Characteristic classes of fiberwise branched surface bundles via arithmetic

TL;DR

This paper explores the cohomology of mapping class groups through arithmetic groups, extending known results to branched surface bundles and deriving new invariants and formulas related to surface covers and group actions.

Contribution

It introduces a families version of the G-index theorem for the signature operator and applies it to compute cohomology, signature, and Toledo invariants for surface bundles with group actions.

Findings

Computed $H^2(Sp^G;Q)$ and $H^2( ext{Gamma};Q)$

Re-derived Hirzebruch's signature formula for branched covers

Calculated Toledo invariants for surface group representations

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.