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.
Contents
Definition
Let
K
{\displaystyle K}
be an abstract simplicial complex.
Suppose that
τ
,
σ
{\displaystyle \tau ,\sigma }
are two simplices of
K
{\displaystyle K}
such that the following two conditions are satisfied:
τ
⊊
σ
,
{\displaystyle \tau \subsetneq \sigma ,}
in particular
dim
τ
<
dim
σ
;
{\displaystyle \dim \tau <\dim \sigma ;}
σ
{\displaystyle \sigma }
is a maximal face of
K
{\displaystyle K}
and no other maximal face of
K
{\displaystyle K}
contains
τ
,
{\displaystyle \tau ,}
then
τ
{\displaystyle \tau }
is called a free face.
A simplicial collapse of
K
{\displaystyle K}
is the removal of all simplices
γ
{\displaystyle \gamma }
such that
τ
⊆
γ
⊆
σ
,
{\displaystyle \tau \subseteq \gamma \subseteq \sigma ,}
where
τ
{\displaystyle \tau }
is a free face. If additionally we have
dim
τ
=
dim
σ
−
1
,
{\displaystyle \dim \tau =\dim \sigma -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
Complexes that do not have a free face cannot be collapsible. Two such interesting examples are R. H. Bing's house with two rooms and Christopher Zeeman's dunce hat; they are contractible (homotopy equivalent to a point), but not collapsible.
Any n-dimensional PL manifold that is collapsible is in fact piecewise-linearly isomorphic to an n-ball.