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Hopf–Whitney Theorem
In mathematics, especially algebraic topology and homotopy theory, the Hopf–Whitney theorem is a result relating the homotopy classes between a CW complex and a multiply connected space with singular cohomology classes of the former with coefficients in the first nontrivial homotopy group of the latter. It can for example be used to calculate Cohomotopy set, cohomotopy as Sphere, spheres are multiply connected. Statement For a n-dimensional CW complex X and a n-1-connected space Y, the well-defined map: : [X,Y]\rightarrow H^n(X,\pi_n(Y)), [f]\mapsto f^*\iota with a certain cohomology class \iota\in H^n(Y,\pi_n(Y)) is an isomorphism. The Hurewicz theorem claims that the well-defined map \pi_n(Y)\rightarrow H_n(Y,\mathbb),[f]\mapsto f_*[S^n] with a fundamental class [S^n]\in H_n(S^n,\mathbb)\cong\mathbb is an isomorphism and that H_(Y,\mathbb)\cong 1, which implies \operatorname_\mathbb^1(H_(Y,\mathbb),\pi_n(Y))\cong 1 for the Ext functor. The Universal coefficient theorem th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ext Functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic structures. The group cohomology, cohomology of groups, Lie algebra cohomology, Lie algebras, and Hochschild cohomology, associative algebras can all be defined in terms of Ext. The name comes from the fact that the first Ext group Ext1 classifies group extension, extensions of one module (mathematics), module by another. In the special case of abelian groups, Ext was introduced by Reinhold Baer (1934). It was named by Samuel Eilenberg and Saunders MacLane (1942), and applied to topology (the universal coefficient theorem for cohomology). For modules over any ring (mathematics), ring, Ext was defined by Henri Cartan and Eilenberg in their 1956 book ''Homological Algebra''. Definition Let R be a ring and let R\text be the category (mathematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Duke Mathematical Journal
''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas. The first issue included a paper by Solomon Lefschetz. Leonard Carlitz served on the editorial board for 35 years, from 1938 to 1973. The current managing editor is Richard Hain (Duke University). Impact According to the journal homepage, the journal has a 2018 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 2.194, ranking it in the top ten mathematics journals in the world. References External links * Mathematics journals Mathematical Journal Academic journals established in 1935 Multilingual journals English-language journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commentarii Mathematici Helvetici
The ''Commentarii Mathematici Helvetici'' is a quarterly peer-reviewed scientific journal in mathematics. The Swiss Mathematical Society (SMG) started the journal in 1929 after a meeting in May of the previous year. The Swiss Mathematical Society still owns and operates the journal; the publishing is currently handled on its behalf by the European Mathematical Society. The scope of the journal includes research articles in all aspects in mathematics. The editors-in-chief have been Rudolf Fueter (1929–1949), J.J. Burckhardt (1950–1981), P. Gabriel (1982–1989), H. Kraft (1990–2005), and Eva Bayer-Fluckiger (2006–present). Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2019 impact factor of 0.854. History The idea for a society-owned research journal emerged in June 1926, when the SMG petitioned the Swiss Confederation for a CHF 3,500 subsidy "to establish its own scientific jour ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Group
In mathematics, topological groups are the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups were studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a construct that can be defined on a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional anal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak Homotopy Equivalence
In mathematics, a weak equivalence is a notion from homotopy theory that in some sense identifies objects that have the same "shape". This notion is formalized in the axiomatic definition of a model category. A model category is a category with classes of morphisms called weak equivalences, fibrations, and cofibrations, satisfying several axioms. The associated homotopy category of a model category has the same objects, but the morphisms are changed in order to make the weak equivalences into isomorphisms. It is a useful observation that the associated homotopy category depends only on the weak equivalences, not on the fibrations and cofibrations. Topological spaces Model categories were defined by Quillen as an axiomatization of homotopy theory that applies to topological spaces, but also to many other categories in algebra and geometry. The example that started the subject is the category of topological spaces with Serre fibrations as fibrations and weak homotopy equivalences as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. Assuming that ''G'' is abelian in the case that n > 1, Eilenberg–MacLane spaces of type K(G,n) always exist, and are all weak homotopy equivalent. Thus, one may consider K(G,n) as referring to a weak homotopy equivalence class of spaces. It is common to refer to any representative as "a K(G,n)" or as "a model of K(G,n)". Moreover, it is comm ... [...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 homeomorphic, but topological spaces that homeomorphic have the same homotopy groups. The notion of homotopy of paths was introduced by Camille Jordan. Introduc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Postnikov Tower
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space by filtering its homotopy type. What this looks like is for a space X there is a list of spaces \_ where\pi_k(X_n) = \begin \pi_k(X) & \text k \leq n \\ 0 & \text k > n \endand there is a series of maps \phi_n: X_n \to X_ that are fibrations with fibers Eilenberg-MacLane spaces K(\pi_n(X),n). In short, we are decomposing the homotopy type of X using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X. Postnikov systems were introduced by, and are named after, Mikhail Postnikov. There is a similar construction called the Whitehead tower (defined below) where instead of having spaces X_n with the homotopy type of X for degrees \leq n, these spaces have null homotopy groups \pi_(X_n) =0 for 1 is classified by a homotopy class : _n\in _,K(\pi_n(X), n+1)\cong H^(X_, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Map
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when is the identity function, the equality is true for all values of to which can be applied. Definition Formally, if is a set, the identity function on is defined to be a function with as its domain and codomain, satisfying In other words, the function value in the codomain is always the same as the input element in the domain . The identity function on is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective. The identity function on is often denoted by . In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or ''diagonal'' of . Algebraic proper ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
