Homotopy types of suspended 4–manifolds

Homotopy types of suspended 4-manifolds PENGCHENG LIGiven a closed, smooth, connected, orientable 4-manifold M whose integral homology groups can have 2-torsion, we determine the homotopy decomposition of the double suspension † 2 M as wedge sums of some elementary A 3 3 -complexes which are 2-connected finite complexes of dimension at most 6.Furthermore, we utilize the Postnikov square (or equivalently Pontryagin square) to find sufficient conditions for the homotopy decompositions of † 2 M to desuspend to that of †M .