# Dehn twist

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).

## 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 and so is homeomorphic to the Cartesian product of

$S^1 \times I,$

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

$e^{{\rm{i}} \theta}$

with

$\theta \in [0,2\pi],$

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

$\displaystyle f(s,t) = (s e^{{\rm{i}} 2 \pi t}, t).$

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.

## 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-$g$ 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."