On locally constructible spheres and balls

• 成果类型: Article
• 署名作者: Benedetti, Bruno; Ziegler, Guenter M.
• 署名单位: Free University of Berlin
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-011-0062-2
• 发表日期: 2011
• 页码: 205-243
• 关键词: nonconstructible simplicial balls; decompositions; triangulations; complexes; polytopes; gravity; entropy

摘要

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.).