  The purpose of this paper is to introduce a version of singular homology
based on smooth mappings of manifolds with corners. Although variants of such a
theory exists in the literature, we felt that certain points were not
adequately addressed. In particular, our goal is to construct a chain level
theory based on smooth mappings of manifolds with corners. In addition, we will
avoid using the fact that smooth manifolds with can be triangulated. As we
shall see, transversality and intersections play a major role in setting up
this theory. From a pedagogical viewpoint, having intersection theory arguments
available from the start facilitates simple and intuitive computations.
