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": Give "A" coordinates ("s", "t") where "s" is a complex number of the form formula_2 with formula_3 and . 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". Example. 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 formula_6. Given the choice of gluing homeomorphism in the figure, a tubular neighborhood of the curve formula_6 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 formula_9 of the annulus, through the homeomorphisms of the annulus to an open cylinder to the neighborhood of formula_6, 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 formula_13  and formula_14, where formula_15 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-formula_16 surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along formula_17 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 formula_18, for formula_19, 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."