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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, 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.
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]
Using a long exact sequence, one can show that each of these statements is equivalent to a vanishing theorem for certain relative topological invariants. In order, these are: In order, these are: The relative singular homology groups H k ( X , Y ; Z ) {\displaystyle H_{k}(X,Y;\mathbb {Z} )} are zero for k ≤ n − 1 {\displaystyle k\leq n-1} .
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.
Singular homology; Cellular homology; Relative homology; Mayer–Vietoris sequence; Excision theorem; Universal coefficient theorem; Cohomology. List of cohomology theories; Cocycle class; Cup product; Cohomology ring; De Rham cohomology; Čech cohomology; Alexander–Spanier cohomology; Intersection cohomology; Lusternik–Schnirelmann ...
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.
Then Borel–Moore homology () is isomorphic to the relative homology H i (Y, S). Under the same assumption on X, the one-point compactification of X is homeomorphic to a finite CW complex. As a result, Borel–Moore homology can be viewed as the relative homology of the one-point compactification with respect to the added point.