Dehn twist

Definition

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 and so is homeomorphic to the Cartesian product of

where I is the unit interval. Give A coordinates (s, t) where s is a complex number of the form

with

and t in the unit interval.

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.

Example

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 γ a .

Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve γ a will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say

in the complex plane.

By extending to the torus the twisting map ( e i θ , t ) ↦ ( e i ( θ + 2 π t ) , t ) of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of γ a , yields a Dehn twist of the torus by a.

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

where [x] are the homotopy classes of the closed curve x in the torus. Notice T a ∗ ( [ a ] ) = [ a ] and T a ∗ ( [ b ] ) = [ b ∗ a ] , where b ∗ a is the path travelled around b then a.

Mapping class group

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- g surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along 3 g − 1 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 2 g + 1 , for g > 1 , which he showed was the minimal number.

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