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On locally constructible spheres and balls

Please always quote using this URN:urn:nbn:de:0296-matheon-6283
  • 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: (1.) Not all simplicial 3-spheres are locally constructible. (This solves a problem by Durhuus and Jonsson.) (2.) There are only exponentially many shellable simplicial 3-spheres with given number of facets. (This answers a question by Kalai.) (3.) All simplicial constructible 3-balls are collapsible. (This answers a question by Hachimori.) (4.) Not every collapsible 3-ball collapses onto its boundary minus a facet. (This property appears in papers by Chillingworth and Lickorish.)

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Metadaten
Author:Bruno Benedetti, Günter M. Ziegler
URN:urn:nbn:de:0296-matheon-6283
Referee:Günter M. Ziegler
Document Type:Preprint, Research Center Matheon
Language:English
Date of first Publication:2009/07/21
Release Date:2009/07/21
Institute:Technische Universität Berlin
Zuse Institute Berlin (ZIB)
MSC-Classification:05-XX COMBINATORICS (For finite fields, see 11Txx) / 05Axx Enumerative combinatorics For enumeration in graph theory, see 05C30 / 05A16 Asymptotic enumeration
52-XX CONVEX AND DISCRETE GEOMETRY / 52Bxx Polytopes and polyhedra / 52B22 Shellability
57-XX MANIFOLDS AND CELL COMPLEXES (For complex manifolds, see 32Qxx) / 57Mxx Low-dimensional topology / 57M25 Knots and links in S3 (For higher dimensions, see 57Q45)
Preprint Number:629
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