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Quotients of Mapping Class Group Iso. to Symplectic Group.

  1. Jul 28, 2011 #1
    Hi, Again:

    I am a bit confused about this result:

    Mg/Mg^(2) ~ Sp(2g,Z) (group iso.)

    Where:

    i) Mg is the mapping class group of the genus-g surface, i.e., the collection of diffeomorphisms: f:Sg-->Sg , up to isotopy.

    ii)Mg^(2) is the subgroup
    of Mg of maps that induce the identity in H_1(Sg,Z/2), and

    iii)Sp(2g,Z) is the symplectic
    group associated with the intersection form over Z, i.e., we consider the pair (Z-module,
    Symplectic form) given by: (H_1(Sg,Z), (a,b)_2), where (a,b)_2 is the intersection form
    in in H_1(Sg,Z/2), so that Sp(2g,Z) is the subgroup of Gl( H_1(Sg,Z)) that preserves
    (, )_2.

    For one thing, the mapping class group is in a different "category" (used in the informal
    sense) than Sp(2g,Z) ; Mg and Mg^2 are maps f,g :Sg-->Sg , and h in Sp(2g,Z) is a linear
    map m: H_1(Sg,Z)-->H_1(Sg,Z) (with (x,y)_2 =(m(x),m(y))_2.

    I understand being in different categories does not make an iso. impossible, but I
    don't see what the isomorphism would be.

    I don't know if these are Lie group isos. or just standard group isos. Any ideas?

    Thanks.
     
  2. jcsd
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