ScalingStacks

Proof. Consider three non-identity homotopy classes of paths $\zeta$, $\zeta_{1}$ and $\zeta_{2}$ in ${\mathbf{R}}$ with $\zeta(0)=\zeta_{1}(0)\neq\zeta(1)=\zeta_{2}(0)$. If $\zeta$ and $\zeta_{2}\circ\zeta\circ\zeta_{1}^{-1}$ have opposite orientations, then $\operatorname{supp}\nolimits(\zeta)\subset\operatorname{supp}\nolimits(\zeta_{1})\cup\operatorname{supp}\nolimits(\zeta_{2})$. We deduce that the lemma holds for $Z=S^{1}$ by using the universal cover of $Z$. As a consequence, the lemma holds when $Z$ is connected and smooth by embedding it in $S^{1}$, hence it holds for $Z$ smooth. Lemma 7.3.24 shows that the lemma holds for any $Z$, since it holds for the non-singular cover of $Z$. ∎