# Collapse (topology)

In topology, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead.[1]

## Definition

Let ${\displaystyle K}$ be an abstract simplicial complex.

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

(i) ${\displaystyle \tau \subset \sigma }$, in particular ${\displaystyle \dim \tau <\dim \sigma }$;

(ii) ${\displaystyle \sigma }$ is a maximal face of K and no other maximal face of K contains ${\displaystyle \tau }$,

then ${\displaystyle \tau }$ is called a free face.

A simplicial collapse of K is the removal of all simplices ${\displaystyle \gamma }$ such that ${\displaystyle \tau \subseteq \gamma \subseteq \sigma }$, where ${\displaystyle \tau }$ 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 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.[2]

## 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.[1]

## References

1. ^ a b Whitehead, J.H.C. (1938) Simplical spaces, nuclei and m-groups, Proceedings of the London Mathematical Society 45, pp 243–327
2. ^ Cohen, M.M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York