  In bounding the homology of a manifold, Forman's Discrete Morse theory
recovers the full precision of classical Morse theory: Given a PL triangulation
of a manifold that admits a Morse function with c_i critical points of index i,
we show that some subdivision of the triangulation admits a boundary-critical
discrete Morse function with c_i interior critical faces of dimension d-i. This
dualizes and extends a recent result by Gallais. Further consequences of our
work are:
  (1) Every simply connected smooth d-manifolds (except possibly when d=4)
admits a locally constructible triangulation. (This solves a problem by
Zivaljevic.)
  (2) Up to refining the subdivision, the classical notion of geometric
connectivity can be translated combinatorially via the notion of collapse
depth.
