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	Orientability - Wikipedia, the free encyclopedia
		It has been suggested that this article or section be merged with orientable manifold.
		For an abstract surface (i.e., a two-dimensional manifold), it is orientable if a consistent concept of clockwise rotation can be defined on the surface in a continuous (intuitively, locally constant) manner.
		Formally, a n-dimensional differentiable manifold is called orientable if it possesses a differential form ω of degree n which is nonzero at every point on the manifold.
	en.wikipedia.org /wiki/Orientability (842 words)

	Orientable manifold - Wikipedia, the free encyclopedia
		In mathematics, a manifold or space is orientable if and only if it is possible to define left- and right-directions globally throughout that space.
		A simply connected two-dimensional space which obeys Euclidean geometry is orientable for two-dimensional objects: it is possible to describe two objects that are reflections of one another but cannot be transformed into one another.
		The best-known non-orientable two-dimensional manifold is the Möbius strip.
	en.wikipedia.org /wiki/Orientable_manifold (652 words)

	PlanetMath: orientation
		The most general, in the sense that it doesn't require any extra structure on the manifold, is based on (co-)homology theory.
		For this article manifold means a connected, topological manifold possibly with boundary.
		One approach (which we will not follow) is to use special kind of homology (for example relative to the boundary for compact manifolds with boundary).
	www.planetmath.org /encyclopedia/Orientation2.html (308 words)

	haken manifolds (Site not responding. Last check: 2007-09-20)
		Haken manifolds are named after Wolfgang Haken, who pioneered the use of incompressible surfaces.
		We will consider only the case of orientable Haken manifolds, as this simplifies the discussion; a regular neighborhood of an orientable surface in an orientable 3-manifold is just a "thickened up" version of the surface.
		Two famous cases of theorems about Haken manifolds being proven in this way is Friedhelm Waldhausen's proof of topological rigidity for Haken manifolds and William Thurston's proof of geometrization for Haken manifolds.
	www.yourencyclopedia.net /Haken_manifolds.html (384 words)

	Orientable (Site not responding. Last check: 2007-09-20)
		Notethat whether the surface is orientable is independent of triangulation; this fact is not obvious, but a standard exercise.
		Another way of thinking about orientability is thinking of it as a choice of "right handedness" vs. "left handedness" at eachpoint in the manifold.
		Formally, a n-dimensional differentiable manifold is called orientable if it possesses a differential form ω of degree nwhich is nonzero at every point on the manifold.
	www.therfcc.org /orientable-189282.html (539 words)

	Orientable manifold (Site not responding. Last check: 2007-09-20)
		In mathematics, a manifold or space is orientable if and only if it is possible to define left- andright-directions globally throughout that space.
		A simply connected two-dimensional space which obeys Euclideangeometry is orientable for two-dimensional objects: it is possible to describe two objects that are reflections of oneanother but cannot be transformed into one another.
		However, such a two-dimensional manifold is non-orientable for any one-dimensional objects: that is, it is impossible todescribe two one-dimensional objects that are reflections of one another but could not be rotated into one another.
	www.therfcc.org /orientable-manifold-220590.html (543 words)

	orientability (Site not responding. Last check: 2007-09-20)
		In geometry and topology, a surface in is called non-orientable, if a figure such as the letter "R" can be moved about on the surface so that it becomes mirror-reversed.
		In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as above) is orientable.
		Formally, a -dimensional differentiable manifold is called orientable if it possesses a differential form of degree which is nonzero at every point on the manifold.
	www.yourencyclopedia.net /orientability.html (677 words)

	Orientability - The Jiggies Reference Guide (Site not responding. Last check: 2007-09-20)
		In geometry and topology, a surface in $\mathbb{R}^3$ is called orientable, if, roughly speaking, it is possible to consistently distinguish between the two sides of the surface.
		In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space (such as $\mathbb{R}^3$ above) is orientable.
		Formally, a $n$ -dimensional differentiable manifold is called orientable if it possesses a differential form $\omega$ of degree $n$ which is nonzero at every point on the manifold.
	www.jiggies.com /reference/Orientable (625 words)

	Arneodo's system (Site not responding. Last check: 2007-09-20)
		In fact, also the stable manifold of this periodic orbit is topologically a Möbius strip.
		The two-dimensional unstable manifold of the equilibrium (
		The unstable manifold of B forms a generic heteroclinic intersection with the non-orientable stable manifold of the periodic orbit.
	www.maths.ex.ac.uk /~hinke/nonorientable (259 words)

	Orientable manifold -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-09-20)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, a (A pipe that has several lateral outlets to or from other pipes) manifold or space is orientable if and only if it is possible to define left- and right-directions globally throughout that space.
		The best-known non-orientable two-dimensional manifold is the (Click link for more info and facts about Möbius strip) Möbius strip.
		The (The 4-dimensional coordinate system (3 dimensions of space and 1 of time) in which physical events are located) space-time manifold of the actual (Everything that exists anywhere) universe is believed to be orientable.
	www.absoluteastronomy.com /encyclopedia/o/or/orientable_manifold.htm (718 words)

	Orientability : Orientable
		Similarly, the Klein bottle isn't orientable, because one cannot distinguish between the inside and the outside.
		In general, the property of being orientable isn't equivalent to being two-sided; however, this holds when the ambient space (such as R^3 above) is orientable.
		Note that whether the surface is orientable is indpendent of triangulation; this fact isn't obvious, but a standard exercise.
	www.fastload.org /or/Orientable.html (638 words)

	Haken manifold (Site not responding. Last check: 2007-09-20)
		Haken manifolds are named after Wolfgang Haken, who pioneered theuse of incompressible surfaces.
		Wewill consider only the case of orientable Haken manifolds, as this simplifiesthe discussion; a regular neighborhood of an orientable surface in an orientable 3-manifold is just a "thickened up" version ofthe surface.
		Two famous cases of theorems about Haken manifolds being proven in this way is Friedhelm Waldhausen 'sproof of topologicalrigidity for Haken manifolds and William Thurston 's proof of geometrization conjecture for Haken manifolds.
	www.therfcc.org /haken-manifold-220596.html (341 words)

	Visualization of the isometry group action on the Fomenko--Matveev--Weeks manifold (ResearchIndex) (Site not responding. Last check: 2007-09-20)
		The smallest known three-dimensional closed orientable hyperbolic manifold M 1, whose volume is equal to 0:94 : : :, was obtained independently by A. Fomenko and S. Matveev and by J. Weeks.
		It is known that the isometry group of the manifold M 1 is isomorphic to the dihedral group D 6 of order 12.
		The aim of the present paper is to describe the lattice of the action of the isometry group Isom(M 1) on the manifold M 1.
	citeseer.ist.psu.edu /149970.html (366 words)

	Encyclopedia: Haken manifold (Site not responding. Last check: 2007-09-20)
		It is a theorem that cutting a Haken manifold along an incompressible surface results in a Haken manifold.
		Friedhelm Waldhausen proved that Haken manifolds are topologically rigid: roughly, any homotopy equivalence of Haken manifolds is homotopic to a homeomorphism.
		More recent is William Thurston's geometrization theorem for Haken manifolds.
	www.nationmaster.com /encyclopedia/Haken-manifold (536 words)

	C Object Oriented (Site not responding. Last check: 2007-09-20)
		This article should be merged with Orientable manifold.
		In general, the property of being orientable is not equivalent to being two-sided; however, this holds when the ambient space(such as above) is orientable.
		Anysurface has a triangulation: a decomposition into triangles such that each edge on a triangle is glued to at most one other edge.We can orient each triangle, by choosing a direction for each edge (think of this as drawing an arrow on each edge) so that th...
	www.swingdancemusic.com /send/42177-c%20object%20oriented.html (703 words)

	Math 423, Fall, 2002 (Site not responding. Last check: 2007-09-20)
		The first is that manifolds, and also more general spaces, can be thought of as being constructed from building-blocks called simplices, points, line segments, triangles, tetrahedra, etc. It is true that any compact manifold is homeomorphic to a union of such objects, in a very specific way.
		If it is possible to do so, then the manifold M is orientable, and an orientation is a choice of such local coordinates, compatible with each other at each point.
		If a manifold is not orientable, all is not lost (though the upcoming definition of integration over M will be).
	www.lehigh.edu /~dlj0/courses/423f02-lect20.html (2384 words)

	Cubical 4-Polytopes
		It has been observed by Stanley and MacPherson that every cubical d-polytope (that is, a convex bounded polyhedron whose facets are combinatorially isomorphic to the (d-1)-dimensional standard cube) determines a PL immersion of an abstract cubical (d-2)-manifold into (the barycentric subdivision of) the boundary of the polytope, as illustrated in the following figure.
		In the case of 4-polytopes, the dual manifolds are surfaces (compact 2-manifolds without boundary).
		is PL-equivalent to a dual manifold immersion of a cubical 4-polytope.
	www.math.tu-berlin.de /~schwartz/c4p (338 words)

	Orientable manifold: Definition and Links by Encyclopedian.com - All about Orientable manifold
		Orientable manifold: Definition and Links by Encyclopedian.com - All about Orientable manifold
		A manifold or space is orientable if and only if it is possible to define left- and right-directions globally throughout that space.
		An example might be a straight line colored red at one and and blue at the other.
	www.encyclopedian.com /or/Orientable-manifold.html (641 words)

	[No title]
		A -orientation of a manifold is understood as a -orientation of its Thom spectrum.
		By [6], the Kervaire invariant of a smooth framed manifold of dimension 2n, where n 6= 2i- 1, is zero.
		For a closed -orientable manifold M2n, there is a Poincar'e triple (M, M, ff) where M is the stable normal bundle and ff 2 ß2n+k(T M) is the normal invariant of M (obtained by the Thom-Pontryagin con- struction.) Definition 2.9.
	hopf.math.purdue.edu /FangF-PanJZ/cl-2-1.txt (4778 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		In case you meant this, note that it is also true that if the base space is orientable, the total space of any principal S1 bundle will also be orientable, and here is the proof: at every point in the n-dimensional base space M, we have an n-form \omega that defines the orientation.
		The base space is the 2-torus, which of course is orientable with H2(M) nontrivial, and in the dimension range you want.
		More generally M may be chosen to be any orientable manifold with nontrivial H1(M;Z/2).
	www.lehigh.edu /~dmd1/ki97.txt (605 words)

	math lessons - Lickorish-Wallace theorem (Site not responding. Last check: 2007-09-20)
		In mathematics, the Lickorish-Wallace theorem in the theory of 3-manifolds states that any closed, orientable, connected 3-manifold may be obtained by performing Dehn surgery on a framed link in the 3-sphere with +/-1 surgery coefficients.
		Lickorish's proof rested on the Lickorish twist theorem, which states that any orientable automorphism of a closed orientable surface is generated by Dehn twists along 3g − 1 specific simple closed curves in the surface, where g denotes the genus of the surface.
		Similar to the orientable case, the surgery can be done in a special way which allows the conclusion that every closed, non-orientable 3-manifold bounds a compact 4-manifold.
	www.mathdaily.com /lessons/Lickorish-Wallace_theorem (230 words)

	Orientable manifold (Site not responding. Last check: 2007-09-20)
		The surface of an flat sheet of paper is an orientable manifold.
		A simply connected two-dimensional space which obeys Euclidean geometry is orientable for two-dimensional objects: it possible to describe two objects that are of one another but cannot be transformed one another.
		However such a two-dimensional manifold is non-orientable any one-dimensional objects: that is it is to describe two one-dimensional objects that are of one another but could not be into one another.
	www.freeglossary.com /Orientable_manifold (612 words)

	Talk:Orientable manifold - Wikipedia, the free encyclopedia
		I think it should go in orientability Tosha 05:13, 22 Aug 2004 (UTC)
		On the small scale, one believes it for obvious reasons (handedness in biological molecules is another good reason).
		An unorientable space-time would have an orientable double cover.
	www.wikipedia.org /wiki/Talk:Orientable_manifold (184 words)

	Orientable Manifold with Boundary - Page 2 - Physics Help and Math Help - Physics Forums
		In that case, you are also using a chart that does not satisfy the requirements given in the definition.
		anyway the case of a one dimensional manifold is a little special, because orientation is usually defined by choosing an ordered basis of the vector space.
		the boundary of a one dimensional manifold however has dimesnino zero, and there is only one basis of that space, the empty basis.
	www.physicsforums.com /showthread.php?t=81202&page=2 (1336 words)

	World War 1 and 2 - Spherical 3-manifold (Site not responding. Last check: 2007-09-20)
		In mathematics, a spherical 3-manifold M is a prime, orientable, closed 3-manifold of the form
		The elliptization conjecture states that if a 3-manifold has finite fundamental group, then it is a spherical manifold.
		A lens space is not determined by its fundamental group, but any other spherical manifold is.
	www.worldwardiary.com /history/Spherical_3-manifold (116 words)

	John M. Bryden - 3-manifold invariants associated to topological quantum field theories
		Let M be a closed orientable 3-manifold obtained from surgery on a framed link.
		Although there has been some progress made in understanding the combinatorial nature of these and other quantum invariants, their geometric nature and their relationship to the fundamental group and to cohomology is not understood.
		An initial attempt to understand these invariants in the context of algebraic topology is to interpret the surgery in the cohomology algebras of Seifert manifolds.
	www.cms.math.ca /Events/summer98/s98-abs/node82.e (261 words)

	[No title] (Site not responding. Last check: 2007-09-20)
		This project will involve such a study for knots in a closed, orientable 3-manifold where we are seeking to understand the topology of the manifold.
		For example, if the property that detects the trivial knot in closed, orientable, irreducible 3-manifolds with cyclic fundamental group is also realizable in such manifolds, then these manifolds are lens spaces; in particular, a simply connected one is the 3-sphere (The Poincare Conjecture).
		Jaco and Rubinstein have shown that under reasonable restrictions a 3-manifold admits a triangulation in which each edge is a knot (one vertex triangulation) and particular edges, ``thick edges," are candidates for realizing knots with the desired properties, depending on the initial restrictions on the 3-manifold.
	www.aimath.org /projects/lopez.html (366 words)

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