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arXiv · 2109.04855

Embedding dimensions of simplicial complexes on few vertices

Florian Frick·Mirabel Hu·Verity Scheel·Steven Simon
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Abstract

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 Erd\H os--Ko--Rado theorem. By analogy with F\'ary'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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Florian Frick, Mirabel Hu, Verity Scheel, Steven Simon. 2021-09-10. Embedding dimensions of simplicial complexes on few vertices. https://doi.org/10.1007/s00026-023-00644-4

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