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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.
If we take a series of short exact sequences linked by chain complexes (that is, a short exact sequence of chain complexes, or from another point of view, a chain complex of short exact sequences), then we can derive from this a long exact sequence (that is, an exact sequence indexed by the natural numbers) on homology by application of the zig ...
Homology groups are similar to homotopy groups in that they can represent "holes" in a topological space. However, homotopy groups are often very complex and hard to compute. In contrast, homology groups are commutative (as are the higher homotopy groups). Hence, it is sometimes said that "homology is a commutative alternative to homotopy". [7]
This follows from the naturality of the sequence produced by the snake lemma. If is a commutative diagram with exact rows, then the snake lemma can be applied twice, to the "front" and to the "back", yielding two long exact sequences; these are related by a commutative diagram of the form
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. [8]
It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). [1] Intuitively, the Puppe sequence allows us to think of homology theory as a functor that takes spaces to long-exact sequences of
is a short exact sequence of abelian groups. By definition, this means that f n is an injection, g n is a surjection, and Im f n = Ker g n. One of the most basic theorems of homological algebra, sometimes known as the zig-zag lemma, states that, in this case, there is a long exact sequence in homology
This will induce a morphism of their long exact sequences of homotopy groups, from which (by applying the Five Lemma, as is done in the Puppe sequence) one can see that the map F → F f is a weak equivalence. Thus the above given construction reproduces the same homotopy type if there already is one.