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 are topological spaces (each with the subspace topology from the bigger one). The long exact sequence of homology of this triple (usually denoted as ) is:

where denotes the relative homology.

Particular cases

If is empty, we get the long exact sequence of homology of a pair, namely the pair .

For various homology theories

For homologies arising from a chain complex

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

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