高度邻近的中心对称球体的新家族

I. Novik, Hailun Zheng
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引用次数: 4

摘要

1995年,Josckusch构造了一个无限族的中心对称(简称cs)三角形,它们是cs- $2$ -邻接的$3$ -球体。最近,Novik和Zheng扩展了Jockusch的构造:对于所有$d$和$n>d$,他们构造了一个具有$2n$个顶点的$d$ -球体的cs三角测量$\Delta^d_n$,即cs- $\lceil d/2\rceil$ -neighborly。这里提供了几个新的cs结构。结果表明,对于所有$k>2$和一个足够大的$n$,存在另一个顶点为$2n$的$(2k-1)$球的cs- $k$相邻的cs三角剖分,而对于$k=2$存在$\Omega(2^n)$这样的成对非同构三角剖分。还证明了对于所有$k>2$和足够大的$n$,具有$2n$顶点cs- $(k-1)$相邻的$(2k-1)$ -球的$\Omega(2^n)$对非同构cs三角剖分。这些构造是在研究$\Delta^d_n$的各个方面的基础上,特别是在一些与Gale均匀性条件精神相似的必要条件和充分条件的基础上进行的。在此过程中,证明了Jockusch的球$\Delta^3_n$是可壳的,并对Murai-Nevo关于$2$堆积可壳球的问题给出了肯定的回答。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
New families of highly neighborly centrally symmetric spheres
In 1995, Josckusch constructed an infinite family of centrally symmetric (cs, for short) triangulations of $3$-spheres that are cs-$2$-neighborly. Recently, Novik and Zheng 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 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.
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