ScalingStacks

Proof. Let us first fix a choice of endpoints $p_{0}\in W_{\mathrm{in}}\setminus\Sigma_{\mathrm{in}}$ and $p_{1}\in W_{\mathrm{out}}\setminus\Sigma_{\mathrm{out}}$. Then any two choices of paths $p_{t}\in$ and $p^{\prime}_{t}$ from $p_{0}$ to $p_{1}$ can be related by isotopy in $W\setminus\Sigma$ or splicing in a little loop linking a component of $\Sigma$. As before, isotopic paths give rise to isotopic surfaces in ${\mathbb{R}}^{3}\times{[0,1]}$, which induce equal maps. Similarly, in the case of a linking loop, we can choose a standard local model and then notice that the sweep-around property from Theorem 1.1 implies that the two paths induce the same map. Finally, the independence from the choice of endpoints $p_{0}\in W_{\mathrm{in}}\setminus\Sigma_{\mathrm{in}}$ and $p_{1}\in W_{\mathrm{out}}\setminus\Sigma_{\mathrm{out}}$ follows as in the proof of Lemma 4.6. ∎