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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), ''Framed'' (1930 film), a pre ... References * Algebraic ... [...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 mo ... [...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] |
Spectrum (homotopy Theory)
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 colors in visible light after passing through a prism. In the optical spectrum, light wavelength is viewed as continuous, and spectral colors are seen to blend into one another smoothly when organized in order of their corresponding wavelengths. As scientific understanding of light advanced, the term came to apply to the entire electromagnetic spectrum, including radiation not visible to the human eye. ''Spectrum'' has since been applied by analogy to topics outside optics. Thus, one might talk about the " spectrum of political opinion", or the "spectrum of activity" of a drug, or the "autism spectrum". In these uses, values within a spectrum may not be associated with precisely quantifiable numbers or definitions. Such uses imply a broad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suspension Spectrum
Suspension or suspended may refer to: Science and engineering * Car suspension * Cell suspension or suspension culture, in biology * Guarded suspension, a software design pattern in concurrent programming suspending a method call and the calling thread until a precondition (guard) is satisfied * Magnetic suspension, a method by which an object is suspended with no support other than magnetic fields * Suspension (topology), in mathematics * Suspension (dynamical systems), in mathematics * Suspension of a ring, in mathematics * Suspension (chemistry), small solid particles suspended in a liquid ** Colloidal suspension * Suspension (mechanics), system allowing a machine to move smoothly with reduced shock * Suspensory behavior, arboreal locomotion of primates * Suspend to disk, also known as hibernation, powering down a computer while retaining its state. * The superstructure of a suspension bridge Temporary revocation of privileges * Suspension (punishment), temporary exclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere (geometry)
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the center (geometry), ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is spherical Earth, often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suspension (mathematics)
In topology, a branch of mathematics, the suspension of a topological space ''X'' is intuitively obtained by stretching ''X'' into a cylinder and then collapsing both end faces to points. One views ''X'' as "suspended" between these end points. The suspension of ''X'' is denoted by ''SX'' or susp(''X''). There is a variant of the suspension for a pointed space, which is called the reduced suspension and denoted by Σ''X''. The "usual" suspension ''SX'' is sometimes called the unreduced suspension, unbased suspension, or free suspension of ''X'', to distinguish it from Σ''X.'' Free suspension The (free) suspension SX of a topological space X can be defined in several ways. 1. SX is the quotient space (X \times ,1/(X\times \)\big/ ( X\times \). In other words, it can be constructed as follows: * Construct the cylinder X \times ,1/math>. * Consider the entire set X\times \ as a single point ("glue" all its points together). * Consider the entire set X\times \ as a single p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism
In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. Very roughly speaking, a topological space is a geometric object, and a homeomorphism results from a continuous deformation of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this description can be misleading. Some continuous deformations do not produce homeomorphisms, such as the deformation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Group
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] |
Localization Of A Topological Space
In mathematics, well-behaved topological spaces can be localized at primes, in a similar way to the localization of a ring at a prime. This construction was described by Dennis Sullivan in 1970 lecture notes that were finally published in . The reason to do this was in line with an idea of making topology, more precisely algebraic topology, more geometric. Localization of a space ''X'' is a geometric form of the algebraic device of choosing 'coefficients' in order to simplify the algebra, in a given problem. Instead of that, the localization can be applied to the space ''X'', directly, giving a second space ''Y''. Definitions We let ''A'' be a subring of the rational numbers, and let ''X'' be a simply connected CW complex. Then there is a simply connected CW complex ''Y'' together with a map from ''X'' to ''Y'' such that *''Y'' is ''A''-local; this means that all its homology groups are modules over ''A'' *The map from ''X'' to ''Y'' is universal for (homotopy classes of) maps from ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chromatic Homotopy Theory
In mathematics, chromatic homotopy theory is a subfield of stable homotopy theory that studies complex-oriented cohomology theory, complex-oriented cohomology theories from the "chromatic" point of view, which is based on Daniel Quillen, 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 Topological modular forms, tmf. Chromatic convergence theorem In algebraic topology, the chromatic convergence theorem states the homotopy limit of the chromatic tower (defined below) of a finite local spectrum, ''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-loca ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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), ''Framed'' (1930 film), a pre-code crime action starring Evelyn Brent, Regis Toomey and Ralf Harolde *Framed (1940 film), ''Framed'' (1940 film), an American crime film *Framed (1947 film), ''Framed'' (1947 film), a film noir starring Glenn Ford and Janis Carter *Framed (1975 film), ''Framed'' (1975 film), a crime drama based on a novel starring Joe Don Baker and Conny Van Dyke *Framed (1990 film), ''Framed'' (1990 film), HBO made-for-television film starring Jeff Goldblum *Framed (TV series), ''Framed'' (TV series), a 1992 drama series of 4 x 1hr episodes, and cut to a 2hr TV film for the US (and elsewhere) in 1993 *Framed (Spider-Man: 1994 TV series), "Framed" (''Spider-Man'': 1994 TV series), a 1996 episode of the animated series *Framed (Dexter's Laboratory), "Fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
