All:
I am trying to figure out the mapping class groupof the torus ; more accurately, I am trying to show that it is equal to $SL(2,\mathbb Z)$.
The method: every homeomorphism h: $ T^2 \rightarrow T^2 $ gives rise to, aka,
induces, an isomorphism g: $ \pi_1(T^2) \rightarrow \pi_1(T^2)$,
and we use the fact that:
i)$\pi_1 (T^2)=\mathbb Z\oplus\mathbb Z$
ii) Aut $ \mathbb Z\oplus\mathbb Z=SL(2,\mathbb Z)$
Now, if we can show that the homomorphism from [the group of homeomorphisms
of $T^2$ to itself ] to $SL(2,\mathbb Z)$ is an isomorphism,
we are done.
Now, it is not too hard (tho, I think not trivial) , to show that $SL(2,\mathbb Z)$ has a generating set with three elements ; the set of transvections (actually a set of four transvections that are generating set for the set of transvections ); the transvections are a generalization of shear maps in linear transformations $T: \mathbb R^n \rightarrow \mathbb R^m$, as maps that add a multiple of a row to another row. A (generating) shear matrix has all diagonal entries$a_{ii}$ identically equal to one, and exactly one non-diagonal entry equal to +/-1
(general shear matrices have all $a_{ii}=1$ and exactly one off-diagonal term with any non-zero value).
So to show the map is onto, I am trying to see that each of the elements of
the generating set are the image of some homeomorphism from the torus to itself, i.e., to show that there are automorphisms of the torus thad induce the basis shear maps, by examing the effect of the shear maps on a standard basis $\{(1,0),(0,1)\}$ of the torus, and trying to construct a self-homeo of the torus that would have the effect on homology described by the effect of the shear maps on the basis .
I was also thinking of using the fact that the mapping class group for $S_g$ is known to have a generating set of size $3g-1$ (best possible) , given by Dehn twists about the basis. I suspect these Dehn twists (about the basis elements) may induce mapsin homology describable as shear matrices. Does any know?
I will try to complete this idea, but I would appreciate some comments on whether
this approach makes sense. I would appreciate your comments/suggestions.
Thanks.
Best Answer
If you use the definition of "mapping class group" as $\pi_0 Diff(S^1\times S^1)$, i.e. isotopy classes of diffeomorphisms of the torus, there's two key technical lemmas that you need.
1) If a closed smooth curve in the torus does not bound an embedded disc, then it is isotopic to a straight line. That means choose $p,q \in \mathbb Z$ coprime and look at the solution set to the equation $px+qy=0$ in $\mathbb R^2$, the image of that under the quotient map $\mathbb R^2 \to \mathbb (S^1)^2$ where you're modding out by the additive subgroup $\mathbb Z^2 \subset \mathbb R^2$.
2) Every diffeomorphism of a disc which is the identity on the boundary can be isotoped to the identity diffeomorphism on the disc, without affecting its behaviour on the boundary.
With those two technical results, plus liberal usage of things like the isotopy extension theorem, the theorem you're trying to prove falls out of the above. Also, you can prove (1) by transversality and the Schoenflies theorem (that a closed curve in the plane bounds a region diffeomorphic to a disc). (2) is more difficult, it's a theorem of Smale's. Its proof is in Thurston's (published) book, and relies on a few observations about ODE's in the plane, like the Poincare-Bendixon theorem.