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. Collapses find applications in computational homology.

Definition

Let $K$ be an abstract simplicial complex.

Suppose that $\tau, \sigma$ are two simplices of $K$ such that the following two conditions are satisfied:
# $\tau \subseteq \sigma,$ 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.Cohen, Marshall M. (1973) ''A Course in Simple-Homotopy Theory'', Springer-Verlag New York

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.

See also

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References

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Algebraic topology
Properties of topological spaces