In topology, a branch of mathematics, a collapse reduces a simplicial complex (or more generally, a CW complex) to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were invented by J. H. C. Whitehead. Collapses find applications in computational homology.
Let be an abstract simplicial complex.
Suppose that such that the following two conditions are satisfied:
- , in particular ;
- 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 R. H. Bing's house with two rooms and Christopher 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) Simplicial spaces, nuclei and m-groups, Proceedings of the London Mathematical Society 45, pp 243–327
- Kaczynski, Tomasz (2004). Computational homology. Mischaikow, Konstantin Michael, Mrozek, Marian,. New York: Springer. ISBN 9780387215976. OCLC 55897585.
- Cohen, Marshall M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York
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