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

Abstract:

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