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

Definition

Let ${\displaystyle K}$ be an abstract simplicial complex.

Suppose that ${\displaystyle \tau ,\sigma }$ are two simplices of ${\displaystyle K}$ such that the following two conditions are satisfied:

1. ${\displaystyle \tau \subsetneq \sigma ,}$ in particular ${\displaystyle \dim \tau <\dim \sigma ;}$
2. ${\displaystyle \sigma }$ is a maximal face of ${\displaystyle K}$ and no other maximal face of ${\displaystyle K}$ contains ${\displaystyle \tau ,}$

then ${\displaystyle \tau }$ is called a free face.

A simplicial collapse of ${\displaystyle K}$ is the removal of all simplices ${\displaystyle \gamma }$ such that ${\displaystyle \tau \subseteq \gamma \subseteq \sigma ,}$ where ${\displaystyle \tau }$ is a free face. If additionally we have ${\displaystyle \dim \tau =\dim \sigma -1,}$ 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.^[1]

See also

• Discrete Morse theory – Combinatorial approach of studying the topology of a manifold
• Shelling (topology) – Mathematical concept

References

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