What is the connection between the mapping class group of a torus and Gl(2,Z)?

[Bacle]

Hi, everyone:

I am trying to understand why the mapping class group of the
torus T^2 (i.e., the group of orientation-preserving self-
diffeomorphisms, up to isotopy) is (iso. to) Gl(2,Z) ( I just
realized this is the name of the group of orientation-preserving
automorphisms). Anyone know what the connection is, between these
two.(or, even better, a proof, or ref. for a proof.)? .
All I can think off is that there may be some connection
with the fact that Pi_1(T^2)=Z(+)Z , but that is all I have.

Thanks for any Ideas.

 

[Bacle]

O.K. I think I made a small break, by finding out (and I came close
to proving) that Aut(Z(+)Z) --which is Pi_1(T^2) --is generated by
three matrices. Now I am trying to find a correspondence between
automorphisms and matrices.

 

[Tinyboss]

Check out "Knots and Links" by Rolfsen...I think he has a series of exercises that lead you through it.