Smoothing discrete Morse theory

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 viceversa is true in all dimensions. This revises and improves a result by Gallais.