Geometry of collapsing and free deformation retraction

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.