## Michigan Mathematical Journal

### Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

Bena Tshishiku

#### 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 $G=\mathbb{Z}/m\mathbb{Z}$ and for a regular $G$-cover $S\rightarrow\bar{S}$ (possibly branched), a finite-index subgroup $\Gamma\lt \operatorname{Mod}(\bar{S})$ acts on $H_{1}(S;\mathbb{Z})$ commuting with the deck group action, thus inducing a homomorphism $\Gamma\rightarrow\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z})$ to an arithmetic group. The induced map $H^{*}(\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z});\mathbb{Q})\rightarrow H^{*}(\Gamma;\mathbb{Q})$ 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 $H^{2}(\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z});\mathbb{Q})\rightarrow H^{2}(\Gamma;\mathbb{Q})$, (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to $\operatorname{SU}(p,q)$ arising from Atiyah–Kodaira constructions, and (iv) describe how classes in $H^{*}(\operatorname{Sp}_{2{g}}^{G}(\mathbb{Z});\mathbb{Q})$ give equivariant cobordism invariants for surface bundles with a fiberwise $G$ action, following Church–Farb–Thibault.

#### Article information

Source
Michigan Math. J., Volume 67, Issue 1 (2018), 31-58.

Dates
Revised: 4 October 2017
First available in Project Euclid: 19 January 2018

https://projecteuclid.org/euclid.mmj/1516330969

Digital Object Identifier
doi:10.1307/mmj/1516330969

Mathematical Reviews number (MathSciNet)
MR3770852

Zentralblatt MATH identifier
06965588

#### Citation

Tshishiku, Bena. Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups. Michigan Math. J. 67 (2018), no. 1, 31--58. doi:10.1307/mmj/1516330969. https://projecteuclid.org/euclid.mmj/1516330969

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