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Rational Homology Sphere
In algebraic topology, a rational homology n-sphere is an n-dimensional manifold with the same rational homology groups as the n-sphere. These serve, among other things, to understand which information the rational homology groups of a space can or cannot measure and which attenuations result from neglecting torsion in comparison to the (integral) homology groups of the space. Definition A rational homology n-sphere is an n-dimensional manifold \Sigma with the same rational homology groups as the n-sphere S^n: : H_k(\Sigma,\mathbb) =H_k(S^n,\mathbb) \cong\begin \mathbb & ;k=0\textk=n \\ 1 & ;\text \end. Properties * Every (integral) homology sphere is a rational homology sphere. * Every simply connected rational homology n-sphere with n\leq 4 is homeomorphic to the n-sphere. Examples * The n-sphere S^n itself is obviously a rational homology n-sphere. * The pseudocircle (for which a weak homotopy equivalence from the circle exists) is a rational homotopy 1-sphere, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Bottle
In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside down. More formally, the Klein bottle is a two-dimensional manifold on which one cannot define a normal vector at each point that varies continuous function, continuously over the whole manifold. Other related non-orientable surfaces include the Möbius strip and the real projective plane. While a Möbius strip is a surface with a Boundary (topology), boundary, a Klein bottle has no boundary. For comparison, a sphere is an orientable surface with no boundary. The Klein bottle was first described in 1882 by the mathematician Felix Klein. Construction The following square is a fundamental polygon of the Klein bottle. The idea is to 'glue' together the corresponding red and blue edges with the arrows matching, as in the diagr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational+homology+sphere
In algebraic topology, a rational homology n-sphere is an n-dimensional manifold with the same rational homology groups as the n-sphere. These serve, among other things, to understand which information the rational homology groups of a space can or cannot measure and which attenuations result from neglecting torsion in comparison to the (integral) homology groups of the space. Definition A rational homology n-sphere is an n-dimensional manifold \Sigma with the same rational homology groups as the n-sphere S^n: : H_k(\Sigma,\mathbb) =H_k(S^n,\mathbb) \cong\begin \mathbb & ;k=0\textk=n \\ 1 & ;\text \end. Properties * Every (integral) homology sphere is a rational homology sphere. * Every simply connected rational homology n-sphere with n\leq 4 is homeomorphic to the n-sphere. Examples * The n-sphere S^n itself is obviously a rational homology n-sphere. * The pseudocircle (for which a weak homotopy equivalence from the circle exists) is a rational homotopy 1-sphere, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Homotopy Sphere
In algebraic topology, a rational homotopy n-sphere is an n-dimensional manifold with the same rational homotopy groups as the n-sphere. These serve, among other things, to understand which information the rational homotopy groups of a space can or cannot measure and which attenuations result from neglecting torsion in comparison to the (integral) homotopy groups of the space. Definition A rational homotopy n-sphere is an n-dimensional manifold \Sigma with the same rational homotopy groups as the n-sphere S^n: : \pi_k(\Sigma)\otimes\mathbb =\pi_k(S^n)\otimes\mathbb \cong\begin \mathbb & ;k=n\textn\text \\ \mathbb & ;k=n,2n-1\textn\text \\ 1 & ;\text \end. Properties * Every (integral) homotopy sphere is a rational homotopy sphere. Examples * The n-sphere S^n itself is obviously a rational homotopy n-sphere. * The Poincaré homology sphere is a rational homology 3-sphere in particular. * The real projective space \R P^n is a rational homotopy sphere for all n>0. The f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Sphere
In algebraic topology, a branch of mathematics, a homotopy sphere is an ''n''-manifold that is homotopy equivalent to the ''n''-sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three .... It thus has the same homotopy groups and the same homology groups as the ''n''-sphere, and so every homotopy sphere is necessarily a homology sphere. The topological generalized Poincaré conjecture is that any ''n''-dimensional homotopy sphere is homeomorphic to the ''n''-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005. The resolution of the smooth Poincaré conjecture in dimensions 5 and larger implies that homotopy spheres in those dimensions are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Groups
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian groups called ''homology groups.'' This operation, in turn, allows one to associate various named ''homologies'' or ''homology theories'' to various other types of mathematical objects. Lastly, since there are many homology theories for topological spaces that produce the same answer, one also often speaks of the ''homology of a topological space''. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.) There is also a related notion of the cohomology of a cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space. Homology of chain complexes To take the homology of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wu Manifold
In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is harder than solving the word problem for groups.. Simply connected compact 5-manifolds were first classified by Stephen Smale and then in full generality by Dennis Barden, while another proof was later given by Aleksey V. Zhubr. This turns out to be easier than the 3- or 4-dimensional case: the 3-dimensional case is the Thurston geometrisation conjecture, and the 4-dimensional case was solved by Michael Freedman (1982) in the topological case, but is a very hard unsolved problem in the smooth case. In dimension 5, the smooth classification of simply connected manifolds is governed by classical algebraic topology. Namely, two simply connected, smooth 5-manifolds are diffeomorphic if and only if there exists an isomorphism of their second homology groups with integer coefficients, prese ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Projective Space
In mathematics, real projective space, denoted or is the topological space of lines passing through the origin 0 in the real space It is a compact, smooth manifold of dimension , and is a special case of a Grassmannian space. Basic properties Construction As with all projective spaces, is formed by taking the quotient of \R^\setminus \ under the equivalence relation for all real numbers . For all in \R^\setminus \ one can always find a such that has norm 1. There are precisely two such differing by sign. Thus can also be formed by identifying antipodal points of the unit -sphere, , in \R^. One can further restrict to the upper hemisphere of and merely identify antipodal points on the bounding equator. This shows that is also equivalent to the closed -dimensional disk, , with antipodal points on the boundary, \partial D^n=S^, identified. Low-dimensional examples * is called the real projective line, which is topologically equivalent to a circle. Thinking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Homotopy Equivalence
In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated homotopy category of a model category has the same objects, but the morphisms are changed in order to make the weak equivalences into isomorphisms. It is a useful observation that the associated homotopy category depends only on the weak equivalences, not on the fibrations and cofibrations. Topological spaces Model categories were defined by Quillen as an axiomatization of homotopy theory that applies to topological spaces, but also to many other categories in algebra and geometry. The example that started the subject is the category of topological spaces with Serre fibrations as fibrations and weak homotopy equivalences as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a Neighbourhood (mathematics), neighborhood that is homeomorphic to an open (topology), open subset of n-dimensional Euclidean space. One-dimensional manifolds include Line (geometry), lines and circles, but not Lemniscate, self-crossing curves such as a figure 8. Two-dimensional manifolds are also called Surface (topology), surfaces. Examples include the Plane (geometry), plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudocircle
The pseudocircle is the finite topological space ''X'' consisting of four distinct points with the following non-Hausdorff topology: \. This topology corresponds to the partial order a [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or hypersphere is an - dimensional generalization of the -dimensional circle and -dimensional sphere to any non-negative integer . The circle is considered 1-dimensional and the sphere 2-dimensional because a point within them has one and two degrees of freedom respectively. However, the typical embedding of the 1-dimensional circle is in 2-dimensional space, the 2-dimensional sphere is usually depicted embedded in 3-dimensional space, and a general -sphere is embedded in an -dimensional space. The term ''hyper''sphere is commonly used to distinguish spheres of dimension which are thus embedded in a space of dimension , which means that they cannot be easily visualized. The -sphere is the setting for -dimensional spherical geometry. Considered extrinsically, as a hypersurface embedded in -dimensional Euclidean space, an -sphere is the locus of points at equal distance (the ''radius'') from a given '' center'' point. Its interior, consisting of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
