Suspension splittings of 5-dimensional Poincaré duality complexes and their applications

Let $X$ be a connected, orientable, 5-dimensional Poincaré duality complex with torsion-free $H_1(X;\mathbb{Z})$. We show that $\mathrm{\Sigma }X$ 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 ${\pi }^3(X;\mathbb{Z}\mathrm{/}k)$ as well as give partial information about the cohomotopy set ${\pi }^2(X)$.