If $X$ is a topological space, and $A$ a subspace, then the inclusion map $A\hookrightarrow X$ makes $C_n(A)$ into a subgroup of $C_n(X)$ . Since the boundary map on $C_*(X)$ restricts to the boundary map on $C_*(A)$ , we can take the quotient complex $C_*(X,A)$ , $$\begin{CD} @<\partial<< C_n(X)/C_n(A) @<\partial<< C_{n+1}(X)/C_{n+1}(A) @<\partial<< \end{CD}$$

The homology groups of this complex $H_n(X,A)$ , are called the relative homology groups of the pair $(X,A)$ . Under relatively mild hypotheses, $H_n(X,A)=H_n(X/A)$ where $X/A$ is the set of equivalence classes of the relation $x\sim y$ if $x=y$ or if $x,y\in A$ , given the quotient topology (this is essentially $X$ , with $A$ reduced to a single point). Relative homology groups are important for a number of reasons, principally for computational ones, since they fit into long exact sequences, which are powerful computational tools in homology.