Square thread form

In topology, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead.

Definition

Let $K$ be an abstract simplicial complex.

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

(i) $\tau \subset \sigma$, in particular $\dim \tau <\dim \sigma$;

(ii) $\sigma$ 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 $\tau \subseteq \gamma \subseteq \sigma$, provided that $\tau$ 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 collapse 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.

Examples

• Complexes that do not have a free face cannot be collapsible. Two such interesting examples are Bing's house with two rooms and 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.

References

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See also

• Shelling (topology)

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