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Homotopical Excision Theorem
In algebraic topology, the homotopy excision theorem offers a substitute for the absence of Excision theorem, excision in homotopy theory. More precisely, let (X; A, B) be an excisive triad with C = A \cap B nonempty, and suppose the pair (A, C) is n-connected, (m-1)-connected, m \ge 2, and the pair (B, C) is (n-1)-connected, n \ge 1. Then the map induced by the inclusion i\colon (A, C) \to (X, B), :i_*\colon \pi_q(A, C) \to \pi_q(X, B), is bijective for q < m+n-2 and is surjective for . A geometric proof is given in a book by Tammo tom Dieck. This result should also be seen as a consequence of the most general form of the Blakers–Massey theorem, which deals with the non-simply-connected case. The most important consequence is the Freudenthal suspension theorem. References Bibliography * J. Peter May, ''A Concise Course in Algebraic Topology'', Chicago University Press. Theorems in homotopy ...[...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 invariants that classify topological spaces up to homeomorphism, though usually most classify up to 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 groups record information about the basic shape, or holes, of a topological space. Homology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excision Theorem
In algebraic topology, a branch of mathematics, the excision theorem is a theorem about relative homology and one of the Eilenberg–Steenrod axioms. Given a topological space X and subspaces A and U such that U is also a subspace of A, the theorem says that under certain circumstances, we can cut out (excise) U from both spaces such that the relative homologies of the pairs (X \setminus U,A \setminus U ) into (X, A) are isomorphic. This assists in computation of singular homology groups, as sometimes after excising an appropriately chosen subspace we obtain something easier to compute. Theorem Statement If U\subseteq A \subseteq X are as above, we say that U can be excised if the inclusion map of the pair (X \setminus U,A \setminus U ) into (X, A) induces an isomorphism on the relative homologies: The theorem states that if the closure of U is contained in the interior of A, then U can be excised. Often, subspaces that do not satisfy this containment criterion still ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories). Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex. In the same vein as above, a " map" is a continuous function, possibly with some extra constraints. Often, one works with a pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Excisive Triad
In topology, a branch of mathematics, an excisive triad is a triple (X; A, B) 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 poin ...s such that ''A'', ''B'' are subspaces of ''X'' and ''X'' is the union of the interior of ''A'' and the interior of ''B''. Note ''B'' is not required to be a subspace of ''A''. See also * Homotopy excision theorem Notes References * * Munkres, James; ''Topology'', Prentice Hall; 2nd edition (December 28, 1999). . {{topology-stub Topology General topology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-connected
In algebraic topology, homotopical connectivity is a property describing a topological space based on the dimension of its holes. In general, low homotopical connectivity indicates that the space has at least one low-dimensional hole. The concept of ''n''-connectedness generalizes the concepts of path-connectedness and simple connectedness. An equivalent definition of homotopical connectivity is based on the homotopy groups of the space. A space is ''n''-connected (or ''n''-simple connected) if its first ''n'' homotopy groups are trivial. Homotopical connectivity is defined for maps, too. A map is ''n''-connected if it is an isomorphism "up to dimension ''n,'' in homotopy". Definition using holes All definitions below consider a topological space ''X''. A hole in ''X'' is, informally, a thing that prevents some suitably-placed sphere from continuously shrinking to a point., Section 4.3 Equivalently, it is a sphere that cannot be continuously extended to a ball. Formally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tammo Tom Dieck
Tammo tom Dieck (29 May 1938, São Paulo) is a German mathematician, specializing in algebraic topology. Tammo tom Dieck studied mathematics from 1957 at the University of Göttingen and at Saarland University, where he received his Promotion (Germany), promotion (Ph.D.) in 1964 under Dieter Puppe with thesis ''Zur K-theory, K-Theorie und ihren Cohomology, Kohomologie-Operationen''. In 1969 tom Dieck received his habilitation at Heidelberg University under Albrecht Dold. From 1970 to 1975 he was a professor at Saarland University. In 1975 he became a professor at the University of Göttingen. Tammo tom Dieck is a world-class expert in algebraic topology and author of several widely-used textbooks in topology. He has done research on Lie groups, G-structures, and cobordism. In the 1990s and 2000s, his research dealt with knot theory (and its algebras) and quantum groups. In 1986 he was an Invited Speaker with talk ''Geometric representation theory of compact Lie groups'' at the In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blakers–Massey Theorem
In mathematics, the first Blakers–Massey theorem, named after Albert Blakers and William S. Massey, gave vanishing conditions for certain triad homotopy groups of spaces. Description of the result This connectivity result may be expressed more precisely, as follows. Suppose ''X'' is a topological space which is the pushout of the diagram : A\xleftarrow C \xrightarrow B, where ''f'' is an ''m''-connected map and ''g'' is ''n''-connected. Then the map of pairs : (A,C)\rightarrow (X,B) induces an isomorphism in relative homotopy groups in degrees k\le (m+n-1) and a surjection in the next degree. However the third paper of Blakers and Massey in this area determines the critical, i.e., first non-zero, triad homotopy group as a tensor product, under a number of assumptions, including some simple connectivity. This condition and some dimension conditions was relaxed in work of Ronald Brown and Jean-Louis Loday. The algebraic result implies the connectivity result, since a ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The London Mathematical Society
The London Mathematical Society (LMS) is one of the United Kingdom's learned societies for mathematics (the others being the Royal Statistical Society (RSS), the Institute of Mathematics and its Applications (IMA), the Edinburgh Mathematical Society and the Operational Research Society (ORS). History The Society was established on 16 January 1865, the first president being Augustus De Morgan. The earliest meetings were held in University College, but the Society soon moved into Burlington House, Piccadilly. The initial activities of the Society included talks and publication of a journal. The LMS was used as a model for the establishment of the American Mathematical Society in 1888. Mary Cartwright was the first woman to be President of the LMS (in 1961–62). The Society was granted a royal charter in 1965, a century after its foundation. In 1998 the Society moved from rooms in Burlington House into De Morgan House (named after the society's first president), at 57†... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freudenthal Suspension Theorem
In mathematics, and specifically in the field of homotopy theory, the Freudenthal suspension theorem is the fundamental result leading to the concept of stabilization of homotopy groups and ultimately to stable homotopy theory. It explains the behavior of simultaneously taking suspensions and increasing the index of the homotopy groups of the space in question. It was proved in 1937 by Hans Freudenthal. The theorem is a corollary of the homotopy excision theorem. Statement of the theorem Let ''X'' be an ''n''-connected pointed space (a pointed CW-complex or pointed simplicial set). The map :X \to \Omega(\Sigma X) induces a map :\pi_k(X) \to \pi_k(\Omega(\Sigma X)) on homotopy groups, where Ω denotes the loop functor and Σ denotes the reduced suspension functor. The suspension theorem then states that the induced map on homotopy groups is an isomorphism if ''k'' ≤ 2''n'' and an epimorphism if ''k'' = 2''n'' + 1. A basic result on loop spaces gives the relation :\p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
