Y-homeomorphism
Encyclopedia
In mathematics
, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism
in non-orientable
surface
s.
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
has no y-homeomorphism.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the y-homeomorphism, or crosscap slide, is a special type of auto-homeomorphism
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
in non-orientable
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space measuring whether or not it is possible to make a consistent choice of surface normal vector at every point. A choice of surface normal allows one to use the right-hand rule to define a "clockwise" direction of loops in the...
surface
Surface
In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R3 — for example, the surface of a ball...
s.
It can be constructed by sliding a Möbius
MOBIUS
MOBIUS is a consortium of libraries in Missouri, United States.The MOBIUS Consortium Office is a Missouri not-for-profit corporation established in 1998 by 50 libraries representing Missouri colleges and universities...
band included on the surface
around an essential
Essential
Essential or essentials can refer to:*An essential property, which defines an entity as being a particular type of entity. See also essentialism, the philosophical view that an entity must have certain characteristics in order to belong to a certain defined group, and its counterpart,...
1-sided
2-sided
In topology, a compact codimension one submanifold F of a manifold M is said to be 2-sided in M when there is an embeddingwith h=x for each x\in F andIn other words, if its normal bundle is trivial....
closed curve until the original position; thus it is necessary that the surfaces have genus
Genus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
greater than one. The projective plane
has no y-homeomorphism.