Collapse (topology)

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

Definition

Let K be a simplicial complex, and suppose that s is a simplex in K. We say that s has a free face t if t is a face of s and t has no other cofaces. We call (s, t) a free pair. If we remove s and t from K, we obtain another simplicial complex, which we call an elementary collapse of K. A sequence of elementary collapses is called a collapse. A simplicial complex that has a collapse to a point, implying all other points were in free pairs, is called collapsible.

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 and in fact simple homotopy equivalent to a point.
• Any n-dimensional PL manifold that is collapsible is in fact piecewise-linearly isomorphic to an n-ball.

References

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

See also

• Shelling (topology)
• Erik Christopher Zeeman

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