Examples
One important use of relative homology is the computation of the homology groups of quotient spaces . In the case that is a subspace of fulfilling the mild regularity condition that there exists a neighborhood of that has as a deformation retract, then the group is isomorphic to . We can immediately use this fact to compute the homology of a sphere. We can realize as the quotient of an n-disk by its boundary, i.e. . Applying the exact sequence of relative homology gives the following:
Because the disk is contractible, we know its homology groups vanish in all dimensions, so the above sequence collapses to the short exact sequence:
Therefore, we get isomorphisms . We can now proceed by induction to show that . Now because is the deformation retract of a suitable neighborhood of, we get that
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