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In this paper we determine the homotopy types of the reduced suspension spaceof certain connected orientable closed smooth $5$-manifolds. As applications,we compute the reduced $K$-groups of $M$ and show that the suspension mapbetween the third cohomotopy set $\pi^3(M)$ and the fourth cohomotopy set$\pi^4(\Sigma M)$ is a bijection.

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