TITLE: Is the topological concept of collapsible useful?
QUESTION [9 upvotes]: I ask this question because in the process of reviewing for my topology comp, I began rereading Alg Topology by Hatcher.  In the introduction is the famous Bing's House of Two Rooms.  I thought this was an interesting example and began reading about it on the web (procrastinating).  Several sites note that Bing's house is contractable (as described in Hatcher) but not collapsible.  The definition of collapsible does not appear in any of my topology or alg topology books (Munkres, Hatcher, Spanier) and the only definition I have found is on wikipedia.  So this brings me to my question, is collapsible a useful topological concept? And can anyone show me why Bing's house is not collapsible (I guess I probably do not fully comprehend the definition)?

REPLY [4 votes]: A small addendum: Collapses seem also useful in some applications, namely practical homology computations, as preprocessing steps for algorithms that compute homology. (Concretely - say you have to compute the homology of a simplicial complex with lots of facets; before starting to write down adjacency matrices etc. you might want to check if you can first "simplify" the complex with collapsing steps.) Sometimes this approach fails, as you see if you start with the Bing house. Yet sometimes one sequence of collapses works much better than another sequence, so it's also a matter of luck. 
However, an advantage of collapsibility over contractibility is that the former is an algorithmically-decidable property. (A non-efficient algorithm to decide collapsibility consists in trying all possible collapsing sequences, and see if one leaves us with a point.) In contrast, contractibility is in some cases algorithmically undecidable (by reduction to the word problem). For more info, cf. the recent preprint by Martin Tancer.