Relative chain complex ($C_{\ast }(X,A)$)

For a subspace $A\subseteq X$, the relative chain complex is the quotient group in each degree,with boundary operator $[c]\mapsto [\partial c]$. This is well defined because $C_{\ast }(A)$ is a chain subcomplex of $C_{\ast }(X)$.

If $L$ is a simplicial subcomplex of $K$, thenIts homology is the simplicial relative homology $H_k(K,L)$.