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Graph Manifold
In topology, a graph manifold (in German: Graphenmannigfaltigkeit) is a 3-manifold which is obtained by gluing some circle bundles. They were discovered and classified by the German topologist Friedhelm Waldhausen Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province, died 2024) was a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory. ... in 1967. This definition allows a very convenient combinatorial description as a graph whose vertices are the fundamental parts and (decorated) edges stand for the description of the gluing, hence the name. Two very important classes of examples are given by the Seifert bundles and the Solv manifolds. This leads to a more modern definition: a graph manifold is either a Solv manifold, a manifold having only Seifert pieces in its JSJ decomposition, or connect sums of the previous two categories. From this perspective, Wald ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
Topology (from the Greek language, Greek words , and ) is the branch of mathematics 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, Torsion (mechanics), 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 List of continuity-related mathematical topics, continuity. Euclidean spaces, and, more generally, metric spaces are examples of topological spaces, 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 to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifold
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (geometry), plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Principles Definition A topological space M is a 3-manifold if it is a second-countable Hausdorff space and if every point in M has a neighbourhood (mathematics), neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, Piecewise linear manifold, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle Bundle
In mathematics, a circle bundle is a fiber bundle where the fiber is the circle S^1. Oriented circle bundles are also known as principal ''U''(1)-bundles, or equivalently, as principal ''SO''(2)-bundles. In physics, circle bundles are the natural geometric setting for electromagnetism. A circle bundle is a special case of a sphere bundle. As 3-manifolds Circle bundles over surfaces are an important example of 3-manifolds. A more general class of 3-manifolds is Seifert fiber spaces, which may be viewed as a kind of "singular" circle bundle, or as a circle bundle over a two-dimensional orbifold. Relationship to electrodynamics The Maxwell equations correspond to an electromagnetic field represented by a 2-form ''F'', with \pi^F being cohomologous to zero, i.e. exact. In particular, there always exists a 1-form ''A'', the electromagnetic four-potential, (equivalently, the affine connection) such that : \pi^F = dA. Given a circle bundle ''P'' over ''M'' and its projection : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedhelm Waldhausen
Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province, died 2024) was a German mathematician known for his work in algebraic topology. He made fundamental contributions in the fields of 3-manifolds and (algebraic) K-theory. Career Waldhausen studied mathematics at the universities of Göttingen, Munich and Bonn. He obtained his Ph.D. in 1966 from the University of Bonn; his advisor was Friedrich Hirzebruch and his thesis was entitled "Eine Klasse von 3-dimensionalen Mannigfaltigkeiten" (A class of 3-dimensional manifolds). After visits to Princeton University, the University of Illinois and the University of Michigan he moved in 1968 to the University of Kiel, where he completed his habilitation (qualified to assume a professorship). In 1969, he was appointed professor at the Ruhr University Bochum before in 1971 becoming a professor at Bielefeld University, an appointment he held until his retirement in 2004. Academic work His early work was mainly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seifert Fiber Spaces
Seifert is a German surname. Notable people with the surname include: * Alfred Seifert (1850–1901), Czech German painter * Alfred Seifert (flax miller) (1877–1945), New Zealand flax-miller * Alwin Seifert (1890–1972), German architect * Benjamin Seifert (born 1982), German cross country skier * Bernhard Seifert (born 1993), German javelin thrower * (born 1955), German entomologist * Bill Seifert (born 1939), American racecar driver * Christian Seifert (born 1969), German entrepreneur * Christopher Seifert (1975–2003), American soldier * Dario Seifert (born 1995), German politician * Else Seifert (1879–1968), German photographer * Emil Seifert (1900–1973), Czech football manager *Ernst Seifert (1855–1928), German organ builder * Frank Seifert (born 1972), German footballer * Friedrich Seifert (born 1941), German mineralogist * George Seifert (born 1940), American football coach * Harald Seifert (born 1953), East German bobsledder * Hartmut Seifert (born 1944), German lab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrization Conjecture
In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries ( Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by as part of his 24 questions, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
JSJ Decomposition
In mathematics, the JSJ decomposition, also known as the toral decomposition, is a topological construct given by the following theorem: : Irreducible orientable closed (i.e., compact and without boundary) 3-manifolds have a unique (up to isotopy) minimal collection of disjointly embedded incompressible tori such that each component of the 3-manifold obtained by cutting along the tori is either atoroidal or Seifert-fibered. The acronym JSJ is for William Jaco, Peter Shalen, and Klaus Johannson. The first two worked together, and the third worked independently. The characteristic submanifold An alternative version of the JSJ decomposition states: :A closed irreducible orientable 3-manifold ''M'' has a submanifold Σ that is a Seifert manifold (possibly disconnected and with boundary) whose complement is atoroidal (and possibly disconnected). The submanifold Σ with the smallest number of boundary tori is called the characteristic submanifold of ''M''; it is unique (u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrization Conjecture
In mathematics, Thurston's geometrization conjecture (now a theorem) states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries ( Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure. The conjecture was proposed by as part of his 24 questions, and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gromov Norm
In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes. Given a closed and oriented manifold, one defines the simplicial norm by minimizing the sum of the absolute values of the coefficients over all singular chains homologous to a given cycle. The simplicial volume is the simplicial norm of the fundamental class.. It is named after Mikhail Gromov, who introduced it in 1982. With William Thurston William Paul Thurston (October 30, 1946August 21, 2012) was an American mathematician. He was a pioneer in the field of low-dimensional topology and was awarded the Fields Medal in 1982 for his contributions to the study of 3-manifolds. Thurst ..., he proved that the simplicial volume of a finite volume hyperbolic manifold is proportional to the hyperbolic volume. The simplicial volume is equal to twice ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Academic journals established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-manifolds
In mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane (a tangent plane) to a small and close enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Principles Definition A topological space M is a 3-manifold if it is a second-countable Hausdorff space and if every point in M has a neighbourhood that is homeomorphic to Euclidean 3-space. Mathematical theory of 3-manifolds The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with say, topological 3-manifolds, or smooth 3-manifolds. Phenomena in three dimensions can be strikingly different from phenomena in other dimensions, and so there is a prevalence of very specialized techniques t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
