Suspension splittings of 5-dimensional Poincar\'{e} duality complexes and their applications

Let $X$ be a connected, orientable, 5-dimensional Poincar\'{e} duality complex with torsion-free $H_1(X;\mathbb{Z})$. We show that $\Sigma X$ is homotopy equivalent to a wedge of recognisable spaces and study to what extent its homotopy type is determined by algebraic data. These results are then used to compute the unstable cohomotopy groups $\pi^3(X)$ and $\pi^3(X;\mathbb{Z}/k)$ as well as give partial information about the cohomotopy set $\pi^2(X)$.