Embedding dimensions of simplicial complexes on few vertices

TL;DR

This paper characterizes when simplicial complexes with few vertices embed into spheres, linking combinatorial properties to topological embeddability, and extends classical theorems with new topological insights.

Contribution

It provides a simple criterion for embedding simplicial complexes into spheres and extends classical combinatorial theorems to a topological setting.

Findings

Characterization of embeddability based on non-face families

Recovery of the van Kampen--Flores theorem

Topological extension of the Erdős–Ko–Rado theorem

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.