Michigan Mathematical Journal

Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups

Bena Tshishiku

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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.

Article information

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

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

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1516330969

Digital Object Identifier
doi:10.1307/mmj/1516330969

Mathematical Reviews number (MathSciNet)
MR3770852

Zentralblatt MATH identifier
06965588

Subjects
Primary: 55R40: Homology of classifying spaces, characteristic classes [See also 57Txx, 57R20] 20J06: Cohomology of groups
Secondary: 57M99: None of the above, but in this section 58J20: Index theory and related fixed point theorems [See also 19K56, 46L80]

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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