For the simplicial complex X and subcomplex a of Exercise 6, form the exact homology sequence of the pair (X, a) and verify by direct computation that this sequence is exact.

Data from Exercises 6

In a computation analogous to Examples 44. 9 and 44.12 of the text, find the relative homology groups H_n( X, a) for the torus X with subcomplex a, as shown in Figs. 42.13 and Fig. 42.14.