Another basic Q. (sorry, I am comfortable with the Euler characteristic, and with filling holes in STL models, but <i>not</i> with homology):<p>Taking chains to be a rough analogue of functions, could we say that cycles on (X,A) serve as a kind of dual space for A? (because if we ignore intermediates, they're basically connecting A<->A?) So from the dual side, we seem to have the case that an appropriate quotient might be A/X?<p>(maybe an alternate view of what I'm trying to ask about: in <i>Mathematics Made Difficult</i>, we find many proofs of [something in A implies something else in A] which, instead of being limited in scope to A and its subfields as is customary, wander leisurely through X to ultimately prove the inclusion in A)
> <i>A cycle on a pair
(X,A)
is a chain on
X
whose boundary is a chain on
A
.</i><p>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?