Tight complexes in 3-space admit perfect discrete Morse functions

TL;DR

This paper proves that all tight simplicial 3-manifolds have perfect discrete Morse functions and that convex simplicial 3-balls are non-evasive, extending classical results to more general settings.

Contribution

It generalizes Chillingworth's theorem to include tight simplicial 3-manifolds and establishes new properties of convex and non-convex 3-balls.

Findings

All tight simplicial 3-manifolds admit perfect discrete Morse functions.

Convex simplicial 3-balls are non-evasive.

Many non-evasive 3-balls are not convex.

Abstract

In 1967, Chillingworth proved that all convex simplicial 3-balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3-manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth's theorem by proving that all convex simplicial 3-balls are non-evasive. In contrast, we show that many non-evasive 3-balls are not convex.