在曲面中嵌入一对图和四维棱柱体的宽度
Embedding a pair of graphs in a surface, and the width of 4-dimensional prismatoids
摘要：
棱柱体是一个多面体，它的所有顶点都包含在两个平行的小平面中，称为它的底。它的宽度是在对偶图中从一个基到另一个基所需的步数。第一作者最近证明了赫希猜想的反例的存在等价于$d$ -宽度大于$d$的棱柱体的存在，并构造了这样的五维棱柱体。在这里，我们证明了同样的事情在四维空间中是不可能的。这一点可以通过观察2-球面上的一对图嵌入来证明，这对图嵌入是由两个基的正规粉丝产生的。
英文摘要：
A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The first author recently showed that the existence of counter-examples to the Hirsch conjecture is equivalent to that of $d$-prismatoids of width larger than $d$, and constructed such prismatoids in dimension five. Here we show that the same is impossible in dimension four. This is proved by looking at the pair of graph embeddings on a 2-sphere that arise from the normal fans of the two bases.
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