where I is the unit interval. Give A coordinates (s, t) where s is a complex number of the form
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 simple closed curve c on S.
Mapping class group
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- 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 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-homeomorphisms."
- Andrew J. Casson, Steven A Bleiler, Automorphisms of Surfaces After Nielsen and Thurston, Cambridge University Press, 1988. ISBN 0-521-34985-0.
- Stephen P. Humphries, Generators for the mapping class group, in: Topology of low-dimensional manifolds (Proc. Second Sussex Conf., Chelwood Gate, 1977), pp. 44–47, Lecture Notes in Math., 722, Springer, Berlin, 1979. MR 0547453
- W. B. R. Lickorish, A representation of orientable combinatorial 3-manifolds. Ann. of Math. (2) 76 1962 531—540. MR 0151948
- 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. MR 0171269