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 reduced 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 itself in , we get that . Another insightful geometric example is given by the relative homology of where . Then we can use the long exact sequence Using exactness of the sequence we can see that contains a loop counterclockwise around the origin. Since the cokernel of fits into the exact sequence it must be isomorphic to . One generator for the cokernel is the -chain since its boundary map is See also Excision Theorem Mayer–Vietoris sequence Notes i.e., the boundary maps to References Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, Specific Homology theory