On locally constructible spheres and balls

TL;DR

This paper characterizes locally constructible spheres and balls, establishing hierarchies with classical properties, and provides new results on the existence, enumeration, and collapsibility of simplicial 3-spheres and 3-balls.

Contribution

It introduces a comprehensive characterization of LC properties for d-spheres and d-balls, linking them to collapsibility, shellability, and constructibility, and resolves several open questions.

Findings

Not all simplicial 3-spheres are locally constructible.

Exponential bounds on the number of shellable 3-spheres.

All constructible 3-balls are collapsible.

Abstract

Durhuus and Jonsson (1995) introduced the class of "locally constructible" (LC) 3-spheres and showed that there are only exponentially-many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres ("the sphere minus a facet collapses to a (d-2)-complex") and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from this study are: 1.) Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.) 2.) There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.) 3.) All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.) 4.) Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)