ScalingStacks

Proof. The statement is true for $Z=S^{1}$ unoriented by Lemmas 3.2.3, 7.4.19 and 7.4.20. It follows from Lemmas 7.4.28 and 7.3.22 that it holds for any connected non-singular $Z$, by embedding it in $S^{1}$. So, the lemma holds for any non-singular $Z$. By realizing an arbitrary $Z$ as a quotient of its non-singular cover, we deduce from Lemmas 7.4.28 and 7.3.22 that the lemma holds for any $Z$. ∎