ScalingStacks

Note that $(m_{c}+m_{\iota(c)})(\beta)=0$ for all but finitely many $c$’s, hence the sum above is finite. More precisely, let $\zeta$ be a non-identity homotopy class of paths in $Z$. We have $\zeta(0+)\neq\zeta(1-)$ and

(7.3.1)

$$(m_{c}+m_{\iota(c)})(\zeta)=\begin{cases}1&\text{if }c\in\{\zeta(0+)\cup\iota(\zeta(0+))\}\text{ and }c{\not\in}\{\zeta(1-)\cup\iota(\zeta(1-))\}\\ -1&\text{if }c\in\{\zeta(1-)\cup\iota(\zeta(1-))\}\text{ and }c{\not\in}\{\zeta(0+)\cup\iota(\zeta(0+))\}\\ 0&\text{otherwise.}\end{cases}$$