Acta Mathematica

On locally constructible spheres and balls

Bruno Benedetti and Günter M. Ziegler

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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.)


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”.

Article information

Acta Math. Volume 206, Number 2 (2011), 205-243.

Received: 22 June 2009
First available in Project Euclid: 31 January 2017

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2011 © Institut Mittag-Leffler


Benedetti, Bruno; Ziegler, Günter M. On locally constructible spheres and balls. Acta Math. 206 (2011), no. 2, 205--243. doi:10.1007/s11511-011-0062-2.

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