Long exact sequence of a pair

Again we see that the identity map induces identity maps, and that the composition of two cellular maps induces the composition of induced maps. Theorem 3.41 is best when one uses the alternative definition of cellular homology using the relative homology groups. It will follow in that context from the naturality of the associated long exact sequence of a pair. We postpone the precise argument until Subsection 5.2.2. 

The Long Exact Sequence of a Pair and Some Applications... 

Proposition 5.14. Any map p between pairs of spaces (X, A) and (X, A) induces a chain complex map between the corresponding homology long exact sequences of these pairs. 

In particular, the composition 9 o ij is a 0-map, since both maps are from the long exact sequence of the pair Hence the composition... 

Proof. Let C denote the cone over A, which has been added to X, and consider the reduced version of the long exact sequence of the pair (Q, C) ... 

For any pair of spaces (X,A) we have a long exact sequence of homology groups... 

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