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.