Barycentric subdivisions of convex complexes are collapsible

TL;DR

This paper proves that any linear subdivision of a convex polytope becomes collapsible after at most one barycentric subdivision, advancing understanding of collapsibility in PL topology.

Contribution

It establishes that linear subdivisions of convex polytopes are collapsible after one barycentric subdivision, and extends results to star-shaped polyhedra with multiple derived subdivisions.

Findings

Linear subdivisions of convex polytopes are collapsible after one barycentric subdivision.

Linear subdivisions of star-shaped polyhedra are collapsible after at most d-2 derived subdivisions.

Progress on longstanding questions in PL topology about collapsibility of subdivisions.

Abstract

A classical question in PL topology, asked among others by Hudson, Lickorish, and Kirby, is whether every linear subdivision of the d-simplex is simplicially collapsible. The answer is known to be positive for d<4. We solve the problem up to one subdivision, by proving that any linear subdivision of any polytope is simplicially collapsible after at most one barycentric subdivision. Furthermore, we prove that any linear subdivision of any star-shaped polyhedron in $\mathbb{R}^d$ is simplicially collapsible after d-2 derived subdivisions at most. This presents progress on an old question by Goodrick.