obtained by adjoining $n$-disks to $P$ along the $f_i$. By construction, $Q$ is $(n-1)$-connected. Moreover, due to our choice of the $f_i$, the van Kampen obstruction of $Q$ is zero.

\[dual-collapsible\] The proof of Theorem \[minors embed\](a) works to establish it for the more general class of dual-collapsible maps. A map is [*dual-collapsible*]{} if it can be triangulated by a simplicial map $f{\colon}K\to L$ such that for each simplex $\tau$ of $L$, its dual cone $\tau^*$ has collapsible preimage $|f^{-1}(\tau^*)|$. It can be shown that composition of dual-collapsible maps is dual-collapsible (which seems not to be the case for collapsible maps, even though it happens to be the case for collapsible retractions [@Co2 8.6]). Many assertions in the present paper involving cell-like maps (directly or through $h$-minors), including the statement of the main theorem, hold for dual-collapsible maps.

Edge-minors
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An [*edge contraction*]{} is a simplicial map $f{\colon}K\to L$ that sends every edge onto an edge, apart from one edge $\{v_1,v_2\}$ which it shrinks onto a vertex. We call $f$ admissible if $\operatorname{lk}\{v_1\}\cap\operatorname{lk}\{v_2\}=\operatorname{lk}\{v_1,v_2\}$. An equivalent condition is that $\{v_1,v_2\}$ is not contained in any “missing face” of $K$, i.e. in an isomorphic copy of $\partial\Delta^n$ in $K$ that does not 