  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
$P*P$ 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
  - $X*K$ embeds in $\mathbb R^{m+2k}$, where $K$ is the $(k-1)$-skeleton of
the $2k$-simplex; or
  - $X*L$ 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 $\mathbb R^{m+2k}$.
