One of the foundational results about mapping class groups of surfaces is that they are generated by Dehn twists. A mapping class is a connected component in a space of diffeomorphisms, so another way of stating this is "any diffeomorphism of a surface is generated by diffeomorphisms each of which is supported in an annulus". This was proven by Dehn, and independently in a stronger form by Lickorish.
I'm teaching a summer reading course, and I am toying with the idea of presenting a proof to this statement. But the proof I know is a bit involved- you use the Birman exact sequence to relate the mapping class group of a surface Σ to the mapping class group of $\Sigma-D^2$ (not so trivial), then you use the fact that the complex of curves on a surface is connected (also non-trivial), and finally that for two non-disjoint connected curves α and β there exists a product of Dehn twists T such that $T(\alpha)=\beta$. This proof looks too involved to present properly in a single lecture.
80 or so years after Dehn's proof, and 47 years after Lickorish's:
Do you know an elegant proof that the mapping class group of a surface is generated by Dehn twists?
