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.

中文翻译：

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CAT(0) 空间的可折叠性

可折叠性是可收缩性的组合强化。我们通过证明 $$\mathrm {CAT}(0)$$ CAT ( 0 ) 具有所有顶点星都是凸的度量的任何复合体的可折叠性将此属性与度量几何相关联。这加强并概括了 Crowley 的结果。我们工作的进一步结果是： (1) 所有 $$\mathrm {CAT}(0)$$ CAT ( 0 ) 立方体复合体都是可折叠的。(2) 任何三角流形允许 $$\mathrm {CAT}(0)$$ CAT ( 0 ) 度量当且仅当它允许可折叠三角剖分。(3) 所有可收缩的 d -流形 ( $$d \ne 4$$ d ≠ 4 ) 允许可折叠 $$\mathrm {CAT}(0)$$ CAT ( 0 ) 三角剖分。这离散了 Ancel-Guilbault 的经典结果。