Strong Homotopy Types,Nerves and Collapses |
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Authors: | Jonathan Ariel Barmak Elias Gabriel Minian |
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Institution: | 1.Departamento de Matemática, FCEyN,Universidad de Buenos Aires,Buenos Aires,Argentina |
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Abstract: | We introduce the theory of strong homotopy types of simplicial complexes. Similarly to classical simple homotopy theory, the
strong homotopy types can be described by elementary moves. An elementary move in this setting is called a strong collapse and it is a particular kind of simplicial collapse. The advantage of using strong collapses is the existence and uniqueness
of cores and their relationship with the nerves of the complexes. From this theory we derive new results for studying simplicial collapsibility with a different point of
view. We analyze vertex-transitive simplicial G-actions and prove a particular case of the Evasiveness conjecture for simplicial complexes. Moreover, we reduce the general
conjecture to the class of minimal complexes. We also strengthen a result of V. Welker on the barycentric subdivision of collapsible complexes. We obtain this and other
results on collapsibility of polyhedra by means of the characterization of the different notions of collapses in terms of
finite topological spaces. |
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