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G-spectrum
In algebraic topology, a G-spectrum is a spectrum A spectrum (: spectra or spectrums) is a set of related ideas, objects, or properties whose features overlap such that they blend to form a continuum. The word ''spectrum'' was first used scientifically in optics to describe the rainbow of co ... with an action of a (finite) group. Let ''X'' be a spectrum with an action of a finite group ''G''. The important notion is that of the homotopy fixed point set X^. There is always :X^G \to X^, a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, X^ is the mapping spectrum F(BG_+, X)^G). Example: \mathbb/2 acts on the complex ''K''-theory ''KU'' by taking the conjugate bundle of a complex vector bundle. Then KU^ = KO, the real ''K''-theory. The cofiber of X_ \to X^ is called the Tate spectrum of ''X''. ''G''-Galois extension in the sense of Rognes This notion is due to J. Rognes . Let ''A'' be an Eā-ring with an action of a finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum (topology)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory\mathcal^*:\text^ \to \text,there exist spaces E^k such that evaluating the cohomology theory in degree k on a space X is equivalent to computing the homotopy classes of maps to the space E^k, that is\mathcal^k(X) \cong \left , E^k\right/math>.Note there are several different categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory. The definition of a spectrum There are many variations of the definition: in general, a ''spectrum'' is any sequence X_n of pointed topological spaces or pointed simplicial sets together with ... [...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 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] |
Group Action
In mathematics, a group action of a group G on a set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformations form a group under function composition; for example, the rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures dra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mapping Spectrum
In 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 t ..., the mapping spectrum F(X, Y) of spectra ''X'', ''Y'' is characterized by : \wedge Y, Z= , F(Y, Z) References {{topology-stub Algebraic topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex K-theory
In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early work on topological -theory is due to Michael Atiyah and Friedrich Hirzebruch. Definitions Let be a compact Hausdorff space and k= \R or \Complex. Then K_k(X) is defined to be the Grothendieck group of the commutative monoid of isomorphism classes of finite-dimensional -vector bundles over under Whitney sum. Tensor product of bundles gives -theory a commutative ring structure. Without subscripts, K(X) usually denotes complex -theory whereas real -theory is sometimes written as KO(X). The remaining discussion is focused on complex -theory. As a first example, note that the -theory of a point is the integers. This is because vector bundles over a point are trivial and thus classified by their rank and the Grothendieck group of the natura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Bundle
In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle E can be promoted to a complex vector bundle, the complexification :E \otimes \mathbb ; whose fibers are E_x\otimes_\R \C. Any complex vector bundle over a paracompact space admits a hermitian metric. The basic invariant of a complex vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X is a complex manifold and if the local trivializations are biholomorphic. Complex structure A complex vector bundle can be thought of as a real vector bundle with an additional structure, the complex structure. By definition, a complex structure is a bundle map between a real vector bundle E and itself: :J: E \to E such that J acts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Vector Bundle
In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle E can be promoted to a complex vector bundle, the complexification :E \otimes \mathbb ; whose fibers are E_x\otimes_\R \C. Any complex vector bundle over a paracompact space admits a hermitian metric. The basic invariant of a complex vector bundle is a Chern class. A complex vector bundle is canonically oriented; in particular, one can take its Euler class. A complex vector bundle is a holomorphic vector bundle if X is a complex manifold and if the local trivializations are biholomorphic. Complex structure A complex vector bundle can be thought of as a real vector bundle with an additional structure, the complex structure. By definition, a complex structure is a bundle map between a real vector bundle E and itself: :J: E \to E such that J ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate Spectrum
Tate is an institution that houses, in a network of four art galleries, the United Kingdom's national collection of British art, and international modern and contemporary art. It is not a government institution, but its main sponsor is the UK Department for Culture, Media and Sport. The name "Tate" is used also as the operating name for the corporate body, which was established by the Museums and Galleries Act 1992 as "The Board of Trustees of the Tate Gallery". The gallery was founded in 1897 as the National Gallery of British Art. When its role was changed to include the national collection of modern art as well as the national collection of British art, in 1932, it was renamed the Tate Gallery after sugar magnate Henry Tate of Tate & Lyle, who had laid the foundations for the collection. The Tate Gallery was housed in the current building occupied by Tate Britain, which is situated in Millbank, London. In 2000, the Tate Gallery transformed itself into the current-day Tate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E-infinity Ring
In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A_\infty-ring is called an E_\infty-ring. While originally motivated by questions of geometric topology and bundle theory, they are today most often used in stable homotopy theory. Background Highly structured ring spectra have better formal properties than multiplicative cohomology theories ā a point utilized, for example, in the construction of topological modular forms, and which has allowed also new constructions of more classical objects such as Morava K-theory. Beside their formal properties, E_\infty-structures are also important in calculations, since they allow for operations in the underlying cohomology theory, analogous to (and generalizing) the well-known Steenrod operations in ordinary cohomology. As not every cohomology theory allows such operations, not every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bousfield Class
In algebraic topology, the Bousfield class of, say, a spectrum ''X'' is the set of all (say) spectra ''Y'' whose smash product with ''X'' is zero: X \otimes Y = 0. Two objects are Bousfield equivalent if their Bousfield classes are the same. The notion applies to module spectra and in that case one usually qualifies a ring spectrum over which the smash product is taken. See also *Bousfield localization In category theory, a branch of mathematics, a (left) Bousfield localization of a model category replaces the model structure with another model structure with the same cofibrations but with more weak equivalences. Bousfield localization is named a ... External linksNcatlab.org References {{topology-stub Topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segal Conjecture
Segal's Burnside ring conjecture, or, more briefly, the Segal conjecture, is a theorem in homotopy theory, a branch of mathematics. The theorem relates the Burnside ring of a finite group ''G'' to the stable cohomotopy of the classifying space ''BG''. The conjecture was made in the mid 1970s by Graeme Segal and proved in 1984 by Gunnar Carlsson. This statement is still commonly referred to as the Segal conjecture, even though it now has the status of a theorem. Statement of the theorem The Segal conjecture has several different formulations, not all of which are equivalent. Here is a weak form: there exists, for every finite group ''G'', an isomorphism :\varprojlim \pi_S^0 \left( BG^_+ \right) \to \widehat(G). Here, lim denotes the inverse limit, S* denotes the stable cohomotopy ring, ''B'' denotes the classifying space, the superscript ''k'' denotes the ''k''-skeleton, and the subscript + denotes the addition of a disjoint basepoint. On the right-hand side, the hat denotes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MathOverflow
MathOverflow is a mathematics question-and-answer (Q&A) website, which serves as an online community of mathematicians. It allows users to ask questions, submit answers, and rate both, all while getting merit points for their activities. It is a part of the Stack Exchange Network, but distinct fromath.stackexchange.com It is primarily for asking questions on mathematics research ā i.e. related to unsolved problems and the extension of knowledge of mathematics into areas that are not yet known ā and does not welcome requests from non-mathematicians for instruction, for example homework exercises. It does welcome various questions on other topics that might normally be discussed among mathematicians, for example about publishing, refereeing, advising, getting tenure, etc. It is generally inhospitable to questions perceived as tendentious or argumentative. Origin and history The website was started by Berkeley graduate students and postdocs Anton Geraschenko, David ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
