Framed flat connections and holonomy ($\mathrm{Hom}({\pi }_1(X),G)$)

For a connected base and a chosen frame at one point, a flat connection gives a representation of the fundamental group by parallel transport. Conversely, $(\tilde{X}\times G)/{\pi }_1(X)$ with monodromy representation has a flat product connection. Based gauge classes correspond to representations, and unbased classes to conjugacy classes. On a fixed underlying bundle only representations whose associated flat bundle has that bundle type are allowed.

// Destination: geometry-and-topology.bigb