aka long exact sequence, or LES
Long exact sequence relates relative homology to absolute homologies ~ if you know two of them you know the third
Long exact sequence for Reduced homology
Theorem 2.13 (book) gives a long exact sequence for the Reduced homology groups of a space, subspace, and quotient space.
Proved using Short exact sequence of chain complexes (where objects in the short exact sequence are Chain complexes, and the maps Chain maps) below. This actually proves the LES for relative homology (below). One then shows that for good pairs (those that satisfy the conditions of theorem 2.13), then this implies the theorem.
Homology long exact sequence for Relative homology
Video for homology long Exact Sequence. when we pass to homology groups, this short exact sequence of chain complexes stretches out into a long exact sequence of homology groups
As said here, in fact the elements of the Chain complexes for the above theorem need not be Vector space (as in Free Abelian groups, Abelian groups are enough.
LES of a triple
Mayer-Vietoris sequence
Others?