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. Collapses find applications in computational homology.

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) $$\tau \subseteq \sigma,$$ 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.

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.