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The abelianization of a symmetric mapping class group

Published online by Cambridge University Press:  01 September 2009

MASATOSHI SATO*
Affiliation:
Graduate School of Mathematical Sciences, The University of Tokyo, Komaba, Tokyo, 153-8914, Japan. e-mail: sato@ms.u-tokyo.ac.jp

Abstract

Let Σg,r be a compact oriented surface of genus g with r boundary components. We determine the abelianization of the symmetric mapping class group (g,r)(p2) of a double unbranched cover p2: Σ2g − 1,2r → Σg,r using the Riemann constant, Schottky theta constant, and the theta multiplier. We also give lower bounds on the order of the abelianizations of the level d mapping class group.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2009

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References

[1]Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J.Geometry of Algebraic Curves. Vol. 1. (Springer-Verlag, 1985).CrossRefGoogle Scholar
[2]Birman, J. S. and Craggs, R.The μ-Invariant of 3-Manifolds and Certain Structural Properties of the Group of Homeomorphisms of a Closed, Oriented 2-Manifold. Trans. Amer. Math. Soc. 237 (1978), 283309.Google Scholar
[3]Birman, J. S. and Hilden, H. M.On isotopies of homeomorphisms of Riemann surfaces. Ann. Math. 97 (3) (1973), 424439.CrossRefGoogle Scholar
[4]Farb, B.Some problems on mapping class groups and moduli space, problems on mapping class groups and related topics. In Proc. Symp. Pure Math. 74 (2006), 1155.CrossRefGoogle Scholar
[5]Farkas, H. M. and Rauch, H. E.Period relations of schottky type on Riemann surfaces. Ann. Math. 92 (3): (1970), 434461.CrossRefGoogle Scholar
[6]Fay, J. D.Theta functions on Riemann surfaces. (Springer, 1973).CrossRefGoogle Scholar
[7]Hain, R. Torelli groups and geometry of moduli spaces of curves. Current Topics in Complex Algebraic Geometry (Clemens, CH and Kollar, J., eds.) MSRI Publications. 28 (1995), 97–143.Google Scholar
[8]Harer, J. L.The second homology group of the mapping class group of an orientable surface. Invent. Math. 72 (2) (1983), 221239.CrossRefGoogle Scholar
[9]Harer, J. L.The rational Picard group of the moduli space of Riemann surfaces with spin structure. Contemp. Math. 150 (1993), 107136.CrossRefGoogle Scholar
[10]Igusa, J.On the graded ring of theta-constants. Amer. J. Math. 86 (1) (1964), 219246.CrossRefGoogle Scholar
[11]Igusa, J.Theta Functions (Springer, 1972).CrossRefGoogle Scholar
[12]Johnson, D.Homeomorphisms of a surface which act trivially on homology. Proc. Amer. Math. Soc. 75 (1) (1979), 119125.CrossRefGoogle Scholar
[13]Johnson, D.An abelian quotient of the mapping class group g. Math. Ann. 249 (3) (1980), 225242.CrossRefGoogle Scholar
[14]Johnson, D.Quadratic forms and the Birman–Craggs homomorphisms. Trans. Amer. Math. Soc. 261 (1) (1980), 235254.CrossRefGoogle Scholar
[15]Johnson, D.The structure of the Torelli Group III: The abelianization of g. Topology, 24 (2) (1985), 127144.CrossRefGoogle Scholar
[16]Lee, R., Miller, E. and Weintraub, S.The Rochlin invariant, theta functions and the holonomy of some determinant line bundle. J. Reine Angew. Math. 392 (1988), 187218.Google Scholar
[17]McCarthy, J. D.On the first cohomology group of cofinite subgroups in surface mapping class groups. Topology 40 (2) (2000), 401418.CrossRefGoogle Scholar
[18]Powell, J.Two theorems on the mapping class group of a surface. Proc. Amer. Math. Soc. 68 (3) (1978), 347350.CrossRefGoogle Scholar
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