Flatness implies holonomy depends only on homotopy class of loops

Let $\pi:P\to M$ be a principal G-bundle with a principal connection $\omega$. Fix a basepoint $x\in M$ and a point $p\in P_x$. For a piecewise smooth loop $\gamma$ based at $x$, horizontal lifting gives an element $\mathrm{Hol}_{p}(\gamma)\in G$, and the subgroup generated by such elements is the holonomy group at $p$. Let $\Omega$ denote the curvature of $\omega$.

Proposition (homotopy invariance of holonomy under flatness)

If $\Omega=0$, then for any two piecewise smooth loops $\gamma_0,\gamma_1$ based at $x$ that are homotopic through based loops, one has

Consequently, the assignment $[\gamma]\mapsto \mathrm{Hol}_{p}(\gamma)$ defines a group homomorphism

and changing the choice of $p\in P_x$ conjugates this homomorphism by an element of $G$.

This is the precise sense in which a flat connection has “holonomy depending only on homotopy class.”

Examples

1. Trivial flat connection. On $P=M\times G$ with the zero connection, every loop has holonomy $e\in G$, so the induced homomorphism $\pi_1(M,x)\to G$ is trivial.
2. Flat $U(1)$-connections on $S^1$. For a connection with constant local 1-form $A=\lambda\,d\theta$ (in a global gauge), curvature is zero and holonomy sends the generator of $\pi_1(S^1)\cong \mathbb Z$ to $e^{2\pi i\lambda}\in U(1)$; dependence is only on the winding number.
3. Möbius bundle as $O(1)$-bundle. The Möbius line bundle corresponds to a principal $O(1)\cong\{\pm 1\}$-bundle with a flat connection whose holonomy representation sends the generator of $\pi_1(S^1)$ to $-1$.