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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| Main Authors: | , , , |
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| Format: | Article (Journal) |
| Language: | English |
| Published: |
22 April 2020
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| In: |
Journal of pure and applied algebra
Year: 2020, Volume: 224, Issue: 11, Pages: 1-49 |
| ISSN: | 1873-1376 |
| DOI: | 10.1016/j.jpaa.2020.106400 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1016/j.jpaa.2020.106400 Verlag, lizenzpflichtig, Volltext: https://www.sciencedirect.com/science/article/pii/S0022404920300992 
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| Author Notes: | Dave Benson, Caterina Campagnolo, Andrew Ranicki, Carmen Rovi |
| Summary: | 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. |
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| Item Description: | Gesehen am 26.03.2021 |
| Physical Description: | Online Resource |
| ISSN: | 1873-1376 |
| DOI: | 10.1016/j.jpaa.2020.106400 |