$SU(2)$-bundles over highly connected $8$-manifolds

In this paper, we analyze the possible homotopy types of the total space of a principal $SU(2)$-bundle over a $3$-connected $8$-dimensional Poincar\'{e} duality complex. Along the way, we also classify the $3$-connected $11$-dimensional complexes $E$ formed from a wedge of $S^4$ and $S^7$ by attaching a $11$-cell.