Comultiplication and suspension (Q1356966)

As is pointed out in the abstract, the authors claim the following two results in this paper: (1) A conjecture by Berstein, Hilton and Ganea: an \((n-1)\)-connected CW complex \(X\) with dim \(X \leq k(n-2)+3\) which has a co-\(A_{k-1}\)-space structure desuspends. (2) A dual result of Segal: given a cosimplicial space \(Y_*\) which is special in the sense that the canonical map \(\bigvee^n_{k=1}Y_1 \rightarrow Y_n\) are homotopy equivalences and \(Y_1\) is \(2\)-connected then there is a functorial desuspension of \(Y_1\).