Y-homeomorphism

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

It can be constructed by sliding a Möbius strip 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 ${\mathbb{ℝ}P}^2$ has no y-homeomorphism.

See also

• Lickorish-Wallace theorem

References

• "On the homeotopy group of a non-orientable surface". Mathematical Proceedings of the Cambridge Philosophical Society 71 (3): 437–448. 1972. doi:10.1017/S0305004100050714. Bibcode: 1972PCPS...71..437B. 
• Chillingworth, D. R. J. (1969). "A finite set of generators for the homeotopy group of a non-orientable surface". Mathematical Proceedings of the Cambridge Philosophical Society 65 (2): 409–430. doi:10.1017/S0305004100044388. Bibcode: 1969PCPS...65..409C. 
• Korkmaz, Mustafa (2002). "Mapping class group of non-orientable surface". Geometriae Dedicata 89: 109–133. doi:10.1023/A:1014289127999. 
• "Homeomorphisms of non-orientable two-manifolds". Mathematical Proceedings of the Cambridge Philosophical Society 59 (2): 307–317. 1963. doi:10.1017/S0305004100036926. Bibcode: 1963PCPS...59..307L.