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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] |
Goro Nishida
was a Japanese mathematician. He was a leading member of the Japanese school of homotopy theory, following in the tradition of Hiroshi Toda. Nishida received his Ph.D. from Kyoto University in 1973, after spending the 1971–72 academic year at the University of Manchester in England. He then became a professor at Kyoto University in 1990. His proof in 1973 of Michael Barratt's conjecture (that positive-degree elements in the stable homotopy ring of spheres are nilpotent) was a major breakthrough: following Frank Adams' solution of the Hopf invariant one problem, it marked the beginning of a new global understanding of algebraic topology. His contributions to the field were celebrated in 2003 at the NishidaFest in Kinosaki, followed by a satellite conference at the Nagoya Institute of Technology; the proceedings were published in ''Geometry and Topology'''s monograph series. In 2000 he was the leading organizer for a concentration year at the Japan–US Mathematics Institut ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotence Theorem
In algebraic topology, the nilpotence theorem gives a condition for an element in the homotopy groups of a ring spectrum to be nilpotent, in terms of the complex cobordism spectrum \mathrm. More precisely, it states that for any ring spectrum R, the kernel of the map \pi_\ast R \to \mathrm_\ast(R) consists of nilpotent elements. It was conjectured by and proved by . Nishida's theorem showed that elements of positive degree of the homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure ... are nilpotent. This is a special case of the nilpotence theorem. See also * Ravenel's conjectures References * * . Open online version.* Further reading Connection of ''X(n)'' spectra to formal group laws Homotopy theory Theorems in algebraic topology {{Topolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivariant Stable Homotopy Theory
In mathematics, more specifically in topology, the equivariant stable homotopy theory is a subfield of equivariant topology that studies a spectrum with 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 ... instead of a space with group action, as in stable homotopy theory. The field has become more active recently because of its connection to algebraic K-theory. See also * Equivariant K-theory * G-spectrum (spectrum with an action of an (appropriate) group ''G'') References External linksCreating Equivariant Stable Homotopy Theory Homotopy theory {{topology-stub ... [...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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams Filtration
In mathematics, especially in the area of algebraic topology known as stable homotopy theory, the Adams filtration and the Adams–Novikov filtration allow a stable homotopy group to be understood as built from layers, the ''n''th layer containing just those maps which require at most ''n'' auxiliary spaces in order to be a composition of homologically trivial maps. These filtrations, named after Frank Adams and Sergei Novikov, are of particular interest because the Adams (–Novikov) spectral sequence converges to them.https://scholar.harvard.edu/files/rastern/files/adamsspectralsequence.pdf&ved=2ahUKEwjI-JDstJOEAxU4WEEAHThuAOI4ChAWegQIGxAB&usg=AOvVaw3_Ga18v-jBF18FS-U4KgPV Definition The group of stable homotopy classes ,Y/math> between two spectra ''X'' and ''Y'' can be given a filtration by saying that a map f\colon X\to Y has filtration ''n'' if it can be written as a composite of maps :X=X_0 \to X_1 \to \cdots \to X_n = Y such that each individual map X_i\to X_ induces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibration Sequence
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in Postnikov systems or obstruction theory. In this article, all mappings are continuous mappings between topological spaces. Formal definitions Homotopy lifting property A mapping p \colon E \to B satisfies the homotopy lifting property for a space X if: * for every homotopy h \colon X \times , 1\to B and * for every mapping (also called lift) \tilde h_0 \colon X \to E lifting h, _ = h_0 (i.e. h_0 = p \circ \tilde h_0) there exists a (not necessarily unique) homotopy \tilde h \colon X \times , 1\to E lifting h (i.e. h = p \circ \tilde h) with \tilde h_0 = \tilde h, _. The following commutative diagram shows the situation: Fibration A fibration (also called Hurewicz fibration) is a mapping p \colon E \to B satisfying the homotopy lifting property for all spaces X. The space B is called base sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cofibration
In mathematics, in particular homotopy theory, a continuous mapping between topological spaces :i: A \to X, is a ''cofibration'' if it has the homotopy extension property with respect to all topological spaces S. That is, i is a cofibration if for each topological space S, and for any continuous maps f, f': A\to S and g:X\to S with g\circ i=f, for any homotopy h : A\times I\to S from f to f', there is a continuous map g':X \to S and a homotopy h': X\times I \to S from g to g' such that h'(i(a),t)=h(a,t) for all a\in A and t\in I. (Here, I denotes the unit interval ,1/math>.) This definition is formally dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton duality in topology. Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of model category as a formal framework for doing homotopy theory in more general categories; a m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Category
A stable is a building in which working animals are kept, especially horses or oxen. The building is usually divided into stalls, and may include storage for equipment and feed. Styles There are many different types of stables in use today; the American-style stable called a barn, for instance, is a large barn with a door at each end and individual stalls inside or free-standing stables with top and bottom-opening doors. The term "stable" is additionally utilised to denote a business or a collection of animals under the care of a single owner, irrespective of their housing or whereabouts. A building with tie stalls is also known as stanchion or stall barn, where animals are tethered by the head or neck to their stall. It is mostly used in the dairy cow industry, but traditionally horses were also tied up. The exterior design of a stable can vary widely based on climate, building materials, historical period and cultural styles of architecture. A wide range of building m ... [...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] |
