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Compactly Supported Homology
In mathematics, a homology theory in algebraic topology is compactly supported if, in every degree ''n'', the relative homology group H''n''(''X'', ''A'') of every pair of spaces :(''X'', ''A'') is naturally isomorphic to the direct limit of the ''n''th relative homology groups of pairs (''Y'', ''B''), where ''Y'' varies over compact subspaces of ''X'' and ''B'' varies over compact subspaces of ''A''.. Singular homology is compactly supported, since each singular chain is a finite sum of simplices, which are compactly supported. Strong homology Strong may refer to: Education * The Strong, an educational institution in Rochester, New York, United States * Strong Hall (Lawrence, Kansas), an administrative hall of the University of Kansas * Strong School, New Haven, Connecticut, United St ... is not compactly supported. If one has defined a homology theory over compact pairs, it is possible to extend it into a compactly supported homology theory in the wider category of Hausdorff ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Homology (mathematics)
In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of Abelian group, abelian groups called ''homology groups.'' This operation, in turn, allows one to associate various named ''homologies'' or ''homology theories'' to various other types of mathematical objects. Lastly, since there are many homology theories for Topological space, topological spaces that produce the same answer, one also often speaks of the ''homology of a topological space''. (This latter notion of homology admits more intuitive descriptions for 1- or 2-dimensional topological spaces, and is sometimes referenced in popular mathematics.) There is also a related notion of the cohomology of a Cochain complexes, cochain complex, giving rise to various cohomology theories, in addition to the notion of the cohomology of a topological space. Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Relative Homology
In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace. Definition Given a subspace A\subseteq X, one may form the short exact sequence :0\to C_\bullet(A) \to C_\bullet(X)\to C_\bullet(X) /C_\bullet(A) \to 0 , where C_\bullet(X) denotes the singular chains on the space ''X''. The boundary map on C_\bullet(X) descends to C_\bullet(A) and therefore induces a boundary map \partial'_\bullet on the quotient. If we denote this quotient by C_n(X,A):=C_n(X)/C_n(A), we then have a complex :\cdots\longrightarrow C_n(X,A) \xrightarrow C_(X,A) \longrightarrow \cdots . By definition, the th relative homology group of the pair of spaces (X,A) is :H_n(X,A) := \ker\partial'_n/\operatorname\partial'_. One s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Pair Of Spaces
In mathematics, more specifically algebraic topology, a pair (X,A) is shorthand for an inclusion of topological spaces i\colon A \hookrightarrow X. Sometimes i is assumed to be a cofibration. A morphism from (X,A) to (X',A') is given by two maps f\colon X\rightarrow X' and g\colon A \rightarrow A' such that i' \circ g =f \circ i . A pair of spaces is an ordered pair where is a topological space and a subspace. The use of pairs of spaces is sometimes more convenient and technically superior to taking a quotient space of by . Pairs of spaces occur centrally in relative homology, homology theory and cohomology theory, where chains in A are made equivalent to 0, when considered as chains in X. Heuristically, one often thinks of a pair (X,A) as being akin to the quotient space X/A. There is a functor from the category of topological spaces to the category of pairs of spaces, which sends a space X to the pair (X, \varnothing). A related concept is that of a triple , with . T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
