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Brown–Gitler Spectrum
In the mathematical discipline of topology, the Brown–Gitler spectrum is a Spectrum (topology), spectrum whose cohomology is a certain cyclic module over the Steenrod algebra. Brown–Gitler spectra are defined by the isomorphism: : \Sigma^n A/ \ A \cong G(n). History The concept was introduced by mathematicians Edgar H. Brown and Samuel Gitler Hammer, Samuel Gitler in a 1973 paper. In topology, Brown–Gitler spectrum is related to the concepts of the Segal conjecture (proven in 1984) and the Burnside ring. Applications Brown–Gitler spectra have had many important applications in homotopy theory. References External links * {{DEFAULTSORT:Brown-Gitler spectrum Topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum (topology)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a 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 , E^k\right/math>.Note there are several different 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 sequence X_n of pointed topological spaces or pointed simplicial sets together with the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology
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 d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclic Module
In mathematics, more specifically in ring theory, a cyclic module or monogenous module is a module over a ring that is generated by one element. The concept is a generalization of the notion of a cyclic group, that is, an Abelian group (i.e. Z-module) that is generated by one element. Definition A left ''R''-module ''M'' is called cyclic if ''M'' can be generated by a single element i.e. for some ''x'' in ''M''. Similarly, a right ''R''-module ''N'' is cyclic if for some . Examples * 2Z as a Z-module is a cyclic module. * In fact, every cyclic group is a cyclic Z-module. * Every simple ''R''-module ''M'' is a cyclic module since the submodule generated by any non-zero element ''x'' of ''M'' is necessarily the whole module ''M''. In general, a module is simple if and only if it is nonzero and is generated by each of its nonzero elements. * If the ring ''R'' is considered as a left module over itself, then its cyclic submodules are exactly its left principal ideals a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steenrod Algebra
In algebraic topology, a Steenrod algebra was defined by to be the algebra of stable cohomology operations for mod p cohomology. For a given prime number p, the Steenrod algebra A_p is the graded Hopf algebra over the field \mathbb_p of order p, consisting of all stable cohomology operations for mod p cohomology. It is generated by the Steenrod squares introduced by for p=2, and by the Steenrod reduced pth powers introduced in and the Bockstein homomorphism for p>2. The term "Steenrod algebra" is also sometimes used for the algebra of cohomology operations of a generalized cohomology theory. Cohomology operations A cohomology operation is a natural transformation between cohomology functors. For example, if we take cohomology with coefficients in a ring R, the cup product squaring operation yields a family of cohomology operations: :H^n(X;R) \to H^(X;R) :x \mapsto x \smile x. Cohomology operations need not be homomorphisms of graded rings; see the Cartan formula below. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word isomorphism is derived from the Ancient Greek: ἴσος ''isos'' "equal", and μορφή ''morphe'' "form" or "shape". The interest in isomorphisms lies in the fact that two isomorphic objects have the same properties (excluding further information such as additional structure or names of objects). Thus isomorphic structures cannot be distinguished from the point of view of structure only, and may be identified. In mathematical jargon, one says that two objects are . An automorphism is an isomorphism from a structure to itself. An isomorphism between two structures is a canonical isomorphism (a canonical map that is an isomorphism) if there is only one isomorphism between the two structures (as it is the case for solutions of a unive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Edgar H
Edgar is a commonly used English given name, from an Anglo-Saxon name ''Eadgar'' (composed of '' ead'' "rich, prosperous" and ''gar'' "spear"). Like most Anglo-Saxon names, it fell out of use by the later medieval period; it was, however, revived in the 18th century, and was popularised by its use for a character in Sir Walter Scott's ''The Bride of Lammermoor'' (1819). People with the given name * Edgar the Peaceful (942–975), king of England * Edgar the Ætheling (c. 1051 – c. 1126), last member of the Anglo-Saxon royal house of England * Edgar of Scotland (1074–1107), king of Scotland * Edgar Angara, Filipino lawyer * Edgar Barrier, American actor * Edgar Baumann, Paraguayan javelin thrower * Edgar Bergen, American actor, radio performer, ventriloquist * Edgar Berlanga, American boxer * Edgar H. Brown, American mathematician * Edgar Buchanan, American actor * Edgar Rice Burroughs, American author, creator of ''Tarzan'' * Edgar Cantero, Spanish author in Catalan, Spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Gitler Hammer
Samuel Carlos Gitler Hammer (July 14, 1933 – September 9, 2014) CINVESTAV, retrieved 2012-05-19.Member biography , Colegio Nacional, retrieved 2012-05-19. was a Mexican mathematician. He was an expert in and is known for the |
Topology (journal)
''Topology'' was a peer-reviewed mathematical journal covering topology and geometry. It was established in 1962 and was published by Elsevier. The last issue of ''Topology'' appeared in 2009. Pricing dispute On 10 August 2006, after months of unsuccessful negotiations with Elsevier about the price policy of library subscriptions, the entire editorial board of the journal handed in their resignation, effective 31 December 2006. Subsequently, two more issues appeared in 2007 with papers that had been accepted before the resignation of the editors. In early January the former editors instructed Elsevier to remove their names from the website of the journal, but Elsevier refused to comply, justifying their decision by saying that the editorial board should remain on the journal until all of the papers accepted during its tenure had been published. In 2007 the former editors of ''Topology'' announced the launch of the '' Journal of Topology'', published by Oxford University Press ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Segal Conjecture
Segal's Burnside ring conjecture, or, more briefly, the Segal conjecture, is a theorem in homotopy theory, a branch of mathematics. The theorem relates the Burnside ring of a finite group ''G'' to the stable cohomotopy of the classifying space ''BG''. The conjecture was made in the mid 1970s by Graeme Segal and proved in 1984 by Gunnar Carlsson. , this statement is still commonly referred to as the Segal conjecture, even though it now has the status of a theorem. Statement of the theorem The Segal conjecture has several different formulations, not all of which are equivalent. Here is a weak form: there exists, for every finite group ''G'', an isomorphism :\varprojlim \pi_S^0 \left( BG^_+ \right) \to \widehat(G). Here, lim denotes the inverse limit, S* denotes the stable cohomotopy ring, ''B'' denotes the classifying space, the superscript ''k'' denotes the ''k''-skeleton, and the subscript + denotes the addition of a disjoint basepoint. On the right-hand side, the hat denotes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burnside Ring
In mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets. The ideas were introduced by William Burnside at the end of the nineteenth century. The algebraic ring structure is a more recent development, due to Solomon (1967). Formal definition Given a finite group ''G'', the generators of its Burnside ring Ω(''G'') are the formal sums of isomorphism classes of finite ''G''-sets. For the ring structure, addition is given by disjoint union of ''G''-sets and multiplication by their Cartesian product. The Burnside ring is a free Z- module, whose generators are the (isomorphism classes of) orbit types of ''G''. If ''G'' acts on a finite set ''X'', then one can write X = \bigcup_i X_i (disjoint union), where each ''X''''i'' is a single ''G''-orbit. Choosing any element ''x''''i'' in ''X''i creates an isomorphism ''G''/''G''''i'' → ''X''''i'', where ''Gi'' is the stabilizer (isotropy) subgro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
