Embeddability of joins and products of polyhedra

Abstract

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