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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The first author was partially supported by NSERC research grant.
Received December 12, 2001; in revised form September 7, 2002 Published online February 28, 2003
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Chigogidze, A., Karasev, A. Topological Model Categories Generated by Finite Complexes. Monatsh. Math. 139, 129–150 (2003). https://doi.org/10.1007/s00605-002-0532-x
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DOI: https://doi.org/10.1007/s00605-002-0532-x