  This manuscript develops a geometric approach to ordinary cohomology of
smooth manifolds, constructing a cochain complex model based on co-oriented
smooth maps from manifolds with corners. Special attention is given to the
pull-back product of such smooth maps, which provides our geometric cochains
with a partially defined product structure inducing the cup product in
cohomology. A parallel treatment of homology is also given allowing for a
geometric unification of the contravariant and covariant theories.
