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Thurston Norm
In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of homology classes represented by surfaces. Definition Let M be a differentiable manifold and c \in H_2(M). Then c can be represented by a smooth embedding S \to M, where S is a (not necessarily connected) surface that is compact and without boundary. The Thurston norm of c is then defined to be :\, c\, _T = \min_ \sum_^n \chi_-(S_i), where the minimum is taken over all embedded surfaces S = \bigcup_i S_i (the S_i being the connected components) representing c as above, and \chi_-(F) = \max(0, -\chi(F)) is the absolute value of the Euler characteristic for surfaces which are not spheres (and 0 for spheres). This function satisfies the following properties: *\, kc \, _T = , k, \cdot \, c \, _T for c \in H_2(M), k \in \Z; * \, c_1 + c_2 \, _T \le \, c_1 \, _T + \, c_2 \, _T for c_1, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homology Group
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] |
Immersion (mathematics)
In mathematics, an immersion is a differentiable function between differentiable manifolds whose differential (or pushforward) is everywhere injective. Explicitly, is an immersion if :D_pf : T_p M \to T_N\, is an injective function at every point ''p'' of ''M'' (where ''TpX'' denotes the tangent space of a manifold ''X'' at a point ''p'' in ''X''). Equivalently, ''f'' is an immersion if its derivative has constant rank equal to the dimension of ''M'': :\operatorname\,D_p f = \dim M. The function ''f'' itself need not be injective, only its derivative must be. A related concept is that of an embedding. A smooth embedding is an injective immersion that is also a topological embedding, so that ''M'' is diffeomorphic to its image in ''N''. An immersion is precisely a local embedding – that is, for any point there is a neighbourhood, , of ''x'' such that is an embedding, and conversely a local embedding is an immersion. For infinite dimensional manifolds, this is sometimes ... [...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] |
Memoirs Of The American Mathematical Society
''Memoirs of the American Mathematical Society'' is a mathematical journal published in six volumes per year, totalling approximately 33 individually bound numbers, by the American Mathematical Society. It is intended to carry papers on new mathematical research between 80 and 200 pages in length. Usually, a bound number consists of a single paper, i.e., it is a monograph. The journal is indexed by Mathematical Reviews, Zentralblatt MATH, Science Citation Index, Research Alert, CompuMath Citation Index, and Current Contents. Other journals from the AMS * ''Bulletin of the American Mathematical Society'' * ''Journal of the American Mathematical Society'' * ''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 appeared in 1953. Each issue of the magazine since ...'' * ''Proceedings of the Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Differential Geometry
The ''Journal of Differential Geometry'' is a peer-reviewed scientific journal of mathematics published by International Press on behalf of Lehigh University in 3 volumes of 3 issues each per year. The journal publishes an annual supplement in book form called ''Surveys in Differential Geometry''. It covers differential geometry and related subjects such as differential equations, mathematical physics, algebraic geometry, and geometric topology. The editor-in-chief is Shing-Tung Yau of Harvard University. History The journal was established in 1967 by Chuan-Chih Hsiung, who was a professor in the Department of Mathematics at Lehigh University at the time. Hsiung served as the journal's editor-in-chief, and later co-editor-in-chief, until his death in 2009. In May 1996, the annual Geometry and Topology conference which was held at Harvard University was dedicated to commemorating the 30th anniversary of the journal and the 80th birthday of its founder. Similarly, in May 2008 Harv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Torus
In mathematics, the mapping torus in topology of a homeomorphism ''f'' of some topological space ''X'' to itself is a particular geometric construction with ''f''. Take the cartesian product of ''X'' with a closed interval ''I'', and glue the boundary components together by the static homeomorphism: :M_f =\frac The result is a fiber bundle whose base is a circle and whose fiber is the original space ''X''. If ''X'' is a manifold, ''Mf'' will be a manifold of dimension one higher, and it is said to "fiber over the circle". As a simple example, let X be the circle, and f be the inversion e^ \mapsto e^ , then the mapping torus is the Klein bottle. Mapping tori of surface homeomorphisms play a key role in the theory of 3-manifolds and have been intensely studied. If ''S'' is a closed surface of genus ''g'' ≥ 2 and if ''f'' is a self-homeomorphism of ''S'', the mapping torus ''Mf'' is a closed 3-manifold that fibers over the circle with fiber ''S''. A deep res ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides (''faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematicians as ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of the real coordinate space R''n'' into the cosets ''x'' + R''p'' of the standardly embedded subspace R''p''. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class ''Cr''), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class ''Cr'' it is usually understood that ''r'' ≥ 1 (otherwise, ''C''0 is a topological foliation). The number ''p'' (the dimension of the leaves) is called the dimension of the foliation and is called its codimension. In some papers on general relativity by mathematical physicists, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a mani ... [...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] |
David Gabai
David Gabai is an American mathematician and the Hughes-Rogers Professor of Mathematics at Princeton University. Focused on low-dimensional topology and hyperbolic geometry, he is a leading researcher in those subjects. Biography David Gabai received his B.S. in mathematics from MIT in 1976 and his Ph.D. in mathematics from Princeton University in 1980. Gabai completed his doctoral dissertation, titled "Foliations and genera of links", under the supervision of William Thurston. After positions at Harvard and University of Pennsylvania, Gabai spent most of the period of 1986–2001 at Caltech, and has been at Princeton since 2001. Gabai was the Chair of the Department of Mathematics at Princeton University from 2012 to 2019. Honours and awards In 2004, David Gabai was awarded the Oswald Veblen Prize in Geometry, given every three years by the American Mathematical Society. He was an invited speaker in the International Congress of Mathematicians 2010, Hyderabad on the top ... [...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] |
