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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 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, Waldhausen's article can be seen as the first breakthrough towards the discovery of JSJ decomposition. One of the numerous consequences of the Thurston-Perelman geometrization theorem is that graph manifolds are precisely the 3-manifolds whose Gromov norm I ... [...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] |
3-manifold
In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Introduction Definition A topological space ''X'' is a 3-manifold if it is a second-countable Hausdorff space and if every point in ''X'' 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 that do not generalize to dimensions g ... [...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. 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 :\pi:P\to M one has the homomorphism :\pi^*:H^2(M,\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friedhelm Waldhausen
Friedhelm Waldhausen (born 1938 in Millich, Hückelhoven, Rhine Province) is 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 on the theory of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Seifert Fiber Spaces
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture. Definition A Seifert manifold is a closed 3-manifold together with a decomposition into a disjoint union of circles (called fibers) such that each fiber has a tubular neighborhood that forms a standard fibered torus. A standard fibered torus corresponding to a pair of coprime integers (a,b) with a>0 is the surface bundle of the automorphism of a disk given by rotation by an angle of 2\pi b/a (with the natural fibering by circles). If a=1 the middle fiber is called ordinary, while if a>1 the middle fiber is called exceptional. A compact Seifert fiber space has only a finite number of exceptional fibers. The set of fibers forms a 2-dimensio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrization Conjecture
In mathematics, Thurston's geometrization conjecture 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 , 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 announced a proof in the 1980s and since then sever ... [...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 (up to iso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometrization Conjecture
In mathematics, Thurston's geometrization conjecture 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 , 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 announced a proof in the 1980s and since then sever ... [...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 certain 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 representing a 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, he proved that the simplicial volume of a finite volume hyperbolic manifold is proportional to the hyperbolic volume In the mathematical field of knot theory, the hyperbolic volume of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological inv .... 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 managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications 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 space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds look like our universe does to a small enough observer. This is made more precise in the definition below. Introduction Definition A topological space ''X'' is a 3-manifold if it is a second-countable Hausdorff space and if every point in ''X'' 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 that do not generalize to dimensions gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
