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Chromatic Homotopy Theory
In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theories from the "chromatic" point of view, which is based on Quillen's work relating cohomology theories to formal groups. In this picture, theories are classified in terms of their "chromatic levels"; i.e., the heights of the formal groups that define the theories via the Landweber exact functor theorem. Typical theories it studies include: complex K-theory, elliptic cohomology, Morava K-theory and tmf. Chromatic convergence theorem In algebraic topology, the chromatic convergence theorem states the homotopy limit of the chromatic tower (defined below) of a finite ''p''-local spectrum X is X itself. The theorem was proved by Hopkins and Ravenel. Statement Let L_ denotes the Bousfield localization with respect to the Morava E-theory and let X be a finite, p-local spectrum. Then there is a tower associated to the localizations :\cdots \rightarrow L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space X, the homotopy groups \pi_(\Sigma^n X) stabilize for n sufficiently large. In particular, the homotopy groups of spheres \pi_(S^n) stabilize for n\ge k + 2. For example, :\langle \text_\rangle = \Z = \pi_1(S^1)\cong \pi_2(S^2)\cong \pi_3(S^3)\cong\cdots :\langle \eta \rangle = \Z = \pi_3(S^2)\to \pi_4(S^3)\cong \pi_5(S^4)\cong\cdots In the two examples above all the maps between homotopy groups are applications of the suspension functor. The first example is a standard corollary of the Hurewicz theorem, that \pi_n(S^n)\cong \Z. In the second example the Hopf map, \eta, is mapped to its suspension \Sigma\eta, which generates \pi_4(S^3)\cong \Z/2. One of the most i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 after Aldridge Bousfield, who first introduced this technique in the context of localization of topological spaces and spectra. Model category structure of the Bousfield localization Given a class ''C'' of morphisms in a model category ''M'' the left Bousfield localization is a new model structure on the same category as before. Its equivalences, cofibrations and fibrations, respectively, are * the ''C''-local equivalences * the original cofibrations of ''M'' and (necessarily, since cofibrations and weak equivalences determine the fibrations) * the maps having the right lifting property with respect to the cofibrations in ''M'' which are also ''C''-local equivalences. In this definition, a ''C''-local equivalence is a map f\colon X \t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams-Novikov Spectral Sequence
In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre. Motivation For everything below, once and for all, we fix a prime ''p''. All spaces are assumed to be CW complexes. The ordinary cohomology groups H^*(X) are understood to mean H^*(X; \Z/p\Z). The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces ''X'' and ''Y''. This is extraordinarily ambitious: in particular, when ''X'' is S^n, these maps form the ''n''th homotopy group of ''Y''. A more reasonable (but still very difficult!) goal is to understand the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Spectral Sequence
In mathematics, the chromatic spectral sequence is a spectral sequence, introduced by , used for calculating the initial term of the Adams spectral sequence for Brown–Peterson cohomology, which is in turn used for calculating the stable homotopy groups of spheres. See also * Chromatic homotopy theory * Adams-Novikov spectral sequence In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is no ... * p-local spectrum References * * Spectral sequences {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moduli Stack Of Formal Group Laws
In algebraic geometry, the moduli stack of formal group laws is a stack classifying formal group laws and isomorphisms between them. It is denoted by \mathcal_. It is a "geometric “object" that underlies the chromatic approach to the stable homotopy theory, a branch of algebraic topology. Currently, it is not known whether \mathcal_ is a derived stack or not. Hence, it is typical to work with stratifications. Let \mathcal^n_ be given so that \mathcal^n_(R) consists of formal group laws over ''R'' of height exactly ''n''. They form a stratification of the moduli stack \mathcal_. \operatorname \overline \to \mathcal^n_ is faithfully flat. In fact, \mathcal^n_ is of the form \operatorname \overline / \operatorname(\overline, f) where \operatorname(\overline, f) is a profinite group called the Morava stabilizer group. The Lubin–Tate theory describes how the strata \mathcal^n_ fit together. References * * Further reading * Topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ravenel Conjectures
In mathematics, the Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel at the end of a paper published in 1984. It was earlier circulated in preprint. The problems involved have largely been resolved, with all but the "telescope conjecture" being proved in later papers by others. The telescope conjecture is now generally believed not to be true, though there are some conflicting claims concerning it in the published literature, and is taken to be an open problem. Ravenel's conjectures exerted influence on the field through the founding of the approach of chromatic homotopy theory. The first of the seven conjectures, then the ''nilpotence conjecture'', was proved in 1988 and is now known as the nilpotence theorem. The telescope conjecture, which was #4 on the original list, remains of substantial interest because of its connection with the convergence of an Adams–Novikov spectral sequence. While opinion has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Redshift Conjecture
In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory K(R) has chromatic level one higher than that of a complex-oriented ring spectrum ''R''. It was formulated by John Rognes in a lecture at Schloss Ringberg Schloss Ringberg (Ringberg Castle) is located in the Bavarian Alps, 50 km south of Munich, on a foothill overlooking the Tegernsee. Not open to the general public, it is a property of the Max Planck Society and used for conferences. Histo ..., Germany, in January 1999, and made more precise by him in a lecture at Mathematische Forschungsinstitut Oberwolfach, Germany, in September 2000. In July 2022, Burklund, Schlank and Yuan announced a solution of a version of the redshift conjecture for arbitrary E_-ring spectra, after Hahn and Wilson did so earlier in the case of the truncated Brown-Peterson spectra BP.Burklund, Schlank, Yuan (2022). ''The Chromatic Nullstellensatz'' References ;Notes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Cohomology
In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms. History and motivation Historically, elliptic cohomology arose from the study of elliptic genera. It was known by Atiyah and Hirzebruch that if S^1 acts smoothly and non-trivially on a spin manifold, then the index of the Dirac operator vanishes. In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerning S^1-actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera. In turn, Witten related these to (conjectural) index theory on free loop spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and Ravenel in the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Spectrum
In stable homotopy theory, a branch of mathematics, the sphere spectrum ''S'' is the monoidal unit in the category of spectra. It is the suspension spectrum of ''S''0, i.e., a set of two points. Explicitly, the ''n''th space in the sphere spectrum is the ''n''-dimensional sphere ''S''''n'', and the structure maps from the suspension of ''S''''n'' to ''S''''n''+1 are the canonical homeomorphisms. The ''k''-th homotopy group of a sphere spectrum is the ''k''-th stable homotopy group of spheres. The localization of the sphere spectrum at a prime number ''p'' is called the local sphere at ''p'' and is denoted by S_. See also * Chromatic homotopy theory * Adams-Novikov spectral sequence *Framed cobordism Framed may refer to: Common meanings *A painting or photograph that has been placed within a picture frame *Someone falsely shown to be guilty of a crime as part of a frameup Film and television * ''Framed'' (1930 film), a pre-code crime action ... References * Algebraic topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic K-theory
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a dis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Morava E-theory
Morava may refer to: Rivers * Great Morava (''Velika Morava''; or only Morava), a river in central Serbia, and its tributaries: ** South Morava (''Južna Morava'') *** Binač Morava (''Binačka Morava'') ** West Morava (''Zapadna Morava'') * Morava (river), a river in the Czech Republic, Austria and Slovakia Places * , a village in the Svishtov Municipality, Bulgaria * Morava (Kočevje), a village in the municipality of Kočevje, Slovenia * Morava (Serbian Cyrillic: Морава), a former name for Gnjilane (Albanian: Gjilan) * Suva Morava ("Dry Morava"), a village in the municipality of Vladičin Han, Serbia * Dolní Morava ("Lower Morava"), a municipality and village in the Ústí nad Orlicí District, Czech Republic * Malá Morava ("Little Morava"), a municipality and village in the Šumperk District, Czech Republic * , a mountain in southeast Albania, near Korçë * Morava Banovina, a province of the Kingdom of Yugoslavia between 1929 and 1941 * Donja Morava ("Lower Morav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Spectrum
Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administration * Local news, coverage of events in a local context which would not normally be of interest to those of other localities * Local union, a locally based trade union organization which forms part of a larger union Arts, entertainment, and media * ''Local'' (comics), a limited series comic book by Brian Wood and Ryan Kelly * ''Local'' (novel), a 2001 novel by Jaideep Varma * Local TV LLC, an American television broadcasting company * Locast, a non-profit streaming service offering local, over-the-air television * ''The Local'' (film), a 2008 action-drama film * '' The Local'', English-language news websites in several European countries Computing * .local, a network address component * Local variable, a variable that is given loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
