Collapse (topology) explained

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.^[1] Collapses find applications in computational homology.^[2]

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.^[3]

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.

Notes and References

1. J. H. C. Whitehead. Whitehead. J.H.C.. 1938. Simplicial spaces, nuclei and m-groups. Proceedings of the London Mathematical Society. 45. 243–327.
2. Book: Kaczynski, Tomasz. Computational homology. 2004. Springer. Mischaikow, Konstantin Michael, Mrozek, Marian. 9780387215976. New York. 55897585.
3. Cohen, Marshall M. (1973) A Course in Simple-Homotopy Theory, Springer-Verlag New York