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Dehn twist

Contents

Contents

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 Σ\Sigma be an orientable surface of genus gg, and consider an embedded circle i:S 1Σi: S^1 \hookrightarrow \Sigma. (For example, one may imagine a circle representing one of 2g2 g generators of H 1(Σ,)H^1(\Sigma, \mathbb{Z}).) The embedded circle has a tubular neighborhood in Σ\Sigma, homeomorphic to an annulus S 1×[0,1]S^1 \times [0, 1]. Representing elements of S 1S^1 by complex numbers zz of norm 1, a twist on the annulus may be defined by sending (z,t)(exp(2πit)z,t)(z, t) \mapsto (\exp(2\pi i t) z, t), which equals the identity on the boundary circles where t=0,t=1t = 0, t = 1. This defines a self-homeomorphism (mod boundary) on the annulus.

The corresponding Dehn twist on Σ\Sigma is obtained by extending the self-homeomorphism on the annulus to the entire surface, by taking a point pp to itself if pp 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)(exp(2πif(t))z,t)(z, t) \mapsto (\exp(2\pi i f(t))z, t) where ff is a smooth bump function such that f(t)=0f(t) = 0 on a small neighborhood of t=0t = 0 and f(t)=1f(t) = 1 on a small neighborhood of t=1t=1. This modification ensures that we get a self-diffeomorphism on the surface Σ\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 Σ\Sigma, of orientation-preserving homeomorphisms considered modulo isotopy. (In fact Lickorish described 3g13g-1 explicit embedded circles for a surface Σ\Sigma of genus gg whose corresponding twists give the generators.)

One example that is easily visualized is the mapping class group MCG(Σ)MCG(\Sigma) of the torus Σ=S 1×S 1\Sigma = S^1 \times S^1. Here the canonical map MCG(Σ)Aut(π 1(Σ))Aut(×)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()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 π 1(Σ)\pi_1(\Sigma); the Wikipedia article has some decent pictures of this action.

References

Last revised on February 8, 2016 at 09:49:35. See the history of this page for a list of all contributions to it.