Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

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) → Sp 2g (Z), under which the stable cohomology of Sp 2g (Z) pulls back to the 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 →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 → Sp G to an arithmetic group Sp G < Sp 2g (Z). The induced map H * (Sp G ; 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 H 2 (Sp G ; Q) → H 2 (Γ; 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.

Arithmeticity of the monodromy of some Kodaira fibrations

“…[BBHIW17] introduced a notion of weakly maximal surface group representations into Hermitian Lie groups based on properties of their Toledo invariants in bounded cohomology. The Atiyah–Kodaira surface group representations have non-zero Toledo invariants (a method for computing these Toledo invariants is described in [Tsh18, § 5.3]), but these representations are not weakly maximal because they are not injective.…”