On embeddability of joins and their `factors'

We present a short and clear proof of the following particular case of a 2006 result of Melikhov-Schepin: Let $K$ be a $k$-dimensional simplicial complex and $K*[3]$ the union of three cones over $K$ along their common bases. If $2d\ge3k+3$ and $K*[3]$ embeds into $\mathbb R^{d+2}$, then $K$ embeds into $\mathbb R^d$. We also present a generalization of this theorem. The proofs are based on the Haefliger-Weber `configuration spaces' embeddability criterion, equivariant suspension theorem and simple properties of joins and cones.