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

Abstract

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 $\pi ^3(M)$ and the fourth cohomotopy set $\pi ^4(\Sigma M)$ is a bijection.