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Topological Considerations


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 tex2html_wrap_inline10826 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 tex2html_wrap_inline10910 is a weak retract of |P| and tex2html_wrap_inline10914 is acyclic, then tex2html_wrap_inline10916 . 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 tex2html_wrap_inline10942 to a retraction onto tex2html_wrap_inline10944 relative to tex2html_wrap_inline10944 is constructed as follows: Let tex2html_wrap_inline10948 and let tex2html_wrap_inline10950 be a contraction. Then for each point tex2html_wrap_inline10952 there are unique tex2html_wrap_inline10954 and tex2html_wrap_inline10956 such that tex2html_wrap_inline10958 and


is a (strong) deformation retraction from |K| to tex2html_wrap_inline10944 . \


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.

  1.   P is dismantlable,
  2.   P is ``dismantlable via removing escamotable points",
  3. P is contractible,
  4.   P is acyclic,
  5. |P| has the topological fixed point property,
  6. K(P) has the fixed simplex property,
  7. tex2html_wrap_inline10982 has the fixed clique property,
  8. P has the fixed point property.

``P is connectedly collapsible" fits in between conditions 1 and 2 (cf. [122]).

next up previous contents index
Next: Cutsets Up: Order vs. Algebraic Topology Previous: (Integer) Homology