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A combinatorial formula for Earle's twisted 1-cocycle on the mapping class group

Published online by Cambridge University Press:  01 January 2009

YUSUKE KUNO
YUSUKE KUNO*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, 3-8-1 Komaba Meguro-ku Tokyo 153-0041, Japan. e-mail: [email protected]
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Abstract

We present a formula expressing Earle's twisted 1-cocycle on the mapping class group of a closed oriented surface of genus ≥ 2 relative to a fixed base point, with coefficients in the first homology group of the surface. For this purpose we compare it with Morita's twisted 1-cocycle which is combinatorial. The key is the computation of these cocycles on a particular element of the mapping class group, which is topologically a hyperelliptic involution.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 146 , Issue 1 , January 2009 , pp. 109 - 118
Copyright
Copyright © Cambridge Philosophical Society 2008

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References

REFERENCES

[1]Earle, C. J.Families of Riemann surfaces and Jacobi varieties. Ann. Math. 107 (1978), 255–286.Google Scholar
[2]Jablow, E. R.Quadratic vector classes on Riemann surfaces. Duke. Math. J. 53, no.1 (1986), 221–232.Google Scholar
[3]Morita, S.Families of Jacobian manifolds and characteristic classes of surface bundles I. Ann. Inst. Fourier 39 (1989), 777–810.Google Scholar
[4]Morita, S.Families of Jacobian manifolds and characteristic classes of surface bundles II. Math. Proc. Camb. Phil. Soc. 105 (1989), 79–101.CrossRefGoogle Scholar
[5]Morita, S.Casson's invariant for homology 3-spheres and characteristic classes of surface bundles I. Topology 28 (1989), 305–323.Google Scholar
[6]Morita, S.Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles. J. Diff. Geom. 47 (1997), 560–599.Google Scholar
[7]Trapp, R.A linear representation of the mapping class group and the theory of winding numbers. Topology Appl. 43 (1992), 47–64.Google Scholar