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

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

1. ${\displaystyle \tau \subset \sigma }$, in particular ${\displaystyle \dim \tau <\dim \sigma }$;
2. ${\displaystyle \sigma }$ 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.[3]