Smoothing discrete Morse theory

TL;DR

This paper establishes an equivalence between PL handle theory and discrete Morse theory, explores implications for manifold triangulations, and advances understanding of Morse functions on spheres and 3-manifolds.

Contribution

It proves the equivalence of PL handle vectors and discrete Morse vectors, and derives new results on triangulations and Morse functions on manifolds.

Findings

Equivalence of discrete Morse vectors and PL handle vectors in all dimensions.

Every simply connected smooth d-manifold (d ≠ 4) admits locally constructible triangulations.

Some non-PL 5-spheres admit discrete Morse functions with only 2 critical faces.

Abstract

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.)