Neighborly cubical spheres and a cubical lower bound conjecture (Q1376054)

Cubical \(d\)-spheres have cells which are \(d\)-cubes. From a combinatorial viewpoint, these bear many resemblances to simplicial spheres; for example, there are Dehn-Sommerville equations relating the numbers of cells of each dimension. The analogue of neighbourliness for a cubical \(d\)-sphere is that its \(\Bigl\lfloor {d-1\over 2} \Bigr\rfloor\)-skeleton be isomorphic to that of a cube. The authors here employ an old idea which they call mirroring (originated by \textit{H. S. M. Coxeter} [Proc. Lond. Math. Soc., II. Ser. 43, 33-62 (1937; Zbl 0016.27101)]), to construct a cubical \(d\)-sphere from a simplicial \((d-1)\)-sphere. Together with a new notion called fissuring, this leads to neighbourly cubical spheres. These techniques are then used to investigate upper and lower bounds conjectures for cubical spheres, proving special cases for the former, and estabishing asymptotic results for the latter.