We discuss a piecewise linear (PL) analogue of Morse theory for PL manifolds. There are several notions of regular and critical points. A point is homologically regular if the homology does not change when passing through its level, it is strongly regular if the function can serve as one coordinate in a chart. Several criteria for strong regularity are presented. In particular we show that in low dimensions d≤4 a homologically regular point on a PL d-manifold is always strongly regular. Examples show that this fails to hold in higher dimensions d≥5. One of our constructions involves an 8-vertex embedding of the dunce hat into a polytopal 4-sphere with 8 vertices such that a regular neighborhood is Mazur's contractible 4-manifold. Finally, decidability questions in this context are discussed.
Last update: February 29, 2024.