Geometry of collapsing and free deformation retraction

TL;DR

This paper establishes a geometric criterion for polyhedron collapsibility via free deformation retraction and explores metric invariants, correcting a previous claim about injective metric spaces.

Contribution

It provides a new characterization of collapsibility using free deformation retraction and addresses inaccuracies in prior work on injective metric spaces.

Findings

Polyhedron collapses are equivalent to free deformation retractions.

Counterexample to Isbell's claim about injective metric spaces.

Partial correction to previous assertions on metric invariants.

Abstract

We show that a compact polyhedron $P$ collapses to a subpolyhedron $Q$ if and only if it admits a piecewise-linear free deformation retraction onto $Q$. We also consider further possibilities for invariant characterisations of collapsibility in terms of metrics; in this connection, we provide a partial correction to Isbell's claim that every injective metric space is freely contractible, and present a counterexample to a step in the original argument.