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Collapse (topology)

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Source updated June 15, 2025

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. Collapses find applications in computational homology.

01Definition

Let $K$ be an abstract simplicial complex.

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

$\tau \subsetneq \sigma ,$ in particular $\dim \tau <\dim \sigma ;$
$\sigma$ is a maximal face of $K$ and no other maximal face of $K$ contains $\tau ,$

then $\tau$ is called a free face.

A simplicial collapse of $K$ is the removal of all simplices $\gamma$ such that $\tau \subseteq \gamma \subseteq \sigma ,$ where $\tau$ is a free face. If additionally we have $\dim \tau =\dim \sigma -1,$ 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.

02Examples

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.

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Sources and credits

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Collapse (topology)

01Definition

02Examples

Sources and credits

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