Suspension Homotopy of $(n-1)$-connected $(2n+2)$-dimensional Poincar\'{e} Duality Complexes

We study the homotopy decompositions of the suspension $\Sigma M$ of an $(n-1)$-connected $(2n+2)$ dimensional Poincar\'{e} duality complex $M$, $n\geq 2$. In particular, we completely determine the homotopy types of $\Sigma M$ of a simply-connected orientable closed (smooth) $6$-manifold $M$, whose integral homology groups can have $2$-torsion. If $3\leq n\leq 5$, we obtain homotopy decompositions of $\Sigma M$ after localization away from $2$.
Comment: 27 pages