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P²-irreducible
In mathematics, a P2-irreducible manifold is a 3-manifold that is irreducible and contains no 2-sided \mathbb RP^2 (real projective plane). An orientable manifold is P2-irreducible if and only if it is irreducible.. Every non-orientable P2-irreducible manifold is a Haken manifold In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible surface. Sometimes one considers only orientable Haken manifolds, in w ..... References 3-manifolds {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haken Manifold
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided incompressible 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. A 3-manifold finitely covered by a Haken manifold is said to be virtually Haken. The Virtually Haken conjecture asserts that every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken. This conjecture was proven by Ian Agol. Haken manifolds were introduced by . proved that Haken manifolds have a hierarchy, where they can be split up into 3-balls along incompressible surfaces. Haken also showed that there was a finite procedure to find an incompressible surface if the 3-manifold had one. gave an algorithm to determine if a 3-manifold was Haken. Normal surfaces are ubiquitous in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Introduction Definition A topological space ''X'' is a 3-manifold if it is a second-countable Hausdorff space and if every point in ''X'' has a neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques that do not generalize to dimensions g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irreducible Manifold
In topology, a branch of mathematics, a prime manifold is an ''n''- manifold that cannot be expressed as a non-trivial connected sum of two ''n''-manifolds. Non-trivial means that neither of the two is an ''n''-sphere. A similar notion is that of an irreducible ''n''-manifold, which is one in which any embedded (''n'' − 1)-sphere bounds an embedded ''n''- ball. Implicit in this definition is the use of a suitable category, such as the category of differentiable manifolds or the category of piecewise-linear manifolds. The notions of irreducibility in algebra and manifold theory are related. An irreducible manifold is prime, although the converse does not hold. From an algebraist's perspective, prime manifolds should be called "irreducible"; however the topologist (in particular the 3-manifold topologist) finds the definition above more useful. The only compact, connected 3-manifolds that are prime but not irreducible are the trivial 2-sphere bundle over the circle S1 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2-sided
In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold F of a manifold M is said to be 2-sided in M when there is an embedding ::h\colon F\times 1,1to M with h(x,0)=x for each x\in F and ::h(F\times 1,1\cap \partial M=h(\partial F\times 1,1. In other words, if its normal bundle is trivial. This means, for example that a curve in a surface is 2-sided if it has a tubular neighborhood which is a cartesian product of the curve times an interval. A submanifold which is not 2-sided is called 1-sided. Examples Surfaces For curves on surfaces, a curve is 2-sided if and only if it preserves orientation, and 1-sided if and only if it reverses orientation: a tubular neighborhood is then a Möbius strip. This can be determined from the class of the curve in the fundamental group of the surface and the orientation character on the fundamental group, which identifies which curves reverse orientation. * An embedded circle in the plane is 2-side ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real projective plane is as the space of lines in passing through the origin. The plane is also often described topologically, in terms of a construction based on the Möbius strip: if one could glue the (single) edge of the Möbius strip to itself in the correct direction, one would obtain the projective plane. (This cannot be done in three-dimensional space without the surface intersecting itself.) Equivalently, gluing a disk along the boundary of the Möbius strip gives the projective plane. Topologically, it has Euler characteristic 1, hence a demigenus (non-orientable genus, Euler genus) of 1. Since the Möbius strip, in turn, can be constructed from a square by glui ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientable Manifold
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 orientable if such a consistent definition exists. In this case, there are two possible definitions, and a choice between them is an orientation of the space. Real vector spaces, Euclidean spaces, and spheres are orientable. A space is non-orientable if "clockwise" is changed into "counterclockwise" after running through some loops in it, and coming back to the starting point. This means that a geometric shape, such as , that moves continuously along such a loop is changed into its own mirror image . A Möbius strip is an example of a non-orientable space. Various equivalent formulations of orientability can be given, depending on the desired application and level of generality. Formulations applicable to general topological manifolds ofte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
