Example of collapsible complex which collapses to a non-collapsible complex  In their paper http://arxiv.org/abs/0907.2954 , Barmak and Minian claim that 
"there exist collapsible complexes which collapse to nontrivial subcomplexes with no free faces"
but unfortunately do not provide a reference.  Where can I find an example of such a complex?  Are there conditions on the complex that will ensure that a collapsible complex will collapse to a point regardless of how you do the collapse? I'm not asking either of these questions "up to barycentric subdivision" but in terms of the actual complex itself.  Thank you.
 A: Check this paper by B. Benedetti and F. Lutz, which gives an explicit example (with 8 vertices) of a triangulated 3-ball that collapses to a dunce hat.

EDIT: Let me also give a conceptual proof. Notice that a simplicial complex $K$ with contractible geometric realisation is simple homotopy equivalent to a point (see M.M. Cohen's book on simple homotopy theory), which means we can transform it into a point by a sequence of elementary expansions and collapses. Moreover, all the expansions may be done first. The expanded complex is collapsible. But by reversing the sequence of expansions you can collapse it into the original contractible complex $K$. Now letting $K$ be any contractible, non-collapsible complex (e.g. the dunce hat) gives you the example you looked for.

Complexes that collapse to a point regardless of the chosen collapsing sequence are called extendably collapsible in the cited paper of Benedetti and Lutz. For example, all triangulated 3-balls with less than 8 vertices are extendably collapsible and all collapsible 1- and 2-dimensional simplicial complexes are extendably collapsible.
