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Classifying Space For SU(n)
In mathematics, the classifying space \operatorname(n) for the special unitary group \operatorname(n) is the base space of the universal \operatorname(n) principal bundle \operatorname(n)\rightarrow\operatorname(n). This means that \operatorname(n) principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into \operatorname(n). The isomorphism is given by pullback. Definition There is a canonical inclusion of complex oriented Grassmannians given by \widetilde\operatorname_n(\mathbb^k)\hookrightarrow\widetilde\operatorname_n(\mathbb^), V\mapsto V\times\. Its colimit is: \operatorname(n) :=\widetilde\operatorname_n(\mathbb^\infty) :=\lim_\widetilde\operatorname_n(\mathbb^k). Since real oriented Grassmannians can be expressed as a homogeneous space by: : \widetilde\operatorname_n(\mathbb^k) =\operatorname(n+k)/(\operatorname(n)\times\operatorname(k)) the group structure carries over to \operatorname(n). Simplest classifyi ... [...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] |
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set. A function is bijective if it is invertible; that is, a function f:X\to Y is bijective if and only if there is a function g:Y\to X, the ''inverse'' of , such that each of the two ways for composing the two functions produces an identity function: g(f(x)) = x for each x in X and f(g(y)) = y for each y in Y. For example, the ''multiplication by two'' defines a bijection from the integers to the even numbers, which has the ''division by two'' as its inverse function. A function is bijective if and only if it is both injective (or ''one-to-one'')—meaning that each element in the codomain is mappe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying+space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of ''G''. It has the property that any ''G'' principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG \to BG. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group ''G'', ''BG'' is a path-connected topological space ''X'' such that the fundamental group of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space For U(n)
In mathematics, the classifying space for the unitary group U(''n'') is a space BU(''n'') together with a universal bundle EU(''n'') such that any hermitian bundle on a paracompact space ''X'' is the pull-back of EU(''n'') by a map ''X'' → BU(''n'') unique up to homotopy. This space with its universal fibration may be constructed as either # the Grassmannian of ''n''-planes in an infinite-dimensional complex Hilbert space; or, # the direct limit, with the induced topology, of Grassmannians of ''n'' planes. Both constructions are detailed here. Construction as an infinite Grassmannian The total space EU(''n'') of the universal bundle is given by :EU(n)=\left \. Here, ''H'' denotes an infinite-dimensional complex Hilbert space, the ''e''''i'' are vectors in ''H'', and \delta_ is the Kronecker delta. The symbol (\cdot,\cdot) is the inner product on ''H''. Thus, we have that EU(''n'') is the space of orthonormal ''n''-frames in ''H''. The Group action (mathematics), group action ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space For SO(n)
In mathematics, the classifying space \operatorname(n) for thspecial orthogonal group'' \operatorname(n) is the base space of the universal \operatorname(n) principal bundle \operatorname(n)\rightarrow\operatorname(n). This means that \operatorname(n) principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into \operatorname(n). The isomorphism is given by pullback. Definition There is a canonical inclusion of real oriented Grassmannians given by \widetilde\operatorname_n(\mathbb^k)\hookrightarrow\widetilde\operatorname_n(\mathbb^), V\mapsto V\times\. Its colimit is:Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151 : \operatorname(n) :=\widetilde\operatorname_n(\mathbb^\infty) :=\lim_\widetilde\operatorname_n(\mathbb^k). Since real oriented Grassmannians can be expressed as a homogeneous space by: : \widetilde\operatorname_n(\mathbb^k) =\operatorname(n+k)/(\operatorname(n)\times\operatorname( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space For O(n)
In mathematics, the classifying space for the orthogonal group O(''n'') may be constructed as the Grassmannian of ''n''-planes in an infinite-dimensional real space \mathbb^\infty. Cohomology ring The cohomology ring of \operatorname(n) with coefficients in the field \mathbb_2 of two elements is generated by the Stiefel–Whitney classes:Hatcher 02, Theorem 4D.4. : H^*(\operatorname(n);\mathbb_2) =\mathbb_2 _1,\ldots,w_n Infinite classifying space The canonical inclusions \operatorname(n)\hookrightarrow\operatorname(n+1) induce canonical inclusions \operatorname(n)\hookrightarrow\operatorname(n+1) on their respective classifying spaces. Their respective colimits are denoted as: : \operatorname :=\lim_\operatorname(n); : \operatorname :=\lim_\operatorname(n). \operatorname is indeed the classifying space of \operatorname. See also * Classifying space for U(''n'') * Classifying space for SO(n) * Classifying space for SU(n) Literature * * * External links * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chern Class
In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, and Gromov–Witten invariants. Chern classes were introduced by . Geometric approach Basic idea and motivation Chern classes are characteristic classes. They are topological invariants associated with vector bundles on a smooth manifold. The question of whether two ostensibly different vector bundles are the same can be quite hard to answer. The Chern classes provide a simple test: if the Chern classes of a pair of vector bundles do not agree, then the vector bundles are different. The converse, however, is not true. In topology, differential geometry, and algebraic geometry, it is often important to count how many linearly independent sect ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...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] |
