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. Usually these manifolds are further required to have a handle decomposition with a single 1-handle, and a single 2-handle; otherwise, they would simply be called contractible manifolds. The boundary of a Mazur manifold is necessarily a homology 3-sphere. History Barry Mazur and Valentin Poénaru discovered these manifolds simultaneously. Selman Akbulut and Robion 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, effectively coining the term `Mazur Manifold.' These results were later generalized to other contractible manifolds by Andrew Casson, John Harer, and Ronald Stern. One of the Mazur manifolds is also an example of an Akbulut cork which can be used to construct exotic 4-manifolds. Mazur manifold ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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: * In ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Notices Of The American Mathematical Society
''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume was published in 1953. Each issue of the magazine since January 1995 is available in its entirety on the journal web site. Articles are peer-reviewed by an editorial board of mathematical experts. Beginning with the January 2025 issue, the editor-in-chief is Mark C. Wilson, succeeding past editor Erica Flapan. The cover regularly features mathematical visualizations. The ''Notices'' is self-described to be the world's most widely read mathematical journal. As the membership journal of the American Mathematical Society, the ''Notices'' is sent to the approximately 30,000 AMS members worldwide, one-third of whom reside outside the United States. By publishing high-level exposition, the ''Notices'' provides opportunities for mathematicians to find out what is going on in the field. Each is ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Double (manifold)
In the subject of manifold theory in mathematics, if M is a topological manifold with boundary, its double is obtained by gluing two copies of M together along their common boundary. Precisely, the double is M \times \ / \sim where (x,0) \sim (x,1) for all x \in \partial M. If M has a smooth structure, then its double can be endowed with a smooth structure thanks to a collar neighbourdhood. Although the concept makes sense for any manifold, and even for some non-manifold sets such as the Alexander horned sphere, the notion of double tends to be used primarily in the context that \partial M is non-empty and M is compact. Doubles bound Given a manifold M, the double of M is the boundary of M \times ,1/math>. This gives doubles a special role in cobordism. Examples The ''n''-sphere is the double of the ''n''-ball. In this context, the two balls would be the upper and lower hemi-sphere respectively. More generally, if M is closed, the double of M \times D^k is M \times ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Schoenflies Problem
In mathematics, the Schoenflies problem or Schoenflies theorem, of geometric topology is a sharpening of the Jordan curve theorem by Arthur Moritz Schoenflies, Arthur Schoenflies. For Camille Jordan, Jordan curves in the Plane (geometry), 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 (mathematics), plane separate the plane into two regions, one (the "inside") bounded set, 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 homeomorphi ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Generalized Poincaré Conjecture
In the mathematical area of topology, the generalized Poincaré conjecture is a statement that a manifold that 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 Smale, Mic ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Rokhlin's Theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, 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 ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Homology Sphere
In algebraic topology, a homology sphere is an ''n''-manifold ''X'' having the homology groups of an ''n''-sphere, for some integer n\ge 1. That is, :H_0(X,\Z) = H_n(X,\Z) = \Z and :H_i(X,\Z) = \ for all other ''i''. Therefore ''X'' is a connected space, with one non-zero higher Betti number, namely, b_n=1. It does not follow that ''X'' is simply connected, only that its fundamental group is perfect (see Hurewicz theorem). A rational homology sphere is defined similarly but using homology with rational coefficients. Poincaré homology sphere The Poincaré homology sphere (also known as Poincaré dodecahedral space) is a particular example of a homology sphere, first constructed by Henri Poincaré. Being a spherical 3-manifold, it is the only homology 3-sphere (besides the 3-sphere itself) with a finite fundamental group. Its fundamental group is known as the binary icosahedral group and has order 120. Since the fundamental group of the 3-sphere is trivial, this sh ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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 cas ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of pure and applied mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. Its ISSN number is 0002-9947. See also * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * '' Memoirs of the American Mathematical Society'' * '' Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' References External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ... American Mathematical Society academic journals Mathematics jo ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Michel Kervaire
Michel André Kervaire (26 April 1927 – 19 November 2007) was a French mathematician who made significant contributions to topology and algebra. He introduced the Kervaire semi-characteristic. He was the first to show the existence of topological ''n''-manifolds with no differentiable structure (using the Kervaire invariant), and (with John Milnor) computed the number of exotic spheres in dimensions greater than four, known as Kervaire–Milnor groups. He is also well known for fundamental contributions to high-dimensional knot theory. The solution of the Kervaire invariant problem was announced by Michael Hopkins in Edinburgh on 21 April 2009. Education He was the son of André Kervaire (a French industrialist) and Nelly Derancourt. After completing high school in France, Kervaire pursued his studies at ETH Zurich (1947–1952), receiving a Ph.D. in 1955. His thesis, entitled ''Courbure intégrale généralisée et homotopie'', was written under the direction of Heinz ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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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]   |
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Ronald Fintushel
Ronald Alan Fintushel (born 1945) is an American mathematician, specializing in low-dimensional geometric topology (specifically of 4-manifolds) and the mathematics of gauge theory. Education and career Fintushel studied mathematics at Columbia University with a bachelor's degree in 1967 and at the University of Illinois at Urbana–Champaign with a master's degree in 1969. In 1975 he received his Ph.D. from the State University of New York at Binghamton with thesis ''Orbit maps of local S^1-actions on manifolds of dimension less than five '' under the supervision of Louis McAuley. Fintushel was a professor at Tulane University and is a professor at Michigan State University. His research deals with geometric topology, in particular of 4-manifolds (including the computation of Donaldson and Seiberg-Witten invariants) with links to gauge theory, knot theory, and symplectic geometry. He works closely with Ronald J. Stern. In 1998 he was an Invited Speaker, with Ronald J. Stern, wit ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |