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Kirby–Siebenmann Class
In mathematics, more specifically in topology, geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a ''PL''-structure. The KS-class For a topological manifold ''M'', the Kirby–Siebenmann class \kappa(M) \in H^4(M;\mathbb/2) is an element of the fourth cohomology group of ''M'' that vanishes if ''M'' admits a piecewise linear structure. It is the only such obstruction, which can be phrased as the weak equivalence TOP/PL \sim K(\mathbb Z/2,3) of ''TOP/PL'' with an Eilenberg–MacLane space. The Kirby-Siebenmann class can be used to prove the existence of topological manifolds that do not admit a PL-structure. Concrete examples of such manifolds are E_8 \times T^n, n \geq 1, where E_8 stands for Michael Freedman, Freedman's E8 manifold. The class is named after Robion Kirby and Larry Siebenmann, who developed the theory of topological and ''PL''-manifolds. See also *Hauptvermutung References Homology theory Geometri ... [...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] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Manifold
In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real ''n''-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable manifolds are topological manifolds equipped with a differential structure). Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure. However, not every topological manifold can be endowed with a particular additional structure. For example, the E8 manifold is a topological manifold which cannot be endowed with a differentiable structure. Formal definition A topological space ''X'' is called locally Euclidean if there is a non-negative integer ''n'' such that every point in ''X'' has a neighborhood which is homeomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Group
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piecewise Linear Structure
In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation. An isomorphism of PL manifolds is called a PL homeomorphism. Relation to other categories of manifolds PL, or more precisely PDIFF, sits between DIFF (the category of smooth manifolds) and TOP (the category of topological manifolds): it is categorically "better behaved" than DIFF — for example, the Generalized Poincaré conjecture is true in PL (with the possible exception of dimension 4, where it is equivalent to DIFF), but is false generally in DIFF — but is "worse behaved" than TOP, as elaborated in surgery theory. Smooth manifolds Smooth manifolds have canonical PL structures — they are uniquely ''triangulizable,'' by Whitehead's theorem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. If n > 1 then ''G'' must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence, therefore any such space is often just called K(G,n). The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. As such, an Eilenberg–MacLane space is a special k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Freedman
Michael Hartley Freedman (born April 21, 1951) is an American mathematician, at Microsoft Station Q, a research group at the University of California, Santa Barbara. In 1986, he was awarded a Fields Medal for his work on the 4-dimensional generalized Poincaré conjecture. Freedman and Robion Kirby showed that an exotic ℝ4 manifold exists. Life and career Freedman was born in Los Angeles, California, in the United States. His father, Benedict Freedman, was an American Jewish aeronautical engineer, musician, writer, and mathematician. His mother, Nancy Mars Freedman, performed as an actress and also trained as an artist. His parents cowrote a series of novels together. . He entered the University of California, Berkeley, but dropped out after two semesters. In the same year he wrote a letter to Ralph Fox, a Princeton professor at the time, and was admitted to graduate school so in 1968 he continued his studies at Princeton University where he received Ph.D. degree in 1973 fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Robion Kirby
Robion Cromwell Kirby (born February 25, 1938) is a Professor of Mathematics at the University of California, Berkeley who specializes in low-dimensional topology. Together with Laurent C. Siebenmann he invented the Kirby–Siebenmann invariant for classifying the piecewise linear structures on a topological manifold. He also proved the fundamental result on the Kirby calculus, a method for describing 3-manifolds and smooth 4-manifolds by surgery on framed links. Along with his significant mathematical contributions, he has over 50 doctoral students and his problem list. He received his Ph.D. from the University of Chicago in 1965. He soon became an assistant professor at UCLA. While there he developed his "torus trick" which enabled him to solve, in dimensions greater than four (with additional joint work with Siebenmann), four of John Milnor's seven most important problems in geometric topology. In 1971, he was awarded the Oswald Veblen Prize in Geometry by the American Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Larry Siebenmann
Laurent Carl Siebenmann (the first name is sometimes spelled Laurence or Larry) (born 1939) is a Canadian mathematician based at the Université de Paris-Sud at Orsay, France. After working for several years as a Professor at Orsay he became a Directeur de Recherches at the Centre national de la recherche scientifique in 1976. He is a topologist who works on manifolds and who co-discovered the Kirby–Siebenmann class. Education Siebenmann's undergraduate studies were at the University of Toronto. He received a Ph.D. from Princeton University under the supervision of John Milnor in 1965 with the dissertation ''The obstruction to finding a boundary for an open manifold of dimension greater than five''. His doctoral students at Orsay included Francis Bonahon and Albert Fathi. Recognition In 1985 he was awarded the Jeffery–Williams Prize by the Canadian Mathematical Society. In 2012 he became a fellow of the American Mathematical Society The American Mathematical Society (AMS ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hauptvermutung
The ''Hauptvermutung'' of geometric topology is a now refuted conjecture asking whether any two Triangulation (topology), triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdivided triangulations are built up in the same combinatorial pattern. It was originally formulated as a conjecture in 1908 by Ernst Steinitz and Heinrich Franz Friedrich Tietze, but it is now known to be false. History The non-manifold version was disproved by John Milnor in 1961 using Analytic torsion, Reidemeister torsion. The manifold version is true in dimensions m\le 3. The cases m = 2 and 3 were mathematical proof, proved by Tibor Radó and Edwin E. Moise in the 1920s and 1950s, respectively. An obstruction to the manifold version was formulated by Andrew Casson and Dennis Sullivan in 1967–69 (originally in the simply-connected case), using the Rochlin invariant and the cohomology group H^3(M;\mathbb/2\mathbb). In dimension m \ge 5, a homeomorphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Theory
In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topology. Similar constructions are available in a wide variety of other contexts, such as abstract algebra, groups, Lie algebras, Galois theory, and algebraic geometry. The original motivation for defining homology groups was the observation that two shapes can be distinguished by examining their holes. For instance, a circle is not a disk because the circle has a hole through it while the disk is solid, and the ordinary sphere is not a circle because the sphere encloses a two-dimensional hole while the circle encloses a one-dimensional hole. However, because a hole is "not there", it is not immediately obvious how to define a hole or how to distinguish different kinds of holes. Homology was originally a rigorous mathematical method for defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
