Dehn twist

File:General Dehn twist on a surface.png

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:

:$c\; \backslash subset\; A\; \backslash cong\; S^1\; \backslash times\; I.$

Give A coordinates (s, t) where s is a complex number of the form $e^\{i\backslash theta\}$ with $\backslash theta\; \backslash in\; [0,\; 2\backslash pi],$ and {{nowrap|t ∈ [0, 1]}}.

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

:$f(s,\; t)\; =\; \backslash left(se^\{i2\backslash pi\; t\},\; t\backslash right).$

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.

File:Dehn twist for the torus.png

File:Dehn twist induced isomorphism.png

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

:$\backslash mathbb\{T\}^2\; \backslash cong\; \backslash mathbb\{R\}^2/\backslash mathbb\{Z\}^2.$

Let a closed curve be the line along the edge a called $\backslash gamma\_a$.

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

:

in the complex plane.

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

:$T\_a:\; \backslash mathbb\{T\}^2\; \backslash to\; \backslash mathbb\{T\}^2$

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

:$\{T\_a\}\_\backslash ast:\; \backslash pi\_1\backslash left(\backslash mathbb\{T\}^2\backslash right)\; \backslash to\; \backslash pi\_1\backslash left(\backslash mathbb\{T\}^2\backslash right):\; [x]\; \backslash mapsto\; \backslash left[T\_a(x)\backslash right]$

where [x] are the homotopy classes of the closed curve x in the torus. Notice $\{T\_a\}\_\backslash ast([a])\; =\; [a]$ and $\{T\_a\}\_\backslash ast([b])\; =\; [b*a]$, where $b*a$ is the path travelled around b then a.

Image:Lickorish Twist Theorem.svg

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."

• Fenchel–Nielsen coordinates
• Lantern relation

• 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. {{MR|0547453}}
• W. B. R. Lickorish, "A representation of orientable combinatorial 3-manifolds." Ann. of Math. (2) 76 1962 531—540. {{MR|0151948}}
• 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. {{MR|0171269}}

Category:Geometric topology

Category:Homeomorphisms