= Framed flat connections and holonomy
{title2=$\operatorname{Hom}(\pi_1(X),G)$}
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, $(\widetilde X\times G)/\pi_1(X)$ 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
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