Collapse (topology)
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]
Definition
Let [math]\displaystyle{ K }[/math] be an abstract simplicial complex.
Suppose that [math]\displaystyle{ \tau, \sigma }[/math] are two simplices of [math]\displaystyle{ K }[/math] such that the following two conditions are satisfied:
- [math]\displaystyle{ \tau \subseteq \sigma, }[/math] in particular [math]\displaystyle{ \dim \tau \lt \dim \sigma; }[/math]
- [math]\displaystyle{ \sigma }[/math] is a maximal face of [math]\displaystyle{ K }[/math] and no other maximal face of [math]\displaystyle{ K }[/math] contains [math]\displaystyle{ \tau, }[/math]
then [math]\displaystyle{ \tau }[/math] is called a free face.
A simplicial collapse of [math]\displaystyle{ K }[/math] is the removal of all simplices [math]\displaystyle{ \gamma }[/math] such that [math]\displaystyle{ \tau \subseteq \gamma \subseteq \sigma, }[/math] where [math]\displaystyle{ \tau }[/math] is a free face. If additionally we have [math]\displaystyle{ \dim \tau = \dim \sigma - 1, }[/math] 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]
Examples
- 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.[1]
See also
- Discrete Morse theory
- Shelling (topology) – Mathematical concept
References
- ↑ 1.0 1.1 Whitehead, J.H.C. (1938). "Simplicial spaces, nuclei and m-groups". Proceedings of the London Mathematical Society 45: 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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