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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 generali ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Topology
In mathematics, differential topology is the field dealing with the topological properties and smooth properties of smooth manifolds. In this sense differential topology is distinct from the closely related field of differential geometry, which concerns the ''geometric'' properties of smooth manifolds, including notions of size, distance, and rigid shape. By comparison differential topology is concerned with coarser properties, such as the number of holes in a manifold, its homotopy type, or the structure of its diffeomorphism group. Because many of these coarser properties may be captured algebraically, differential topology has strong links to algebraic topology. The central goal of the field of differential topology is the classification of all smooth manifolds up to diffeomorphism. Since dimension is an invariant of smooth manifolds up to diffeomorphism type, this classification is often studied by classifying the ( connected) manifolds in each dimension separately: * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Akbulut Cork
In topology, an Akbulut cork is a structure that is frequently used to show that in 4-dimensions, the smooth h-cobordism theorem fails. It was named after Turkish mathematician Selman Akbulut. A compact contractible Stein 4-manifold C with involution F on its boundary is called an Akbulut cork, if F extends to a self-homeomorphism but cannot extend to a self-diffeomorphism inside (hence a cork is an exotic copy of itself relative to its boundary). A cork (C,F) is called a cork of a smooth 4-manifold X, if removing C from X and re-gluing it via F changes the smooth structure of X (this operation is called "cork twisting"). Any exotic copy X' of a closed simply connected 4-manifold X differs from X by a single cork twist. The basic idea of the Akbulut cork is that when attempting to use the h-corbodism theorem in four dimensions, the cork is the sub-cobordism that contains all the exotic properties of the spaces connected with the cobordism, and when removed the two spaces become t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schoenflies Problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Schoenflies. For Jordan curves in the plane it is often referred to as the Jordan–Schoenflies theorem. Original formulation The original formulation of the Schoenflies problem states that not only does every simple closed curve in the plane separate the plane into two regions, one (the "inside") bounded and the other (the "outside") unbounded; but also that these two regions are homeomorphic to the inside and outside of a standard circle in the plane. An alternative statement is that if C \subset \mathbb R^2 is a simple closed curve, then there is a homeomorphism f : \mathbb R^2 \to \mathbb R^2 such that f(C) is the unit circle in the plane. Elementary proofs can be found in , , and . The result can first be proved for polygons when the homeomorphism can be taken to be piecewise linear and the identity map off some compact set; the case ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Poincaré Conjecture
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold which is a homotopy sphere a sphere. More precisely, one fixes a category of manifolds: topological (Top), piecewise linear (PL), or differentiable (Diff). Then the statement is :Every homotopy sphere (a closed ''n''-manifold which is homotopy equivalent to the ''n''-sphere) in the chosen category (i.e. topological manifolds, PL manifolds, or smooth manifolds) is isomorphic in the chosen category (i.e. homeomorphic, PL-isomorphic, or diffeomorphic) to the standard ''n''-sphere. The name derives from the Poincaré conjecture, which was made for (topological or PL) manifolds of dimension 3, where being a homotopy sphere is equivalent to being simply connected and closed. The generalized Poincaré conjecture is known to be true or false in a number of instances, due to the work of many distinguished topologists, including the Fields medal awardees John Milnor, Steve Sma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rokhlin Invariant
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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Sphere
Homology may refer to: Sciences Biology *Homology (biology), any characteristic of biological organisms that is derived from a common ancestor *Sequence homology, biological homology between DNA, RNA, or protein sequences * Homologous chromosomes, chromosomes in a biological cell that pair up (synapse) during meiosis *Homologous recombination, genetic recombination in which nucleotide sequences are exchanged between molecules of DNA *Homologous desensitization, a receptor decreases its response to a signalling molecule when that agonist is in high concentration * Homology modeling, a method of protein structure prediction Chemistry * Homology (chemistry), the relationship between compounds in a homologous series *Homologous series, a series of organic compounds having different quantities of a repeated unit * Homologous temperature, the temperature of a material as a fraction of its absolute melting point * Homologation reaction, a chemical reaction which produces the ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-cobordism
In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M \hookrightarrow W \quad\mbox\quad N \hookrightarrow W are homotopy equivalences. The ''h''-cobordism theorem gives sufficient conditions for an ''h''-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder ''M'' × , 1 Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds. The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture. Background Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trans
Trans- is a Latin prefix meaning "across", "beyond", or "on the other side of". Used alone, trans may refer to: Arts, entertainment, and media * Trans (festival), a former festival in Belfast, Northern Ireland, United Kingdom * ''Trans'' (film), a 1998 American film * Trans Corp, an Indonesian business unit of CT Corp in the fields of media, lifestyle, and entertainment ** Trans Media, a media subsidiary of Trans Corp *** Trans TV, an Indonesian television network *** Trans7, an Indonesian television network Literature * '' Trans: Gender and Race in an Age of Unsettled Identities'', a 2016 book by Rogers Brubaker * '' Trans: When Ideology Meets Reality'', a 2021 book by Helen Joyce Music * ''Trans'' (album), by Neil Young * ''Trans'' (Stockhausen), a 1971 orchestral composition Places * Trans, Mayenne, France, a commune * Trans, Switzerland, a village Science and technology * Trans effect in inorganic chemistry, the increased lability of ligands that are trans to c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pacific J
The Pacific Ocean is the largest and deepest of Earth's five oceanic divisions. It extends from the Arctic Ocean in the north to the Southern Ocean (or, depending on definition, to Antarctica) in the south, and is bounded by the continents of Asia and Oceania in the west and the Americas in the east. At in area (as defined with a southern Antarctic border), this largest division of the World Ocean—and, in turn, the hydrosphere—covers about 46% of Earth's water surface and about 32% of its total surface area, larger than Earth's entire land area combined .Pacific Ocean . '' Britannica Concise.'' 2008: Encyclopædia Britannica, Inc. The centers of both the [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i.e. that the space not exclude any ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barry Mazur
Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology. Life Born in New York City, Mazur attended the Bronx High School of Science and MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. He was nonetheless accepted for graduate studies at Princeton University, from where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titled "On embeddings of spheres." He then became a Junior Fellow at Harvard University from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard. He is the brother of Joseph Mazur and the father o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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