Embedding dimensions of simplicial complexes on few vertices
arXiv:2109.04855 · doi:10.1007/s00026-023-00644-4
Abstract
We provide a simple characterization of simplicial complexes on few vertices that embed into the -sphere. Namely, a simplicial complex on vertices embeds into the -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 Fáry's theorem for planar graphs, we show in addition that such complexes satisfy the rigidity property that continuous and linear embeddability are equivalent.
8 pages