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]
Let be an abstract simplicial complex.
Suppose that are two simplices of such that the following two conditions are satisfied:
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. [3]
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]
Let be an abstract simplicial complex.
Suppose that are two simplices of such that the following two conditions are satisfied:
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. [3]