examples and their modifications constructed using Corollary \[quasi-embedding\]) are not going to be $(n-1)$-connected. Relevance of $(n-1)$-connected $n$-polyhedra is ensured by the following observation.

\[connected\] Let $P$ be an $n$-polyhedron, $n\ne 2$, that embeds in $S^{2n}$. Then $P$ embeds in an $(n-1)$-connected $n$-polyhedron $Q$ such that $Q$ embeds in $S^{2n}$.

We shall show that more generally if $P$ is an $n$-polyhedron, $n\ge 2$, with vanishing van Kampen obstruction, then $P$ embeds in a polyhedron $Q$ with vanishing van Kampen obstruction. The van Kampen obstruction is well-known to be complete for $n\ne 2$ (see [@M2] and Remark \[erratum2\] below).

Let $K$ be some triangulation of $P$. By general position $P\cup |CK^{(n-2)}|$ embeds in $S^{2n}$. So without loss of generality $P$ is $(n-2)$-connected. Then $P$ is homotopy equivalent to a wedge of $(n-1)$-spheres. If $n=2$, $\pi_1(P)$ is finitely generated (and even finitely presented). If $n\ge 3$, $P$ is simply-connected, hence $\pi_{n-1}(P)$ is finitely generated over $\Z$ (see [@tD 20.6.2(3)] for a new simple proof of this classical result). Let $f_i{\colon}S^{n-1}\to P$ represent the free homotopy classes of some finite basis of $\pi_{n-1}(P)$. We may amend the $f_i$ so that the image of each $f_i$ meets each $n$-simplex of $K$. Let $Q$ be 