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 and for a regular -cover (possibly branched), a finite-index subgroup acts on commuting with the deck group action, thus inducing a homomorphism to an arithmetic group. The induced map can be understood using index theory. To this end, we describe a families version of the -index theorem for the signature operator and apply this to (i) compute , (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to arising from Atiyah–Kodaira constructions, and (iv) describe how classes in give equivariant cobordism invariants for surface bundles with a fiberwise action, following Church–Farb–Thibault.
"Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups." Michigan Math. J. 67 (1) 31 - 58, March 2018. https://doi.org/10.1307/mmj/1516330969