The dunce hat in a minimal non-extendably collapsible 3-ball

TL;DR

This paper constructs a minimal 8-vertex 3-ball that is collapsible but collapses onto a non-collapsible dunce hat, demonstrating the smallest such example and exploring its topological properties.

Contribution

It provides the first explicit geometric realization of a minimal 8-vertex triangulation of the dunce hat and shows the existence of a minimal collapsible 3-ball collapsing onto it.

Findings

The minimal 8-vertex 3-ball is collapsible.

The dunce hat is not collapsible.

The Alexander dual of the dunce hat is collapsible.

Abstract

We obtain a geometric realization of a minimal 8-vertex triangulation of the dunce hat in Euclidean 3-space. We show there is a simplicial 3-ball with 8 vertices that is collapsible, but also collapses onto the dunce hat, which is not collapsible. This 3-ball is as small as possible, because all triangulated 3-balls with fewer vertices are extendably collapsible. As we will see, the Alexander dual of the dunce hat is collapsible.