Let be an abstract simplicial complex.
Suppose that such that the following two conditions are satisfied:
(i) , in particular ;
(ii) is a maximal face of K and no other maximal face of K contains ,
then is called a free face.
A simplicial collapse of K is the removal of all simplices such that , where is a free face. If additionally we have dim τ = dim σ-1, then this is called an elementary collapse.
A simplicial complex that has a sequence of collapses leading to a point is called collapsible. Every collapsible complex is contractible, but the converse is not true.
- Complexes that do not have a free face cannot be collapsible. Two such interesting examples are Bing's house with two rooms and Zeeman's dunce hat; they are contractible (homotopy equivalent to a point), but not collapsible.
- Any n-dimensional PL manifold that is collapsible is in fact piecewise-linearly isomorphic to an n-ball.
- Whitehead, J.H.C. (1938) Simplical spaces, nuclei and m-groups, Proceedings of the London Mathematical Society 45, pp 243–327
- Cohen, M.M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York
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