Metric Geometry and Collapsibility
Karim Adiprasito, Bruno Benedetti · arXiv (Cornell University) · 2011
Cheeger’s finiteness theorem bounds the number of diffeomorphism types of manifolds with bounded curvature, diameter and volume; the Hadamard–Cartan theorem, as popularized by Gromov, shows the contractibility of all non-positively curved simply connected metric length spaces. We establish a discrete version of Cheeger’s theorem (“In terms of the number of facets, there are only exponentially many geometric triangulations of Riemannian man-ifolds with bounded geometry”), and a discrete version of the Hadamard–Cartan theorem (“Every complex that is CAT(0) with a metric for which all vertex stars are convex, is col-lapsible”). The first theorem has applications to discrete quantum gravity; the second shows that Forman’s discrete Morse theory may be even sharper than classical Morse theory, in bounding the homology of a manifold. In fact, although Whitehead proved in 1939 that all PL collapsible manifolds are balls, we show that some collapsible manifolds are not balls. Further central consequences of our work are: (1) Every flag connected complex in which all links are strongly connected, is Hirsch. (This strengthens a result by Provan–Billera.) (2) Any linear subdivision of the d-simplex collapses simplicially, after d − 2 barycentric subdivisions. (This presents progress on an old question by Kirby and Lickorish.) (3) There are exponentially many geometric triangulations of Sd. (This interpolates between the result that polytopal d-spheres are exponentially many, and the conjecture that all triangulations of Sd are exponentially many.) (4) If a vertex-transitive simplicial complex is CAT(0) with the equilateral flat metric, then it is a simplex. (This connects metric geometry with the evasiveness conjecture.) (5) The space of phylogenetic trees is collapsible. (This connects discrete Morse theory to mathematical biology.)