Collapse (topology)
In topology, a branch of mathematics, a collapse reduces a simplicial complex to a homotopy-equivalent subcomplex. Collapses, like CW complexes themselves, were invented by J. H. C. Whitehead. Collapses find applications in computational homology.
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
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.
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, but not collapsible.
- Any n-dimensional PL manifold that is collapsible is in fact piecewise-linearly isomorphic to an n-ball.