# 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*:

Let *f* be the map from *S* to itself which is the identity outside of *A* and inside *A* we have

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*

This self homeomorphism acts on the closed curve along *b*. In the tubular neighborhood it takes the curve of *b* once along the curve ofÂ *a*.

A homeomorphism between topological spaces induces a natural isomorphism between their fundamental groups. Therefore one has an automorphism

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