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 poin

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.