少量点上简单复合体的嵌入维数

Pub Date : 2023-03-28 DOI:10.1007/s00026-023-00644-4
Florian Frick, Mirabel Hu, Verity Scheel, Steven Simon
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引用次数: 3

摘要

我们提供了嵌入d球面的几个顶点上的单纯复形的一个简单表征。也就是说,在\(d+3\)顶点上的单纯复形嵌入到d球面中,当且仅当其非面不形成相交族。作为直接结果,我们恢复了经典的van Kampen–Flores定理,并提供了Erdõs–Ko–Rado定理的拓扑扩展。通过与平面图的Fáry定理的类比,我们还证明了这种复形满足连续可嵌入性和线性可嵌入性等价的刚性性质。
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Embedding Dimensions of Simplicial Complexes on Few Vertices

We provide a simple characterization of simplicial complexes on few vertices that embed into the d-sphere. Namely, a simplicial complex on \(d+3\) vertices embeds into the d-sphere if and only if its non-faces do not form an intersecting family. As immediate consequences, we recover the classical van Kampen–Flores theorem and provide a topological extension of the Erdős–Ko–Rado theorem. By analogy with Fáry’s theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.

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