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On locally constructible spheres and balls

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Acta Mathematica

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.)

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Authors and Affiliations

  1. Institute of Mathematics, Freie Universität Berlin, Arnimallee 2, DE-14195, Berlin, Germany

    Bruno Benedetti & Günter M. Ziegler

Authors
  1. Bruno Benedetti
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  2. Günter M. Ziegler
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Correspondence to Günter M. Ziegler.

Additional information

B. B. was supported by DFG via the Berlin Mathematical School, G. M. Z. was partially supported by DFG. Both authors are supported by ERC Advanced Grant No. 247029 “SDModels”.

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Benedetti, B., Ziegler, G.M. On locally constructible spheres and balls. Acta Math 206, 205–243 (2011). https://doi.org/10.1007/s11511-011-0062-2

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