Collapse (topology)


In topology, a branch of mathematics, a collapse reduces a simplicial 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 be an abstract simplicial complex.
Suppose that are two simplices of such that the following two conditions are satisfied:
  1. , in particular ;
  2. is a maximal face of and no other maximal face of contains,
then is called a free face.
A simplicial collapse of is the removal of all simplices such that, where is a free face. If additionally we have, 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