|
J-homomorphism
In mathematics, the ''J''-homomorphism is a mapping from the homotopy groups of the special orthogonal groups to the homotopy groups of spheres. It was defined by , extending a construction of . Definition Whitehead's original homomorphism is defined geometrically, and gives a homomorphism :J \colon \pi_r (\mathrm(q)) \to \pi_(S^q) of abelian groups for integers ''q'', and r \ge 2. (Hopf defined this for the special case q = r+1.) The ''J''-homomorphism can be defined as follows. An element of the special orthogonal group SO(''q'') can be regarded as a map :S^\rightarrow S^ and the homotopy group \pi_r(\operatorname(q))) consists of homotopy classes of maps from the ''r''-sphere to SO(''q''). Thus an element of \pi_r(\operatorname(q)) can be represented by a map :S^r\times S^\rightarrow S^ Applying the Hopf construction to this gives a map :S^= S^r*S^\rightarrow S( S^) =S^q in \pi_(S^q), which Whitehead defined as the image of the element of \pi_r(\operatorname(q)) under t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Groups Of Spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle () and the ordinary sphere (). The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th ''homotopy group'' summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere . This summary does not distinguish between two mappings if one can be continuously deformed to the oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Groups Of Spheres
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle () and the ordinary sphere (). The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th ''homotopy group'' summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere . This summary does not distinguish between two mappings if one can be continuously deformed to the oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Stem
In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle () and the ordinary sphere (). The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th ''homotopy group'' summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere . This summary does not distinguish between two mappings if one can be continuously deformed to the ot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bott Periodicity
In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable complex vector bundles, as well as the stable homotopy groups of spheres. Bott periodicity can be formulated in numerous ways, with the periodicity in question always appearing as a period-2 phenomenon, with respect to dimension, for the theory associated to the unitary group. See for example topological K-theory. There are corresponding period-8 phenomena for the matching theories, (real number, real) KO-theory and (quaternionic) KSp-theory, associated to the real orthogonal group and the quaternionic symplectic group, respectively. The J-homomorphism is a homomorphism from the homotopy groups of orthogonal groups to stable homotopy groups of spheres, which causes the period 8 Bott periodicity to be visible in the stable homotopy groups of sph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Construction
In algebraic topology, the Hopf construction constructs a map from the join X*Y of two spaces ''X'' and ''Y'' to the suspension ''SZ'' of a space Z out of a map from X\times Y to ''Z''. It was introduced by in the case when ''X'' and ''Y'' are spheres. used it to define the J-homomorphism. Construction The Hopf construction can be obtained as the composition of a map :''X*Y\rightarrow S(X\times Y)'' and the suspension :''S(X\times Y)\rightarrow SZ'' of the map from X\times Y to ''Z''. The map from X*Y to ''S(X\times Y)'' can be obtained by regarding both sides as a quotient of X\times Y\times I where ''I'' is the unit interval. For X*Y one identifies (x,y,0) with (z,y,0) and (x,y,1) with (x,z,1), while for ''S(X\times Y)'' one contracts all points of the form (x,y,0) to a point and also contracts all points of the form (x,y,1) to a point. So the map from X\times Y\times I to ''S(X\times Y)'' factors through X*Y. References * *{{Citation , last1=Whitehead , first1=George W. , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exotic Sphere
In an area of mathematics called differential topology, an exotic sphere is a differentiable manifold ''M'' that is homeomorphic but not diffeomorphic to the standard Euclidean ''n''-sphere. That is, ''M'' is a sphere from the point of view of all its topological properties, but carrying a smooth structure that is not the familiar one (hence the name "exotic"). The first exotic spheres were constructed by in dimension n = 7 as S^3- bundles over S^4. He showed that there are at least 7 differentiable structures on the 7-sphere. In any dimension showed that the diffeomorphism classes of oriented exotic spheres form the non-trivial elements of an abelian monoid under connected sum, which is a finite abelian group if the dimension is not 4. The classification of exotic spheres by showed that the oriented exotic 7-spheres are the non-trivial elements of a cyclic group of order 28 under the operation of connected sum. Introduction The unit ''n''-sphere, S^n, is the set of all ('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Sphere
In algebraic topology, a branch of mathematics, a ''homotopy sphere'' is an ''n''-manifold that is homotopy equivalent to the ''n''-sphere. It thus has the same homotopy groups and the same homology groups as the ''n''-sphere, and so every homotopy sphere is necessarily a homology sphere. The topological generalized Poincaré conjecture is that any ''n''-dimensional homotopy sphere is homeomorphic to the ''n''-sphere; it was solved by Stephen Smale in dimensions five and higher, by Michael Freedman in dimension 4, and for dimension 3 (the original Poincaré conjecture) by Grigori Perelman in 2005. The resolution of the smooth Poincaré conjecture in dimensions 5 and larger implies that homotopy spheres in those dimensions are precisely exotic spheres. It is still an open question () whether or not there are non-trivial smooth homotopy spheres in dimension 4. References See also *Homology sphere *Homotopy groups of spheres *Poincaré conjecture In the mathematics, mathema ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
H-cobordism
In geometric topology and differential topology, an (''n'' + 1)-dimensional cobordism ''W'' between ''n''-dimensional manifolds ''M'' and ''N'' is an ''h''-cobordism (the ''h'' stands for homotopy equivalence) if the inclusion maps : M \hookrightarrow W \quad\mbox\quad N \hookrightarrow W are homotopy equivalences. The ''h''-cobordism theorem gives sufficient conditions for an ''h''-cobordism to be trivial, i.e., to be C-isomorphic to the cylinder ''M'' × , 1 Here C refers to any of the categories of smooth, piecewise linear, or topological manifolds. The theorem was first proved by Stephen Smale for which he received the Fields Medal and is a fundamental result in the theory of high-dimensional manifolds. For a start, it almost immediately proves the generalized Poincaré conjecture. Background Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book ''Disquisitiones Arithmeticae'', published in 1801. A familiar use of modular arithmetic is in the 12-hour clock, in which the day is divided into two 12-hour periods. If the time is 7:00 now, then 8 hours later it will be 3:00. Simple addition would result in , but clocks "wrap around" every 12 hours. Because the hour number starts over at zero when it reaches 12, this is arithmetic ''modulo'' 12. In terms of the definition below, 15 is ''congruent'' to 3 modulo 12, so "15:00" on a 24-hour clock is displayed "3:00" on a 12-hour clock. Congruence Given an integer , called a modulus, two integers and are said to be congruent modulo , if is a divisor of their difference (that is, if there is an integer such that ). Congruence modulo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57–5 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cokernel
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the name: the kernel is a subobject of the domain (it maps to the domain), while the cokernel is a quotient object of the codomain (it maps from the codomain). Intuitively, given an equation that one is seeking to solve, the cokernel measures the ''constraints'' that must satisfy for this equation to have a solution – the obstructions to a solution – while the kernel measures the ''degrees of freedom'' in a solution, if one exists. This is elaborated in intuition, below. More generally, the cokernel of a morphism in some category (e.g. a homomorphism between groups or a bounded linear operator between Hilbert spaces) is an object and a morphism such that the composition is the zero morphism of the category, and furthermore is universa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology (journal)
''Topology'' was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of ''Topology'' appeared in 2009. Pricing dispute On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published. In 2007 the former editors of ''Topology'' announced the launch of the ''Journal of Topology'', published by Oxford University Press ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
