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

For each integer $n > 0$, we give a distinct closed model category structure to the category of pointed spaces $\mathop {\mathrm {Top}}\nolimits _{\star }$ such that the corresponding localized category $\mathop {\mathrm {Ho}}\nolimits ( \mathop {\mathrm {Top}}\nolimits _{\star }^{n})$ is equivalent to the standard homotopy category of $(n-1)$-connected CW-complexes. The structure of closed model category given by Quillen to $\mathop {\mathrm {Top}}\nolimits _{\star }$ 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 $q\ge n$ .