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 be an abstract simplicial complex.
Suppose that are two simplices of such that the following two conditions are satisfied:
- , in particular ;
- 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
- 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
References
- 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