Dehn twist
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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 a 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. B. 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
- Andrew J Casson, Steven A Bleiler, Automorphisms of Surfaces After Nielson and Thurston, Cambridge University Press, 1988. ISBN 0521349850.