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