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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,Z).

The method: every homeomorphism h: <\tex> T^2 -->T^2<tex> gives rise to, aka,

induces an isomorphism g: <\tex> \mathbb pi_1(T^2)-->\mathbb pi_1(T^2)<tex>,

and we use the fact that:

i)<\tex>\mathbb Pi_1 (T^2)=\mathbb Z(+)\mathbb Z<tex>

ii) Aut <\tex> \mathbb Z(+)\mathbb Z=SL(2,\mathbb Z)<tex>

Now, if we can show that the homomorphism from [the group of homeomorphisms

of <\tex>T^2<tex> to itself ] to <\tex>SL(2,\mathbb Z)<tex> is an isomorphism,

we are done.

Now, it is not too hard (tho, I think not trivial) , to show that <\tex>SL(2,\mathbb Z)<tex> has a generating set with three elements ; the set of transvections (actually a generating set for the set of transvections ); the transvections are a generalization of

shear maps in linear transformations <\tex>T: \mathbb R^n -->\mathbb R^m<tex>, as

maps that add a multiple of a row to another row. A (generating) shear matrix has all diagonal entries identically equal to one, and exactly one non-diagonal entry equal to +/-1

(general shear matrices have all <\tex>a_ii=1<tex> 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 that effect on homology .

I will try to complete this idea, but I would appreciate some comments on whether

this approach makes sense.

Thanks.

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# Mapping Class Group of the Torus.

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