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Adams Resolution
In mathematics, specifically algebraic topology, there is a resolution analogous to free resolutions of Spectrum (topology), spectra yielding a tool for constructing the Adams spectral sequence. Essentially, the idea is to take a connective spectrum of finite type X and iteratively resolve with other spectra that are in the homotopy kernel of a map resolving the cohomology classes in H^*(X;\mathbb/p) using Eilenberg–Maclane spectrum, Eilenberg–MacLane spectra. This construction can be generalized using a spectrum E, such as the Brown–Peterson cohomology, Brown–Peterson spectrum BP, or the complex cobordism spectrum MU, and is used in the construction of the Adams–Novikov spectral sequencepg 49. Construction The mod p Adams resolution (X_s,g_s) for a spectrum X is a certain "chain-complex" of spectra induced from recursively looking at the fibers of maps into generalized Eilenberg–MacLane spectra, Eilenberg–Maclane spectra giving generators for the cohomology of resol ... [...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] |
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] |
Hopf Algebroid
In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf ''k''-algebroids. If ''k'' is a field, a commutative ''k''-algebroid is a cogroupoid object in the category of ''k''-algebras; the category of such is hence dual to the category of groupoid ''k''-schemes. This commutative version has been used in 1970-s in algebraic geometry and stable homotopy theory. The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000). They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf alg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cobar Complex
Cobar is a town in Outback New South Wales, Australia, whose economy is based mainly upon base metals and gold mining. The town is by road northwest of the state capital, Sydney. It is at the crossroads of the Kidman Way and Barrier Highway. The town and the local government area, the Cobar Shire, are on the eastern edge of the Outback. At the 2021 census, the town of Cobar had a population of 3,369. The Shire has a population of approximately 4,700 and an area of . Many sights of cultural interest can be found in and around Cobar. The town retains much of its colonial 19th-century architecture. The Towsers Huts, 3 km south of town but currently inaccessible to the public, are ruins of very simple colonial dwellings from around 1870. The ancient Aboriginal rock paintings at Mount Grenfell are some of the largest and most important in Australia. The Cobar Sound Chapel opened in April 2022. History Indigenous origins The Cobar area is part of the traditional territor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Spectrum
In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map :''μ'': ''E'' ∧ ''E'' → ''E'' and a unit map : ''η'': ''S'' → ''E'', where ''S'' is the sphere spectrum. These maps have to satisfy associativity and unitality conditions up to homotopy, much in the same way as the multiplication of a ring is associative and unital. That is, : ''μ'' (id ∧ ''μ'') ~ ''μ'' (''μ'' ∧ id) and : ''μ'' (id ∧ ''η'') ~ id ~ ''μ''(''η'' ∧ id). Examples of ring spectra include singular homology with coefficients in a ring, complex cobordism, K-theory In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometr ..., and Morava K-theory. See also * Highly structured ring spectrum References * Algebraic topology Spectra (topology) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monomorphism
In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2. Monomorphisms are a categorical generalization of injective functions (also called "one-to-one functions"); in some categories the notions coincide, but monomorphisms are more general, as in the examples below. In the setting of posets intersections are idempotent: the intersection of anything with itself is itself. Monomorphisms generalize this property to arbitrary categories. A morphism is a monomorphism if it is idempotent with respect to pullbacks. The categorical dual of a monomorphism is an epimorphism, that is, a monomorphism in a categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Steenrod Algebra
Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual number, a number system used in automatic differentiation * Dual (grammatical number), a grammatical category used in some languages * Dual county, a Gaelic games county which competes in both Gaelic football and hurling * Dual diagnosis, a psychiatric diagnosis of co-occurrence of substance abuse and a mental problem * Dual fertilization, simultaneous application of a P-type and N-type fertilizer * Dual impedance, electrical circuits that are the dual of each other * Dual SIM cellphone supporting use of two SIMs * Aerochute International Dual a two-seat Australian powered parachute design Acronyms and other uses * Dual (brand), a manufacturer of Hifi equipment * DUAL (cognitive architecture), an artificial intelligence design model * DUAL algorit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Fiber
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f:A \to B. It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups\cdots \to \pi_(B) \to \pi_n(\text(f)) \to \pi_n(A) \to \pi_n(B) \to \cdotsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangleC(f)_\bullet 1\to A_\bullet \to B_\bullet \xrightarrowgives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber. Construction The homotopy fiber has a simple description for a continuous map f:A \to B. If we replace f by a fibr ... [...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] |
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] |
Complex Cobordism
In mathematics, complex cobordism is a generalized cohomology theory related to cobordism of manifolds. Its spectrum is denoted by MU. It is an exceptionally powerful cohomology theory, but can be quite hard to compute, so often instead of using it directly one uses some slightly weaker theories derived from it, such as Brown–Peterson cohomology or Morava K-theory, that are easier to compute. The generalized homology and cohomology complex cobordism theories were introduced by using the Thom spectrum. Spectrum of complex cobordism The complex bordism MU^*(X) of a space X is roughly the group of bordism classes of manifolds over X with a complex linear structure on the stable normal bundle. Complex bordism is a generalized homology theory, corresponding to a spectrum MU that can be described explicitly in terms of Thom spaces as follows. The space MU(n) is the Thom space of the universal n-plane bundle over the classifying space BU(n) of the unitary group U(n). The natural i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
