On the embeddability of $[3]\ast K$

Abstract

We relate the embeddability of the simplicial complex $[3]\ast K$ into ${\mathbb{R}}^{n+2}$ to that of $K$ into ${\mathbb{R}}^n$. In brief, the embeddability of $K$ into ${\mathbb{R}}^n$, in the metastable range $2n\ge 3(d+1)$, is equivalent to the embeddability of $[3]\ast K$ into ${\mathbb{R}}^{n+2}$. We show moreover than the van Kampen obstruction of $K$ vanishes if and only if the van Kampen obstruction of $[3]\ast K$ vanishes. It follows that for $d=2$, embeddabilty of $[3]\ast K$ is equivalent with the vanishing of the van Kampen obstruction for $K$, but not with the embeddability of $K$.

Main part of this paper is merged with arXiv:2003.12285