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.
Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)
Arbarello, E., Cornalba, M., Griffiths, P. A. and Harris, J.Geometry of Algebraic Curves. Vol. 1. (Springer-Verlag, 1985).CrossRefGoogle Scholar
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), 283–309.Google Scholar
Birman, J. S. and Hilden, H. M.On isotopies of homeomorphisms of Riemann surfaces. Ann. Math.97(3) (1973), 424–439.CrossRefGoogle Scholar
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), 11–55.CrossRefGoogle Scholar
Farkas, H. M. and Rauch, H. E.Period relations of schottky type on Riemann surfaces. Ann. Math.92(3): (1970), 434–461.CrossRefGoogle Scholar