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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:

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

  2. 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).

  3. 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?

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If you either collapse or de-collapse (the term expand is used here) a 2-dimensional polyhedron that contains a Bing's house, using only strata of dimension <=2, you end up with another 2-dimensional polyhedron that still contains a Bing's house. For that reason you never end up with a point.

The reason for this is intuitive: there is no way to eliminate a piece of Bing's house via collapsing, because there is no place to start a collapse: Bing's house has no "boundary", that is the Bing's house does not contain any point whose link is a segment.

It seems to me that this argument extends to your proposed "collapse of a sub-disc" move: just replace "a Bing's house" with "a subpolyhedron without boundary". This is an object that remains after the moves and never disappears, although it is undetected by any homotopy or homology invariant.

One nice introductory book for this subject is Metzler - Hog-Angeloni - Sieradski "Two-Dimensional Homotopy and Combinatorial Group Theory"

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  • $\begingroup$ Thank you for your answer! Are you sure this argument actually extends on generalized collapse? Take a look on $1$-dimensional complexes, for instance: any graph without leaves is a "polyhedron without boundary", but any graph is equivalent to the bouquet of $n$ circles under this generalized collapse... Generalized collapse reduces to the chain of few regular collapses and expansions, but in dimension n+1, I think. $\endgroup$
    – Lev Soukhanov
    Commented Feb 26, 2019 at 19:16

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