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Nilpotent Space
In topology, a branch of mathematics, a nilpotent space, first defined by Emmanuel Dror (1969), is a based topological space ''X'' such that * the fundamental group \pi = \pi_1 (X) is a nilpotent group; * \pi acts nilpotently on the higher homotopy groups \pi_i (X), i \ge 2, i.e., there is a central series \pi_i (X) = G^i_1 \triangleright G^i_2 \triangleright \dots \triangleright G^i_ = 1 such that the induced action of \pi on the quotient group G^i_k/G^i_ is trivial for all k. Simply connected spaces and simple spaces are (trivial) examples of nilpotent spaces; other examples are connected loop spaces. The homotopy fiber of any map between nilpotent spaces is a disjoint union of nilpotent spaces. Moreover, the null component of the pointed mapping space \operatorname_*(K,X), where ''K'' is a pointed, finite-dimensional CW complex and ''X'' is any pointed space, is a nilpotent space. The odd-dimensional real projective spaces are nilpotent spaces, while the projective plane is n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topology
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such as Stretch factor, stretching, Twist (mathematics), twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set (mathematics), set endowed with a structure, called a ''Topology (structure), topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity (mathematics), 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 homotopy, homotopies. A property that is invariant under such deformations is a topological property. Basic exampl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Loop Space
In topology, a branch of mathematics, the loop space Ω''X'' of a pointed topological space ''X'' is the space of (based) loops in ''X'', i.e. continuous pointed maps from the pointed circle ''S''1 to ''X'', equipped with the compact-open topology. Two loops can be multiplied by concatenation. With this operation, the loop space is an ''A''∞-space. That is, the multiplication is homotopy-coherently associative. The set of path components of Ω''X'', i.e. the set of based-homotopy equivalence classes of based loops in ''X'', is a group, the fundamental group ''π''1(''X''). The iterated loop spaces of ''X'' are formed by applying Ω a number of times. There is an analogous construction for topological spaces without basepoint. The free loop space of a topological space ''X'' is the space of maps from the circle ''S''1 to ''X'' with the compact-open topology. The free loop space of ''X'' is often denoted by \mathcalX. As a functor, the free loop space construction is righ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adams Spectral Sequence
In mathematics, the Adams spectral sequence is a spectral sequence introduced by which computes the stable homotopy groups of topological spaces. Like all spectral sequences, it is a computational tool; it relates homology theory to what is now called stable homotopy theory. It is a reformulation using homological algebra, and an extension, of a technique called 'killing homotopy groups' applied by the French school of Henri Cartan and Jean-Pierre Serre. Motivation For everything below, once and for all, we fix a prime ''p''. All spaces are assumed to be CW complexes. The ordinary cohomology groups H^*(X) are understood to mean H^*(X; \Z/p\Z). The primary goal of algebraic topology is to try to understand the collection of all maps, up to homotopy, between arbitrary spaces ''X'' and ''Y''. This is extraordinarily ambitious: in particular, when ''X'' is S^n, these maps form the ''n''th homotopy group of ''Y''. A more reasonable (but still very difficult!) goal is to understand ... [...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] |
Rational Homotopy Theory
In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by and . This simplification of homotopy theory makes certain calculations much easier. Rational homotopy types of simply connected spaces can be identified with (isomorphism classes of) certain algebraic objects called Sullivan minimal models, which are commutative differential graded algebras over the rational numbers satisfying certain conditions. A geometric application was the theorem of Sullivan and Micheline Vigué-Poirrier (1976): every simply connected closed Riemannian manifold ''X'' whose rational cohomology ring is not generated by one element has infinitely many geometrically distinct closed geodesics. The proof used rational homotopy theory to show that the Betti numbers of the free loop space of ''X'' are unbounded. The theorem then follows from a 1969 r ... [...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] |
Whitehead Theorem
In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping ''f'' between CW complexes ''X'' and ''Y'' induces isomorphisms on all homotopy groups, then ''f'' is a homotopy equivalence. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of algebraic topology, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping. Statement In more detail, let ''X'' and ''Y'' be topological spaces. Given a continuous mapping :f\colon X \to Y and a point ''x'' in ''X'', consider for any ''n'' ≥ 1 the induced homomorphism :f_*\colon \pi_n(X,x) \to \pi_n(Y,f(x)), where π''n''(''X'',''x'') denotes the ''n''-th homotopy group of ''X'' with base point ''x''. (For ''n'' = 0, π0(''X'') just means the set of path components o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lecture Notes In Mathematics
''Lecture Notes in Mathematics'' is a book series in the field of mathematics, including articles related to both research and teaching. It was established in 1964 and was edited by A. Dold, Heidelberg and B. Eckmann, Zürich. Its publisher is Springer Science+Business Media (formerly Springer-Verlag). The intent of the series is to publish not only lecture notes, but results from seminars and conferences, more quickly than the several-years-long process of publishing polished journal papers in mathematics. In order to speed the publication process, early volumes of the series (before electronic publishing) were reproduced photographically from typewritten manuscripts. According to Earl Taft it has been "enormously successful" and "is considered a very valuable service to the mathematical community". there have been 2232 volumes in this series. See also * ''Lecture Notes in Physics'' * ''Lecture Notes in Computer Science ''Lecture Notes in Computer Science'' is a series of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CW Complex
A CW complex (also called cellular complex or cell complex) is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead (open access) to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation (often with a much smaller complex). The ''C'' stands for "closure-finite", and the ''W'' for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces\emptyset = X_ \subset X_0 \subset X_1 \subset \cdotssuch that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to D^k, to X_ by continuous gluing maps g^k_\alpha: \partial e^k_\alpha \to X_. The maps are also called attaching maps. Each X_k is called the k-skeleton of the complex. The topology of X = \cup_ X_ ... [...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 fibra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simple Space
In algebraic topology, a branch of mathematics, a simple space is a connected topological space that has a homotopy type of a CW complex and whose fundamental group is abelian and acts trivially on the homotopy and homology of the universal covering space. Though not all authors include the assumption on the homotopy type. Examples Topological groups For example, any topological group is a simple space (provided it satisfies the condition on the homotopy type). Eilenberg-Maclane spaces Most Eilenberg-Maclane spaces K(A,n) are simple since the only nontrivial homotopy group is in degree n. This means the only non-simple spaces are K(G,1) for G nonabelian. Universal covers Every connected topological space X has an associated (universal) simple space from the universal cover \pi:U_X \to X; indeed, \pi_1(U_X) = * and the universal cover is its own universal cover. References *Dennis Sullivan Dennis Parnell Sullivan (born February 12, 1941) is an American mathematicia ... [...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] |
