Topological Model Categories Generated by Finite Complexes |
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Authors: | Alex Chigogidze and Alexandr Karasev |
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Affiliation: | (1) Fakultät für Mathematik, Universität Bielefeld, Universitätsstrasse 25, D-33615 Bielefeld, Germany |
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Abstract: | Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S n+1 , n ⩾ 0. |
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