Signature cocycles on the mapping class group and symplectic groups

JOURNAL OF PURE AND APPLIED ALGEBRA

卷:224

Article

Benson, Dave¹  Campagnolo, Caterina²  Ranicki, Andrew³  Rovi, Carmen⁴ 

[1] Univ Aberdeen, Inst Math, Aberdeen AB24 3UE, Scotland

[2] Karlsruhe Inst Technol, Dept Math, D-76128 Karlsruhe, Germany

[3] Univ Edinburgh, Sch Math, Edinburgh EH9 3FD, Midlothian, Scotland

[4] Heidelberg Univ, Excellence Cluster Struct, D-69120 Heidelberg, Germany

关键词: Cocycles;    Signature;    Fibre bundle;    Symplectic group;   

DOI  :  10.1016/j.jpaa.2020.106400

来源: Elsevier

【 摘 要 】

Werner Meyer constructed a cocycle in H-2(Sp(2g, 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. (C) 2020 Elsevier B.V. All rights reserved.

【 授权许可】

Free