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 $K$ 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.^[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]

References

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

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[Whitehead-1] 1 2 Whitehead, J.H.C. (1938) Simplical spaces, nuclei and m-groups, Proceedings of the London Mathematical Society 45, pp 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

Collapse (topology)

Definition

Examples

See also

References