Metric Geometry and Collapsibility

Details Metric Geometry and Collapsibility Metric Geometry and Collapsibility

2011-07-28

arxiv.org/abs/1107.5789v2

Mathematics

Metric Geometry

35 pages, 5 figures. Rewritten and expanded version, with several new results, especially on complexes with convex geometric r

Scientific paper

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 manifolds 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 collapsible"). 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 S^d. (This interpolates between the result that polytopal d-spheres are exponentially many, and the conjecture that all triangulations of S^d 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.)

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