Focus Area 3: Topological connectivity and diameter of Discrete Structures 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.)