Dehn twist
In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
Definition
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>c \subset A \cong S^1 \times I.</math>
Give A coordinates (s, t) where s is a complex number of the form <math>e^{i\theta}</math> with <math>\theta \in [0, 2\pi],</math> and Template:Nowrap.
Let f be the map from S to itself which is the identity outside of A and inside A we have
- <math>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
Consider the torus represented by a fundamental polygon with edges a and b
- <math>\mathbb{T}^2 \cong \mathbb{R}^2/\mathbb{Z}^2.</math>
Let a closed curve be the line along the edge a called <math>\gamma_a</math>.
Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve <math>\gamma_a</math> will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say
- <math>a(0; 0, 1) = \{z \in \mathbb{C}: 0 < |z| < 1\}</math>
in the complex plane.
By extending to the torus the twisting map <math>\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>\gamma_a</math>, yields a Dehn twist of the torus by a.
- <math>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>{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>{T_a}_\ast([a]) = [a]</math> and <math>{T_a}_\ast([b]) = [b*a]</math>, where <math>b*a</math> is the path travelled around b then a.
Mapping class group
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>g</math> surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along <math>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>2g + 1</math>, for <math>g > 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
References
- Andrew J. Casson, Steven A Bleiler, Automorphisms of Surfaces After Nielsen and Thurston, Cambridge University Press, 1988. Template:ISBN.
- 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. Template:MR
- W. B. R. Lickorish, "A representation of orientable combinatorial 3-manifolds." Ann. of Math. (2) 76 1962 531—540. Template:MR
- 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. Template:MR