Dehn twist

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 \subset A \cong S^1 \times I. Give A coordinates (s, t) where s is a complex number of the form e^{i\theta} with \theta \in [0, 2\pi], and . Let f be the map from S to itself which is the identity outside of A and inside A we have :f(s, t) = \left(se^{i2\pi t}, t\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. ==Example==

Consider the torus represented by a fundamental polygon with edges a and b :\mathbb{T}^2 \cong \mathbb{R}^2/\mathbb{Z}^2. Let a closed curve be the line along the edge a called \gamma_a. Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve \gamma_a will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say :a(0; 0, 1) = \{z \in \mathbb{C}: 0 in the complex plane. By extending to the torus the twisting map \left(e^{i\theta}, t\right) \mapsto \left(e^{i\left(\theta + 2\pi t\right)}, t\right) of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of \gamma_a, yields a Dehn twist of the torus by a. :T_a: \mathbb{T}^2 \to \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}_\ast: \pi_1\left(\mathbb{T}^2\right) \to \pi_1\left(\mathbb{T}^2\right): [x] \mapsto \left[T_a(x)\right] where [x] are the homotopy classes of the closed curve x in the torus. Notice {T_a}_\ast([a]) = [a] and {T_a}_\ast([b]) = [b*a], where b*a 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-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." ==See also==