  After surveying classical notions of PL topology of the Seventies, we clarify
the relation between Morse theory and its discretization by Forman. We show
that PL handles theory and discrete Morse theory are equivalent, in the sense
that every discrete Morse vector on some PL triangulation is also a PL handle
vector, and conversely, every PL handle vector is also a discrete Morse vector
on some PL triangulation. It follows that in dimension up to 7, every discrete
Morse vector on some PL triangulation is also a smooth Morse vector; the vice
versa is true in all dimensions. This revises and improves a result by Gallais.
  Some further consequences of our work are:
  (1) For $d \ne 4$, every simply connected smooth d-manifold admits locally
constructible triangulations. In contrast, the Mazur 4-manifold has no locally
constructible triangulation. (This solves a question by Zivaljevic and
completes work by the author and Ziegler.)
  (2) The Heegaard genus of 3-manifolds can be characterized as the smallest
integer g for which some triangulation of the manifold has discrete Morse
vector (1,g,g,1). (This allows for heuristics to bound the Heegaard genus of
any 3-manifold.)
  (3) Some non-PL 5-spheres admit discrete Morse functions with only 2 critical
faces. (This result, joint with Adiprasito, completes the Sphere Theorem by
Forman.)
