Theorem 4.25 can be viewed as a homological analogue of Theorem 3.4. Since the assumptions only require that the retraction is a weak retraction, it is now natural to look for conditions that ensure the existence of such a weak retraction. Most of the work in that direction has been done by Constantin and Fournier in [18]. The connection to topology is established through the topological realization of a simplicial complex:
Clearly the topological realization of a finite simplicial complex can be embedded as a subspace in with the usual topology. Simplicial maps are extended to affine maps by affine interpolation.
From Lemmas 4.22 and 4.24 we can easily infer that if is a weak retract of |P| and is acyclic, then . This can be seen as an underlying fact to the work in [18] in which contractibility (through Proposition 4.39) plays a central role.
Proof: The homotopy of to a retraction onto relative to is constructed as follows: Let and let be a contraction. Then for each point there are unique and such that and
is a (strong) deformation retraction from |K| to . \
For any finite ordered set all the properties below make sense. They are listed such that the lower-numbered properties imply the higher-numbered ones. Similar lists can be made up for graphs and simplicial complexes.
``P is connectedly collapsible" fits in between conditions 1 and 2 (cf. [122]).