Collapsibility of CAT(0) spaces

TL;DR

This paper establishes that certain CAT(0) complexes with convex vertex stars are collapsible, leading to broad implications for the structure of CAT(0) spaces, cube complexes, and triangulated manifolds.

Contribution

It proves the collapsibility of CAT(0) complexes with convex vertex stars, generalizing previous results and connecting metric geometry with combinatorial topology.

Findings

All CAT(0) cube complexes are collapsible.

A triangulated manifold admits a CAT(0) metric iff it has collapsible triangulations.

All contractible d-manifolds (d ≠ 4) admit collapsible CAT(0) triangulations.

Abstract

Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is CAT(0) with a metric for which all vertex stars are convex. This strengthens and generalizes a result by Crowley. Further consequences of our work are: (1) All CAT(0) cube complexes are collapsible. (2) Any triangulated manifold admits a CAT(0) metric if and only if it admits collapsible triangulations. (3) All contractible d-manifolds ($d \ne 4$) admit collapsible CAT(0) triangulations. This discretizes a classical result by Ancel--Guilbault.