English

Van Kampen-Flores theorem for cell complexes

Algebraic Topology 2023-08-07 v2

Abstract

The van Kampen-Flores theorem states that the nn-skeleton of a (2n+2)(2n+2)-simplex does not embed into R2n\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 nn-skeleton of a (2n+2)(2n+2)-simplex into R2n+1\mathbb{R}^{2n+1}.

Cite

@article{arxiv.2109.09919,
  title  = {Van Kampen-Flores theorem for cell complexes},
  author = {Daisuke Kishimoto and Takahiro Matsushita},
  journal= {arXiv preprint arXiv:2109.09919},
  year   = {2023}
}

Comments

10 pages, some of the results (especially Theorem 1.4 and Corollary 1.5) were improved, final version, to appear in Discrete & Computational Geometry

R2 v1 2026-06-24T06:09:58.328Z