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Chain Homotopy Equivalence
This is a glossary of properties and concepts in algebraic topology in mathematics. See also: glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differential geometry and topology, Timeline of manifolds. *Convention: Throughout the article, ''I'' denotes the unit interval, ''S''''n'' the ''n''-sphere and ''D''''n'' the ''n''-disk. Also, throughout the article, spaces are assumed to be reasonable; this can be taken to mean for example, a space is a CW complex or compactly generated weakly Hausdorff space. Similarly, no attempt is made to be definitive about the definition of a spectrum. A simplicial set is not thought of as a space; i.e., we generally distinguish between simplicial sets and their geometric realizations. *Inclusion criterion: As there is no glossary of homological algebra in Wikipedia right now, this glossary also includes a few concepts in homological algebra (e.g., chain homotopy); some concepts in geometric topol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 of algebraic topology 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 gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Model Category
In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called ' weak equivalences', ' fibrations' and 'cofibrations' satisfying certain axioms relating them. These abstract from the category of topological spaces or of chain complexes (derived category theory). The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic ''K''-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results. Motivation Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets. Another model category is the category of chain complexes of ''R''-modules for a commutative ring ''R''. Homotopy th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah Duality
Atiyyah ( ar, عطية ''‘aṭiyyah''), which generally implies "something (money or goods given as regarded) received as a gift" or also means "present, gift, benefit, boon, favor, granting, giving"''.'' The name is also spelt Ateah, Atiyeh, Attiah, Attieh, Atieh, Atiya, Atiyya, Attiya, Attiyah, Attyé or Ateya, Attua, Antuya, Atia. It may refer to: Surname Academics * Aziz Suryal Atiya (1898-1988), Coptic historian and scholar and an expert in Islamic and Crusades studies * Sir Michael Atiyah (1929–2019), British mathematician, brother of Patrick * Patrick Atiyah (1931-2018), English barrister and legal writer, brother of Michael * George N. Atiyeh (1923–2008), Lebanese librarian Authors and journalists * Jarir ibn Atiyah (c. 650 – c. 728), Arab poet and satirist * Edward Atiyah (1903–1964), Lebanese born writer, father of Michael and Patrick * Karen Attiah (born August 12, 1986), writer, journalist and editor Arts and entertainment * Assane Attyé (born 198 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded the Fields Medal in 1966 and the Abel Prize in 2004. Life Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at the University of Oxford and the University of Cambridge and in the United States at the Institute for Advanced Study. He was the President of the Royal Society (1990–1995), founding director of the Isaac Newton Institute (1990–1996), master of Trinity College, Cambridge (1990–1997), chancellor of the University of Leicester (1995–2005), and the President of the Royal Society of Edinburgh (2005–2008). From 1997 until his death, he was an honorary professor in the University of Edinburgh. Atiyah's mathematical collaborators included Raoul Bott, Friedrich Hirzebruch and Isadore Sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aspherical Space
In topology, a branch of mathematics, an aspherical space is a topological space with all homotopy groups \pi_n(X) equal to 0 when n>1. If one works with CW complexes, one can reformulate this condition: an aspherical CW complex is a CW complex whose universal cover is contractible. Indeed, contractibility of a universal cover is the same, by Whitehead's theorem, as asphericality of it. And it is an application of the exact sequence of a fibration that higher homotopy groups of a space and its universal cover are same. (By the same argument, if ''E'' is a path-connected space and p\colon E \to B is any covering map, then ''E'' is aspherical if and only if ''B'' is aspherical.) Each aspherical space ''X'' is, by definition, an Eilenberg–MacLane space of type K(G,1), where G = \pi_1(X) is the fundamental group of ''X''. Also directly from the definition, an aspherical space is a classifying space for its fundamental group (considered to be a topological group when endowed with t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold Approximate Fibration
In algebraic topology, a branch of mathematics, an approximate fibration is a sort of fibration such that the homotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from ... holds only approximately. The notion was introduced by Coram and Duvall in 1977. A manifold approximate fibration is a proper approximate fibration between manifolds. Some authors believe that manifold approximate fibrations are the "correct bundle theory for topological manifolds and singular spaces". References * Further reading nLab - approximate fibration Algebraic topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Fibration
In algebraic topology, a branch of mathematics, an approximate fibration is a sort of fibration such that the homotopy lifting property In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from ... holds only approximately. The notion was introduced by Coram and Duvall in 1977. A manifold approximate fibration is a proper approximate fibration between manifolds. Some authors believe that manifold approximate fibrations are the "correct bundle theory for topological manifolds and singular spaces". References * Further reading nLab - approximate fibration Algebraic topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism Group
In mathematics, particularly topology, the homeomorphism group of a topological space is the group consisting of all homeomorphisms from the space to itself with function composition as the group operation. Homeomorphism groups are very important in the theory of topological spaces and in general are examples of automorphism groups. Homeomorphism groups are topological invariants in the sense that the homeomorphism groups of homeomorphic topological spaces are isomorphic as groups. Properties and examples There is a natural group action of the homeomorphism group of a space on that space. Let X be a topological space and denote the homeomorphism group of X by G. The action is defined as follows: \begin G\times X &\longrightarrow X\\ (\varphi, x) &\longmapsto \varphi(x) \end This is a group action since for all \varphi,\psi\in G, \varphi\cdot(\psi\cdot x)=\varphi(\psi(x))=(\varphi\circ\psi)(x) where \cdot denotes the group action, and the identity element of G (which is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Trick
Alexander's trick, also known as the Alexander trick, is a basic result in geometric topology, named after J. W. Alexander. Statement Two homeomorphisms of the ''n''-dimensional ball D^n which agree on the boundary sphere S^ are isotopic. More generally, two homeomorphisms of ''D''''n'' that are isotopic on the boundary are isotopic. Proof Base case: every homeomorphism which fixes the boundary is isotopic to the identity relative to the boundary. If f\colon D^n \to D^n satisfies f(x) = x \text x \in S^, then an isotopy connecting ''f'' to the identity is given by : J(x,t) = \begin tf(x/t), & \text 0 \leq \, x\, 0 the transformation J_t replicates f at a different scale, on the disk of radius t, thus as t\rightarrow 0 it is reasonable to expect that J_t merges to the identity. The subtlety is that at t=0, f "disappears": the germ at the origin "jumps" from an infinitely stretched version of f to the identity. Each of the steps in the homotopy could be smoothed (smoot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams Operations
In mathematics, an Adams operation, denoted ψ''k'' for natural numbers ''k'', is a cohomology operation in topological K-theory, or any allied operation in algebraic K-theory or other types of algebraic construction, defined on a pattern introduced by Frank Adams. The basic idea is to implement some fundamental identities in symmetric function theory, at the level of vector bundles or other representing object in more abstract theories. Adams operations can be defined more generally in any λ-ring. Adams operations in K-theory Adams operations ψ''k'' on K theory (algebraic or topological) are characterized by the following properties. # ψ''k'' are ring homomorphisms. # ψ''k''(l)= lk if l is the class of a line bundle. # ψ''k'' are functorial. The fundamental idea is that for a vector bundle ''V'' on a topological space ''X'', there is an analogy between Adams operators and exterior powers, in which :ψ''k''(''V'') is to Λ''k''(''V'') as :the power sum Σ α''k'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams E-invariant
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] |
Adams Conjecture
Adams may refer to: * For persons, see Adams (surname) Places United States *Adams, California *Adams, California, former name of Corte Madera, California *Adams, Decatur County, Indiana *Adams, Kentucky *Adams, Massachusetts, a New England town **Adams (CDP), Massachusetts, the central village in the town *Adams, Minnesota * Adams, North Dakota *Adams, Nebraska *Adams, New Jersey * Adams (town), New York ** Adams (village), New York, within the town *Adams, Oklahoma *Adams, Oregon * Adams, Pennsylvania, a former community in Armstrong County *Adams, Tennessee *Adams, Wisconsin, city in Adams County *Adams, Adams County, Wisconsin, town *Adams, Green County, Wisconsin, town *Adams, Jackson County, Wisconsin, town *Adams, Walworth County, Wisconsin, unincorporated community *Adams Center, Wisconsin, a ghost town Elsewhere *Adams (lunar crater) *Adams (Martian crater) *Adams Island, New Zealand, one of the Auckland Islands *Adams, Ilocos Norte Transportation ;Vehicles *Adams (1903 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
