Dehn twist

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

To be precise, suppose that g is a simple closed curve in an closed, orientable surface S. Let A be a regular neighborhood of g. Then A is an annulus and so is homeomorphic to the Cartesian product of

<math>S^1 \times I,<math>

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

<math>e^{{\rm{i}} \theta}<math>

with

<math>\theta \in [0,2\pi],<math>

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

<math> f(s,t) = (s e^{{\rm{i}} 2 \pi t}, t) <math>.

Then f is a Dehn twist. It is a theorem of Max Dehn (and W. R. Lickorish independently) that maps of this form generate the mapping class group of any closed, orientable surface. Lickorish showed that Dehn twists along <math>3g-1<math> curves could generate the mapping class group; this was later improved by Stephen P. Humphries to <math>2g+1<math>, which he showed was the minimal number. Lickorish also showed an analogous result for non-orientable surfaces which require not only Dehn twists, but "Y-homeomorphisms."

References

For more information on this subject, look at the book by Andrew Casson and Steve Bleiler entitled Automorphisms of surfaces after Nielsen and Thurston.