ScalingStacks

Proof. We have to show that the parallel transport isomorphisms associated to closed loops $p_{t}$ in $S\setminus L$ are identity maps. Suppose first that $p_{t}$ is a contractible loop. Then the pair $(S\times{[0,1]}\setminus\{(p_{t},t)\}_{t\in{[0,1]}},L\times{[0,1]})$ is diffeomorphic to a pair $({\mathbb{R}}^{3}\times{[0,1]},\Sigma)$ where $\Sigma$ is isotopic to an identity link cobordism, which implies that the parallel transport isomorphism $\mathrm{KhR}_{N}(p_{t})$ is the identity. This also implies that the parallel transport isomorphisms associated to isotopic paths between two points $p_{0}$ and $p_{1}$ in $S\setminus L$ are equal. Now suppose that the loop $p_{t}$ is a small meridian around a component of $L$. Then the pair $(S\times{[0,1]}\setminus\{(p_{t},t)\}_{t\in{[0,1]}},L\times{[0,1]})$ is diffeomorphic to a pair $({\mathbb{R}}^{3}\times{[0,1]},\Sigma)$ where $\Sigma$ is a sweep-around cobordism as in (1.1). By Theorem 1.1, it follows that the parallel transport isomorphism $\mathrm{KhR}_{N}(p_{t})$ is the identity. Since $\pi_{1}(S\setminus L)$ is generated by such small meridian loops, it follows that the parallel transport isomorphism for every loop $p_{t}$ is the identity. ∎