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

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

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

1. ${\displaystyle \tau \subsetneq \sigma ,}$ in particular ${\displaystyle \dim \tau <\dim \sigma ;}$
2. ${\displaystyle \sigma }$ is a maximal face of ${\displaystyle K}$ and no other maximal face of ${\displaystyle K}$ contains ${\displaystyle \tau ,}$

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

A simplicial collapse of ${\displaystyle K}$ 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 ${\displaystyle \dim \tau =\dim \sigma -1,}$ 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

• 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