On locally constructible spheres and balls

Bruno Benedetti, Günter M. Ziegler

Research output: Contribution to journalArticle

16 Scopus citations

Abstract

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.)

Original languageEnglish (US)
Pages (from-to)205-243
Number of pages39
JournalActa Mathematica
Volume206
Issue number2
DOIs
StatePublished - Jun 1 2011
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)

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