  We discuss a 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
$d \leq 4$ a homologically regular point on a PL $d$-manifold is always
strongly regular. Examples show that this fails to hold in higher dimensions $d
\geq 5$. 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.
