A closed model category for (𝑛-1)-connected spaces

Abstract

For each integer n > 0 n > 0 , we give a distinct closed model category structure to the category of pointed spaces Top ⋆ \operatorname {Top}_\star such that the corresponding localized category Ho ⁡ ( Top ⋆ n ) \operatorname {Ho}(\operatorname {Top}_\star ^n) is equivalent to the standard homotopy category of ( n − 1 ) (n-1) -connected CW-complexes. The structure of closed model category given by Quillen to Top ⋆ \operatorname {Top}_\star is based on maps which induce isomorphisms on all homotopy group functors π q \pi _q and for any choice of base point. For each n > 0 n>0 , the closed model category structure given here takes as weak equivalences those maps that for the given base point induce isomorphisms on π q \pi _q for q ≥ n q\ge n .