Y-homeomorphism

From Wikipedia, the free encyclopedia

In mathematics, the y-homeomorphism is a special type of auto-homeomorphism in non-orientable surfaces.

It can be constructed by sliding a Möbius band included on the surface around an essential 1-sided closed curve until the original position; thus it is necessary that the surfaces have genus greater than one. The projective plane ${\mathbb RP}^2$ has no y-homeomorphism.

For any other non-orientable surface the possible y-homeomorphisms are isotopic. Then, in the extended mapping class group of the surface there is only one isotopy class different from the classes represented by Dehn twists. This isotopy class has period two.

[edit] See also

Lickorish-Wallace theorem

[edit] References

1. J.S. Birman, D.R.J. Chillingworth, On the homeotopy group of a non-orientable surface, Trans. Amer. Math. Soc. 247 (1979), 87-124.
2. D.R.J. Chillingworth, A finite set of generators for the homeotopy group of a non-orientable surface, Proc. Camb. Phil. Soc. 65 (1969), 409-430.
3. M. Korkmaz, Mapping class group of non-orientable surface, Geometriae Dedicata 89 (2002), 109-133.
4. W.B.R. Lickorish, Homeomorphisms of non-orientable two-manifolds, Math. Proc. Camb. Phil. Soc. 59 (1963), 307-317.