The homotopy decomposition of the suspension of a non-simply-connected five-manifold

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $five$ -manifolds. As applications, we compute the reduced K -groups of M and show that the suspension map between the third cohomotopy set π ³(M) and the fourth cohomotopy set π ⁴(Σ M) is a bijection.