Suspension Homotopy of $(n-1)$-connected $(2n+2)$-dimensional Poincaré Duality Complexes

We study the homotopy decompositions of the suspension $\mathrm{\Sigma }M$ of an $(n-1)$-connected $(2n+2)$ dimensional Poincaré duality complex $M$, $n\ge 2$. In particular, we completely determine the homotopy types of $\mathrm{\Sigma }M$ of a simply-connected orientable closed (smooth) $6$-manifold $M$, whose integral homology groups can have $2$-torsion. If $3\le n\le 5$, we obtain homotopy decompositions of $\mathrm{\Sigma }M$ after localization away from $2$.