$\partial\Delta^4$ is not an edge-minor of $S$; see [@Ne Example 6.1].

The preceding results now imply the following version of the Halin–Jung theorem:

A simplicial complex $K$ has no edge-minors among the seven self-dual subcomplexes of $\partial\Delta^4$ along with $K_{3,3}$ iff $|K|$ embeds in $S^2$.

Since a given (finite) simplicial complex only has finitely many minors, this yields an algorithm deciding embeddability of $|K|$ in $S^2$. A presumably faster, but more elaborate algorithm is discussed in [@MTW].

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

\(a) $|L|$ embeds in $S^{m-2}$;

\(b) the embeddings of $|L|$ in $S^{m-1}$ induced by $j$ and $hj$ are equivalent, where $j$ is the inclusion of $|K|$ in $S^{m-1}\bydef|\partial\Delta^{m_1}*\dots*\partial\Delta^{m_r}|$, and $h$ is an orientation-reversing homeomorphism of $S^{m-1}$.

The proof, given in §\[edge-minors2\], is reminiscent of the above argument towards Theorem \[self-dual\].

Hemi-icosahedron and hemi-dodecahedron {#semi}
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The central symmetry of $\R^3$ in the origin yields a simplicial free involution on the boundary of a regular icosahedron centered at the origin. Its quotient by this involution is a simplicial complex $\R P^2_\triangle$ with $6$ vertices, triangulating the real projective plane. It 