the following general result already implies that the prohibited subcomplexes that occur in the Kuratowski, MacLane–Adkisson and Halin–Jung theorems as well as in part (a) of the Robertson–Seymour–Thomas theorem [*must*]{} occur there (possibly along with some additional ones).

\[self-dual\] Let $K=K_1*\dots*K_r$, where each $K_i$ is a self-dual subcomplex of $\partial\Delta^{m_i}$, where $m_1+\dots+m_r=m$. Then

\(a) [(Schild, 1993)]{} $|K|$ does not embed in $S^{m-2}$; but $|L|$ embeds in $S^{m-2}$ for every proper subcomplex $L$ of $K$;

\(b) every embedding $g$ of $|K|$ in $S^{m-1}$ is inequivalent to $hg$, where $h$ is an orientation-reversing homeomorphism of $S^{m-1}$; but for every proper subcomplex $L$ of $K$, the restrictions of $j$ and $hj$ to $|L|$ are equivalent, where $j$ denotes the inclusion of $|K|$ in $|\partial\Delta^{m_1}*\dots*\partial\Delta^{m_r}|=S^{m-1}$.

A simple proof of the assertions on proper subcomplexes in (a) and (b) is given below. A simple proof of the non-embeddability in (a) and the inequivalence in (b) is given in §\[embeddings\], where we also elaborate on historic/logical antecedents of Theorem \[self-dual\].

Let $\sigma$ be a maximal simplex of $K$ that is not in $L$. Then $\sigma=\sigma_1*\dots*\sigma_r$, where each $\sigma_i$ is a maximal (and in particular nonempty) simplex of $K_i$. Since $K_i$ is self-dual, the complementary simplex $\tau_i$ 