To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Collapse (topology)

From Wikipedia, the free encyclopedia

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 be an abstract simplicial complex.

Suppose that are two simplices of such that the following two conditions are satisfied:

  1. , in particular ;
  2. is a maximal face of and no other maximal face of contains ,

then is called a free face.

A simplicial collapse of is the removal of all simplices such that , where is a free face. If additionally we have , 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

See also

References

  1. ^ a b Whitehead, J.H.C. (1938). "Simplicial spaces, nuclei and m-groups". Proceedings of the London Mathematical Society. 45: 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
This page was last edited on 22 December 2019, at 13:32
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.