Homology action of a pair-of-pants homeomorphism
= Homology action of a pair-of-pants homeomorphism
Orient the three boundary classes $c_1,c_2,c_3$ so that $c_1+c_2+c_3=0$ in $H_1(N;\mathbb Z)$. A homeomorphism with orientation sign $\epsilon$ and boundary permutation $\sigma$ sends $c_i$ to $\epsilon c_{\sigma(i)}$. Its trace on the rank-two quotient of $\mathbb Z^3$ by $c_1+c_2+c_3$ is therefore $\epsilon(\#\operatorname{Fix}\sigma-1)$.