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Pullback (cohomology)
In algebraic topology, given a continuous map ''f'': ''X'' → ''Y'' of topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...s and a ring ''R'', the pullback along ''f'' on cohomology theory is a grade-preserving ''R''-algebra homomorphism: :f^*: H^*(Y; R) \to H^*(X; R) from the cohomology ring of ''Y'' with coefficients in ''R'' to that of ''X''. The use of the superscript is meant to indicate its contravariant nature: it reverses the direction of the map. For example, if ''X'', ''Y'' are manifolds, ''R'' the field of real numbers, and the cohomology is de Rham cohomology, then the pullback is induced by the pullback of differential forms. The homotopy invariance of cohomology states that if two maps ''f'', ''g'': ''X'' → ''Y'' are homotopic to each other, then th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points, along with an additional structure called a topology, which can be defined as a set of neighbourhoods for each point that satisfy some axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a mathematical space that allows for the definition of limits, continuity, and connectedness. Common types of topological spaces include Euclidean spaces, metric spaces and manifolds. Although very general, the concept of topological spaces is fundamental, and used in virtually every branch of modern mathematics. The study of topological spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomology Ring
In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is also present in other theories such as de Rham cohomology. It is also functorial: for a continuous mapping of spaces one obtains a ring homomorphism on cohomology rings, which is contravariant. Specifically, given a sequence of cohomology groups ''H''''k''(''X'';''R'') on ''X'' with coefficients in a commutative ring ''R'' (typically ''R'' is Z''n'', Z, Q, R, or C) one can define the cup product, which takes the form :H^k(X;R) \times H^\ell(X;R) \to H^(X; R). The cup product gives a multiplication on the direct sum of the cohomology groups :H^\bullet(X;R) = \bigoplus_ H^k(X; R). This multiplication turns ''H''•(''X'';''R'') into a ring. In fact, it is naturally an N-graded ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ \stackrel\ \Omega^1(M)\ \stackrel\ \Omega^2(M)\ \stackrel\ \Omega^3(M) \to \cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration-along-fibers
In differential geometry, the integration along fibers of a ''k''-form yields a (k-m)-form where ''m'' is the dimension of the fiber, via "integration". It is also called the fiber integration. Definition Let \pi: E \to B be a fiber bundle over a manifold with compact oriented fibers. If \alpha is a ''k''-form on ''E'', then for tangent vectors ''w''''i'''s at ''b'', let : (\pi_* \alpha)_b(w_1, \dots, w_) = \int_ \beta where \beta is the induced top-form on the fiber \pi^(b); i.e., an m-form given by: with \widetilde lifts of w_i to E, :\beta(v_1, \dots, v_m) = \alpha(v_1, \dots, v_m, \widetilde, \dots, \widetilde). (To see b \mapsto (\pi_* \alpha)_b is smooth, work it out in coordinates; cf. an example below.) Then \pi_* is a linear map \Omega^k(E) \to \Omega^(B). By Stokes' formula, if the fibers have no boundaries(i.e. ,\int0), the map descends to de Rham cohomology: :\pi_*: \operatorname^k(E; \mathbb) \to \operatorname^(B; \mathbb). This is also called the fiber i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Complexes
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to form t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
