Characteristic classes of fiberwise branched surface bundles via arithmetic

Abstract

This paper is about cohomology of mapping class groups from the perspective of arithmetic groups. For a closed surface $S$ of genus $g$, the mapping class group $Mod(S)$ admits a well-known arithmetic quotient $Mod(S)\rightarrow Sp(2g, Z)$, under which the stable cohomology of $Sp(2g,Z)$ pulls back to algebra generated by the odd MMM classes of $Mod(S)$. We extend this example to other arithmetic groups associated to mapping class groups and explore some of the consequences for surface bundles. For $G=Z/mZ$ and for a regular $G$-cover $S\rightarrow S'$ (possibly branched), a finite index subgroup $\Gamma< Mod( S')$ admits a homomorphism to an arithmetic group $Sp^G<Sp(2g,Z)$. The induced map on cohomology 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 $H^2(Sp^G;Q)\rightarrow H^2(\Gamma;Q)$, (ii) re-derive Hirzebruch's formula for signature of a branched cover (in the case of a surface bundle), (iii) compute Toledo invariants of surface group representations to $SU(p,q)$ arising from Atiyah--Kodaira constructions, and (iv) describe how classes in $H^*(Sp^G;Q)$ give equivariant cobordism invariants for surface bundles with a fiberwise $G$ action, following Church--Farb--Thibault.