Cubulating the sphere with many facets

Abstract

For each $d\ge 3$ we construct cube complexes homeomorphic to the $d$-sphere with $n$ vertices in which the number of facets (assuming $d$ constant) is $\hat{I}\text{c◯}(n^{5/4})$. This disproves a conjecture of Kalai's stating that the number of faces (of all dimensions) of cubical spheres is maximized by the boundaries of neighbourly cubical polytopes. The conjecture was already known to be false for $d=3$, $n=64$. Our construction disproves it for all $d\ge 3$ and $n$ sufficiently large. Moreover, since neighborly cubical polytopes have roughly $n(\log n{)}^{d/2}$ facets, we show that even the order of growth (at least for the number of facets) in the conjecture is wrong.