If is a topological space, and a subspace, then the inclusion map makes into a subgroup of . Since the boundary map on restricts to the boundary map on , we can take the quotient complex ,

The homology groups of this complex , are called the relative homology groups of the pair . Under relatively mild hypotheses, where is the set of equivalence classes of the relation if or if , given the quotient topology (this is essentially , with 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.