  We use a vector field flow defined through a cubulation of a closed manifold
to reconcile the partially defined commutative product on geometric cochains
with the standard cup product on cubical cochains, which is fully defined and
commutative only up to coherent homotopies. The interplay between intersection
and cup product dates back to the beginnings of homology theory, but, to our
knowledge, this result is the first to give an explicit cochain level
comparison between these approaches.
