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Topic: Non-orientable

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Orientable manifold - Wikipedia, the free encyclopedia However, such a two-dimensional manifold is non-orientable for any one-dimensional objects: that is, it is impossible to describe two one-dimensional objects that are reflections of one another but could not be rotated into one another. The best-known non-orientable two-dimensional manifold is the Möbius strip. 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. en.wikipedia.org /wiki/Orientable_manifold   (652 words)

Arneodo's system The unstable manifold of B forms a generic heteroclinic intersection with the non-orientable stable manifold of the periodic orbit. This means that one of its Floquet multipliers moved through -1 and the unstable manifold is, therefore, non-orientable. The two-dimensional unstable manifold of the equilibrium ( www.maths.ex.ac.uk /~hinke/nonorientable   (259 words)

PlanetMath: orientation An oriented manifold is a (necessarily orientable) manifold -manifold is called orientable if its top homology group is isomorphic to the integers. The most general, in the sense that it doesn't require any extra structure on the manifold, is based on (co-)homology theory. www.planetmath.org /encyclopedia/Orientation2.html   (308 words)

Encyclopedia: Haken manifold 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. Sometimes one considers only orientable Haken manifolds, in which case a Haken manifold is a compact, orientable, irreducible 3-manifold that contains an orientable, incompressible surface. Haken manifolds are named after Wolfgang Haken, who pioneered the use of incompressible surfaces. www.nationmaster.com /encyclopedia/Haken-manifold   (536 words)

Cubical 4-Polytopes The immersed manifold is orientable if and only if the 2-skeleton of the cubical d-polytope (d > 2) is ``edge orientable'' in the sense of Hetyei [2]. is PL-equivalent to a dual manifold immersion of a cubical 4-polytope. 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. www.math.tu-berlin.de /~schwartz/c4p   (338 words)

cl-2-1.txt 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. 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. hopf.math.purdue.edu /FangF-PanJZ/cl-2-1.txt   (4778 words)

Visualization of the isometry group action on the Fomenko--Matveev--Weeks manifold (ResearchIndex) 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)

Math 423, Fall, 2002 If a manifold is not orientable, all is not lost (though the upcoming definition of integration over M will be). 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. 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. www.lehigh.edu /~dlj0/courses/423f02-lect20.html   (2384 words)

World War 1 and 2 - Spherical 3-manifold 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)

ki97.txt 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). 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. www.lehigh.edu /~dmd1/ki97.txt   (605 words)

Arneodo: animation The unstable manifold of the periodic orbit for www.maths.ex.ac.uk /~hinke/nonorientable/perwu.html   (17 words)

Comparing Heegaard And Jsj Structures Of Orientable 3-Manifolds - Scharlemann, Schultens (ResearchIndex) The Heegaard genus g of an irreducible closed orientable 3-manifold puts a limit on the number and complexity of the pieces that arise in the JacoShalen -Johannson decomposition of the manifold by its canonical tori. Scharlemann and J. Schultens, Comparing Heegaard and JSJ structures on orientable 3manifolds, MSRI preprint 1998-007. 3 Comparing Heegaard splittings of non-Haken 3- manifolds - th.. citeseer.ist.psu.edu /scharlemann98comparing.html   (429 words)

The Alexander polynomial of a 3-manifold and the Thurston norm on cohomology - McMullen (ResearchIndex) Abstract: Let M be a connected, compact, orientable 3-manifold with b1 (M) > 1, whose boundary (if any) is a union of tori. 3 The Alexander polynomial of a three-dimensional manifold (context) - Turaev - 1975 1 manifolds with inequivalent symplectic forms and 3-manifolds.. citeseer.ist.psu.edu /mcmullen01alexander.html   (717 words)

lopez.html 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). 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. 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)

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)

Talk:Orientable manifold - Wikipedia, the free encyclopedia An unorientable space-time would have an orientable double cover. Out of curiosity, can someone point out a reference for why the space-time manifold is believed to be orientable? On the small scale, one believes it for obvious reasons (handedness in biological molecules is another good reason). www.wikipedia.org /wiki/Talk:Orientable_manifold   (184 words)

Re: Spinors, 4-pi rotation & orientation in topology? What you demonstratete here is that to a non-orientable manifold, like the Moebius band, there exists a two-fold cover which is orientable. This has nothing directly to do with spin (except that the obstructions to orientability and spinnability are cohomology classes with values in Z_2). SPINnability is a HIGHER kind of orientability, and SPINORS are in fact modules ("representations") of CLIFFORD ALGEBRAS and do not really require knowledge about orientations or spins, only about metrics. www.lns.cornell.edu /spr/2004-07/msg0062162.html   (360 words)

AGT 2 (2002) Paper 21 (Abstract) This follows by showing that every orientable 3-manifold M admits a codimension one foliation F such that the holonomy cover of every leaf is contractible. We show that every orientable 3-manifold is a classifying space B\Gamma where \Gamma is a groupoid of germs of homeomorphisms of R. The F we construct can be taken to be C^1 but not C^2. www.maths.warwick.ac.uk /agt/AGTVol2/agt-2-21.abs.html   (113 words)

math lessons - Lickorish-Wallace theorem 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. 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. 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. www.mathdaily.com /lessons/Lickorish-Wallace_theorem   (230 words)

polyhed.min AB- Let F be a closed, orientable surface of genus g, and let e(g) be the minimal number of vertices of a triangulation of F by an embedding polyhedron. AB- A polyhedron on S, a compact 2-manifold, is called a triangulation if each face is a triangle with 3 distinct vertices and the intersection of any two distinct triangles is either empty, a single vertex, or a single edge. However, I take it this is not possible at least for the projective plane: #9 - minimal polyhedral immersion of RP^2 (using 9 vertices) Some authors extend the problem in different ways, for example: #1 - "minimal" torus, klein bottle, KB minus one face, but with slightly new definition of "minimal". www.math.niu.edu /~rusin/known-math/95/polyhed.min   (1539 words)

Orientable Manifold with Boundary - Page 2 - Physics Help and Math Help - Physics Forums 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. Orientability implies there exists an atlas with coordinate charts whose Jacobians have pos. www.physicsforums.com /showthread.php?t=81202&page=2   (1336 words)

Citations: Computational complexity of combinatorial surfaces - Vegter, Yap (ResearchIndex) A well known exotic example of a surface is the Klein bottle which is non orientable and cannot be physically realized in three dimensional space. A well known exotic surface is the Klein bottle which is non orientable and cannot be physically realized in three dimensional space. ....canonical, polygonal schema for any triangulated orientable manifold without boundary in O(n) time. citeseer.ist.psu.edu /context/16308/0   (1765 words)

All smooth orientable 4-dimensional manifolds embed in R^7 (ResearchIndex) 2 the imbedding of orientable manifolds in a euclidean space (context) - Wu - 1963 Abstract: The purpose of this note is to show how a recent result of Donaldson combined with Boechat and Haefliger's work and the classification of integral indefinite unimodular bilinear forms prove that all closed di#erentiable orientable 4-manifold embed in R 7 ; a result that has gone largely unnoticed. All smooth orientable 4-dimensional manifolds embed in R^7 citeseer.ist.psu.edu /269803.html   (364 words)

AI & Robotics Seminar Free-form handles method allows to interactively create multi-segment, curved handles between two star-shaped faces of an orientable 2-manifold mesh or to connect two 2-manifold meshes along such faces. Very high genus manifold surfaces are common in art and are also useful in industrial applications since many man-made objects can have high genus. High genus manifold surfaces are functional models and therefore can be used in physical simulations. parasol-www.cs.tamu.edu /seminar/airobotics/spring02/abstracts/akleman.shtml   (243 words)

Conference in Goemetric Topology By taking $\bf X$ with the canonical well-order, several types of closed orientable prime 3-manifolds are ordered and identified with the corresponding simple minimal link types. In this talk, we consider the set $\bf X$ of integral vectors of finite length as any well-ordered set, although a canonical well-order is in mind. Let $\bf M$ be the set of unoriented types of closed connected orientable 3-manifolds. www.math.uiowa.edu /~wu/gtc/abs/AkioKawa.htm   (245 words)

GI'98 Online Papers A novel algorithm for the encoding of orientable manifold triangle mesh geometry is presented. Mesh connectivity is encoded in a lossless manner. Use of our algorithm may lead to significant reduction of bandwidth required for the transmission of VRML files over the Internet. www.graphicsinterface.org /proceedings/1998/107   (78 words)

Víctor Núñez If M is a non-orientable Seifert manifold such that M cannot be expressed as an S But if M is a non-orientable Seifert manifold which does admit an S -bundle structure with fiber a closed orientable surface, then M has tri-genus of the form (0,g,g), with g a very big number. www.utm.edu /staff/jschomme/topology/c/a/a/k/48.htm   (218 words)

intro.html ShadowStrife is a stratgic online game where you try to expand the territory you control in a twodimensional orientable manifold (aka a torus). But trade also plays an important role in the game, so often it is a good idea to ally with your neighbours and fight an enemy further away. The easiest way to do this is off course to beat your neighbour in the head with a really big stick. www.srcf.ucam.org /~jn226/strife/intro.html   (72 words)

► » Cross product of two orientable smooth manifolds orientable, then the pullbacks of the tangent bundles of M, N are I have a question about the cross product of two orientable smooth manifolds. Proof that M and N are two orientable smooth manifolds if and only if MxN is an www.science-chat.org /detail-6053042.html   (642 words)

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