ScalingStacks

Proof. Lemma 4.1 (1) implies that we have such parallel transport maps $\mathrm{KhR}_{N}(\phi_{t})$ between the fibers over any pair of points $\phi_{0}$ and $\phi_{1}$ in the base. Note that $\pi_{1}(M(B),\phi_{0})\cong\pi_{1}(SO(3))\cong{\mathbb{Z}}/2{\mathbb{Z}}$ is generated by the class of the loop obtained from $\phi_{0}$ by rotating through $360$ degrees around the $z$-axis, to which $\mathrm{KhR}_{N}$ assigns the identity map, see also Section 6.4. Lemma 4.1 (2) and Theorem 2.4 thus imply that the parallel transport maps between the fibers do not depend on the choice of the path $\phi_{t}$. ∎