Embeddability of joins and products of polyhedra

We present a short proof of S. Parsa’s theorem that there exists a compact $n$-polyhedron $P$, $n\ge 2$, non-embeddable in ${\mathbb{R}}^{2n}$, such that $P\ast P$ embeds in ${\mathbb{R}}^{4n+2}$. This proof can serve as a showcase for the use of geometric cohomology. We also show that a compact $n$-polyhedron $X$ embeds in ${\mathbb{R}}^m$, $m\ge \frac{3(n+1)}{2}$, if either - $X\ast K$ embeds in ${\mathbb{R}}^{m+2k}$, where $K$ is the $(k-1)$-skeleton of the $2k$-simplex; or - $X\ast L$ embeds in ${\mathbb{R}}^{m+2k}$, where $L$ is the join of $k$ copies of the $3$-point set; or - $X$ is acyclic and $X\times {\text{(triod)}}^k$ embeds in ${\mathbb{R}}^{m+2k}$.