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Fat cellulations of the 3-sphere
Geometry/Topology| Speaker: | Greg Kuperberg, UC Davis |
| Location: | 693 Kerr |
| Start time: | Wed, May 29 2002, 4:10PM |
Description
It is easy to tell when a cellulation of the 2-sphere arises as the face
structure of a convex polyhedron. It is necessary and sufficient that
each cell is embedded and that the intersection of two cells is a cell;
such a cellulation is called strongly regular. This fact is part of
a collection of results that say a lot about the combinatorial
structure of convex polyhedra in R^3.
In comparison convex polytopes in R^4 are poorly understood. It is
known that not every strongly regular cellulation can be made convex.
But it is plausible to conjecture that strongly regular cellulations
are at least numerically equivalent to convex polytopes, in the sense
of having the same number of faces in every dimension. One measure of
the numerical profile of a polytope or cellulation is fatness, defined
as the number of edges plus ridges divided by the number of vertices
plus facets. I will describe a family of cellulations of the 3-sphere
with unbounded fatness. It is reasonable to conjecture that fatness
is bounded for convex 4-polytopes. If so, the numerics of a strongly
regular cellulation of the 3-sphere can be very different from that
of any convex polytope.
(Joint work with David Eppstein and Gunter Ziegler)
arXiv reference: math.CO/0204007