Lemma 7.1.21 (0P8X) · ScalingStacks

0P8X

Lemma 7.1.21. Let $\gamma$ be a path in $X$ such that $\gamma^{-1}(x)$ has finitely many connected components, none of which contain $0$ or $1$ in the closure of their interior.

We have $\partial(\gamma^{-1}(x))=\bigcup_{c\in C(x)}(I_{c}^{+}(\gamma)\cup I_{c}^{-}(\gamma))$ and $|I_{c}^{+}(\gamma)|-|I_{c}^{-}(\gamma)|=m_{c}([\gamma])$ for all $c\in C(x)$ .

0P8Y

Proof. The first statement is clear. Let us now prove the second statement. That statement is clear if $\gamma((0,1))\cap(X_{exc}\cup\{x\})=\emptyset$ .

The left side of the equality is additive under compositions of paths, and so is the right side by Lemma 7.1.22 below.

Assume now $\gamma^{-1}(X_{exc}\cup\{x\})$ is finite. The path $\gamma$ is a (finite) composition of paths mapping $(0,1)$ into the complement of $X_{exc}\cup\{x\}$ , hence the statement holds for $\gamma$ .

Consider now the general case. The proof of Lemma 7.1.16 for $E=X_{exc}\cup\{x\}$ produces a path $\gamma^{\prime}$ homotopic to $\gamma$ such that $\gamma^{\prime-1}(E)$ is finite and such that $|I_{c}^{+}(\gamma)|-|I_{c}^{-}(\gamma)|=|I_{c}^{+}(\gamma^{\prime})|-|I_{c}^{-}(\gamma^{\prime})|$ . Since the statement holds for $\gamma^{\prime}$ , it follows that it holds for $\gamma$ . ∎