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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] |
Generalized Cohomology Theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are function (mathematics), functions on the group of chain (algebraic topology), chains in homology theory. From its start in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and abstract algebra, algebra. The terminology tends to hide the fact that cohomology, a Covariance and contravariance of functors, c ... [...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] |
Hopf Algebra
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations. Hopf algebras occur naturally in algebraic topology, where they originated and are related to the H-space concept, in group scheme theory, in group theory (via the concept of a group ring), and in numerous other places, making them probably the most familiar type of bialgebra. Hopf algebras are also studied in their own right, with much work on specific classes of examples on the one hand and classification problems o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
\beta 1, \beta 2, \ldots
Beta (, ; uppercase , lowercase , or cursive ; or ) is the second letter of the Greek alphabet. In the system of Greek numerals, it has a value of 2. In Ancient Greek, beta represented the voiced bilabial plosive . In Modern Greek, it represents the voiced bilabial fricative while in borrowed words is instead commonly transcribed as μπ. Letters that arose from beta include the Roman letter and the Cyrillic letters and . Name Like the names of most other Greek letters, the name of beta was adopted from the acrophonic name of the corresponding letter in Phoenician, which was the common Semitic word ('house', compare and ). In Greek, the name was , pronounced in Ancient Greek. It is spelled in modern monotonic orthography and pronounced . History The letter beta was derived from the Phoenician letter beth . The letter Β had the largest number of highly divergent local forms. Besides the standard form (either rounded or pointed, ), there were forms as varied as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cf 1, Cf 2, \ldots
The Latin tag ''cf.'', much used in text books, means 'compare'. CF, Cf, cf and similar may refer to: Arts and entertainment * "Commercial film", the term for a television advertisement in South Korea * Captain Flamingo, a Canadian animated television series * Christopher Forgues, an artist and musician based in Providence, Rhode Island, United States; known as C.F. Computing * .cf, the Top-Level Domain for Central African Republic * .NET Compact Framework, a version of the .NET Framework for mobile and embedded devices * Adobe ColdFusion, a web application development platform * Collaborative filtering, a method of making predictions by collecting information from many users * CompactFlash, a type of memory card * Compact Floppy, a variation of floppy disk * Consolidation function, in computer science * Core Foundation, a C application programming interface in Mac OS X * Coupling Facility, an IBM mainframe feature * CurseForge, a website offering mods for various video games ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series Ring
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, of the form \sum_^\infty a_nx^n=a_0+a_1x+ a_2x^2+\cdots, where the a_n, called ''coefficients'', are numbers or, more generally, elements of some ring, and the x^n are formal powers of the symbol x that is called an indeterminate or, commonly, a variable. Hence, power series can be viewed as a generalization of polynomials where the number of terms is allowed to be infinite, and differ from usual power series by the absence of convergence requirements, which implies that a power series may not represent a function of its variable. Formal power series are in one to one correspondence with their sequences of coefficients, but the two concepts must not be confused, since th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brown–Peterson Cohomology
In mathematics, Brown–Peterson cohomology is a generalized cohomology theory introduced by , depending on a choice of prime ''p''. It is described in detail by . Its representing spectrum is denoted by BP. Complex cobordism and Quillen's idempotent Brown–Peterson cohomology BP is a summand of MU(''p''), which is complex cobordism MU localized at a prime ''p''. In fact MU''(p)'' is a wedge product of suspensions of BP. For each prime ''p'', Daniel Quillen showed there is a unique idempotent map of ring spectra ε from MUQ(''p'') to itself, with the property that ε( P''n'' is P''n''if ''n''+1 is a power of ''p'', and 0 otherwise. The spectrum BP is the image of this idempotent ε. Structure of BP The coefficient ring \pi_*(\text) is a polynomial algebra over \Z_ on generators v_n in degrees 2(p^n-1) for n\ge 1. \text_*(\text) is isomorphic to the polynomial ring \pi_*(\text) _1, t_2, \ldots/math> over \pi_*(\text) with generators t_i in \text_(\text) of degrees 2 (p^i-1) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazard's Universal Ring
In mathematics, Lazard's universal ring is a ring introduced by Michel Lazard in over which the universal commutative one-dimensional formal group law is defined. There is a universal commutative one-dimensional formal group law over a universal commutative ring defined as follows. We let :F(x,y) be :x+y+\sum_ c_ x^i y^j for indeterminates c_, and we define the universal ring ''R'' to be the commutative ring generated by the elements c_, with the relations that are forced by the associativity and commutativity laws for formal group laws. More or less by definition, the ring ''R'' has the following universal property: :For every commutative ring ''S'', one-dimensional formal group laws over ''S'' correspond to ring homomorphisms from ''R'' to ''S''. The commutative ring ''R'' constructed above is known as Lazard's universal ring. At first sight it seems to be incredibly complicated: the relations between its generators are very messy. However Lazard proved that it has a ver ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Group Law
In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology. Definitions A one-dimensional formal group law over a commutative ring ''R'' is a power series ''F''(''x'',''y'') with coefficients in ''R'', such that # ''F''(''x'',''y'') = ''x'' + ''y'' + terms of higher degree # ''F''(''x'', ''F''(''y'',''z'')) = ''F''(''F''(''x'',''y''), ''z'') (associativity). The simplest example is the additive formal group law ''F''(''x'', ''y'') = ''x'' + ''y''. The idea of the definition is that ''F'' should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutative Ring Spectrum
In algebraic topology, a commutative ring spectrum, roughly equivalent to a E_\infty-ring spectrum, is a commutative monoid in a goodsymmetric monoidal with respect to smash product and perhaps some other conditions; one choice is the category of symmetric spectra category of spectra. The category of commutative ring spectra over the field \mathbb of rational numbers is Quillen equivalent to the category of differential graded algebras over \mathbb. Example: The Witten genus may be realized as a morphism of commutative ring spectra MString → tmf. See also: simplicial commutative ring, highly structured ring spectrum and derived scheme. Terminology Almost all reasonable categories of commutative ring spectra can be shown to be Quillen equivalent to each other. Thus, from the point view of the 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
