Michigan Mathematical Journal
- Michigan Math. J.
- Volume 67, Issue 1 (2018), 31-58.
Characteristic Classes of Fiberwise Branched Surface Bundles via Arithmetic Groups
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 and for a regular -cover (possibly branched), a finite-index subgroup acts on commuting with the deck group action, thus inducing a homomorphism to an arithmetic group. The induced map can be understood using index theory. To this end, we describe a families version of the -index theorem for the signature operator and apply this to (i) compute , (ii) rederive Hirzebruch’s formula for signature of a branched cover, (iii) compute Toledo invariants of surface group representations to arising from Atiyah–Kodaira constructions, and (iv) describe how classes in give equivariant cobordism invariants for surface bundles with a fiberwise action, following Church–Farb–Thibault.
Michigan Math. J., Volume 67, Issue 1 (2018), 31-58.
Received: 26 July 2016
Revised: 4 October 2017
First available in Project Euclid: 19 January 2018
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
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]
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