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Abstract: Durhuus and Jonsson 1995 introduced the class of -locally constructible-LC 3-spheres and showed that there are only exponentially-many combinatorialtypes of simplicial LC 3-spheres. Such upper bounds are crucial for theconvergence of models for 3D quantum gravity.We characterize the LC property for d-spheres -the sphere minus a facetcollapses to a d-2-complex- and for d-balls. In particular, we link it tothe classical notions of collapsibility, shellability and constructibility, andobtain hierarchies of such properties for simplicial balls and spheres. Themain corollaries from this study are:1. Not all simplicial 3-spheres are locally constructible. This solves aproblem by Durhuus and Jonsson.2. There are only exponentially many shellable simplicial 3-spheres withgiven number of facets. This answers a question by Kalai.3. All simplicial constructible 3-balls are collapsible. This answers aquestion by Hachimori.4. Not every collapsible 3-ball collapses onto its boundary minus a facet.This property appears in papers by Chillingworth and Lickorish.

Author: Bruno Benedetti, Günter M. Ziegler

Source: https://arxiv.org/

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