Collapsibility of CAT(0) spaces

Collapsibility is a combinatorial strengthening of contractibility. We relate this property to metric geometry by proving the collapsibility of any complex that is $$\mathrm {CAT}(0)$$ 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 $$\mathrm {CAT}(0)$$ CAT ( 0 ) cube complexes are collapsible. (2) Any triangulated manifold admits a $$\mathrm {CAT}(0)$$ CAT ( 0 ) metric if and only if it admits collapsible triangulations. (3) All contractible d -manifolds ( $$d \ne 4$$ d ≠ 4 ) admit collapsible $$\mathrm {CAT}(0)$$ CAT ( 0 ) triangulations. This discretizes a classical result by Ancel–Guilbault.

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