$S^{2n}$ (see Theorem \[self-dual\]), and $g_r$ is finite-collapsible, $P_r$ does not embed in $S^{2n}$. If $Q$ is a proper subpolyhedron of $P_r$, then $R\bydef f_r^{-1}(Q)$ is a proper subpolyhedron of $P_0$, and $f_r|_R{\colon}R\to Q$ is clearly finite-collapsible. Since $R$ embeds in $S^{2n}$ (see Remark \[van Kampen\]), so does $Q$.

A map between polyhedra is called [*collapsible*]{}, resp. [*cell-like*]{} if every its point-inverse is collapsible, resp. contractible (and so, in particular, nonempty). The following generalization of Corollary \[finite-collapsible\] is a relatively easy consequence of well-known classical results.

\[minors embed\] (a) If $f{\colon}P\to Q$ is a collapsible map between polyhedra, and $P$ embeds in a manifold $M$, then $Q$ embeds in $M$.

\(b) If $f{\colon}P\to Q$ is a cell-like map between $n$-polyhedra, and $P$ embeds in an $m$-manifold $M$, where $m\ge n+3$, then $Q$ embeds in $M$.

The proof is not hard and quite instructive, but as this introduction is getting a bit too involved we defer it until §\[collapsing\]. There we also elaborate on the following

\[quasi-embedding\] If $f{\colon}X\to Y$ is a map between $n$-polyhedra whose nondegenerate point-inverses lie in a subpolyhedron of dimension $\le m-n-2$, and $X$ embeds in $S^m$, then $Y$ embeds in $S^m$.

The given embedding of $X$ 