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Eilenberg–Zilber Theorem
In mathematics, specifically in algebraic topology, the Eilenberg–Zilber theorem is an important result in establishing the link between the homology groups of a product space X \times Y and those of the spaces X and Y. The theorem first appeared in a 1953 paper in the American Journal of Mathematics by Samuel Eilenberg and Joseph A. Zilber. One possible route to a proof is the acyclic model theorem. Statement of the theorem The theorem can be formulated as follows. Suppose X and Y are topological spaces, Then we have the three chain complexes C_*(X), C_*(Y), and C_*(X \times Y) . (The argument applies equally to the simplicial or singular chain complexes.) We also have the ''tensor product complex'' C_*(X) \otimes C_*(Y), whose differential is, by definition, :\partial_( \sigma \otimes \tau) = \partial_X \sigma \otimes \tau + (-1)^p \sigma \otimes \partial_Y\tau for \sigma \in C_p(X) and \partial_X, \partial_Y the differentials on C_*(X),C_*(Y). Then the theorem says that we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Graded Coalgebra
Differential may refer to: Mathematics * Differential (mathematics) comprises multiple related meanings of the word, both in calculus and differential geometry, such as an infinitesimal change in the value of a function * Differential algebra * Differential calculus ** Differential of a function, represents a change in the linearization of a function *** Total differential is its generalization for functions of multiple variables ** Differential (infinitesimal) (e.g. ''dx'', ''dy'', ''dt'' etc.) are interpreted as infinitesimals ** Differential topology * Differential (pushforward) The total derivative of a map between manifolds. * Differential exponent, an exponent in the factorisation of the different ideal * Differential geometry, exterior differential, or exterior derivative, is a generalization to differential forms of the notion of differential of a function on a differentiable manifold * Differential (coboundary), in homological algebra and algebraic topology, one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proceedings Of The Cambridge Philosophical Society
''Mathematical Proceedings of the Cambridge Philosophical Society'' is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society. It aims to publish original research papers from a wide range of pure and applied mathematics. The journal, formerly titled ''Proceedings of the Cambridge Philosophical Society'', has been published since 1843. See also *Cambridge Philosophical Society The Cambridge Philosophical Society (CPS) is a scientific society at the University of Cambridge. It was founded in 1819. The name derives from the medieval use of the word philosophy to denote any research undertaken outside the fields of law ... External linksofficial website Academic journals associated with learned and professional societies Cambridge University Press academic journals Mathematics education in the United Kingdom Mathematics journals {{math-journal-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Pure And Applied Algebra
The ''Journal of Pure and Applied Algebra'' is a monthly peer-reviewed scientific journal covering that part of algebra likely to be of general mathematical interest: algebraic results with immediate applications, and the development of algebraic theories of sufficiently general relevance to allow for future applications. Its founding editors-in-chief were Peter J. Freyd (University of Pennsylvania) and Alex Heller (City University of New York). The current managing editors are Eric Friedlander (University of Southern California), Charles Weibel (Rutgers University), and Srikanth Iyengar (University of Utah). Abstracting and indexing The journal is abstracted and indexed in Current Contents/Physics, Chemical, & Earth Sciences, Mathematical Reviews, PASCAL, Science Citation Index, Zentralblatt MATH, and Scopus. According to the ''Journal Citation Reports'', the journal has a 2016 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acyclic Model
In algebraic topology, a discipline within mathematics, the acyclic models theorem can be used to show that two homology theories are isomorphic. The theorem was developed by topologists Samuel Eilenberg and Saunders MacLane. They discovered that, when topologists were writing proofs to establish equivalence of various homology theories, there were numerous similarities in the processes. Eilenberg and MacLane then discovered the theorem to generalize this process. It can be used to prove the Eilenberg–Zilber theorem; this leads to the idea of the model category. Statement of the theorem Let \mathcal be an arbitrary category and \mathcal(R) be the category of chain complexes of R-modules over some ring R. Let F,V : \mathcal \to \mathcal(R) be covariant functors such that: * F_i = V_i = 0 for i 0 and all M \in \mathcal_k \cup \mathcal_. Then the following assertions hold: * Every natural transformation \varphi : H_0(F) \to H_0(V) induces a natural chain map f : F \to V. * I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Künneth Theorem
In mathematics, especially in homological algebra and algebraic topology, a Künneth theorem, also called a Künneth formula, is a statement relating the homology of two objects to the homology of their product. The classical statement of the Künneth theorem relates the singular homology of two topological spaces ''X'' and ''Y'' and their product space X \times Y. In the simplest possible case the relationship is that of a tensor product, but for applications it is very often necessary to apply certain tools of homological algebra to express the answer. A Künneth theorem or Künneth formula is true in many different homology and cohomology theories, and the name has become generic. These many results are named for the German mathematician Hermann Künneth. Singular homology with coefficients in a field Let ''X'' and ''Y'' be two topological spaces. In general one uses singular homology; but if ''X'' and ''Y'' happen to be CW complexes, then this can be replaced by cellular hom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ronald Brown (mathematician)
Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles. Education and career Born on 4 January 1935 in London, Brown attended Oxford University, obtaining a B.A. in 1956 and a D.Phil. in 1962. Brown began his teaching career during his doctorate work, serving as an assistant lecturer at the University of Liverpool before assuming the position of Lecturer. In 1964, he took a position at the University of Hull, serving first as a Senior Lecturer and then as a Reader before becoming a Professor of pure mathematics at Bangor University, then a part of the University of Wales, in 1970. Brown served as Professor of Pure Mathematics for 30 years; he also served during the 1983–84 term as a Professor for one month at Louis Pasteur University in Strasbourg. In 1999, Brown took a half-time research professorship until he became Professor Emeritus in 2001. He was ele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free action of ''G''. It has the property that any ''G'' principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle ''EG'' → ''BG''. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group ''G'', ''BG'' is, roughly speaking, a path-connected topological space ''X'' such that the fundam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-abelian Group
In mathematics, and specifically in group theory, a non-abelian group, sometimes called a non-commutative group, is a group (''G'', ∗) in which there exists at least one pair of elements ''a'' and ''b'' of ''G'', such that ''a'' ∗ ''b'' ≠ ''b'' ∗ ''a''. This class of groups contrasts with the abelian groups. (In an abelian group, all pairs of group elements commute). Non-abelian groups are pervasive in mathematics and physics. One of the simplest examples of a non-abelian group is the dihedral group of order 6. It is the smallest finite non-abelian group. A common example from physics is the rotation group SO(3) in three dimensions (for example, rotating something 90 degrees along one axis and then 90 degrees along a different axis is not the same as doing them in reverse order). Both discrete groups and continuous groups may be non-abelian. Most of the interesting Lie groups are non-abelian, and these play an important role in gauge theory. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded-commutative Ring
In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx, where , ''x'' , and , ''y'' , denote the degrees of ''x'' and ''y''. A commutative (non-graded) ring, with trivial grading, is a basic example. An exterior algebra is an example of a graded-commutative ring that is not commutative in the non-graded sense. A cup product on cohomology satisfies the skew-commutative relation; hence, a cohomology ring is graded-commutative. In fact, many examples of graded-commutative rings come from algebraic topology and homological algebra. References * David Eisenbud, ''Commutative Algebra. With a view toward algebraic geometry'', Graduate Texts in Mathematics, vol 150, Springer-Verlag, New York, 1995. * See also *DG algebra *graded-symmetric algebra *alternating algebra *supercommutative algebra In mathematics, a supercommuta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cup Product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree ''p'' and ''q'' to form a composite cocycle of degree ''p'' + ''q''. This defines an associative (and distributive) graded commutative product operation in cohomology, turning the cohomology of a space ''X'' into a graded ring, ''H''∗(''X''), called the cohomology ring. The cup product was introduced in work of J. W. Alexander, Eduard Čech and Hassler Whitney from 1935–1938, and, in full generality, by Samuel Eilenberg in 1944. Definition In singular cohomology, the cup product is a construction giving a product on the graded cohomology ring ''H''∗(''X'') of a topological space ''X''. The construction starts with a product of cochains: if \alpha^p is a ''p''-cochain and \beta^q is a ''q''-cochain, then :(\alpha^p \smile \beta^q)(\sigma) = \alpha^p(\sigma \circ \iota_) \cdot \beta^q(\sigma \circ \iota_) where σ is a singular (''p'' + ''q'') -simplex and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
