Dehn twist

File:Dehn Twist Animation.webm

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

[math]\displaystyle{ c \subset A \cong S^1 \times I. }[/math]

Give A coordinates (s, t) where s is a complex number of the form [math]\displaystyle{ e^{i\theta} }[/math] with [math]\displaystyle{ \theta \in [0, 2\pi], }[/math] 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

[math]\displaystyle{ f(s, t) = \left(se^{i2\pi t}, t\right). }[/math]

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

[math]\displaystyle{ \mathbb{T}^2 \cong \mathbb{R}^2/\mathbb{Z}^2. }[/math]

Let a closed curve be the line along the edge a called [math]\displaystyle{ \gamma_a }[/math].

Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve [math]\displaystyle{ \gamma_a }[/math] will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say

[math]\displaystyle{ a(0; 0, 1) = \{z \in \mathbb{C}: 0 \lt |z| \lt 1\} }[/math]

in the complex plane.

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

[math]\displaystyle{ T_a: \mathbb{T}^2 \to \mathbb{T}^2 }[/math]

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

[math]\displaystyle{ {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] }[/math]

where [x] are the homotopy classes of the closed curve x in the torus. Notice [math]\displaystyle{ {T_a}_\ast([a]) = [a] }[/math] and [math]\displaystyle{ {T_a}_\ast([b]) = [b*a] }[/math], where [math]\displaystyle{ b*a }[/math] 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-[math]\displaystyle{ g }[/math] surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along [math]\displaystyle{ 3g - 1 }[/math] 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 [math]\displaystyle{ 2g + 1 }[/math], for [math]\displaystyle{ g \gt 1 }[/math], 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

• Lantern relation

References

• Andrew J. Casson, Steven A Bleiler, Automorphisms of Surfaces After Nielsen and Thurston, Cambridge University Press , 1988. ISBN:0-521-34985-0.
• Stephen P. Humphries, "Generators for the mapping class group," in: Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47, Lecture Notes in Math., 722, Springer, Berlin, 1979. MR0547453
• W. B. R. Lickorish, "A representation of orientable combinatorial 3-manifolds." Ann. of Math. (2) 76 1962 531—540. MR0151948
• W. B. R. Lickorish, "A finite set of generators for the homotopy group of a 2-manifold", Proc. Cambridge Philos. Soc. 60 (1964), 769–778. MR0171269