Long exact sequence of homology of a triple

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This article defines a long exact sequence of homology groups, for topological spaces or pairs of topological spaces

Definition

Suppose A \subset B \subset C are topological spaces (each with the subspace topology from the bigger one). The long exact sequence of homology of this triple (usually denoted as (C;B,A)) is:

\ldots \to H_n(B,A) \to H_n(C,A) \to H_n(C,B) \to H_{n-1}(B,A) \to \ldots

where H_n(X,Y) denotes the relative homology.

Particular cases

If A is empty, we get the long exact sequence of homology of a pair, namely the pair (C,B).

For various homology theories

For homologies arising from a chain complex

If the homology theory involves homology of a chain complex Chain_., then the above can be interpreted as the long exact sequence of homology arising from the following short exact sequence of relative chain complexes:

0 \to Chain_.(B,A) \to Chain_.(C,A) \to Chain_.(C,B) \to 0

In particular, this description works for singular homology, cellular homology and simplicial homology.