Collapsibility of CAT(0) spaces

Karim Adiprasito; Bruno Benedetti

doi:10.1007/s10711-019-00481-x

Collapsibility of CAT(0) spaces

Karim Adiprasito, Bruno Benedetti^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

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
Pages (from-to)	181-199
Number of pages	19
Journal	Geometriae Dedicata
Volume	206
Issue number	1
DOIs	https://doi.org/10.1007/s10711-019-00481-x
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

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10.1007/s10711-019-00481-x

Cite this

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title = "Collapsibility of CAT(0) spaces",

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.",

keywords = "CAT (0) spaces, Collapsibility, Convexity Evasiveness, Discrete Morse theory, Triangulations",

author = "Karim Adiprasito and Bruno Benedetti",

note = "Funding Information: We are grateful to Anders Bj{\"o}rner, Elmar Vogt, Federico Ardila, Frank Lutz, G{\"u}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. 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: {\textcopyright} 2019, Springer Nature B.V.",

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N2 - 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.

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KW - Convexity Evasiveness

KW - Discrete Morse theory

KW - Triangulations

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IS - 1

ER -

Collapsibility of CAT(0) spaces

Abstract

Bibliographical note

Keywords

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