maps
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In this subsection we assume familiarity with collapsing and regular neighborhoods (see e.g. [@Hu]). The following fact is well-known.

\[trivial\] If $Q$ is a collapsible subpolyhedron of a manifold $M$, then the quotient $M/Q$ is homeomorphic to $M$.

Let $N$ be a regular neighborhood of $Q$ in $M$. Since $Q$ is collapsible, $N$ is a ball. On the other hand, $N/Q$ is homeomorphic rel $\partial N$ to $pt*\partial N$, which is also a ball. This yields a homeomorphism $N\to pt*\partial N$ keeping $\partial N$ fixed, which extends by the identity to a homeomorphism $M\to M/Q$.

Let us call a map between polyhedra [*finite-collapsible*]{} if it is the composition of a sequence of quotient maps, each shrinking a collapsible subpolyhedron to a point.

\[finite-collapsible\] Let $f{\colon}P\to Q$ be a finite-collapsible map between polyhedra. If $P$ embeds in $S^m$, then so does $Q$.

We use this nearly trivial observation to give a simple proof of a result by Zaks [@Za] (see also [@Um2], [@Sa1 3.7.1]); as a byproduct, we also get a slightly stronger statement:

\[Zaks\] For each $n\ge 2$ there exists an infinite list of pairwise non-homeomorphic compact $(n-1)$-connected $n$-polyhedra $P_i$ such that each $P_i$ does not embed in 