already on $|L|$ for some subcomplex $L$ of $K$, isomorphic to $\Delta^2$ or $(\Delta^2)^{(0)}*\Delta^0$ or a subdivision of $\Delta^0\sqcup\partial\Delta^2$.

It is quite obvious also that $|K|$ embeds in $S^1$ iff $K$ contains no subcomplex isomorphic to $\Delta^2$ or $(\Delta^2)^{(0)}*\Delta^0$ or a subdivision of $\Delta^0\sqcup\partial\Delta^2$.

Halin and Jung were able to go one dimension higher and gave a list of prohibited subcomplexes for the problem of embedding of a $2$-polyhedron in the plane [@HJ]. We again restate it as applied to the problem of embedding a polyhedron of an arbitrary dimension in $S^2$:

If $K$ is a simplicial complex, $|K|$ embeds in $S^2$ iff it does not contain a subcomplex, isomorphic to a subdivision of $K_5$, $K_{3,3}$, or one of the following complexes: $$\begin{gathered}
HJ_0\bydef\Delta^3,\\
HJ_1\bydef\Delta^0\sqcup\partial\Delta^3,\\
HJ_2\bydef\Delta^0*(\Delta^0\sqcup\partial\Delta^2),\\
HJ_3\bydef\Delta^1*\partial\Delta^1*\emptyset\cup
\partial\Delta^1*\emptyset*\Delta^0\cup\emptyset*\Delta^1*\emptyset,\\
HJ_4\bydef(\Delta^2)^{(0)}*\Delta^1,\\
HJ_5\bydef\Delta^2*\emptyset\cup(\Delta^2)^{(0)}*\partial\Delta^1.\\\end{gathered}$$

See [@MTW Appendix A] for pictures of $HJ_1$ through $HJ_5$. We note that $|HJ_0|$, $|HJ_3|$, $|HJ_4|$ and $|HJ_5|$ each contain a subpolyhedron homeomorphic to one of $|K_5|$, $|K_{3,3}|$, $|HJ_1|$ and $|HJ_2|$. This leads to a shorter list of prohibited [*subpolyhedra*]{} for the problem of embedding a polyhedron in $S^2$. A proof of this weak version of the Halin–Jung theorem is rather easy modulo the Kuratowski theorem (see [@Sk-A1]).

A statement 