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Borel Conjecture
In geometric topology, the Borel conjecture (named for Armand Borel) asserts that an aspherical closed manifold is determined by its fundamental group, up to homeomorphism. It is a rigidity conjecture, asserting that a weak, algebraic notion of equivalence (namely, homotopy equivalence) should imply a stronger, topological notion (namely, homeomorphism). Precise formulation of the conjecture Let M and N be closed and aspherical topological manifolds, and let :f \colon M \to N be a homotopy equivalence. The Borel conjecture states that the map f is homotopic to a homeomorphism. Since aspherical manifolds with isomorphic fundamental groups are homotopy equivalent, the Borel conjecture implies that aspherical closed manifolds are determined, up to homeomorphism, by their fundamental groups. This conjecture is false if topological manifolds and homeomorphisms are replaced by smooth manifolds and diffeomorphisms; counterexamples can be constructed by taking a connected sum wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy, ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lens Space
A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-manifold case, a lens space can be visualized as the result of gluing two solid tori together by a homeomorphism of their boundaries. Often the 3-sphere and S^2 \times S^1, both of which can be obtained as above, are not counted as they are considered trivial special cases. The three-dimensional lens spaces L(p;q) were introduced by Heinrich Tietze in 1908. They were the first known examples of 3-manifolds which were not determined by their homology and fundamental group alone, and the simplest examples of closed manifolds whose homeomorphism type is not determined by their homotopy type. J. W. Alexander in 1919 showed that the lens spaces L(5;1) and L(5;2) were not homeomorphic even though they have isomorphic fundamental groups and the same homology, though they do not have ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjectures
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 1012 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphisms
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation of a line ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Topology
In mathematics, geometric topology is the study of manifolds and Map (mathematics)#Maps as functions, maps between them, particularly embeddings of one manifold into another. History Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy, ''simple'' homotopy theory. The use of the term geometric topology to describe these seems to have originated rather recently. Differences between low-dimensional and high-dimensional topology Manifolds differ radically in behavior in high and low dimension. High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2. Dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfgang Lück
Wolfgang Lück (born 19 February 1957 in Herford) is a German mathematician who is an internationally recognized expert in algebraic topology. Life and work After receiving his '' Abitur'' from the Ravensberger Gymnasium in Herford in 1975, he studied at the University of Göttingen where he obtained his Diplom in 1981 and his doctoral degree under Tammo tom Dieck in 1984. His thesis was entitled ''Eine allgemeine Beschreibung für Faserungen auf projektiven Klassengruppen und Whiteheadgruppen''. From 1982 on he was research assistant and from 1985 on he was assistant in Göttingen. In 1989 Lück received his Habilitation. From 1990 to 1991, he was associate professor at the University of Kentucky in Lexington. From 1991 until 1996, he was professor at the University of Mainz, and from 1996 until 2010 he taught at the University of Münster. Since 2010 he has been a professor at the University of Bonn. In 2003, he was awarded the Max Planck Research Award, in 2008 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matthias Kreck
Matthias Kreck (born 22 July 1947, in Dillenburg) is a German mathematician who works in the areas of Algebraic Topology and Differential topology. From 1994 to 2002 he was director of the Oberwolfach Research Institute for Mathematics and from October 2006 to September 2011 he was the director of the Hausdorff Center for Mathematics at the University of Bonn, where he is currently a professor. Life and work Kreck grew up as the son of the theologian in Herborn and studied mathematics and physics from 1966 to 1970, and business administration at the Universities of Bonn, Berlin and Regensburg. In 1970 he submitted his diploma in Mathematics in Bonn and in 1972 he received his doctorate there under the supervision of Friedrich Hirzebruch, with a thesis titled ''An invariant for stably parallelized manifolds''. From 1972 to 1976 he studied Protestant theology in Bonn: in a similar period he was also assistant from 1970 to 1976 to professor Hirzebruch. In 1977 he completed his hab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
International Centre For Theoretical Physics
The Abdus Salam International Centre for Theoretical Physics (ICTP) is a research center for physical and mathematical sciences, located in Trieste, Friuli-Venezia Giulia, Italy. The center operates under a tripartite agreement between the Government of Italy, Italian Government, UNESCO, and the International Atomic Energy Agency. It is located near the Miramare, Miramare Park, about 10 kilometres from the downtown of Trieste city, Italy. The centre was founded in 1964 by Pakistani Nobel Laureate Abdus Salam. ICTP is part of the Trieste System, a network of national and international scientific institutes in Trieste, promoted by the Italian physicist Paolo Budinich. Mission * Foster the growth of advanced studies and research in physical and mathematical sciences, especially in support of excellence in developing countries; * Develop high-level scientific programmes keeping in mind the needs of developing countries, and provide an international forum of scientific contact for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a Lemon (geometry), spindle torus (or ''self-crossing torus'' or ''self-intersecting torus''). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-sphere
In mathematics, a hypersphere or 3-sphere is a 4-dimensional analogue of a sphere, and is the 3-dimensional n-sphere, ''n''-sphere. In 4-dimensional Euclidean space, it is the set of points equidistant from a fixed central point. The interior of a 3-sphere is a 4-ball. It is called a 3-sphere because topologically, the surface itself is 3-dimensional, even though it is curved into the 4th dimension. For example, when traveling on a 3-sphere, you can go north and south, east and west, or along a 3rd set of cardinal directions. This means that a 3-sphere is an example of a 3-manifold. Definition In coordinates, a 3-sphere with center and radius is the set of all points in real, Four-dimensional space, 4-dimensional space () such that :\sum_^3(x_i - C_i)^2 = ( x_0 - C_0 )^2 + ( x_1 - C_1 )^2 + ( x_2 - C_2 )^2+ ( x_3 - C_3 )^2 = r^2. The 3-sphere centered at the origin with radius 1 is called the unit 3-sphere and is usually denoted : :S^3 = \left\. It is often convenient to r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Poincaré Conjecture
In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured by Henri Poincaré in 1904, the theorem concerns spaces that locally look like ordinary three-dimensional space but which are finite in extent. Poincaré hypothesized that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. Attempts to resolve the conjecture drove much progress in the field of geometric topology during the 20th century. The eventual proof built upon Richard S. Hamilton's program of using the Ricci flow to solve the problem. By developing a number of new techniques and results in the theory of Ricci flow, Grigori Perelman was able to modify and complete Hamilton's program. In papers posted to the arXiv reposi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Novikov Conjecture
The Novikov conjecture is one of the most important unsolved problems in topology. It is named for Sergei Novikov who originally posed the conjecture in 1965. The Novikov conjecture concerns the homotopy invariance of certain polynomials in the Pontryagin classes of a manifold, arising from the fundamental group. According to the Novikov conjecture, the ''higher signatures'', which are certain numerical invariants of smooth manifolds, are homotopy invariants. The conjecture has been proved for finitely generated abelian groups. It is not yet known whether the Novikov conjecture holds true for all groups. There are no known counterexamples to the conjecture. Precise formulation of the conjecture Let G be a discrete group and BG its classifying space, which is an Eilenberg–MacLane space of type K(G,1), and therefore unique up to homotopy equivalence as a CW complex. Let :f\colon M\rightarrow BG be a continuous map from a closed oriented n-dimensional manifold M to BG, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
