To S. Parsa's theorem on embeddability of joins

TL;DR

This paper clarifies and references a result by S. Parsa showing that certain simplicial complexes do not embed into specific Euclidean spaces individually, but their join can embed into a higher-dimensional Euclidean space.

Contribution

It provides a reliable reference for Parsa's theorem on the embeddability of joins of simplicial complexes, clarifying a previously complex result.

Findings

Existence of complexes not embeddable in certain dimensions

Join of complexes embeds into higher-dimensional space

Clarification of Parsa's theorem on embeddability

Abstract

The purpose of this short note is to guide a reader to a reliable reference for the following result of S. Parsa: For any $k,l\ge2$ there exist simplicial complexes $K, L$ of dimensions $k,l$ such that $K$ does not embed into $\mathbb R^{2k}$, and $L$ does not embed into $\mathbb R^{2l}$, but the join $K*L$ embeds into $\mathbb R^{2(k+l+1)}$.