nLab Dehn twist

Idea

A Dehn twist is a type of self-homeomorphism on a surface (a 2-dimensional topological manifold), especially an orientable closed manifold (of dimension 2).

Definition

Let $\Sigma$ be an orientable surface of genus $g$, and consider an embedded circle $i: S^1 \hookrightarrow \Sigma$. (For example, one may imagine a circle representing one of $2 g$ generators of $H^1(\Sigma, \mathbb{Z})$.) The embedded circle has a tubular neighborhood in $\Sigma$, homeomorphic to an annulus $S^1 \times [0, 1]$. Representing elements of $S^1$ by complex numbers $z$ of norm 1, a twist on the annulus may be defined by sending $(z, t) \mapsto (\exp(2\pi i t) z, t)$, which equals the identity on the boundary circles where $t = 0, t = 1$. This defines a self-homeomorphism (mod boundary) on the annulus.

The corresponding Dehn twist on $\Sigma$ is obtained by extending the self-homeomorphism on the annulus to the entire surface, by taking a point $p$ to itself if $p$ is outside the annulus. (If we are working in the category of smooth manifolds, one may modify the twist on the annulus by taking $(z, t) \mapsto (\exp(2\pi i f(t))z, t)$ where $f$ is a smooth bump function such that $f(t) = 0$ on a small neighborhood of $t = 0$ and $f(t) = 1$ on a small neighborhood of $t=1$. This modification ensures that we get a self-diffeomorphism on the surface $\Sigma$ as smooth manifold.)

Properties

The following theorem was first proved by MaxDehn, with later simplifications by Lickorish:

Theorem

(“Lickorish twist”) The Dehn twists generate the mapping class group of $\Sigma$, of orientation-preserving homeomorphisms considered modulo isotopy. (In fact Lickorish described $3g-1$ explicit embedded circles for a surface $\Sigma$ of genus $g$ whose corresponding twists give the generators.)

One example that is easily visualized is the mapping class group $MCG(\Sigma)$ of the torus $\Sigma = S^1 \times S^1$. Here the canonical map $MCG(\Sigma) \to Aut(\pi_1(\Sigma)) \cong Aut(\mathbb{Z} \times \mathbb{Z})$ is an isomorphism (where automorphisms of $\mathbb{Z}$ are required to preserve orientation, i.e., are elements of the group $SL_2(\mathbb{Z})$; cf. modular group). Thus in this case the Dehn twists can be visualized in terms of their action on loops representing elements of $\pi_1(\Sigma)$; the Wikipedia article has some decent pictures of this action.

Related concepts

• mapping class group

• Dehn surgery

References

• Wikipedia, Dehn twist