# Collapse (topology)

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

## Definition

Let be an abstract simplicial complex.

Suppose that such that the following two conditions are satisfied:

(ii) is a maximal face of K and no other maximal face of K contains ,

A simplicial **collapse** of K is the removal of all simplices such that , provided that 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.^{[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

- ↑
^{1.0}^{1.1}Whitehead, J.H.C. (1938)*Simplical spaces, nuclei and m-groups*, Proceedings of the London Mathematical Society 45, pp 243–327 - ↑ Cohen, M.M. (1973)
*A Course in Simple-Homotopy Theory*, Springer-Verlag New York