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Obstruction Theory
In mathematics, obstruction theory is a name given to two different mathematical theories, both of which yield cohomological invariants. In the original work of Stiefel and Whitney, characteristic classes were defined as obstructions to the existence of certain fields of linear independent vectors. Obstruction theory turns out to be an application of cohomology theory to the problem of constructing a cross-section of a bundle. In homotopy theory The older meaning for obstruction theory in homotopy theory relates to the procedure, inductive with respect to dimension, for extending a continuous mapping defined on a simplicial complex, or CW complex. It is traditionally called ''Eilenberg obstruction theory'', after Samuel Eilenberg. It involves cohomology groups with coefficients in homotopy groups to define obstructions to extensions. For example, with a mapping from a simplicial complex ''X'' to another, ''Y'', defined initially on the 0-skeleton of ''X'' (the vertices of ''X'' ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
0-skeleton
In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the . These subspaces increase with . The is a discrete space, and the a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when has infinite dimension, in the sense that the do not become constant as In geometry In geometry, a of P (functionally represented as skel''k''(''P'')) consists of all elements of dimension up to ''k''. For example: : skel0(cube) = 8 vertices : skel1(cube) = 8 vertices, 12 edges : skel2(cube) = 8 vertices, 12 edges, 6 square faces For simplicial sets The abov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Postnikov System
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X. Postnikov systems were introduced by, and are named after, Mikhail Postnikov. Definition A Postnikov system of a path-connected space X is an inverse system of spaces :\cdots \to X_n \xrightarrow X_\xrightarrow \cdots \xrightarrow X_2 \xrightarrow X_1 \xrightarrow * with a sequence of maps \phi_n\colon X \to X_n compatible with the inverse system such that # The map \phi_n\colon X \to X_n induces an isomorphism \pi_i(X) \to \pi_i(X_n) for every i\leq n. # \pi_i(X_n) = 0 for i > n. # Each map p_n\colon X_n \to X_ is a fibration, and so the fiber F_n is an Eilenberg–MacLane space, K(\pi_n(X),n). The first two conditions imply that X_1 is also a K(\pi_1(X),1)-space. More generally, i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Fiber
In mathematics, especially homotopy theory, the homotopy fiber (sometimes called the mapping fiber)Joseph J. Rotman, ''An Introduction to Algebraic Topology'' (1988) Springer-Verlag ''(See Chapter 11 for construction.)'' is part of a construction that associates a fibration to an arbitrary continuous function of topological spaces f:A \to B. It acts as a homotopy theoretic kernel of a mapping of topological spaces due to the fact it yields a long exact sequence of homotopy groups\cdots \to \pi_(B) \to \pi_n(\text(f)) \to \pi_n(A) \to \pi_n(B) \to \cdotsMoreover, the homotopy fiber can be found in other contexts, such as homological algebra, where the distinguished triangleC(f)_\bullet 1\to A_\bullet \to B_\bullet \xrightarrowgives a long exact sequence analogous to the long exact sequence of homotopy groups. There is a dual construction called the homotopy cofiber. Construction The homotopy fiber has a simple description for a continuous map f:A \to B. If we replace f by a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space. # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a princi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplicial Homology
In algebraic topology, simplicial homology is the sequence of homology groups of a simplicial complex. It formalizes the idea of the number of holes of a given dimension in the complex. This generalizes the number of connected components (the case of dimension 0). Simplicial homology arose as a way to study topological spaces whose building blocks are ''n''- simplices, the ''n''-dimensional analogs of triangles. This includes a point (0-simplex), a line segment (1-simplex), a triangle (2-simplex) and a tetrahedron (3-simplex). By definition, such a space is homeomorphic to a simplicial complex (more precisely, the geometric realization of an abstract simplicial complex). Such a homeomorphism is referred to as a ''triangulation'' of the given space. Many topological spaces of interest can be triangulated, including every smooth manifold (Cairns and Whitehead). Simplicial homology is defined by a simple recipe for any abstract simplicial complex. It is a remarkable fact that simpl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Equivalent
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deformation being called a homotopy (, ; , ) between the two functions. A notable use of homotopy is the definition of homotopy groups and cohomotopy groups, important invariants in algebraic topology. In practice, there are technical difficulties in using homotopies with certain spaces. Algebraic topologists work with compactly generated spaces, CW complexes, or spectra. Formal definition Formally, a homotopy between two continuous functions ''f'' and ''g'' from a topological space ''X'' to a topological space ''Y'' is defined to be a continuous function H: X \times ,1\to Y from the product of the space ''X'' with the unit interval , 1to ''Y'' such that H(x,0) = f(x) and H(x,1) = g(x) for all x \in X. If we think of the second ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contractible Space
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. Properties A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a topological space ''X'' the following are all equivalent: *''X'' is contractible (i.e. the identity map is null-homotopic). *''X'' is homotopy equivalent to a one-point space. *''X'' deformation retracts onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.) *For any space ''Y'', any two maps ''f'',''g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-sphere
In mathematics, an -sphere or a hypersphere is a topological space that is homeomorphic to a ''standard'' -''sphere'', which is the set of points in -dimensional Euclidean space that are situated at a constant distance from a fixed point, called the ''center''. It is the generalization of an ordinary sphere in the ordinary three-dimensional space. The "radius" of a sphere is the constant distance of its points to the center. When the sphere has unit radius, it is usual to call it the unit -sphere or simply the -sphere for brevity. In terms of the standard norm, the -sphere is defined as : S^n = \left\ , and an -sphere of radius can be defined as : S^n(r) = \left\ . The dimension of -sphere is , and must not be confused with the dimension of the Euclidean space in which it is naturally embedded. An -sphere is the surface or boundary of an -dimensional ball. In particular: *the pair of points at the ends of a (one-dimensional) line segment is a 0-sphere, *a circle, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-skeleton
In mathematics, particularly in algebraic topology, the of a topological space presented as a simplicial complex (resp. CW complex) refers to the subspace that is the union of the simplices of (resp. cells of ) of dimensions In other words, given an inductive definition of a complex, the is obtained by stopping at the . These subspaces increase with . The is a discrete space, and the a topological graph. The skeletons of a space are used in obstruction theory, to construct spectral sequences by means of filtrations, and generally to make inductive arguments. They are particularly important when has infinite dimension, in the sense that the do not become constant as In geometry In geometry, a of P (functionally represented as skel''k''(''P'')) consists of all elements of dimension up to ''k''. For example: : skel0(cube) = 8 vertices : skel1(cube) = 8 vertices, 12 edges : skel2(cube) = 8 vertices, 12 edges, 6 square faces For simplicial sets The above d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Section (fiber Bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi. In other words, if E is a fiber bundle over a base space, B: : \pi \colon E \to B then a section of that fiber bundle is a continuous map, : \sigma \colon B \to E such that : \pi(\sigma(x)) = x for all x \in B . A section is an abstract characterization of what it means to be a graph. The graph of a function g\colon B \to Y can be identified with a function taking its values in the Cartesian product E = B \times Y , of B and Y : :\sigma\colon B\to E, \quad \sigma(x) = (x,g(x)) \in E. Let \pi\colon E \to B be the projection onto the first factor: \pi(x,y) = x . Then a graph is any function \sigma for which \pi(\sigma(x)) = x . The language of fibre bundles allows this notion of a section to be generalized to the case when E is not necessarily a Cartesian product. If \pi\colon E \to B is a fibre bundle, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibration
The notion of a fibration generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in postnikov-systems or obstruction theory. In this article, all mappings are continuous mappings between topological spaces. Formal definitions Homotopy lifting property A mapping p \colon E \to B satisfies the homotopy lifting property for a space X if: * for every homotopy h \colon X \times , 1\to B and * for every mapping (also called lift) \tilde h_0 \colon X \to E lifting h, _ = h_0 (i.e. h_0 = p \circ \tilde h_0) there exists a (not necessarily unique) homotopy \tilde h \colon X \times , 1\to E lifting h (i.e. h = p \circ \tilde h) with \tilde h_0 = \tilde h, _. The following commutative diagram shows the situation:^ Fibration A fibration (also called Hurewicz fibration) is a mapping p \colon E \to B satisfying the homotopy lifting property for all spaces X. The space B is called bas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
