Homology, Triple Homology Sequence

Triple Homology Sequence

Let C be a chain complex, and let B be a subchain of C, and let A be a subchain of B. Now B embeds in C, with quotient C/B, and the same relationship holds if we mod out by A. In other words, the following sequence is short exact.

0 → B/A → C/A → C/B → 0

convert this to a long exact sequence. The result is called the triple homology sequence. This is a relationship among relative homologies. On every row, h(B/A) → h(C/A) → h(C/B) is exact.

If you happen to know that B is a summand of C, then the long exact sequence is split exact down the middle, and the homology of C/A is the direct product of the homology of B/A and the homology of C/B.