NE-Reduction and Collapses

NE-reduction can be used to define an interesting equivalence relation on the set of all abstract simplicial complexes. 

Definition 13.24. LetX andY be abstract simplicial complexes. Recursively, we say that X ne Y if X ne Y orY ym X, or if there exists an abstract simplicial complex Z such that X Z and Y ne 

Clearly, if X is nonevasive, then X ne pt but is the opposite true The answer to that is no. As one example, consider the standard instance of a space 

Independently of a particular triangulation, the space H is not collapsible hence it is evasive. On the other hand, we leave it to the reader to see that it is possible to triangulate the filled cylinder C given by the equations z 1, x +y 1, so that C ne H. 

Recall from Section 6.4 that the analogous equivalence relation, where ne and yoE are replaced by and y, is called the simple homotopy type, and that the celebrated Whitehead theorem implies that the simplicial complexes with the simple homotopy type of a point are precisely those that are contractible see Theorem 6.16. Therefore, the class of simplicial complexes that are NE-equivalent to a point relates to nonevasiveness in the same way as con-tractibility relates to collapsibility. Clearly, this means that this class should constitute an interesting object of study. 

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