A Dehn twist is a type of self-homeomorphism on a surface (a 2-dimensional topological manifold), especially an orientable closed manifold (of dimension 2).
Let be an orientable surface of genus , and consider an embedded circle . (For example, one may imagine a circle representing one of generators of .) The embedded circle has a tubular neighborhood in , homeomorphic to an annulus . Representing elements of by complex numbers of norm 1, a twist on the annulus may be defined by sending , which equals the identity on the boundary circles where . This defines a self-homeomorphism (mod boundary) on the annulus.
The corresponding Dehn twist on is obtained by extending the self-homeomorphism on the annulus to the entire surface, by taking a point to itself if is outside the annulus. (If we are working in the category of smooth manifolds, one may modify the twist on the annulus by taking where is a smooth bump function such that on a small neighborhood of and on a small neighborhood of . This modification ensures that we get a self-diffeomorphism on the surface as smooth manifold.)
The following theorem was first proved by MaxDehn, with later simplifications by Lickorish:
(“Lickorish twist”) The Dehn twists generate the mapping class group of , of orientation-preserving homeomorphisms considered modulo isotopy. (In fact Lickorish described explicit embedded circles for a surface of genus whose corresponding twists give the generators.)
One example that is easily visualized is the mapping class group of the torus . Here the canonical map is an isomorphism (where automorphisms of are required to preserve orientation, i.e., are elements of the group ; cf. modular group). Thus in this case the Dehn twists can be visualized in terms of their action on loops representing elements of ; the Wikipedia article has some decent pictures of this action.
Last revised on February 8, 2016 at 14:49:35. See the history of this page for a list of all contributions to it.