Given a finite simplicial complex (as a topological space) $\Delta$ and a face $\tau$, suppose we delete the interior of $\tau$ (a point if $\tau$ is a vertex, otherwise homeomorphic to an open ball of some dimension) and look at the remaining space $\Delta \setminus \mbox{int}(\tau)$ and see if the homotopy type changes. In particular, I am interested in a contractible $\Delta$. Now if we have $\Delta$ contractible and faces $\tau_1,...,\tau_n$ such that $\Delta \setminus \mbox{int}(\tau_i)$ is contractible, is it always the case that $\Delta \setminus (\mbox{int}(\tau_1) \cup... \cup \mbox{int}(\tau_n))$ is also contractible?
Note that none of the $\tau$'s can be maximal faces of $\Delta$, so we will never consider them. For some simple examples, one can remove any number of (nonmaximal) open faces of a simplex and still preserve contractiblility. Given the wedge of a 2-simplex with a 1-simplex, the only open face that can't be deleted is the common vertex, but deleting any combination of the other open faces is okay.
One idea I have is to consider the link of a nonmaximal face (with respect to $\Delta$), and if the link is contractible then deleting the interior of the face should preserve contractibility of the entire space, but I don't see a simple way to use this to prove the overall question.
I hope what I am asking is clear, and if any clarificiation is needed please let me know.
