# 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.[1] Collapses find applications in computational homology.[2]

## 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 ,

then
is called a **free face**.

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.[3]

## 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.[1]

## See also

## References

- Whitehead, J.H.C. (1938). "Simplicial spaces, nuclei and
*m*-groups".*Proceedings of the London Mathematical Society*.**45**: 243–327. - Kaczynski, Tomasz (2004).
*Computational homology*. Mischaikow, Konstantin Michael, Mrozek, Marian,. New York: Springer. ISBN 9780387215976. OCLC 55897585.CS1 maint: extra punctuation (link) - Cohen, Marshall M. (1973)
*A Course in Simple-Homotopy Theory*, Springer-Verlag New York