Closed model categories for uniquely S-divisible spaces

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 πkX is uniquely S-divisible. This equivalence of categories is given by an (S,n)-colocalizationfunctor 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 πk 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 with coefficients in for k n. We note that the category is the homotopycategory of rational 1-connected spaces.