Fat 3-Spheres, 4-Polytopes and 5-Lattices

Günter M. Ziegler
TU Berlin/UC Berkeley
ziegler@math.tu-berlin.de

The following three classes contain very similar objects -- in a topological resp. geometric resp. combinatorial model:

• $3$-dimensional CW-spheres with the intersection property,
• $4$-dimensional convex polytopes, and
• Eulerian lattices of rank $5$.

We introduce and study the parameter of fatness, $\frac{f_1+f_2}{f_0+f_3}$ for these three classes -- which seems to be a key indicator to show how little we know. So, it is not clear whether fatness is bounded at all on any of these classes. Here we construct examples of

• rational $4$-dimensional convex polytopes of fatness larger than $5-\varepsilon$,
• $4$-dimensional convex polytopes of fatness larger than $5.01$, and
• $3$-dimensional CW-spheres with the intersection property of fatness larger than $6-\varepsilon$.

This implies counter-examples to conjectured $f$-vector inequalities of Bayer (1988) and of Billera & Ehrenborg (1999).

Most of our examples are constructed using the ``Eppstein construction'': as the convex hull of a $4$-polytope with all ridges tangent to $S^3$, and its polar. This construction has a close connection with ball packings in $S^3$. Their study should lead to an infinite family of $2$-simple $2$-simplicial polytopes.

(Joint work with David Eppstein, UC Irvine)