Collapse (topology)

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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.[1] Collapses find applications in computational homology.[2]


Let be an abstract simplicial complex.

Suppose that such that the following two conditions are satisfied:

  1. , in particular ;
  2. 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.

This definition can be extended to CW-complexes and is the basis for the concept of simple-homotopy equivalence.[3]


See also[edit]


  1. ^ a b Whitehead, J.H.C. (1938) Simplicial spaces, nuclei and m-groups, Proceedings of the London Mathematical Society 45, pp 243–327
  2. ^ Kaczynski, Tomasz (2004). Computational homology. Mischaikow, Konstantin Michael, Mrozek, Marian,. New York: Springer. ISBN 9780387215976. OCLC 55897585.
  3. ^ Cohen, Marshall M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York