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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.
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.
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 groups called homology groups.
where the homology groups of L, M, and N cyclically follow each other, and δ n are certain homomorphisms determined by f and g, called the connecting homomorphisms. Topological manifestations of this theorem include the Mayer–Vietoris sequence and the long exact sequence for relative homology.
In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary.Such a formulation was introduced by Solomon Lefschetz (), at the same time introducing relative homology, for application to the Lefschetz fixed-point theorem. [1]
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 nth relative homology groups of pairs (Y, B), where Y varies over compact subspaces of X and B varies over compact subspaces of A.
The relative SFT of this pair is a differential graded algebra; Ng derives a powerful knot invariant from a combinatorial version of the zero-th degree part of the homology. It has the form of a finitely presented tensor algebra over a certain ring of multivariable Laurent polynomials with integer coefficients.
Pairs of spaces occur centrally in relative homology, [1] homology theory and cohomology theory, where chains in are made equivalent to 0, when considered as chains in . Heuristically, one often thinks of a pair ( X , A ) {\displaystyle (X,A)} as being akin to the quotient space X / A {\displaystyle X/A} .