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.
Mapping class group of the Torus (intuitive/visual proof)
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algebraic-topology
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0Yes, I think so. – 2011-04-12
1 Answers
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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.