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Novikov's Compact Leaf Theorem
{{Short description, Result about foliation of compact 3-manifolds In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that : ''A codimension-one foliation of a compact 3-manifold whose universal covering space is not contractible must have a compact leaf.'' Novikov's compact leaf theorem for ''S''3 Theorem: ''A smooth codimension-one foliation of the 3-sphere'' ''S''3 ''has a compact leaf. The leaf is a torus'' ''T''2 ''bounding a solid torus with the Reeb foliation.'' The theorem was proved by Sergei Novikov in 1964. Earlier Charles Ehresmann had conjectured that every smooth codimension-one foliation on ''S''3 had a compact leaf, which was known to be true for all known examples; in particular, the Reeb foliation has a compact leaf that is ''T''2. Novikov's compact leaf theorem for any ''M''3 In 1965, Novikov proved the compact leaf theorem for any ''M''3: Theorem: ''Let'' ''M''3 ''be a closed 3-manifold with a smooth codimensi ... [...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] |
Sergei Novikov (mathematician)
Sergei Petrovich Novikov (also Serguei) (Russian: Серге́й Петро́вич Но́виков) (born 20 March 1938) is a Soviet and Russian mathematician, noted for work in both algebraic topology and soliton theory. In 1970, he won the Fields Medal. Early life Novikov was born on 20 March 1938 in Gorky, Soviet Union (now Nizhny Novgorod, Russia). He grew up in a family of talented mathematicians. His father was Pyotr Sergeyevich Novikov, who gave a negative solution to the word problem for groups. His mother, Lyudmila Vsevolodovna Keldysh, and maternal uncle, Mstislav Vsevolodovich Keldysh, were also important mathematicians. In 1955 Novikov entered Moscow State University, from which he graduated in 1960. Four years later he received the Moscow Mathematical Society Award for young mathematicians. In the same year he defended a dissertation for the ''Candidate of Science in Physics and Mathematics'' degree (equivalent to the PhD) at Moscow State University. In 196 ... [...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] |
Universal Covering Space
A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete space D and for every x \in X an Neighbourhood (mathematics), open neighborhood U \subset X, such that \pi^(U)= \displaystyle \bigsqcup_ V_d and \pi, _:V_d \rightarrow U is a homeomorphism for every d \in D . Often, the notion of a covering is used for the covering space E as well as for the map \pi : E \rightarrow X. The open sets V_ are called sheets, which are uniquely determined up to a homeomorphism if U is Connected space, connected. For each x \in X the discrete subset \pi^(x) is called the fiber of x. The degree of a covering is the cardinality of the space D. If E is Path connected, path-connected, then the covering \pi : E \rightarrow X is denoted as a path-connected covering. Examples * For every topological space X there exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It may be embedded in 4-dimensional Euclidean space as the set of points equidistant from a fixed central point. Analogous to how the boundary of a ball in three dimensions is an ordinary sphere (or 2-sphere, a two-dimensional surface), the boundary of a ball in four dimensions is a 3-sphere (an object with three dimensions). A 3-sphere is an example of a 3-manifold and an ''n''-sphere. Definition In coordinates, a 3-sphere with center and radius is the set of all points in real, 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 regard as the space with 2 complex dimensions () or the quaternions (). The unit 3-sphere is then given by :S^3 = \left\ or :S^3 = \left\. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Solid Torus
In mathematics, a solid torus is the topological space formed by sweeping a disk around a circle. It is homeomorphic to the Cartesian product S^1 \times D^2 of the disk and the circle, endowed with the product topology. A standard way to visualize a solid torus is as a toroid, embedded in 3-space. However, it should be distinguished from a torus, which has the same visual appearance: the torus is the two-dimensional space on the boundary of a toroid, while the solid torus includes also the compact interior space enclosed by the torus. Topological properties The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary. The boundary is homeomorphic to S^1 \times S^1, the ordinary torus. Since the disk D^2 is contractible, the solid torus has the homotopy type of a circle, S^1.. Therefore the fundamental group and homology groups are isomorphic to those of the circle: \begin \pi_1\left(S^1 \times D^2\right) &\cong \pi_1\left(S^1\right) \cong \mathbb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reeb Foliation
In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993). It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividing torus surface forms one more leaf. By Novikov's compact leaf theorem {{Short description, Result about foliation of compact 3-manifolds In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that : ''A codimension-one foliation of a compact 3-manifold whose universal covering space is n ..., every smooth foliation of the 3-sphere includes a compact torus leaf, bounding a solid torus foliated in the same way. Illustrations References * * * External links * {{topology-stub Foliations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charles Ehresmann
Charles Ehresmann (19 April 1905 – 22 September 1979) was a German-born French mathematician who worked in differential topology and category theory. He was an early member of the Bourbaki group, and is known for his work on the differential geometry of smooth fiber bundles, notably the introduction of the concepts of Ehresmann connection and of jet bundles, and for his seminar on category theory. Life Ehresmann was born in Strasbourg (at the time part of the German Empire) to an Alsatian-speaking family; his father was a gardener. After World War I, Alsace returned part of France and Ehresmann was taught in French at Lycée Kléber. Between 1924 and 1927 he studied at the École Normale Supérieure (ENS) in Paris and obtained agrégation in mathematics. After one year of military service, in 1928-29 he taught at a French school in Rabat, Morocco. He studied further at the University of Göttingen during the years 1930–31, and at Princeton University in 1932–34. He co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of the topological space. The fundamental group is the first and simplest homotopy group. The fundamental group is a homotopy invariant—topological spaces that are homotopy equivalent (or the stronger case of homeomorphic) have isomorphic fundamental groups. The fundamental group of a topological space X is denoted by \pi_1(X). Intuition Start with a space (for example, a surface), and some point in it, and all the loops both starting and ending at this point— paths that start at this point, wander around and eventually return to the starting point. Two loops can be combined in an obvious way: travel along the first loop, then along the second. Two loops are considered equivalent if one can be deformed into the other without breakin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or ''holes'', of a topological space. To define the ''n''-th homotopy group, the base-point-preserving maps from an ''n''-dimensional sphere (with base point) into a given space (with base point) are collected into equivalence classes, called homotopy classes. Two mappings are homotopic if one can be continuously deformed into the other. These homotopy classes form a group, called the ''n''-th homotopy group, \pi_n(X), of the given space ''X'' with base point. Topological spaces with differing homotopy groups are never equivalent ( homeomorphic), but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (algebra)
In algebra, the kernel of a homomorphism (function that preserves the structure) is generally the inverse image of 0 (except for groups whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the kernel of a linear map. The kernel of a matrix, also called the ''null space'', is the kernel of the linear map defined by the matrix. The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.See and . For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as normal subgroup for groups and two-sided ideals for r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Genus (mathematics)
In mathematics, genus (plural genera) has a few different, but closely related, meanings. Intuitively, the genus is the number of "holes" of a surface. A sphere has genus 0, while a torus has genus 1. Topology Orientable surfaces The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It is equal to the number of handles on it. Alternatively, it can be defined in terms of the Euler characteristic ''χ'', via the relationship ''χ'' = 2 − 2''g'' for closed surfaces, where ''g'' is the genus. For surfaces with ''b'' boundary components, the equation reads ''χ'' = 2 − 2''g'' − ''b''. In layman's terms, it's the number of "holes" an object has ("holes" interpreted in the sense of doughnut holes; a hollow sphere would be considered as having zero holes in this sense). A torus has 1 such h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
