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 isomorphi ...

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 i ...

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

s. It can be constructed by sliding a Möbius band included on the surface around an essential 1-sided

closed curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...

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 extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

greater than one. The projective plane

^2

has no y-homeomorphism.

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 William Bernard Raymond Lickorish (born 19 February 1938) is a mathematician. He is emeritus professor of geometric topology in the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and also an emeritus fello ...

, ''Homeomorphisms of non-orientable two-manifolds'', Math. Proc. Camb. Phil. Soc. 59 (1963), 307–317. Geometric topology Homeomorphisms {{Geometry-stub

See also

References