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Collapse (topology)

{{Other uses|Collapse (disambiguation){{!}}Collapse}}

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.{{cite journal|authorlink=J. H. C. Whitehead|last=Whitehead|first= J.H.C.| year=1938|title=Simplicial spaces, nuclei and m-groups|journal= Proceedings of the London Mathematical Society|volume= 45|pages=243–327}} Collapses find applications in computational homology.{{Cite book|title=Computational homology|first=Tomasz|last=Kaczynski|date=2004|publisher=Springer|others=Mischaikow, Konstantin Michael, Mrozek, Marian|isbn=9780387215976|location=New York|oclc=55897585}}

Definition

Let K be an abstract simplicial complex.

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

  1. \tau \subsetneq \sigma, in particular \dim \tau < \dim \sigma;
  2. \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 \tau = \dim \sigma - 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.Cohen, Marshall M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York

Examples

See also

  • {{annotated link|Discrete Morse theory}}
  • {{annotated link|Shelling (topology)}}

References

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{{Topology}}

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Category:Algebraic topology

Category:Properties of topological spaces