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

General Dehn twist on a compact surface represented by a n-gon.

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

S^1 \times I,

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

e^{{\rm{i}} \theta}

with

\theta \in [0,2\pi],

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

\displaystyle f(s,t) = (s e^{{\rm{i}} 2 \pi t}, t).

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 (e^{i\theta},t)\mapsto(e^{i(\theta+2\pi t)},t) 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(\mathbb{T}^2)\to\pi_1(\mathbb{T}^2):[x]\mapsto[T_a(x)]

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

The 3g − 1 curves from the twist theorem, shown here for g = 3.

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."

See also

References