---
abstract: |
    We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of $6/7$ of the statement of the Robertson–Seymour–Thomas intrinsic linking criterion. Let us call a cell complex [*dichotomial*]{} if to every nonempty cell there corresponds a unique nonempty cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two $3$-dimensional dichotomial cell complexes, and their $1$-skeleta are $K_5$ and $K_{3,3}$; and precisely six $4$-dimensional ones, and their $1$-skeleta all but one graphs of the Petersen family.

    In higher dimensions $n\ge 3$, we observe that in order to characterize those compact $n$-polyhedra that embed in $\R^{2n}$ in terms of finitely many “prohibited minors”, it suffices to establish finiteness of the list of all $(n-1)$-connected $n$-dimensional finite cell complexes that do not embed in $\R^{2n}$ yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the $n$-skeleta of $(2n+1)$-dimensional dichotomial cell complexes. The $2$-skeleta of $5$-dimensional dichotomial cell complexes include (apart from the three joins of the $i$-skeleta of $(2i+2)$-simplices) at least ten non-simplicial complexes.
address: 'Steklov Mathematical Institute of the Russian Academy 