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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] |
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 functions on the group of chains in homology theory. From its beginning 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 algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum (topology)
In algebraic topology, a branch of mathematics, a spectrum is an object representable functor, representing a Cohomology#Generalized cohomology theories, generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory\mathcal^*:\text^ \to \text,there exist spaces E^k such that evaluating the cohomology theory in degree k on a space X is equivalent to computing the homotopy classes of maps to the space E^k, that is\mathcal^k(X) \cong \left[X, E^k\right].Note there are several different category (mathematics), categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory. The definition of a spectrum There are many variations of the definition: in general, a ''spectrum'' is any s ... [...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 inclu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Localization Of A 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 fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wedge Product
A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converting a force applied to its blunt end into forces perpendicular (normal) to its inclined surfaces. The mechanical advantage of a wedge is given by the ratio of the length of its slope to its width..''McGraw-Hill Concise Encyclopedia of Science & Technology'', Third Ed., Sybil P. Parker, ed., McGraw-Hill, Inc., 1992, p. 2041. Although a short wedge with a wide angle may do a job faster, it requires more force than a long wedge with a narrow angle. The force is applied on a flat, broad surface. This energy is transported to the pointy, sharp end of the wedge, hence the force is transported. The wedge simply transports energy in the form of friction and collects it to the pointy end, consequently breaking the item. History Wedges have exi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Suspension (topology)
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 variation of the suspension for 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 \, 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 point ("g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daniel Quillen
Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 1978. From 1984 to 2006, he was the Waynflete Professor of Pure Mathematics at Magdalen College, Oxford. Education and career Quillen was born in Orange, New Jersey, and attended Newark Academy. He entered Harvard University, where he earned both his AB, in 1961, and his PhD in 1964; the latter completed under the supervision of Raoul Bott, with a thesis in partial differential equations. He was a Putnam Fellow in 1959. Quillen obtained a position at the Massachusetts Institute of Technology after completing his doctorate. He also spent a number of years at several other universities. He visited France twice: first as a Sloan Fellow in Paris, during the academic year 1968–69, where he was greatly influenced by Grothendieck, and the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idempotent
Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of places in abstract algebra (in particular, in the theory of projector (linear algebra), projectors and closure operators) and functional programming (in which it is connected to the property of referential transparency). The term was introduced by American mathematician Benjamin Peirce in 1870 in the context of elements of algebras that remain invariant when raised to a positive integer power, and literally means "(the quality of having) the same power", from + ''wikt:potence, potence'' (same + power). Definition An element x of a set S equipped with a binary operator \cdot is said to be ''idempotent'' under \cdot if : . The ''binary operation'' \cdot is said to be ''idempotent'' if : . Examples * In the monoid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring Spectra
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, and Morava K-theory. See also *Highly structured ring spectrum In mathematics, a highly structured ring spectrum or A_\infty-ring is an object in homotopy theory encoding a refinement of a multiplicative structure on a cohomology theory. A commutative version of an A_\infty-ring is called an E_\infty-ring. W ... References * Algebraic topology Homotopy theo ... [...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 a ... [...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 th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 of spheres viewed as topological spaces, forgetting about their precise geometry. Unlike homology groups, which are also topological invariants, the homotopy groups are surprisingly complex and difficult to compute. The -dimensional unit sphere — called the -sphere for brevity, and denoted as — generalizes the familiar circle () and the ordinary sphere (). The -sphere may be defined geometrically as the set of points in a Euclidean space of dimension located at a unit distance from the origin. The -th ''homotopy group'' summarizes the different ways in which the -dimensional sphere can be mapped continuously into the sphere . This summary does not distinguish between two mappings if one can be continuously deformed to the oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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