# Dehn twist A positive Dehn twist applied to a cylinder about the red curve c modifies the green curve as shown.

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

## 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, homeomorphic to the Cartesian product of a circle and a unit interval I:

$c\subset A\cong S^{1}\times I.$ Give A coordinates (s, t) where s is a complex number of the form $e^{i\theta }$ with $\theta \in [0,2\pi ],$ and t ∈ [0, 1].

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

$f(s,t)=\left(se^{i2\pi t},t\right).$ 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 An example of a Dehn twist on the torus, along the closed curve a, in blue, where a is an edge of the fundamental polygon representing the torus. The automorphism on the fundamental group of the torus induced by the self-homeomorphism of the Dehn twist along one of the generators of the torus.

Consider the torus represented by a fundamental polygon with edges a and b

$\mathbb {T} ^{2}\cong \mathbb {R} ^{2}/\mathbb {Z} ^{2}.$ Let a closed curve be the line along the edge a called $\gamma _{a}$ .

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

$a(0;0,1)=\{z\in \mathbb {C} :0<|z|<1\}$ in the complex plane.

By extending to the torus the twisting map $\left(e^{i\theta },t\right)\mapsto \left(e^{i\left(\theta +2\pi t\right)},t\right)$ of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of $\gamma _{a}$ , yields a Dehn twist of the torus by a.

$T_{a}:\mathbb {T} ^{2}\to \mathbb {T} ^{2}$ 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

${T_{a}}_{\ast }:\pi _{1}\left(\mathbb {T} ^{2}\right)\to \pi _{1}\left(\mathbb {T} ^{2}\right):[x]\mapsto \left[T_{a}(x)\right]$ where [x] are the homotopy classes of the closed curve x in the torus. Notice ${T_{a}}_{\ast }([a])=[a]$ and ${T_{a}}_{\ast }([b])=[b*a]$ , 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 $3g-1$ 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 $2g+1$ , 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."