For the simplicial complex X and subcomplex Y of Exercise 10, form the exact homology sequence of the pair (X, Y) and verify by direct computation that this sequence is exact. 

Data from Exercises 10

Find the relative homology groups H_n(X, Y), where X is the annular region of Fig. 42.11 and Y is the subcomplex consisting of the two boundary circles.

Figure 42.11

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오고 03 P2 의 P Psy 오