Collapse (topology)

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In topology, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead.[1]

Definition[edit]

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.

This definition can be extended to CW-complexes and is the basis for the concept of simple-homotopy equivalence.[2]

Examples[edit]

  • 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.[1]

References[edit]

  1. ^ a b Whitehead, J.H.C. (1938) Simplical spaces, nuclei and m-groups, Proceedings of the London Mathematical Society 45, pp 243–327
  2. ^ Cohen, M.M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York

See also[edit]