Co-\(H\)-spaces and localization (Q616511)

If a simply connected \(CW\)-complex \(X\) is a co-\(H\)-space, then so is the \(p\)-localization \(X_{(p)}\) for each prime \(p\). However, the opposite is not true: An infinite space constructed by Roitberg is not a co-\(H\)-space but its \(p\)-localization is a co-\(H\)-space for each prime \(p\) [\textit{J. Roitberg} and \textit{P. Touhey}, Topology 39, 95--101 (2000; Zbl 0957.55009)]. In the paper under review, the author shows that the opposite is also true under the assumption that the space \(X\) is finite.