Collapse (topology)

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

Definition

Let K be an abstract simplicial complex.

Suppose that  \tau, \sigma \in K such that the following two conditions are satisfied:

(i)  \tau \subset  \sigma, in particular  \dim \tau <\dim  \sigma;

(ii)   \sigma is a maximal face of K and no other maximal face of K contains  \tau  ,

then  \tau is called a free face.

A simplicial collapse of K is the removal of all simplices \gamma such that  \tau \subseteq\gamma \subseteq  \sigma, where \tau 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

References

  1. 1 2 Whitehead, J.H.C. (1938) Simplical spaces, nuclei and m-groups, Proceedings of the London Mathematical Society 45, pp 243327
  2. Cohen, M.M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York

See also


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