PL Morse theory in low dimensions

Abstract

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

24 pages, 3 figures

References in corpus (2)

• Hardness of embedding simplicial complexes in ${\mathbb{R}}^d$
• On the definition of homological critical value

Cited by in corpus (1)

• Multiparameter Discrete Morse Theory