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
Views: 5210
Created: 22nd Feb 2021 at 14:34
Last updated: 5th Mar 2024 at 21:24
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