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Leray Spectral Sequence
In mathematics, the Leray spectral sequence was a pioneering example in homological algebra, introduced in 1946 by Jean Leray. It is usually seen nowadays as a special case of the Grothendieck spectral sequence. Definition Let f:X\to Y be a continuous map of topological spaces, which in particular gives a functor f_* from sheaves of abelian groups on X to sheaves of abelian groups on Y. Composing this with the functor \Gamma of taking sections on \text_\text(Y) is the same as taking sections on \text_\text(X), by the definition of the direct image functor f_*: :\mathrm (X) \xrightarrow \mathrm(Y) \xrightarrow \mathrm. Thus the derived functors of \Gamma \circ f_* compute the sheaf cohomology for X: : R^i (\Gamma \cdot f_*)(\mathcal)=H^i(X,\mathcal). But because f_* and \Gamma send injective objects in \text_\text(X) to \Gamma- acyclic objects in \text_\text(Y), there is a spectral sequencepg 33,19 whose second page is : E^_2=(R^p\Gamma \cdot R^q f_*)(\mathcal)=H^p(Y,R^qf ... [...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] |
Smooth Manifolds
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K3 Surface
In mathematics, a complex analytic K3 surface is a compact connected complex manifold of dimension 2 with trivial canonical bundle and irregularity zero. An (algebraic) K3 surface over any field means a smooth proper geometrically connected algebraic surface that satisfies the same conditions. In the Enriques–Kodaira classification of surfaces, K3 surfaces form one of the four classes of minimal surfaces of Kodaira dimension zero. A simple example is the Fermat quartic surface :x^4+y^4+z^4+w^4=0 in complex projective 3-space. Together with two-dimensional compact complex tori, K3 surfaces are the Calabi–Yau manifolds (and also the hyperkähler manifolds) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved del Pezzo surfaces (which are easy to classify) and the negatively curved surfaces of general type (which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lefschetz Hyperplane Theorem
In mathematics, specifically in algebraic geometry and algebraic topology, the Lefschetz hyperplane theorem is a precise statement of certain relations between the shape of an algebraic variety and the shape of its subvarieties. More precisely, the theorem says that for a variety ''X'' embedded in projective space and a hyperplane section ''Y'', the homology (mathematics), homology, cohomology, and homotopy groups of ''X'' determine those of ''Y''. A result of this kind was first stated by Solomon Lefschetz for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories. A far-reaching generalization of the hard Lefschetz theorem is given by the decomposition theorem. The Lefschetz hyperplane theorem for complex projective varieties Let ''X'' be an ''n''-dimensional complex projective algebraic variety in CP''N'', and let ''Y'' be a hyperplane section of ''X'' s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hurewicz Theorem
In mathematics, the Hurewicz theorem is a basic result of algebraic topology, connecting homotopy theory with homology theory via a map known as the Hurewicz homomorphism. The theorem is named after Witold Hurewicz, and generalizes earlier results of Henri Poincaré. Statement of the theorems The Hurewicz theorems are a key link between homotopy groups and homology groups. Absolute version For any path-connected space ''X'' and positive integer ''n'' there exists a group homomorphism :h_* \colon \pi_n(X) \to H_n(X), called the Hurewicz homomorphism, from the ''n''-th homotopy group to the ''n''-th homology group (with integer coefficients). It is given in the following way: choose a canonical generator u_n \in H_n(S^n), then a homotopy class of maps f \in \pi_n(X) is taken to f_*(u_n) \in H_n(X). The Hurewicz theorem states cases in which the Hurewitz homomorphism is an isomorphism. * For n\ge 2, if ''X'' is (n-1)-connected (that is: \pi_i(X)= 0 for all ''i''2 there exists ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pierre Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 196 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre Spectral Sequence
In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homological algebra, the singular (co)homology of the total space ''X'' of a (Serre) fibration in terms of the (co)homology of the base space ''B'' and the fiber ''F''. The result is due to Jean-Pierre Serre in his doctoral dissertation. Cohomology spectral sequence Let f\colon X\to B be a Serre fibration of topological spaces, and let ''F'' be the (path-connected) fiber. The Serre cohomology spectral sequence is the following: : E_2^ = H^p(B, H^q(F)) \Rightarrow H^(X). Here, at least under standard simplifying conditions, the coefficient group in the E_2-term is the ''q''-th integral cohomology group of ''F'', and the outer group is the singular cohomology of ''B'' with coefficients in that group. Strictly speaking, what is meant is cohomology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a mani ... [...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] |
Simply Connected Space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question. The fundamental group of a topological space is an indicator of the failure for the space to be simply connected: a path-connected topological space is simply connected if and only if its fundamental group is trivial. Definition and equivalent formulations A topological space X is called if it is path-connected and any loop in X defined by f : S^1 \to X can be contracted to a point: there exists a continuous map F : D^2 \to X such that F restricted to S^1 is f. Here, S^1 and D^2 denotes the unit circle and closed unit disk in the Euclidean plane respectively. An equivalent formulation is this: X is simply connected if and only if it is path-connected, and whene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiley (publisher)
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orange, New Jer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
