The homotopy decomposition of the suspension of a non-simply-connected $5$-manifold

In this paper we determine the homotopy types of the reduced suspension space of certain connected orientable closed smooth $5$-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 ${\pi }^4(\mathrm{\Sigma }M)$ is a bijection.