Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

Source 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=Z/mZ and for a regular G-cover S→S¯ (possibly branched), a finite-index subgroup Γ<Mod(S¯) acts on H1(S;Z) commuting with the deck group action, thus inducing a homomorphism Γ→Sp2gG(Z) to an arithmetic group. The induced map H∗(Sp2gG(Z);Q)→H∗(Γ;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 H2(Sp2gG(Z);Q)→H2(Γ;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∗(Sp2gG(Z);Q) give equivariant cobordism invariants for surface bundles with a fiberwise G action, following Church–Farb–Thibault.