Embeddability of joins and products of polyhedra - Melikhov, Sergey A. (Topological Methods in Nonlinear Analysis, 2022)
We present a short proof of S. Parsa's theorem that there existsa 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}