationship between representation spaces and mapping spaces via Chern-Weil theory.

Provides new insights into the topology of moduli spaces of flat connections.

Abstract

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 $$\text{Hom}(\pi_{1}(M), G)\rightarrow\text{Map}_{*}(M,BG)$$ 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.