Connection transport is invariant under thin homotopy

On one closed curvilinear triangle $\mathrm{\Delta }$ of the subdivision, pull the connection back by the smooth restriction of $H$ and write its two coefficient matrices as $A_s=\omega (H_s)$ and $A_t=\omega (H_t)$. For a smooth family of paths inside $\mathrm{\Delta }$, let $U_t=-A_{t}U$, $U(s,0)=I$, and put $K=U_s+A_{s}U$. Smooth dependence in [F4] permits this differentiation, and [F5] gives uniqueness for the linear equation satisfied by $K$. The product rule gives $K_t+A_{t}K=-({\mathrm{\partial }}_s{A}_t-{\mathrm{\partial }}_t{A}_s+[A_s,A_t])U=-H^{\ast }F({\mathrm{\partial }}_s,{\mathrm{\partial }}_t)U$. Here [F3] identifies the pulled-back two-form as the curvature of the supplier connection.

Since $\mathrm{rank}(dH)\le 1$ on $\mathrm{\Delta }$, $H_s,H_t$ are linearly dependent and the alternating two-form satisfies $H^{\ast }F({\mathrm{\partial }}_s,{\mathrm{\partial }}_t)=F(H_s,H_t)=0$. Fix a vertex $q_0$ and join it to each $q\in \mathrm{\Delta }$ by the image of the straight radial path in the standard-simplex parametrization. The associated transports $Q(q)$ exist globally by [F5] and depend smoothly on $q$: apply [F4] locally in the endpoint parameter and continue over finitely many subintervals of the compact path interval, using uniqueness on overlaps. For each radial path, [F3] and the ODE give $(Q^{\mathsf{T}}\eta Q{)}^{\mathrm{\prime }}=0$, hence $Q(q{)}^{\mathsf{T}}\eta Q(q)=\eta$ and $Q(q)$ is invertible. Step 1.1 on this radial family with fixed start gives $dQ=-H^{\ast }\omega Q$. Thus transport from $q_1$ to $q_2$ in $\mathrm{\Delta }$ is $Q(q_2)Q(q_1{)}^{-1}$ by [F5], and the triangle boundary transport is identity.

The finite curvilinear-triangle subdivision is a finite connected CW complex $K$; choose the lower-left corner as basepoint and a maximal tree in its one-skeleton. The outer boundary loop contracts to that basepoint by the straight-line contraction of the parameter square, so [F6] makes its fundamental-group class the identity. By [F7], ${\pi }_1(K)$ has generators given by non-tree edge loops closed along the tree and relators given by based triangular face boundaries. For each free generator $x_e$, assign the inverse transport of its based edge loop and extend to a homomorphism $\mathrm{\Phi }$ from the free group; [F2]'s path-composition law gives $\mathrm{\Phi }(w)=U_w^{-1}$ for every edge word $w$. These transports are invertible because along each segment [F3] and the ODE give $(U^{\mathsf{T}}\eta U{)}^{\mathrm{\prime }}=-U^{\mathsf{T}}({\omega }^{\mathsf{T}}\eta +\eta \omega )U=0$, hence $U^{\mathsf{T}}\eta U=\eta$; reversing the ODE gives $U_{\bar{\alpha }}=U_{\alpha }^{-1}$. A based face relator is a tree path, a triangular boundary and the reverse tree path, so step 2.1 makes its transport and its $\mathrm{\Phi }$-image the identity. The boundary word is identity in the presentation by [F6]–[F8], and [F9] writes it as a finite product of conjugates of face relators and their inverses; all of these map to $I$, so $U_{\lambda }=I$. The side edges of the square are constant by fixed endpoints, giving $U_{\lambda }=U_{{\gamma }_1}^{-1}{U}_{{\gamma }_0}$ and $U_{{\gamma }_0}(1)=U_{{\gamma }_1}(1)$.