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March 2018 Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups
Bena Tshishiku
Michigan Math. J. 67(1): 31-58 (March 2018). DOI: 10.1307/mmj/1516330969

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=Z/mZ and for a regular G-cover SS¯ (possibly branched), a finite-index subgroup Γ<Mod(S¯) acts on H1(S;Z) commuting with the deck group action, thus inducing a homomorphism ΓSp2gG(Z) to an arithmetic group. The induced map H(Sp2gG(Z);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 H2(Sp2gG(Z);Q)H2(Γ;Q), (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to SU(p,q) arising from Atiyah–Kodaira constructions, and (iv) describe how classes in H(Sp2gG(Z);Q) give equivariant cobordism invariants for surface bundles with a fiberwise G action, following Church–Farb–Thibault.

Citation

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Bena Tshishiku. "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

Information

Received: 26 July 2016; Revised: 4 October 2017; Published: March 2018
First available in Project Euclid: 19 January 2018

zbMATH: 06965588
MathSciNet: MR3770852
Digital Object Identifier: 10.1307/mmj/1516330969

Subjects:
Primary: 20J06, 55R40
Secondary: 57M99, 58J20

Rights: Copyright © 2018 The University of Michigan

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Vol.67 • No. 1 • March 2018
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