I don't know much about simple homotopy theory, so maybe my question is quite trivial:

How does one prove that some complexes are contractible but non-collapsible?

*there are few refinements of this question:*

I will use the term "decollapse" for the inverse to the collapse operation.

I understand that if one allows to increase dimension of the complex, that the sequence of decollapses and collapses is the same as simple homotopy equivalence, and so any contractible complex is collapsible in this sense (Whitehead torsion is forced to vanish).

I've also heard that there are a lot of examples of 2-dimensional non-collapsible but contractible complexes, one of the most famous examples being Bing's House. I understand that there is no possible collapse of Bing's House, but it is not clear for me how does one prove that it is not equivalent to a point using decollapses and collapses (of dimension at most 2).

I also would like to ask whether it is possible to prove even for stronger operations: I'd like to be able to collapse any (polyhedrally) embedded closed disk. I suspect that even for these stronger operations Bing's House is still not equivalent to a point if we do not allow to increase the dimension of the complex.

Are there any known invariants for these problem?