In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the y-homeomorphism, or crosscap slide, is a special type of auto-
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isom ...
in
non-orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
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
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
greater than one. The projective plane
has no y-homeomorphism.
See also
*
Lickorish-Wallace theorem
References
*J. S. Birman, D. R. J. Chillingworth, ''On the homeotopy group of a non-orientable surface'', Trans. Amer. Math. Soc. 247 (1979), 87-124.
*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.
*M. Korkmaz, ''Mapping class group of non-orientable surface'', Geometriae Dedicata 89 (2002), 109–133.
*
W. B. R. Lickorish, ''Homeomorphisms of non-orientable two-manifolds'', Math. Proc. Camb. Phil. Soc. 59 (1963), 307–317.
Geometric topology
Homeomorphisms
{{Geometry-stub