On homotopy rigidity of the functor \(\varSigma\varOmega\) on co-\( H\)-spaces (Q2253804)

Let \(Top\) denote the category of topological spaces, and let \(\mathcal{C}\) be a subcategory. Then the functor \(F:\mathcal{C}\to Top\) is called \textit{homotopy rigid on} \(\mathcal{C}\) if for any objects \(X\) and \(Y\) in \(\mathcal{C}\), \(F(X)\) and \(F(Y)\) have the same homotopy type iff \(X\) and \(Y\) have the same homotopy type. In this paper, the authors consider the subcategory \(\mathcal{C}_p\) of of simply connected \(p\)-local finite co-\(H\)-spaces for a prime \(p\), and they prove that the two functors \(\Sigma\Omega\) and \(\Omega\) are homotopy rigid on \(\mathcal{C}_p\). Their proof is based on the homotopy decomposition of co-\(H\)-spaces and suspension splittings of smash products of looped co-\(H\)-spaces with the modular representation theory.