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 \(\mathrm {CAT}(0)\) 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. (1)

    All \(\mathrm {CAT}(0)\) cube complexes are collapsible.

  2. (2)

    Any triangulated manifold admits a \(\mathrm {CAT}(0)\) metric if and only if it admits collapsible triangulations.

  3. (3)

    All contractible d-manifolds (\(d \ne 4\)) admit collapsible \(\mathrm {CAT}(0)\) triangulations. This discretizes a classical result by Ancel–Guilbault.

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We are grateful to Anders Björner, Elmar Vogt, Federico Ardila, Frank Lutz, Günter Ziegler, Tadeusz Januszkiewicz and Victor Chepoi, for useful suggestions. Karim Adiprasito acknowledges support by a Minerva fellowship of the Max Planck Society, an NSF Grant DMS 1128155, an ISF Grant 1050/16 and ERC StG 716424 - CASe. Bruno Benedetti acknowledges support by NSF Grants 1600741 and 1855165, the DFG Collaborative Research Center TRR109, and the Swedish Research Council VR 2011-980. Part of this work was supported by the National Science Foundation under Grant No. DMS-1440140 while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Fall 2017 semester.

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Adiprasito, K., Benedetti, B. Collapsibility of CAT(0) spaces. Geom Dedicata 206, 181–199 (2020).

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  • \(\mathrm {CAT}(0)\) spaces
  • Collapsibility
  • Discrete Morse theory
  • Convexity Evasiveness
  • Triangulations

Mathematics Subject Classification

  • 52B70
  • 52A20
  • 57Q10
  • 53C23