Closed model categories for uniquely S-divisible spaces

Abstract

For each integer n>1 and a multiplicative system S of non-zero integers, we give a distinct closed model category structure to the category of pointed spaces Top ★ and we prove that the corresponding localized category Ho(Top ★ ( S, n) ), obtained by inverting the weak equivalences, is equivalent to the standard homotopy category of uniquely ( S, n)-divisible, ( n−1)-connected spaces. A space X is said to be uniquely ( S, n)-divisible if for k⩾ n the homotopy group π k X is uniquely S-divisible. This equivalence of categories is given by an ( S, n)-colocalization functor that carries a pointed space X to a space X ( S, n) . There is also a natural map X ( S, n) → X which is (finally) universal among all the maps Z→ X with Z a uniquely ( S, n)-divisible, ( n−1)-connected space. The structure of closed model category given by Quillen to Top ★ is based on maps which induce isomorphisms on all homotopy group functors π k and for any choice of base point. For each pair ( S, n), the closed model category structure given here take as weak equivalences those maps that for the given base point induce isomorphisms on the homotopy groups functors π k( Z[S −1];−) with coefficients in Z[S −1] for k⩾ n. We note that the category Ho( Top ★ ( Z−{0},2) ) is the homotopy category of rational 1-connected spaces.