  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:
  1.) Not all simplicial 3-spheres are locally constructible. (This solves a
problem by Durhuus and Jonsson.)
  2.) There are only exponentially many shellable simplicial 3-spheres with
given number of facets. (This answers a question by Kalai.)
  3.) All simplicial constructible 3-balls are collapsible. (This answers a
question by Hachimori.)
  4.) Not every collapsible 3-ball collapses onto its boundary minus a facet.
(This property appears in papers by Chillingworth and Lickorish.)
