Mapping Class Group of the Torus.

[Bacle]

Hi, 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,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.

 

[Bacle]

I am making a new comment , since the previous post seemed long-enough.

I was also thinking of using the fact that the mapping class group Mg of Sg (the genus-g surface )is generated by 3g-1 twists ; for g=1, this means two twists generate
Mg, and I am pretty sure these two are twists about a meridian and a parallel respectively (right?). So it seems like, in the basis {(1,0),(0,1)} , these twists
may have a representation as a shear matrix . Is this correct?


EDIT:

Just wanted to say that the map between the group of homeomorphisms of T^2 and
the group of automorphisms of SL(2,Z) will _not_ be an injection, for the simple reason
that any two homotopic maps induce the same map on homology.
Thanks.

 

[lavinia]

Don't know anything about mapping class groups but from the definition the argument for the the 2 torus might go like this.

- Any homeomorphism induces a group isomorphism of ZxZ and so must be Z-linear and map the generators (1,0) and (0,1) to another set of generator. this is an element of SL(2,Z) and this map is independent of the homotopy class of the homeomorphism.

- conversely an element of SL(2,Z) acts on the Euclidean plane as a linear isomorphism that preserves the standard lattice.

not sure how to get 1-1. Will think on it.

1-1 follows from showing that a homeomorphism of the torus that is the identity on the fundamental group is homotopic to the identity map.

 

[lavinia]

In the first part of my answer I should have said that the homeomorphism is orientation preserving in order for the induced map on the fundamental group to be in SL(2,Z).

 

[blue_raver22]

Hello,

Your approach to showing that the mapping class group of the torus is equal to SL(2,Z) seems reasonable. As you mentioned, the key is to show that the homomorphism from the group of homeomorphisms of T^2 to itself to SL(2,Z) is an isomorphism. To do this, you are trying to show that each element of the generating set of SL(2,Z) is the image of a homeomorphism of the torus.

Your idea to examine the effect of the shear maps on a standard basis of the torus and try to construct a self-homeomorphism with that effect on homology is a good approach. It may also be helpful to consider the fundamental group of the torus and how it is affected by the shear maps. This can give insight into how to construct the desired homeomorphism.

Keep working on your approach and don't hesitate to ask for feedback or assistance if needed. Good luck!