Signature cocycles on the mapping class group and symplectic groups

Abstract:

Werner Meyer constructed a cocycle in H^2(Sp_2(g,Z); Z) which computes the signature of a closed oriented surface bundle over a surface. By studying properties of this cocycle, he also showed that the signature of such a surface bundle is a multiple of 4. In this paper, we study signature cocycles both from the geometric and algebraic points of view. We present geometric constructions which are relevant to the signature cocycle and provide an alternative to Meyer's decomposition of a surface bundle. Furthermore, we discuss the precise relation between the Meyer and Wall-Maslov index. The main theorem of the paper, Theorem 6.6, provides the necessary group cohomology results to analyze the signature of a surface bundle modulo any integer N. Using these results, we are able to give a complete answer for N=2,4 and 8, and based on a theorem of Deligne, we show that this is the best we can hope for using this method.

SEEK ID: https://publications.h-its.org/publications/1219

DOI: 10.1016/j.jpaa.2020.106400

Research Groups: Groups and Geometry

Publication type: Journal

Journal: Journal of Pure and Applied Algebra

Citation: Journal of Pure and Applied Algebra 224(11):106400

Date Published: 1st Nov 2020

Registered Mode: by DOI

Authors: Dave Benson, Caterina Campagnolo, Andrew Ranicki, Carmen Rovi

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Citation
Benson, D., Campagnolo, C., Ranicki, A., & Rovi, C. (2020). Signature cocycles on the mapping class group and symplectic groups. In Journal of Pure and Applied Algebra (Vol. 224, Issue 11, p. 106400). Elsevier BV. https://doi.org/10.1016/j.jpaa.2020.106400
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Created: 22nd Feb 2021 at 14:34

Last updated: 5th Mar 2024 at 21:24

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