Dehn twist

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).

Suppose that c is a simple closed curve in a closed, orientable surface S. Let A be a tubular neighborhood of c. Then A is an annulus, homeomorphic to the Cartesian product of a circle and a unit interval I: Give A coordinates (s, t) where s is a complex number of the form

Let f be the map from S to itself which is the identity outside of A and inside A we have Then f is a Dehn twist about the curve c. Dehn twists can also be defined on a non-orientable surface S, provided one starts with a 2-sided simple closed curve c on S. Consider the torus represented by a fundamental polygon with edges a and b Let a closed curve be the line along the edge a called

Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve

This neighborhood is homeomorphic to an annulus, say in the complex plane.

By extending to the torus the twisting map

, yields a Dehn twist of the torus by a.

This self homeomorphism acts on the closed curve along b.

A homeomorphism between topological spaces induces a natural isomorphism between their fundamental groups.

Therefore one has an automorphism where [x] are the homotopy classes of the closed curve x in the torus.

It is a theorem of Max Dehn that maps of this form generate the mapping class group of isotopy classes of orientation-preserving homeomorphisms of any closed, oriented genus-

W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along

explicit curves generate the mapping class group (this is called by the punning name "Lickorish twist theorem"); this number was later improved by Stephen P. Humphries to

Lickorish also obtained an analogous result for non-orientable surfaces, which require not only Dehn twists, but also "Y-homeomorphisms."