# Characteristic classes of fiberwise branched covers via arithmetic groups

### Michigan Math. J. 67 (2018), 31–58.

This paper is about cohomology of mapping class groups of surfaces. One source of cohomology classes is the symplectic representation R: Mod(S) –> Sp(2g,Z). The image of the induced map on cohomology – in the stable range and with rational coefficients – is the algebra generated by the odd Miller-Morita-Mumford (MMM) classes. In this paper we extend this example to the centralizer of a finite cyclic subgroup G of the mapping class group Mod(S)^G –> Sp(2g,Z)^G. The main result of this paper is a computation of the induced map on cohomology in degree 2 when G is a finite cyclic group. We apply this computation to compute Toledo invariants of surface group representations to SU(p,q) arising from the Atiyah-Kodaira construction, and we show that cohomology classes in the image give equivariant cobordism invariants for surface bundles with fiberwise G action, following Church-Farb-Thibault.