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On locally constructible spheres and balls
Freie Universität Berlin, Germany.
(Mathematisches Institut, Freie Universitat, Berlin)
2011 (English)In: Acta Mathematica, ISSN 0001-5962, E-ISSN 1871-2509, Vol. 206, no 2, 205-243 p.Article in journal (Refereed) Published
Abstract [en]

Durhuus and Jonsson (1995) introduced the class of “locally constructible” (LC) 3-spheres and showed that there are only exponentially many combinatorial types of simplicial LC 3-spheres. Such upper bounds are crucial for the convergence of models for 3D quantum gravity. We characterize the LC property for d-spheres (“the sphere minus a facet collapses to a (d−2)-complex”) and for d-balls. In particular, we link it to the classical notions of collapsibility, shellability and constructibility, and obtain hierarchies of such properties for simplicial balls and spheres. The main corollaries from this study are: – Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.) There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.) – All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.) – Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)

Place, publisher, year, edition, pages
2011. Vol. 206, no 2, 205-243 p.
Keyword [en]
Nonconstructible Simplicial Balls, Convex Polyhedra, Quantum-Gravity, Triangulations, Complexes, Decompositions, Polytopes, Entropy, 3-Balls
National Category
URN: urn:nbn:se:kth:diva-93317DOI: 10.1007/s11511-011-0062-2ISI: 000291691900001OAI: diva2:515673
QC 20120627Available from: 2012-04-14 Created: 2012-04-14 Last updated: 2012-06-27Bibliographically approved

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Benedetti, BrunoZiegler, Günter
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