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I am giving a talk in a graduate seminar soon and would like show that the mapping class group of the torus is $SL_2(\mathbb{Z})$.The proof I was going to present lacks geometric intuition and is kind of long. I was wondering if anyone had an intuitive/visual proof of this fact or knew where I might be able to find such a proof.

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An easy proof that the mapping class group contains $SL_2(Z)$ is by looking at what an orientation preserving diffeomorphism must do to $H^1(T^2)$. – Jason DeVito Apr 11 '11 at 17:33
@Jason, this is just a result of the intersection pairing, correct? – confusedmath Apr 12 '11 at 2:34
Yes, I think so. – Jason DeVito Apr 12 '11 at 3:08

Stillwell's book "Classical topology and combinatorial group theory" gives a proof, with pictures, starting on page 206. It is possible to summarize the proof (use "curve straightening") but I thnk that Stillwell will be a good (or best) place to start.

By the way, the mapping class group is $GL(2,Z)$ if you allow orientation reversing homeomorphisms in the definition.

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