Sergey Avvakumov and Alfredo Hubard Construct Cubical Spheres with Many Facets!

In this post, I discuss a remarkable new paper

Cubulating the sphere with many facets by Sergey Avvakumov and Alfredo Hubard

Abstract: For each we construct cube complexes homeomorphic to the -sphere with vertices in which the number of facets (assuming constant) is .

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 , . Our construction disproves it for all and sufficiently large. Moreover, since neighborly cubical polytopes have roughly facets, we show that even the order of growth (at least for the number of facets) in the conjecture is wrong.

Avvakumov and Hubard’s result about cubical two dimensional manifolds

A crucial ingredient in the proof is a theorem about cubical 2-dimensional manifolds. Sergey and Alfredo proved that for any there is a “cubulation” of a closed orientable 2-surface with vertices and squares. This result appears related to the 1890 Heawood Conjecture for the existence of triangulated surfaces with “many” triangles.

The paper is accompanied by beautiful, illuminating figures.

Updates regarding previous posts.

1. Noga Alon and Shakhar Smorodinsky uploaded to the arXiv their paper Extended VC-dimension, and Radon and Tverberg type theorems for unions of convex sets. We briefly talked about the Radon type result (an old question of mine) in this post. Moving from a Radon-type theorem to a Tverberg-type theorem required an extended notion of VC dimension that is of independent interest and may have further applications. There are also various remaining questions that I find interesting.  

2. To my previous post I added a Trivia question –  which mathematician is connected to both Abraham Fraenkel and Akitsugu Kawaguchi’s famous works (on different topics).  Hint: chess.

Some background and history for Avvakumov and Hubard’s New Result

A cubical complex is a collection of cubes (called “faces”) of various dimensions so that:

(1) A face of a cube in also belongs to ,

(2) The intersection of two faces of is a  face of both.

Alternatively you can consider a cubical complex as a poset with the lattice property so that every proper lower interval  is isomorphic to the face lattice of a cube.

A cubical complex realizes a topological space denoted by . If is a sphere we call a cubical sphere. Special cases of cubical spheres are the boundary complexes of cubical polytopes.

All these definitions are analogous to the definitions for simplicial complexes, simplicial spheres, and simplicial polytopes. However, the theory of simplicial polytopes and spheres is much more developed compared to the theory of cubical polytopes and spheres.

Questions about Cubical Polytopes and Spheres from the early-mid 90s.

In the early 90s Ron Adin proposed his well known and still open “generalized lower bound inequalities” for cubical polytopes. Those include, as a special case, conjectured lower bounds for the number of facets as a function of the dimension and the number of vertices. (Those lower bounds are also still open.) We discussed Adin’s conjecture in Item D) of this post.

The upper bound theorem for simplicial polytopes (and spheres) asserts that the number of facets (and -faces) for a simplicial -polytope (and a simplicial -sphere) is maximized by the cyclic -polytope with vertices. When is fixed the number of facets behaves like . What is the upper bound for the number of facets in the cubical case?

Proposition: Let be a cubical polytope (or, in fact cubical complex) with vertices and -dimensional faces. Then

.

Proof: For every -face , , consider its (large) diagonals. Every pair of (non-adjacent) vertices can occur only once as such a large diagonal.

Here are two conjectures that I proposed in the mid 90s  regarding upper bounds.

Conjecture 1: There are cubical convex -polytopes (and cubical -spheres) with vertices where , that have the same -skeleton as , the dimensional cube, .

For cubical spheres, this conjecture was proved by Eric Babson, Lou Billera and Clara Chan in 1997. Neighborly cubical polytopes were constructed by Michael Joswig and Günter Ziegler in 2000. 

Conjecture 2: The objects from Conjecture 1 attain the maximum number of k-faces among all cubical -polytopes (and even -dimensional spheres).

In their same paper Joswig and Ziegler disproved conjecture 2 for . Conjecture 2 would imply that the number of facets of a -polytope with vertices is bounded by .

The new paper by Sergey and Alfredo shows that the number of facets of cubical spheres can be much much larger – .  This is a major advance in our state of knowledge!

It would be very interesting to find cubical polytopes where the number of facets is more than . Also, as far as I know, there is no known upper bound smaller than . 

Jockusch’s 1993 Result 

Conjectures 1 and 2 were motivated by a 1993 paper by William Jockusch who showed that there are cubical 4-polytopes where the number of facets is larger than 5/4 times the number of vertices. This disproved an earlier conjecture that for every cubical polytope has fewer facets than vertices (or at least, perhaps the ratio is bounded).  This early conjecture in its stronger form was motivated by the interesting fact that there is no cubical sphere whose dual is also a cubical sphere. Until Jockusch’s paper, cubical zonotopes looked like candidates for the upper bound and for them the number of facets is always smaller than the number of vertices.  The papers by Babson, Billera, and Chan, and by Joswig and Ziegler showed that the ratio between the number of facets and the number of vertices can be unbounded.