Proposition 7.4.30 (0PBT) · ScalingStacks

0PBT

Proposition 7.4.30. Let $\theta\in\operatorname{Hom}\nolimits_{{\mathcal{P}}^{\bullet}(Z)}(I,J)$ . We have $i(\theta)=\sum_{\Omega\in\pi_{0}(Z)}|(L(\theta)\cap\Omega)/\mathrm{inv}|e_{\Omega}$ . In particular, $L(\theta)$ is finite.

0PBU

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 $Z$ . ∎