List Of Algebraic Topology Topics

	List Of Algebraic Topology Topics This is a list of algebraic topology topics. Homology (mathematics) Simplex Simplicial complex Polytope Triangulation Barycentric subdivision Simplicial approximation theorem Abstract simplicial complex Simplicial set ** Simplicial category Chain (algebraic topology) Betti number Euler characteristic Genus * Riemann–Hurwitz formula Singular homology Cellular homology Relative homology Mayer–Vietoris sequence Excision theorem Universal coefficient theorem Cohomology List of cohomology theories * Cocycle class Cup product Cohomology ring De Rham cohomology Čech cohomology Alexander–Spanier cohomology Intersection cohomology ** Lusternik–Schnirelmann category Poincaré duality Fundamental class ''Applications'' Jordan curve theorem Brouwer fixed point theorem * Invariance of domain Lefschetz fixed-point theorem Hairy ball theorem Degree of a continuous mapping Borsuk–Ulam theorem Ham sandwich theorem Homol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Algebraic Topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cellular Homology In mathematics, cellular homology in algebraic topology is a homology theory for the category of CW-complexes. It agrees with singular homology, and can provide an effective means of computing homology modules. Definition If X is a CW-complex with ''n''-skeleton X_ , the cellular-homology modules are defined as the homology groups ''Hi'' of the cellular chain complex : \cdots \to (X_,X_) \to (X_,X_) \to (X_,X_) \to \cdots, where X_ is taken to be the empty set. The group : (X_,X_) is free abelian, with generators that can be identified with the n -cells of X . Let e_^ be an n -cell of X , and let \chi_^: \partial e_^ \cong \mathbb^ \to X_ be the attaching map. Then consider the composition : \chi_^: \mathbb^ \, \stackrel \, \partial e_^ \, \stackrel \, X_ \, \stackrel \, X_ / \left( X_ \setminus e_^ \right) \, \stackr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Intersection Cohomology In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them over the next few years. Intersection cohomology was used to prove the Kazhdan–Lusztig conjectures and the Riemann–Hilbert correspondence. It is closely related to ''L''2 cohomology. Goresky–MacPherson approach The homology groups of a compact, oriented, connected, ''n''-dimensional manifold ''X'' have a fundamental property called Poincaré duality: there is a perfect pairing : H_i(X,\Q) \times H_(X,\Q) \to H_0(X,\Q) \cong \Q. Classically—going back, for instance, to Henri Poincaré—this duality was understood in terms of intersection theory. An element of :H_j(X) is represented by a ''j''-dimensional cycle. If an ''i''-dimensional and an (n-i)-dimensional cycle are in general position, then their intersection is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Alexander–Spanier Cohomology In mathematics, particularly in algebraic topology, Alexander–Spanier cohomology is a cohomology theory for topological spaces. History It was introduced by for the special case of compact metric spaces, and by for all topological spaces, based on a suggestion of Alexander D. Wallace. Definition If ''X'' is a topological space and ''G'' is an ''R''- module where ''R'' is a ring with unity, then there is a cochain complex ''C'' whose ''p''-th term C^p is the set of all functions from X^ to ''G'' with differential d\colon C^ \to C^ given by :df(x_0,\ldots,x_p)= \sum_i(-1)^if(x_0,\ldots,x_,x_,\ldots,x_p). The defined cochain complex C^(X;G) does not rely on the topology of X. In fact, if X is a nonempty space, G\simeq H^(C^(X;G)) where G is a graded module whose only nontrivial module is G at degree 0. An element \varphi\in C^p(X) is said to be ''locally zero'' if there is a covering \ of X by open sets such that \varphi vanishes on any (p+1)-tuple of X which lies in some ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
picture info	Čech Cohomology In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open set, open cover (topology), covers of a topological space. It is named for the mathematician Eduard Čech. Motivation Let ''X'' be a topological space, and let \mathcal be an open cover of ''X''. Let N(\mathcal) denote the nerve of a covering, nerve of the covering. The idea of Čech cohomology is that, for an open cover \mathcal consisting of sufficiently small open sets, the resulting simplicial complex N(\mathcal) should be a good combinatorial model for the space ''X''. For such a cover, the Čech cohomology of ''X'' is defined to be the simplicial homology, simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of ''X'', ordered by Open cover#Refinement, refinement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	De Rham Cohomology In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every Closed and exact differential forms, exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of Hole#In mathematics, "holes" in the manifold, and the de Rham cohomology groups comprise a set of Topological invariant, topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cohomology Ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. Specifically, given a sequence of cohomology groups ''H''''k''(''X'';''R'') on ''X'' with coefficients in a commutative ring ''R'' (typically ''R'' is Z''n'', Z, Q, R, or C) one can define the cup product, which takes the form :H^k(X;R) \times H^\ell(X;R) \to H^(X; R). The cup product gives a multiplication on the direct sum of the cohomology groups :H^\bullet(X;R) = \bigoplus_ H^k(X; R). This multiplication turns ''H''•(''X'';''R'') into a ring. In fact, it is naturally an N-graded r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Cup Product In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p+q. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space X into a graded ring, H^(X), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944. Definition In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring H^(X) of a topological space X. The construction starts with a product of cochains: if \alpha^p is a p-cochain and \beta^q is a q-cochain, then :(\alpha^p \smile \beta^q)(\sigma) = \alpha^p(\sigma \circ \iota_) \cdot \beta^q(\sigma \circ \iota_) where \sigma is a singular (p+q)- simplex and \iota_S , S \subset \ is the canonical embeddi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Group Cohomology In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''- simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	List Of Cohomology Theories This is a list of some of the ordinary and generalized cohomology theory, generalized (or extraordinary) homology and cohomology theories in algebraic topology that are defined on the categories of CW complexes or spectrum (homotopy theory), spectra. For other sorts of homology theories see the #See also, links at the end of this article. Notation S=\pi=S^0 is the sphere spectrum. S^n is the spectrum of the n-dimensional sphere S^nY=S^n\land Y is the nth suspension (topology), suspension of a spectrum Y. [X,Y] is the abelian group of morphisms from the spectrum X to the spectrum Y, given (roughly) as homotopy classes of maps. [X,Y]_n=[S^nX,Y] [X,Y]_* is the graded abelian group given as the sum of the groups [X,Y]_n. \pi_n(X)=[S^n,X]=[S,X]_n is the nth stable homotopy group of X. \pi_(X) is the sum of the groups \pi_n(X), and is called the coefficient ring of X when X is a ring spectrum. X\land Y is the smash product of two spectra. If X is a spectrum, then it defines ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Cohomology In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are function (mathematics), functions on the group of chain (algebraic topology), chains in homology theory. From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and abstract algebra, algebra. The terminology tends to hide the fact that cohomology, a Covariance and contravariance of functors, c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Universal Coefficient Theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': :H_i(X,\Z) completely determine its ''homology groups with coefficients in'' , for any abelian group : :H_i(X,A) Here H_i might be the simplicial homology, or more generally the singular homology. The usual proof of this result is a pure piece of homological algebra about chain complexes of free abelian groups. The form of the result is that other coefficients may be used, at the cost of using a Tor functor. For example, it is common to take A to be \Z/2\Z, so that coefficients are modulo 2. This becomes straightforward in the absence of 2- torsion in the homology. Quite generally, the result indicates the relationship that holds between the Betti numbers b_i of X and the Betti numbers b_ with coefficients in a field F. These can differ, but o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]