Theorem. Graph connections are classified by holonomy [fdip-006F]
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  • November 1, 2026

Let \(\Gamma \) be a finite connected graph with chosen edge orientations and root \(o\). For a path traversing \(e_1,\ldots ,e_k\), use inverse edge values when traversing backwards and define its holonomy by \[U_{e_k}\cdots U_{e_1}.\] Thus the rightmost factor transports first. Use the same composition convention for loops: \(\beta \alpha \) first traverses \(\alpha \) and then \(\beta \). The gauge classes of \(G\)-valued connections are \[\operatorname {Hom}(\pi _1(\Gamma ,o),G)/G \;\cong \;G^{b_1}/G,\qquad b_1=|E|-|V|+1,\] where \(G\) acts by simultaneous conjugation. The second identification depends on a spanning tree and orientations of the remaining edges. This classifies group-valued connections, not their possibly nonfaithful representation images or the spectra of their precisions.

Proof. 

  • November 1, 2026

Choose a spanning tree \(T\). Each vertex \(v\) has a unique tree path from \(o\); let \(t_v\in G\) be its transport, with \(t_o=1\). Use the gauge \(g_v=t_v^{-1}\). If a tree edge runs from \(s\) to \(h\), its path transports satisfy \(t_h=U_et_s\), with the equivalent inverse identity when the rooted path traverses the edge backwards. Hence \[(gU)_e=t_h^{-1}U_et_s=1\qquad (e\in T).\] Every connection therefore has a representative with all tree values one. The remaining \(|E|-|T|=b_1\) edge values are unrestricted.

A gauge transformation carrying one such representative to another satisfies \(g_hg_s^{-1}=1\) on every tree edge. Tree connectivity forces all \(g_v\) to equal one element \(a\in G\). It conjugates every remaining edge value by \(a\). Conversely every simultaneous conjugation comes from this constant gauge. This proves the classification by \(G^{b_1}/G\), including the case of a tree, where \(b_1=0\).

Traversing a path and then its inverse gives cancelling group factors, so transport is unchanged by removing backtracking. The loops that run from \(o\) through the tree to the source of a non-tree edge, traverse it and return through the tree are a free basis of \(\pi _1(\Gamma ,o)\). This is [hatcher2002algebraic, Proposition 1A.2 and its proof, pp. 84–85]: contract the tree to obtain a wedge of \(b_1\) circles. Assigning arbitrary values to this free basis gives every group homomorphism. In tree gauge these values are precisely the non-tree edge values. A general gauge conjugates each rooted loop holonomy by \(g_o\), proving the first identification as well.

With two-cells attached, call a connection flat when the ordered holonomy around every attaching boundary is the identity in \(G\). Equivalently the graph homomorphism kills the normal subgroup generated by those boundaries. It therefore factors through the fundamental group of the resulting space. A representation \(\rho \) can send a nonidentity group element to the identity operator; operator flatness alone is therefore insufficient when \(\rho \) is nonfaithful.

In the representation on \(F\), a parallel section is determined by its root value \(x\in F\). Tree transport defines its other values, and each non-tree edge condition says that its rooted holonomy fixes \(x\). Consequently evaluation at \(o\) identifies the space of parallel sections with the common fixed space of the holonomy representation. This proves the massless-kernel criterion of Definition [fdip-006D] in terms of cycle data.