$S^{2n+1}$ from the fact that the $(n+1)$-skeleton of the $(2n+4)$-simplex has nonzero van Kampen obstruction even modulo $2$ (thereby reducing a certain $L_n$ to a certain strengthened $E_{n+1}$ by an algebraic argument with configuration spaces) [@M2 Example 4.7]. (iii) An odd-dimensional version of the van Kampen obstruction $\theta_{2n}$ to an $E_n$ can be identified as a complete obstruction $\theta_{2n+1}$ to an $L_n$ [@M2].

Arguments which might relate to a possible common higher dimensional generalization of the Kuratowski and Robertson–Seymour–Thomas theorems could include, apart from those in §\[commensurability\], those in [@Sa0], [@Sa1] (see also [@Um1]), [@Sk-A2]. Unfortunately, the present paper gives little clue to understanding [*proofs*]{} of the Kuratowski and Robertson–Seymour–Thomas theorems, but it does attempt to provide a better understanding of their [*statements*]{}.

To this end let us first look at a more statistically representative selection of cases.

Self-dual complexes
-------------------

MacLane and Adkisson proved that two embeddings of a $1$-polyhedron $\Gamma$ in the plane are inequivalent iff they differ already on a copy of $S^1$ or on a copy of the triod contained in $\Gamma$ [@AdM]. We restate this as follows.

If $K$ is a simplicial complex, then two embeddings of $|K|$ in $S^2$ are inequivalent iff they differ 