Finite BF partition function is a topological invariant

For each connected component $K_j$, start with any flat assignment and set the root gauge value at $v_j$ to the identity. Along the unique tree paths in $T_j$ from $v_j$, choose the gauge value at each new vertex so that the transformed label on the traversed tree edge is the identity; when that edge is traversed against its chosen orientation, use its inverse label. This recursively produces a tree-normalized assignment on every component. It is unique among gauges whose value at each $v_j$ is the identity, since the condition that each transformed tree label remain $1$ determines each next vertex value.

In a tree-normalized assignment on $K_j$, every edge of $T_j$ has label $1$. Assign to each generator $x_e$ of ${\pi }_1(K_j,v_j)$ the inverse of the label on its oriented non-tree edge. By [F4], this is a homomorphism for the traversal-first product on loops: holonomy of a concatenation is the reverse-ordered product, and reversing the path in the Statement reverses that order. Flatness says exactly that every relator $r_f$ maps to $1$. By the presentation in [F2], the normalized assignments on $K_j$ are in bijection with homomorphisms $\rho :{\pi }_1(K_j,v_j)\to G$.

For a connected subcomplex $L_i\subseteq K_{j(i)}$ and base data as in the Statement, the pointed inclusion at $v_i$ induces a well-defined homomorphism on fundamental groups by [F5]. The base-path change map $c_{{\gamma }_i}^{K_{j(i)}}:{\pi }_1(K_{j(i)},v_i)\to {\pi }_1(K_{j(i)},w_{j(i)})$, given by $[\alpha ]\mapsto [{\gamma }_i\ast \alpha \ast {\bar{\gamma }}_i]$, is a well-defined isomorphism: endpoint-fixed homotopies remain endpoint-fixed after adjoining ${\gamma }_i,{\bar{\gamma }}_i$, insertion of the retraced path ${\bar{\gamma }}_i\ast {\gamma }_i$ contracts in a product of loop classes, and $c_{{\bar{\gamma }}_i}^{K_{j(i)}}$ is its inverse. Thus ${\mathrm{ȷ}}_{{\gamma }_i,\ast }=c_{{\gamma }_i}^{K_{j(i)}}\circ {\mathrm{ȷ}}_{\ast }$ is well defined. For a flat assignment $A$ and an edge loop $\mathrm{\ell }$ in $L_i$, the edge-product convention gives ${\mathrm{\Phi }}_{K_{j(i)}}(A)({\mathrm{ȷ}}_{{\gamma }_i,\ast }[\mathrm{\ell }])=H_A({\bar{\gamma }}_i)H_A(\bar{\mathrm{\ell }})H_A({\gamma }_i)=H_A({\bar{\gamma }}_i){\mathrm{\Phi }}_{L_i}(A{\mathrm{\mid }}_{L_i})([\mathrm{\ell }])H_A({\bar{\gamma }}_i{)}^{-1}$. If a gauge arrow $h:A\to A^{\mathrm{\prime }}$ is applied, its path-product rule gives $H_{A^{\mathrm{\prime }}}({\bar{\gamma }}_i)=h(w_{j(i)})H_A({\bar{\gamma }}_i)h(v_i{)}^{-1}$; this is exactly the naturality condition for the stated conjugation arrows. Hence inverse path holonomy intertwines restriction with the componentwise precomposition functor up to that natural isomorphism.

On a fixed component $K_j$, a gauge transformation taking one tree-normalized assignment to another must have equal values at the two endpoints of every edge of $T_j$, because both edge labels are $1$. Since $T_j$ is connected, it is constant on all vertices of $K_j$. Conversely, each constant gauge value $a\in G$ preserves tree normalization and conjugates every generator label by $a$. Thus the full subgroupoid on tree-normalized objects over $K_j$ is identified with $\mathrm{Hom}({\pi }_1(K_j,v_j),G)\mathrm{/}\mathrm{/}G$. Step 1.1 makes its inclusion into the full flat-connection groupoid essentially surjective, and the inclusion is fully faithful because it is full. The root-one gauge used in step 1.1 leaves inverse path holonomy at $v_j$ unchanged, so the object map ${\mathrm{\Phi }}_{K_j}$ in the Statement agrees with normalization followed by this identification. For any gauge arrow $h:A\to A^{\mathrm{\prime }}$, the gauge factors telescope along each based edge loop $\delta$ at $v_j$ to give $H_{A^{\mathrm{\prime }}}(\bar{\delta })=h(v_j)H_A(\bar{\delta })h(v_j{)}^{-1}$, so ${\mathrm{\Phi }}_{K_j}(A^{\mathrm{\prime }})=h(v_j){\mathrm{\Phi }}_{K_j}(A)h(v_j{)}^{-1}$. Conversely, after normalizing $A$ and $A^{\mathrm{\prime }}$ by root-one gauges, each conjugating arrow between their representations is the unique constant gauge arrow between the normalized assignments; conjugating back gives the unique gauge arrow between $A$ and $A^{\mathrm{\prime }}$ with that value at $v_j$. Hence the stated functor is fully faithful as well as essentially surjective, and is an equivalence.

For each connected component $K_j$, the subgroup of vertex gauges equal to $1$ at $v_j$ acts freely and transitively on each fiber of the map from flat assignments to their tree-normalized representative, by step 1.1 and its uniqueness. Its order is $\mathrm{\mid }G{\mathrm{\mid }}^{\mathrm{\mid }(K_j{)}_0\mathrm{\mid }-1}$, so $\mathrm{\mid }\mathrm{Flat}(K_j,G)\mathrm{\mid }=\mathrm{\mid }G{\mathrm{\mid }}^{\mathrm{\mid }(K_j{)}_0\mathrm{\mid }-1}\mathrm{\mid }\mathrm{Hom}({\pi }_1(K_j,v_j),G)\mathrm{\mid }$. Substituting this count into [F1] gives the connected normalization. Equivalently, the action-groupoid cardinality in step 4.1 sums $1\mathrm{/}\mathrm{\mid }\mathrm{Stab}(\rho )\mathrm{\mid }$ over conjugacy classes; no freeness of the conjugation action is assumed.

For $K=K_1\bigsqcup \cdots \bigsqcup K_m$, the defining edge-label and vertex-gauge data split componentwise, so ${\mathcal{C}}_G(K)={\prod }_j{\mathcal{C}}_G(K_j)$. Applying step 4.1 on each component gives the product equivalence ${\mathrm{\Phi }}_K:{\mathcal{C}}_G(K)\simeq {\mathcal{R}}_G(K)$, and applying step 3.1 componentwise gives the stated restriction naturality. Automorphism orders multiply in a product groupoid and its isomorphism classes are componentwise tuples, so its groupoid cardinality is the product of the connected cardinalities from step 5.1. When $m=0$, the empty product is terminal with cardinality $1$.