# Collapse (topology)

In topology, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead.

## Definition

Suppose that $\tau ,\sigma \in K$ such that the following two conditions are satisfied:

A simplicial collapse of K is the removal of all simplices $\gamma$ such that $\tau \subseteq \gamma \subseteq \sigma$ , provided that $\tau$ is a free face. If additionally we have dim τ = dim σ-1, then this is called an elementary collapse.

A simplicial complex that has a collapse 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 Bing's house with two rooms and 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.