PL Morse theory in low dimensions

TL;DR

This paper develops a piecewise-linear Morse theory for PL manifolds, introducing notions of regularity, and shows that in low dimensions homologically regular points are strongly regular, with counterexamples in higher dimensions.

Contribution

It introduces criteria for strong regularity in PL Morse theory and demonstrates the dimensional threshold where these properties hold or fail.

Findings

Homologically regular points are always strongly regular in dimensions d ≤ 4.

Counterexamples exist in dimensions d ≥ 5.

Constructed an embedding of the dunce hat related to Mazur's manifold.

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 $d \leq 4$ 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.