relative homology groups

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 tothe boundary map on $C_{*}(A)$, we can take the quotient complex $C_{*}(X,A)$,

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