Random collapsibility and 3-sphere recognition

TL;DR

This paper investigates the complexity of collapsing 3-sphere triangulations using discrete Morse theory, introduces a new estimation method based on spanning trees, and classifies minimal triangulations of certain complexes.

Contribution

It presents a novel method to estimate collapsing difficulty of 3-sphere triangulations and classifies all minimal triangulations of the dunce hat and small non-collapsible complexes.

Findings

22 out of all 3-sphere triangulations with ≤8 vertices admit non-collapsing sequences.

Classified all minimal triangulations of the dunce hat.

Developed a heuristic for generating 3-sphere triangulations with complex collapsing properties.

Abstract

A triangulation of a $3$-manifold can be shown to be homeomorphic to the $3$-sphere by describing a discrete Morse function on it with only two critical faces, that is, a sequence of elementary collapses from the triangulation with one tetrahedron removed down to a single vertex. Unfortunately, deciding whether such a sequence exist is believed to be very difficult in general. In this article we present a method, based on uniform spanning trees, to estimate how difficult it is to collapse a given $3$-sphere triangulation after removing a tetrahedron. In addition we show that out of all $3$-sphere triangulations with eight vertices or less, exactly $22$ admit a non-collapsing sequence onto a contractible non-collapsible $2$-complex. As a side product we classify all minimal triangulations of the dunce hat, and all contractible non-collapsible $2$-complexes with at most $18$ triangles. This is complemented by large scale experiments on the collapsing difficulty of $9$- and $10$-vertex spheres. Finally, we propose an easy-to-compute characterisation of $3$-sphere triangulations which experimentally exhibit a low proportion of collapsing sequences, leading to a heuristic to produce $3$-sphere triangulations with difficult combinatorial properties.