extends by general position to an embedding of $X\cup_Z MC(f|_Z)$, where $Z$ is the given subpolyhedron and $MC$ denotes the mapping cylinder. On the other hand, the point-inverses of the projection $X\cup_Z MC(f|_Z)\to Y$ are cones, so Theorem \[minors embed\](a) yields an embedding of $Y$.

Let us call a polyhedron $Z$ an [*$h$-minor*]{} of a polyhedron $X$, if there exists a subpolyhedron $Y$ of $X$ and a cell-like map $Y\to Z$. (We note that composition of cell-like maps is obviously cell-like, cf. [@Sm]; see also [@Hat comment to Corollary 2.3] for a combinatorial proof.) By Theorem \[minors embed\](a), all $h$-minors of an $n$-polyhedron embeddable in $S^m$, $m-n\ge 3$, also embed in $S^m$.

One could hope that using $h$-minors instead of subpolyhedra enables one to prevent Theorem \[Zaks\] from “driving us from the paradise” which Theorem \[self-dual\] might seem to promise. This is not so: using Corollary \[quasi-embedding\], it is easy to construct an infinite list $P_0,P_1,\dots$ of pairwise non-homeomorphic $n$-polyhedra, $n\ge 3$, such that each $P_i$ does not embed in $S^{2n}$, but every its proper (in any reasonable sense) $h$-minor embeds in $S^{2n}$. In fact, the original examples of Zaks [@Za] work (compare [@Ne 6.5]).

However, all such examples (Zaks’ 