Dehn twist
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In geometric topology, a branch of mathematics, a Dehn twist is a certain type of self-homeomorphism of a surface (two-dimensional manifold).
To be precise, suppose that c is a simple closed curve in a closed, orientable surface S. Let A be a regular neighborhood of c. Then A is an annulus and so is homeomorphic to the Cartesian product of
where I is the unit interval. Give A coordinates (s, t) where s is a complex number of the form
- eiθ
with
![\theta \in [0,2\pi],](../../../math/9/7/5/975fe98a9bfdf20e875fdb8121c06184.png)
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
- f(s,t) = (sei2πt,t).
Then f is a Dehn twist about the curve c. It is a theorem of Max Dehn (and W. B. R. Lickorish independently) that maps of this form generate the mapping class group of any closed, orientable genus-g surface. Lickorish showed that Dehn twists along 3g − 1 curves could generate the mapping class group; this was later improved by Stephen P. Humphries to 2g + 1, 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."
[edit] References
- Andrew J. Casson, Steven A Bleiler, Automorphisms of Surfaces After Nielsen and Thurston, Cambridge University Press, 1988. ISBN 0-521-34985-0.
- W. B. R. Lickorish, A finite set of generators for the homeotopy group of a 2-manifold, Proc. Cambridge Philos. Soc. 60 (1964), 769–778.
