Collapse (topology)

Definition

Let K be an abstract simplicial complex.

Suppose that τ , σ ∈ K such that the following two conditions are satisfied:

(i) τ ⊂ σ , in particular dim ⁡ τ < dim ⁡ σ ;

(ii) σ is a maximal face of K and no other maximal face of K contains τ ,

then τ is called a free face.

A simplicial collapse of K is the removal of all simplices γ such that τ ⊆ γ ⊆ σ , where τ 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.

Examples