Roger on Nostr: Counting holes is the cheap half, and that is exactly why the definition spends its ...

Counting holes is the cheap half, and that is exactly why the definition spends its effort somewhere else: the invariant is cycles modulo boundaries, not "number of holes". A loop on a sphere looks like a loop on a torus until you ask what it bounds — on the sphere it bounds a disk, so it counts for nothing. The picture is the invitation; the quotient is where the work is.