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E8 Manifold
In mathematics, the ''E''8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the ''E''8 lattice. History The E_8 manifold was discovered by Michael Freedman in 1982. Rokhlin's theorem shows that it has no smooth structure (as does Donaldson's theorem), and in fact, combined with the work of Andrew Casson on the Casson invariant, this shows that the E_8 manifold is not even triangulable as a simplicial complex. Construction The manifold can be constructed by first plumbing together disc bundles of Euler number 2 over the sphere, according to the Dynkin diagram for E_8. This results in P_, a 4-manifold with boundary equal to the Poincaré homology sphere. Freedman's theorem on fake 4-balls then says we can cap off this homology sphere with a fake 4-ball to obtain the E_8 manifold. See also * * * References * * {{DEFAULTSORT:E8 Manifold 4-manifolds Geometric topology Manifold In mathematics, a manifold is a ... [...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] |
Triangulation (topology)
In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic to a simplicial complex are called triangulable. Triangulation has various uses in different branches of mathematics, for instance in algebraic topology, in complex analysis or in modeling. Motivation On the one hand, it is sometimes useful to forget about superfluous information of topological spaces: The replacement of the original spaces with simplicial complexes may help to recognize crucial properties and to gain a better understanding of the considered object. On the other hand, simplicial complexes are objects of combinatorial character and therefore one can assign them quantities rising from their combinatorial pattern, for instance, the Euler characteristic. Triangulation allows now to assign such quantities to topological spaces. Investigations concerning the exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-manifolds
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are important in physics because in General Relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. Topological 4-manifolds The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of implies that the homeomorphism type of the manifold only depends on this intersection form, and on a \Z/2\Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and Kirb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential in in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called ''Surveys in Differential Geometry''. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University. History The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009. In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Harv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fake 4-ball
In mathematics, a fake 4-ball is a compact contractible topological 4-manifold. Michael Freedman proved that every three-dimensional homology sphere bounds a fake 4-ball. His construction involves the use of Casson handle In 4-dimensional topology, a branch of mathematics, a Casson handle is a 4-dimensional topological 2-handle constructed by an infinite procedure. They are named for Andrew Casson, who introduced them in about 1973. They were originally called "fle ...s and so does not work in the smooth category. References * Alexandru Scorpan, ''The Wild World of 4-Manifolds'', American Mathematical Society, 4-manifolds Geometric topology {{topology-stub ... [...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] |
Dynkin Diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the classification of semisimple Lie algebras over algebraically closed fields, in the classification of Weyl groups and other finite reflection groups, and in other contexts. Various properties of the Dynkin diagram (such as whether it contains multiple edges, or its symmetries) correspond to important features of the associated Lie algebra. The term "Dynkin diagram" can be ambiguous. In some cases, Dynkin diagrams are assumed to be directed, in which case they correspond to root systems and semi-simple Lie algebras, while in other cases they are assumed to be undirected, in which case they correspond to Weyl groups. In this article, "Dynkin diagram" means ''directed'' Dynkin diagram, and ''undirected'' Dynkin diagrams will be explicitly so named. Classification of semisimple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre (geometry), centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. spherical Earth, The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Number (topology)
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent. It is commonly denoted by \chi ( Greek lower-case letter chi). The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids. It was stated for Platonic solids in 1537 in an unpublished manuscript by Francesco Maurolico. Leonhard Euler, for whom the concept is named, introduced it for convex polyhedra more generally but failed to rigorously prove that it is an invariant. In modern mathematics, the Euler characteristic arises from homology and, more abstractly, homological algebra. Polyhedra The Euler characteristic \chi was classically defined for the surfaces of polyhedra, acc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial from an abstract simplicial complex, the former is often called a geometric simplicial complex.'', Section 4.3'' Definitions A simplicial complex \mathcal is a set of simplices that satisfies the following conditions: :1. Every face of a simplex from \mathcal is also in \mathcal. :2. The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry. A simplicial ''k''-complex \ ... [...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] |
