  A Morse function f on a manifold with corners M allows the characterization
of the Morse data for a critical point by the Morse index. In fact, a modified
gradient flow allows a proof of the Morse theorems in a manner similar to that
of classical Morse theory. It follows that M is homotopy equivalent to a
CW-complex with one cell of dimension \lambda for each essential critical point
of index \lambda. The goal of this article is to determine the boundary maps of
this CW-complex, in the case where M is compact and orientable.
  First, the boundary maps are defined in terms of the modified gradient flow.
Then a transversality condition is imposed which insures that the attaching map
is non-degenerate in a neighborhood of each critical point. The degree of this
map is then interpreted as a sum of trajectories connecting two critical points
each counted with a multiplicity determined by a choice of orientations on the
tangent spaces of the unstable manifold at each critical point.
