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Signature cocycles on the mapping class group and symplectic groups

Werner Meyer constructed a cocycle in H2(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...

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Hauptverfasser: Benson, David J. (VerfasserIn) , Campagnolo, Caterina (VerfasserIn) , Ranicki, Andrew (VerfasserIn) , Rovi, Carmen (VerfasserIn)
Dokumenttyp: Article (Journal)
Sprache:Englisch
Veröffentlicht: 22 April 2020
In: Journal of pure and applied algebra
Year: 2020, Jahrgang: 224, Heft: 11, Pages: 1-49
ISSN:1873-1376
DOI:10.1016/j.jpaa.2020.106400
Online-Zugang:Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1016/j.jpaa.2020.106400
Verlag, lizenzpflichtig, Volltext: https://www.sciencedirect.com/science/article/pii/S0022404920300992
Volltext
Verfasserangaben:Dave Benson, Caterina Campagnolo, Andrew Ranicki, Carmen Rovi
Beschreibung
Zusammenfassung:Werner Meyer constructed a cocycle in H2(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.
Beschreibung:Gesehen am 26.03.2021
Beschreibung:Online Resource
ISSN:1873-1376
DOI:10.1016/j.jpaa.2020.106400

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