Relative homology

Summary

Relative homology assigns to a space $X$ and a subspace $A$ the groups $H_n(X, A)$. They measure the homology of $X$ while ignoring everything that happens inside $A$. It is built from the singular chains of $X$ modulo the chains of $A$. Relative groups fit into the long exact sequence of the pair and are the setting of the excision theorem. This makes them the main tool for computing singular homology.