Cubulating the sphere with many facets

TL;DR

The paper addresses the problem of face numbers in cube complexes by constructing cubulations of the $d$-sphere for all $d\ge3$ with many facets. It introduces an inductive approach that starts from a base cubulation, removes a facet to form a $d$-ball, and then increases dimension via products with an interval and handlegluing, achieving a lower bound of $c(d)\cdot n^{5/4}$ facets on at most $n$ vertices. This proves Kalai's conjecture is false in general, as neighborly cubical polytopes have facet growth around $n(\log n)^{d/2}$, while the new construction attains a fundamentally different rate. The work also notes a simple upper bound of $n(n-1)/2^{d}$ on the number of facets and discusses the need for stronger topological hypotheses to obtain subquadratic bounds.

Abstract

For each $d\geq 3$ we construct cube complexes homeomorphic to the $d$-sphere with $n$ vertices in which the number of facets (assuming $d$ constant) is $Ω(n^{5/4})$. 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 $d\geq 3$ 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.

Figures (10)

• Figure 1: Subdivision of the cone $u_1 * (\{v\} \times K_{m,m})$ for $m=2$.
• Figure 2: Inserting a square, insetting a cube, a subdivison of a square in ten squares.
• Figure 3: Proof of Lemma \ref{['lemma:short-skeleton-basis']}.
• Figure 4: Regularizing neighborhoods of the basis curves.
• Figure 5: Proof of Theorem \ref{['theorem:main']}.

Theorems & Definitions (13)

• Theorem 1.1
• Proposition 1.2
• proof
• Lemma 2.1
• Remark 2.2
• Definition 2.3: Canonical homological basis
• Lemma 2.4
• proof
• Lemma 2.5
• proof