# Metric Geometry, Convexity and Collapsibility

## Karim Adiprasito, Bruno Benedetti – 2011

Collapsibility is a classical notion introduced by Whitehead as part of his simple homotopy theory. We provide several results relating it to metric geometry and convexity. (1) Every complex that is CAT(0) with a metric for which all vertex stars are convex is collapsible. Focus Area 3: Topological connectivity and diameter of Discrete Structures (2) Any linear subdivision of any polytope is simplicially collapsible after one barycentric subdivision. This solves up to one derived subdivision a classical question by Lickorish. (3) Any linear subdivision of any star-shaped polyhedron in R^s is simplicially collapsible after d-2 barycentric subdivisions at most. This presents progress on an old question by Goodrick. We furthermore provide the following applications: (1) Any simplicial complex admits a CAT(0) metric if and only if it admits collapsible triangulations. (2) All contractible manifolds (except for some 4-dimensional ones) admit collapsible CAT(0) triangulations. This provides a polyhedral version of a classical result of Ancel and Guilbault. (3) There are exponentially many geometric triangulations of S^d. This interpolates between the known result that boundaries of simplicial (d+1)-polytopes are exponentially many, and the conjecture that d-spheres are more than exponentially many. (4) In terms of the number of facets, there are only exponentially many geometric triangulations of space forms with bounded geometry. This establishes a discrete version of Cheeger's finiteness theorem.