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

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