On polytopal upper bound spheres

Abstract

Generalizing a result (the case $k=1$) due to M. A. Perles, we show that any polytopal upper bound sphere of odd dimension $2k+1$ belongs to the generalized Walkup class ${\mathcal{K}}_k(2k+1)$, i.e., all its vertex links are $k$-stacked spheres. This is surprising since the $k$-stacked spheres minimize the face-vector (among all polytopal spheres with given $f_0,...,f_{k-1}$) while the upper bound spheres maximize the face vector (among spheres with a given $f_0$). It has been conjectured that for $d\ne 2k+1$, all $(k+1)$-neighborly members of the class ${\mathcal{K}}_k(d)$ are tight. The result of this paper shows that, for every $k$, the case $d=2k+1$ is a true exception to this conjecture.

4 pages

References in corpus (3)

• On stellated spheres, shellable balls, lower bounds and a combinatorial criterion for tightness
• On stellated spheres and a tightness criterion for combinatorial manifolds
• On the generalized lower bound conjecture for polytopes and spheres

Cited by in corpus (3)

• Separation index of graphs and stacked 2-spheres
• On stacked triangulated manifolds
• Tight and stacked triangulations of manifolds