In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
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:
Give A coordinates (s, t) where s is a complex number of the form e i θ θ --> {\displaystyle e^{i\theta }} with θ θ --> ∈ ∈ --> [ 0 , 2 π π --> ] , {\displaystyle \theta \in [0,2\pi ],} and t ∈ [0, 1].
Let f be the map from S to itself which is the identity outside of A and inside A we have
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.
Consider the torus represented by a fundamental polygon with edges a and b
Let a closed curve be the line along the edge a called γ γ --> a {\displaystyle \gamma _{a}} .
Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve γ γ --> a {\displaystyle \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 ( e i θ θ --> , t ) ↦ ↦ --> ( e i ( θ θ --> + 2 π π --> t ) , t ) {\displaystyle \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 γ γ --> a {\displaystyle \gamma _{a}} , yields a Dehn twist of the torus by a.
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
where [x] are the homotopy classes of the closed curve x in the torus. Notice T a ∗ ∗ --> ( [ a ] ) = [ a ] {\displaystyle {T_{a}}_{\ast }([a])=[a]} and T a ∗ ∗ --> ( [ b ] ) = [ b ∗ ∗ --> a ] {\displaystyle {T_{a}}_{\ast }([b])=[b*a]} , where b ∗ ∗ --> a {\displaystyle b*a} is the path travelled around b then a.
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 {\displaystyle g} surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along 3 g − − --> 1 {\displaystyle 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 2 g + 1 {\displaystyle 2g+1} , for g > 1 {\displaystyle 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."
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