Van Kampen-Flores theorem for cell complexes

TL;DR

This paper generalizes the van Kampen-Flores theorem to continuous maps from skeletons of regular CW complexes into Euclidean spaces, extending known non-embeddability results and chirality properties.

Contribution

It provides two proofs for the generalized van Kampen-Flores theorem and extends results on the chirality of embeddings for certain complexes.

Findings

Generalization of van Kampen-Flores theorem to CW complexes

Two distinct proofs for the generalized theorem

Extension of chirality results for embeddings into Euclidean space

Abstract

The van Kampen-Flores theorem states that the $n$-skeleton of a $(2n+2)$-simplex does not embed into $\mathbb{R}^{2n}$. We give two proofs for its generalization to a continuous map from a skeleton of a certain regular CW complex (e.g. a simplicial sphere) into a Euclidean space. We will also generalize Frick and Harrison's result on the chirality of embeddings of the $n$-skeleton of a $(2n+2)$-simplex into $\mathbb{R}^{2n+1}$.