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> A cycle on a pair (X,A) is a chain on X whose boundary is a chain on A .

is this roughly saying that a cycle on (X,A) is a chain on X that would be a cycle on the quotient X/A?

Yes, this is true (or nearly) for simplicial complexes. A cycle on (X, A) is the same as a cycle on (X/A, A/A) = (X/A, single point), and this is a useful point of view, since X/A can be much simpler to visualize.

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