Quotients of Mapping Class Group Iso. to Symplectic Group.

[Bacle]

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.

 

[pat_connell]

The isomorphism in question is between the mapping class group of the genus-g surface and the symplectic group associated with the intersection form over Z. This means that there is an isomorphism between the groups of diffeomorphisms of the surface (Mg) and linear maps of the first homology group (Sp(2g,Z)) which preserve the intersection form (, )_2. This isomorphism is a standard group isomorphism and not a Lie group isomorphism, as it does not involve any Lie algebra structures. It is an example of an algebraic group isomorphism, as it involves an algebraic structure (in this case, the intersection form). The isomorphism itself is given by the composition of two maps: one from Mg to Mg^2, and the other from Mg^2 to Sp(2g,Z). The first map is the natural inclusion, i.e., all diffeomorphisms of the surface are mapped to those which induce the identity on homology. The second map maps the elements of the mapping class group to linear maps which preserve the intersection form: if f is an element of Mg^2, then its image in Sp(2g,Z) is the linear map h_f which maps x to f*x, where f* is the induced homomorphism on homology. In other words, the isomorphism is given by mapping each element f of Mg^2 to its induced homomorphism f* on homology, and then mapping f* to the corresponding linear map h_f in Sp(2g,Z).