Collapse (topology)
Encyclopedia
In topology
, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead
.
, and suppose that s is a simplex
in K. We say that s has a free face t if t is a face of s and t has no other cofaces. We call (s, t) a free pair. If we remove s and t from K, we obtain another simplicial complex, which we call an elementary collapse of K. A sequence of elementary collapses is called a collapse. A simplicial complex that has a collapse to a point, implying all other points were in free pairs, is called collapsible.
This definition can be extended to CW-complexes and is the basis for the concept of simple-homotopy equivalence
.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, a branch of mathematics, collapse is a concept due to J. H. C. Whitehead
J. H. C. Whitehead
John Henry Constantine Whitehead FRS , known as Henry, was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai , in India, and died in Princeton, New Jersey, in 1960....
.
Definition
Let K be a simplicial complexSimplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
, and suppose that s is a simplex
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
in K. We say that s has a free face t if t is a face of s and t has no other cofaces. We call (s, t) a free pair. If we remove s and t from K, we obtain another simplicial complex, which we call an elementary collapse of K. A sequence of elementary collapses is called a collapse. A simplicial complex that has a collapse to a point, implying all other points were in free pairs, is called collapsible.
This definition can be extended to CW-complexes and is the basis for the concept of simple-homotopy equivalence
Simple-homotopy equivalence
In mathematics, particularly the area of topology, a simple-homotopy equivalence is a refinement of the concept of homotopy equivalence. Two CW-complexes are simple-homotopy equivalent if they are related by a sequence of collapses and expansions , and a homotopy equivalence is a simple homotopy...
.
Examples
- Complexes that do not have a free face cannot be collapsible. Two such interesting examples are BingRH BingR. H. Bing was an American mathematician who worked mainly in the areas of geometric topology and continuum theory...
's house with two roomsHouse with two roomsHouse with two rooms or Bing's house is a particular contractible 2-complex that is not collapsible.The name was given by R. H. Bing.-External links:*...
and Zeeman's dunce hatDunce hat (topology)For the item of clothing designed to be humiliating, now rarely used, see dunce cap.In topology, the dunce hat is a compact topological space formed by taking a solid triangle and gluing all three sides together, with the orientation of one side reversed...
; they are contractible and in fact simple homotopy equivalent to a point. - Any n-dimensional PL manifold that is collapsible is in fact piecewise-linearly isomorphic to an n-ball.