Closed model categories for uniquely \(S\)-divisible spaces (Q1399144)

For each integer \(n> 1\) and each multiplication system \(S\) of non-zero integers the authors define a new closed model category structure on the category of pointed spaces \(\text{Top}_*\). The weak equivalences are the maps that induce isomorphisms on the homotopy groups \(\pi_k(\mathbb{Z}[S^{-1}]; -)\) for \(k\geq n\). The corresponding localized category \(H_0(\text{Top}_*)\) is equivalent to the homotopy category of uniquely \((S,n)\)-divisible (i.e., \(\pi_k(-)\) is uniquely \(S\)-divisible for \(k\geq n\)) \((n-1)\)-connected spaces. When \(S= \mathbb{Z}\setminus\{0\}\), we recover the usual homotopy category of rational \(1\)-connected spaces.