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. Collapses find applications in computational homology.
Definition Let K {\displaystyle K} be an abstract simplicial complex. Suppose that τ , σ {\displaystyle \tau ,\sigma } are two simplices of K {\displaystyle K} such that the following two conditions are satisfied:
τ ⊊ σ , {\displaystyle \tau \subsetneq \sigma ,} in particular dim τ < dim σ ; {\displaystyle \dim \tau <\dim \sigma ;}
σ {\displaystyle \sigma } is a maximal face of K {\displaystyle K} and no other maximal face of K {\displaystyle K} contains τ , {\displaystyle \tau ,}
then τ {\displaystyle \tau } is called a free face. A simplicial collapse of K {\displaystyle K} is the removal of all simplices γ {\displaystyle \gamma } such that τ ⊆ γ ⊆ σ , {\displaystyle \tau \subseteq \gamma \subseteq \sigma ,} where τ {\displaystyle \tau } is a free face. If additionally we have dim τ = dim σ − 1 , {\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.
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.
See also Discrete Morse theory – Combinatorial approach of studying the topology of a manifold Shelling (topology) – Mathematical concept
References