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Rokhlin's Theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold ''M'' has a spin structure (or, equivalently, the second Stiefel–Whitney class w_2(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H^2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952. Examples *The intersection form on ''M'' ::Q_M\colon H^2(M,\Z)\times H^2(M,\Z)\rightarrow \mathbb :is unimodular on \Z by Poincaré duality, and the vanishing of w_2(M) implies that the intersection form is even. By a theorem of Cahit Arf, any even unimodular lattice has signature divisible by 8, so Rokhlin's theorem forces one extra factor of 2 to divide the signature. *A K3 surface is compact, 4 dimensional, and w_2(M) vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem. *A complex surface in \mathbb^3 of degree d is spin if and only if d is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simply Connected Space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Manifold
In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical physics, in particular to quantum field theory where they are an essential ingredient in the definition of any theory with uncharged fermions. They are also of purely mathematical interest in differential geometry, algebraic topology, and K theory. They form the foundation for spin geometry. Overview In geometry and in field theory, mathematicians ask whether or not a given oriented Riemannian manifold (''M'',''g'') admits spinors. One method for dealing with this problem is to require that ''M'' has a spin structure. This is not always possible since there is potentially a topological obstruction to the existence of spin structures. Spin structures will exist if and only if the second Stiefel–Whitney class ''w''2(''M'') ∈ H2(''M'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Armand Borel
Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in algebraic topology, in the theory of Lie groups, and was one of the creators of the contemporary theory of linear algebraic groups. Biography He studied at the ETH Zürich, where he came under the influence of the topologist Heinz Hopf and Lie-group theorist Eduard Stiefel. He was in Paris from 1949: he applied the Leray spectral sequence to the topology of Lie groups and their classifying spaces, under the influence of Jean Leray and Henri Cartan. With Hirzebruch, he significantly developed the theory of characteristic classes in the early 1950s. He collaborated with Jacques Tits in fundamental work on algebraic groups, and with Harish-Chandra on their arithmetic subgroups. In an algebraic group ''G'' a ''Borel subgroup'' ''H'' is one mini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arf Invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf invariant is the substitute, in characteristic 2, for the discriminant for quadratic forms in characteristic not 2. Arf used his invariant, among others, in his endeavor to classify quadratic forms in characteristic 2. In the special case of the 2-element field F2 the Arf invariant can be described as the element of F2 that occurs most often among the values of the form. Two nonsingular quadratic forms over F2 are isomorphic if and only if they have the same dimension and the same Arf invariant. This fact was essentially known to , even for any finite field of characteristic 2, and Arf proved it for an arbitrary perfect field. The Arf invariant is particularly applied in geometric topology, where it is primarily used to define an invari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Casson Invariant
In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson. Kevin Walker (1992) found an extension to rational homology 3-spheres, called the Casson–Walker invariant, and Christine Lescop (1995) extended the invariant to all closed oriented 3-manifolds. Definition A Casson invariant is a surjective map λ from oriented integral homology 3-spheres to Z satisfying the following properties: *λ(S3) = 0. *Let Σ be an integral homology 3-sphere. Then for any knot ''K'' and for any integer ''n'', the difference ::\lambda\left(\Sigma+\frac\cdot K\right)-\lambda\left(\Sigma+\frac\cdot K\right) :is independent of ''n''. Here \Sigma+\frac\cdot K denotes \frac Dehn surgery on Σ by ''K''. *For any boundary link ''K'' ∪ ''L'' in Σ the following expression is zero: ::\lambda\left(\Sigma+\frac\cdot K+\frac\cdot L\right) -\lambda\left(\Sigma+\fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mazur Manifold
In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold (with boundary) which is not diffeomorphic to the standard 4-ball. The boundary of a Mazur manifold is necessarily a homology 3-sphere. Frequently the term ''Mazur manifold'' is restricted to a special class of the above definition: 4-manifolds that have a handle decomposition containing exactly three handles: a single 0-handle, a single 1-handle and single 2-handle. This is equivalent to saying the manifold must be of the form S^1 \times D^3 union a 2-handle. An observation of Mazur's shows that the double of such manifolds is diffeomorphic to S^4 with the standard smooth structure. History Barry Mazur and Valentin Poenaru discovered these manifolds simultaneously. Akbulut and Kirby showed that the Brieskorn homology spheres \Sigma(2,5,7) , \Sigma(3,4,5) and \Sigma(2,3,13) are boundaries of Mazur manifolds. These results were later generalized ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Homology Sphere
Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Lucien Poincaré (1862–1920), physicist, brother of Raymond and cousin of Henri * Raymond Poincaré (1860–1934), French Prime Minister or President ''inter alia'' from 1913 to 1920, cousin of Henri See also *List of things named after Henri Poincaré In physics and mathematics, a number of ideas are named after Henri Poincaré: * Euler–Poincaré characteristic * Hilbert–Poincaré series * Poincaré–Bendixson theorem * Poincaré–Birkhoff theorem * Poincaré–Birkhoff–Witt theorem, usu .... * * {{DEFAULTSORT:Poincare French-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah–Singer Index Theorem
In differential geometry, the Atiyah–Singer index theorem, proved by Michael Atiyah and Isadore Singer (1963), states that for an elliptic differential operator on a compact manifold, the analytical index (related to the dimension of the space of solutions) is equal to the topological index (defined in terms of some topological data). It includes many other theorems, such as the Chern–Gauss–Bonnet theorem and Riemann–Roch theorem, as special cases, and has applications to theoretical physics. History The index problem for elliptic differential operators was posed by Israel Gel'fand. He noticed the homotopy invariance of the index, and asked for a formula for it by means of topological invariants. Some of the motivating examples included the Riemann–Roch theorem and its generalization the Hirzebruch–Riemann–Roch theorem, and the Hirzebruch signature theorem. Friedrich Hirzebruch and Armand Borel had proved the integrality of the  genus of a spin manifold, and At ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Group Of Spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle () and the ordinary sphere (). The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th ''homotopy group'' summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere . This summary does not distinguish between two mappings if one can be continuously deformed to the oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsion (algebra)
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is torsion-free if its torsion submodule comprises only the zero element. This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements. This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is allowed by the fact that the abelian groups are the modules over the ring of integers (in fact, this is the origin of the terminology, that has been introduced for abelian groups before being generalized to modules). In the case of groups that are noncommutative, a ''torsion element'' is an element of finite order. Contrary to the commuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
