Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

Abstract

This paper is about the cohomology of certain finite-index subgroups of mapping class groups and its relation to the cohomology of arithmetic groups. For $G=\mathbb{Z}/m\mathbb{Z}$ and for a regular $G$-cover $S\to \overline{S}$ (possibly branched), a finite-index subgroup $\Gamma <Mod\left(\overline{S}\right)$ acts on $H_1(S;\mathbb{Z})$ commuting with the deck group action, thus inducing a homomorphism $\Gamma \to {Sp}_{2g}^G\left(\mathbb{Z}\right)$ to an arithmetic group. The induced map $H^{\ast }\left({Sp}_{2g}^G\right(\mathbb{Z});\mathbb{Q})\to H^{\ast }(\Gamma ;\mathbb{Q})$ can be understood using index theory. To this end, we describe a families version of the $G$-index theorem for the signature operator and apply this to (i) compute $H^2\left({Sp}_{2g}^G\right(\mathbb{Z});\mathbb{Q})\to H^2(\Gamma ;\mathbb{Q})$, (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to $SU(p,q)$ arising from Atiyah–Kodaira constructions, and (iv) describe how classes in $H^{\ast }\left({Sp}_{2g}^G\right(\mathbb{Z});\mathbb{Q})$ give equivariant cobordism invariants for surface bundles with a fiberwise $G$ action, following Church–Farb–Thibault.