# 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.