Surjection Between Mapping Class Grp. and Symplectic Matrices

[Bacle]

Hi, Everyone:

I am reading a paper that refers to a "natural surjection" between M_g

and the group of symplectic 2gx2g-matrices. All I know is this map is related to some

action of M_g on H₁(S_g,Z). I think this

action is define by/as the induced maps on homology by the D_i , i.e.,

the Dehn twists that generate S_g. I think the kernel is the Torelli

group, but I am not sure.

Any Ideas/Refs.?

Thanks.

 

[lavinia]

I don't know anything about this subject but this Wikipedia article seems to briefly answer your question.

http://en.wikipedia.org/wiki/Mapping_class_group#Torelli_group

Look at the section on the Torelli group.

I gather from the blurbs in Wikipedia that the mapping class group of an orientable surface acts on the first Z- cohomology of the surface which through the cup product is a linear symplectic manifold. The action preserves the cup product and hence the linear symplectic form that the cup product determines. The reason that the cup product determines a symplectic form is because for a 2 manifold the bilinear form is antisymmetric.

The kernel of the action is called the Torelli group.

 

[Bacle]

Thanks, Lavinia:

I just had a dumb confusion; the issue is that every automorphism in Mg gives

rise to an automorphism in homology, and Tg is the kernel of this general assignment

( of the homology functor, I guess) , i.e, the subgroup of automorphisms

that induce the identity map in homology. Sorry, just to make up for my dumb question;

I was confused because I believed since the induced maps (by a diffeo.) are isomorphisms,

that they fixed homology, i.e., were the identity, but this is clearly not the case.

Phew... feels good to get it out of my system. Now, I got to go deal with

ugly level-2 prime congruence subgroups of Sp(2g,Z). Later.