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≠ 4) admit collapsible CAT (0) triangulations. This discretizes a classical result by Ancel–Guilbault.
Original language | American English |
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Pages (from-to) | 181-199 |
Number of pages | 19 |
Journal | Geometriae Dedicata |
Volume | 206 |
Issue number | 1 |
DOIs | |
State | Published - 1 Jun 2020 |
Bibliographical note
Funding Information: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.
Funding Information:
We are grateful to Anders Bj?rner, Elmar Vogt, Federico Ardila, Frank Lutz, G?nter 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.
Publisher Copyright:
© 2019, Springer Nature B.V.
Keywords
- CAT (0) spaces
- Collapsibility
- Convexity Evasiveness
- Discrete Morse theory
- Triangulations