Topological Model Categories Generated by Finite Complexes

. 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.