Subdivisions, shellability, and collapsibility of products

TL;DR

This paper proves that certain derived subdivisions of convex polytope triangulations are shellable, characterizes PL properties via subdivisions, and shows that contractible complexes can be made collapsible through product operations, advancing topological combinatorics.

Contribution

It introduces a new relative shellability concept, characterizes PL triangulations via subdivisions, and demonstrates how products with intervals can make complexes collapsible.

Findings

Second derived subdivision of any convex polytope triangulation is shellable.

First derived subdivision of convex 3D polytope triangulation is shellable.

Contractible complexes can be made collapsible by taking products with an interval.

Abstract

We prove that the second derived subdivision of any rectilinear triangulation of any convex polytope is shellable. Also, we prove that the first derived subdivision of every rectilinear triangulation of any convex 3-dimensional polytope is shellable. This complements Mary Ellen Rudin's classical example of a non-shellable rectilinear triangulation of the tetrahedron. Our main tool is a new relative notion of shellability that characterizes the behavior of shellable complexes under gluing. As a corollary, we obtain a new characterization of the PL property in terms of shellability: A triangulation of a sphere or of a ball is PL if and only if it becomes shellable after sufficiently many derived subdivisions. This improves on results by Whitehead, Zeeman and Glaser, and answers a question by Billera and Swartz. We also show that any contractible complex can be made collapsible by repeatedly taking products with an interval. This strengthens results by Dierker and Lickorish, and resolves a conjecture of Oliver. Finally, we give an example that this behavior extends to non-evasiveness, thereby answering a question of Welker.