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. Collapses find applications in computational homology. Definition. Let formula_1 be an abstract simplicial complex. Suppose that formula_2 are two simplices of formula_1 such that the following two conditions are satisfied: then formula_10 is called a free face. A simplicial collapse of formula_1 is the removal of all simplices formula_12 such that formula_13, where formula_9 is a free face. If additionally we have formula_15 , 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.