In geometric topology

Geometric topology

In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.- Topics :...

, a branch of mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a Dehn twist is a certain type of self-homeomorphism

Homeomorphism

In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...

 of a surface

Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...

 (two-dimensional manifold

Manifold

In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....

).

Definition

Suppose that c is a simple closed curve

Curve

In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...

 in a closed, orientable

Orientability

In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...

 surface S. Let A be a tubular neighborhood

Tubular neighborhood

In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle.The idea behind a tubular neighborhood can be explained in a simple example. Consider a smooth curve in the plane without self-intersections. On each point on the curve...

 of c. Then A is an annulus

Annulus (mathematics)

In mathematics, an annulus is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. Or, it is the area between two concentric circles...

 and so is homeomorphic to the Cartesian product

Cartesian product

In mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...

 of


where I is the unit interval

Unit interval

In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1...

. Give A coordinates (s, t) where s is a complex number of the form


with


and t in the unit interval.

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

2-sided

In topology, a compact codimension one submanifold F of a manifold M is said to be 2-sided in M when there is an embeddingwith h=x for each x\in F andIn other words, if its normal bundle is trivial....

 simple closed curve c on S.

Mapping class group

It is a theorem of Max Dehn

Max Dehn

Max Dehn was a German American mathematician and a student of David Hilbert. He is most famous for his work in geometry, topology and geometric group theory...

 that maps of this form generate the mapping class group of isotopy

Homotopy

In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...

 classes of orientation-preserving homeomorphisms of any closed, oriented genus

Genus (mathematics)

In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...

- surface. W. B. R. Lickorish later rediscovered this result with a simpler proof and in addition showed that Dehn twists along explicit curves generate the mapping class group (this is called by the punning name "Lickorish twist

Twist (sweets)

Twist is a bite-sized confectionery produced by Freia/Kraft Foods, primarily for the Scandinavian market. Twist is sold in bags, and each bag contains a variety of pieces, mostly consisting of chocolates with some sort of filling in the center. New pieces are added and old ones discontinued...

 theorem"); this number was later improved by Stephen P. Humphries to , for , 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-homeomorphism

Y-homeomorphism

In mathematics, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism in non-orientable surfaces.It can be constructed by sliding a Möbius band included on the surface...

s."