# Dehn twist

Dehn twist (Redirected from Lickorish twist theorem) Jump to navigation Jump to search A positive Dehn twist applied to a cylinder about the red curve c modifies the green curve as shown.

In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).

Contents 1 Definition 2 Example 3 Mapping class group 4 See also 5 References Definition General Dehn twist on a compact surface represented by a n-gon.

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: {displaystyle csubset Acong S^{1}times I.} Give A coordinates (s, t) where s is a complex number of the form {displaystyle e^{itheta }} with {displaystyle theta in [0,2pi ],} 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 {displaystyle f(s,t)=left(se^{i2pi t},tright).} 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 An example of a Dehn twist on the torus, along the closed curve a, in blue, where a is an edge of the fundamental polygon representing the torus. The automorphism on the fundamental group of the torus induced by the self-homeomorphism of the Dehn twist along one of the generators of the torus.

Consider the torus represented by a fundamental polygon with edges a and b {displaystyle mathbb {T} ^{2}cong mathbb {R} ^{2}/mathbb {Z} ^{2}.} Let a closed curve be the line along the edge a called {displaystyle gamma _{a}} .

Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve {displaystyle gamma _{a}} will look like a band linked around a doughnut. This neighborhood is homeomorphic to an annulus, say {displaystyle a(0;0,1)={zin mathbb {C} :0<|z|<1}} in the complex plane. By extending to the torus the twisting map {displaystyle left(e^{itheta },tright)mapsto left(e^{ileft(theta +2pi tright)},tright)} of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of {displaystyle gamma _{a}} , yields a Dehn twist of the torus by a. {displaystyle 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 {displaystyle {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 {displaystyle {T_{a}}_{ast }([a])=[a]} and {displaystyle {T_{a}}_{ast }([b])=[b*a]} , where {displaystyle b*a} is the path travelled around b then a. Mapping class group The 3g − 1 curves from the twist theorem, shown here for g = 3. 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- {displaystyle g} surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along {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 {displaystyle 2g+1} , for {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."

See also Lantern relation References 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. MR0547453 W. B. R. Lickorish, "A representation of orientable combinatorial 3-manifolds." Ann. of Math. (2) 76 1962 531—540. MR0151948 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. MR0171269 Categories: Geometric topologyHomeomorphisms

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