Suppose we have a connected simplicial complex $X$ such that we can partition the vertex set of $X$ into disjoint $M_1,\ldots,M_k$ such that for all $i$, the induced simplicial complex on the vertices in $M_i$ is a simplex. Then we can compose maps $f_1,\ldots,f_k$ such that $f_i$ deformation retracts $M_i$ to a point. The result is some complex $C$ that is homotopy equivalent to $X$, with $k$ vertices, one for each $M_i$.
What I would like is for $C$ on the vertex set $\{M_1,\ldots,M_k\}$ to be a simplicial complex with the following description. For any $I\subseteq [k]$, there is a face $\{M_i : i\in I\}$ in the complex $C$ precisely if there exist $x_i\in M_i$ for all $i\in I$ such that $\{x_i : i\in I\}$ is a face in $X$.
Intuitively, this should happen if whenever we retract something down to an edge, the thing we retracted was a simplex (thus we don't get double edges, or double faces in general).
Are there any nice sufficient conditions for this to happen? Of course I'm asking this question because I have a specific example in mind. Thus I would also accept an answer that deals with the following special case. It arose out of a discussion on MathOverflow, but I'll simplify the description to exactly what's related to my question here.
Special case. Fix some integer $n\ge 4$ and let $V$ be the set of all composite numbers in $[n]$ as well as all primes in the range $[2,n/2]$. Let $X$ be the simplicial complex on this vertex set, with a face for any subset of the vertex set in which no two elements are coprime. It can be shown that $X$ is connected.
Enumerate the primes $p_1,\ldots,p_k$ in $[2,n/2]$ and let $M_k$ be the set of all elements of $V$ whose smallest prime factor is $p_k$. It is clear that $M_k$ is a disjoint cover of $V$, and each $M_k$ is a simplex, since everything in it is divisible by $p_k$. So we can retract each $M_k$ down to a point to obtain a simpler complex $C$ with the same homotopy type as $X$. I would like to show that $C$ is contractible, which I think would follow if we could prove that $C$ is a simplicial complex such that for $I\subseteq [k]$, a set $\{M_i: i\in I\}$ is a (single) face in $C$ if and only if there are choices $x_i\in M_i$ for $i\in I$ such that $\{x_i : i\in I\}$ was a face in $X$. (Because then we could take the set $\{2,2\cdot p_2, \ldots, 2\cdot p_k\}$, which would show that $C$ is just a simplex.)
Thanks in advance for any help! Like I said, any insight on the problem would be appreciated, but especially sufficient conditions on $X$ that are satisfied by my example above!