New families of highly neighborly centrally symmetric spheres

Abstract:

Jockusch [J. Combin. Theory Ser. A 72 (1995), pp. 318–321] constructed an infinite family of centrally symmetric (cs, for short) triangulations of $3$-spheres that are cs-$2$-neighborly. Recently, Novik and Zheng [Adv. Math. 370 (2020), 16 pp.] extended Jockusch’s construction: for all $d$ and $n>d$, they constructed a cs triangulation of a $d$-sphere with $2n$ vertices, $\Delta ^d_n$, that is cs-$\lceil d/2\rceil$-neighborly. Here, several new cs constructions, related to $\Delta ^d_n$, are provided. It is shown that for all $k>2$ and a sufficiently large $n$, there is another cs triangulation of a $(2k-1)$-sphere with $2n$ vertices that is cs-$k$-neighborly, while for $k=2$ there are $\Omega (2^n)$ such pairwise non-isomorphic triangulations. It is also shown that for all $k>2$ and a sufficiently large $n$, there are $\Omega (2^n)$ pairwise non-isomorphic cs triangulations of a $(2k-1)$-sphere with $2n$ vertices that are cs-$(k-1)$-neighborly. The constructions are based on studying facets of $\Delta ^d_n$, and, in particular, on some necessary and some sufficient conditions similar in spirit to Gale’s evenness condition. Along the way, it is proved that Jockusch’s spheres $\Delta ^3_n$ are shellable and an affirmative answer to Murai–Nevo’s question about $2$-stacked shellable balls is given.