Bruno Benedetti: Metric Geometry and Collapsibility

How to define a distance on a simplicial complex? The natural way is to put a metric on each simplex, so that metrics on intersecting simplices agree on the intersection. For example, the EQUILATERAL FLAT metric puts on each simplex the metric of the regular Euclidean simplex. Creative solutions are also welcome: For example, we may use the metric of a regular spherical simplex. Or dropping regularity, we may give different lengths to different edges, maintaining some assumptions on the angles: This way one gets ACUTE metrics, or NON-OBTUSE metrics, that are not necessarily equilateral.

We play these games in relation with the serious notion of non-positive curvature (and of CAT(0) spaces), introduced by Alexandrov and popularized by Gromov. We show that if a complex is CAT(0) with a non-obtuse metric, then it is combinatorially collapsible. This is a discrete analog of the classical theorem that CAT(0) complexes are contractible.

As an application, we construct some manifolds different than balls that admit a collapsible triangulation. This contrasts a 1939 result of Whitehead. Also, it shows that surprisingly discrete Morse theory can be sharper than smooth Morse theory in bounding the homology of a manifold. If time permits, we discuss applications to Hirsch diameter bounds.

This is joint work with Karim Adiprasito (arxiv.org/abs/1107.5789).