Relative Homology and the Associated Long Exact Sequence

1 Relative Homology and the Associated Long Exact Sequence 

In the next definition we encounter the first and simplest case of a filtration. These will be investigated in greater depth in Chapter 16. 

Definition 5.8. Let X be an arbitrary topological space, and let A be a subspace of X. Then we call the ordered pair (X,A) a pair of spaces. When X is a CW complex and A is its CW subcomplex, the pair (X,A) is called a CW pair. When X and A are simplicial complexes, the pair (X, A) is called a simplicial pair. 

It is easy to see that when X, A) is a pair of spaces, the singular chain complex of A is a subcomplex of the singular chain complex of X. 

Definition 5.9. Let X, A) be a pair, and let TL be a commutative ring with unit. The relative homology of this pair with coefficients in TZ is denoted by H, X,A, TZ), and it is defined to be the homology of the quotient chain complex Tl)/Cf A- 7 )). 

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