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 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 ($d \ne 4$) admit collapsible CAT(0) triangulations. This discretizes a classical result by Ancel--Guilbault. 
Karim Adiprasito and Bruno Benedetti
math.MG math.CO math.DG