Event

Talk
On triangulated 3-spheres
14 Jun 2010, 15:00 – 16:00

The unsolved question whether there are only exponentially-many combinatorial types of simplicial 3-spheres is crucial for the convergence of models for 3D quantum gravity. Working towards this question, Durhuus and Jonsson (1995) introduced the restriction to "locally constructible" (LC) 3-spheres, and showed that there are only exponentially-many LC 3-spheres. 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 the LC-property on the one hand to Knot Theory, on the other hand to Combinatorial Topology concepts such as collapsibility, shellability, and constructibility. Thus we obtain strict hierarchies of such properties for simplicial balls and spheres.

Two 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.)

(Joint work with Bruno Benedetti)

Related to:

• Günter M. Ziegler