of his paper in fact contains this remark, which was also pointed out by his referee. We note that Proposition \[nevo\] along with assertion (2) above yields a proof of Conjecture 1.3 in [@Ne]: if $L$ is a Nevo minor of $K$, and $|K|$ embeds in $S^m$, then $|L|$ embeds in $S^m$.

By assertion (1) above, it suffices to show that if $f{\colon}K\to L$ is a Nevo-admissible edge contraction, then $L$ is a minor of $K$. Suppose that $f$ shrinks an edge $e=\{v_1,v_2\}$. If $f$ is not admissible, $\operatorname{lk}(v_1)\cap\operatorname{lk}(v_2)$ is the union of $\operatorname{lk}(e)$ with a nonempty collection of $(n-1)$-simplices $\sigma_1,\dots,\sigma_k$. Let $K^+$ be the subcomplex of $K$ obtained by removing the $n$-simplices $\{v_0\}*\sigma_i$ from $K$. Then $f|_{K^+}{\colon}K^+\to L$ is an admissible edge contraction. Hence $L$ is a minor of $K$.

\[Steinitz\] A simplicial complex of dimension $\le 2$ is an edge-minor of every its subdivision.

This is proved in §\[edge-minors2\] using innermost circle arguments.

Since every triangulation $T$ of $S^2$ is easily seen to be a subdivision of $\partial\Delta^3$, Lemma \[Steinitz\] implies the result of Steinitz (1934) that $\partial\Delta^3$ is a minor of $T$; see [@Zo] and [@Su].

In contrast, there exists a subdivision $S$ of $\partial\Delta^4$ such that 