Collapse (topology)

In topology, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead.^[1]

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 $\tau \subseteq\gamma \subseteq \sigma$ , where $\tau$ 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.^[2]

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.^[1]

References

1 2 Whitehead, J.H.C. (1938) Simplical spaces, nuclei and m-groups, Proceedings of the London Mathematical Society 45, pp 243–327
↑ Cohen, M.M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York

Collapse (topology)

Definition

Examples

References

See also