Comultiplication and suspension

We prove the conjecture of Berstein, Hilton, and Ganea that an (n − 1)-connected CW-complexX with dim X ⩽ k(n − 2) + 3 which has a co-Ak − 1-space structure desuspends. Moreover, we show the following dual of a result of Segal: given a cosimplicial space Y∗ which is special in the sense that the canonical maps ⋁k = 1n Y1 → Yn are homotopy equivalences and Y1 is 2-connected then there is a functorial desuspension of Y1.