  We calculate the homology groups of certain 2-connected 7-manifolds admitting
quasi-regular Sasaki-Einstein metrics. These manifolds are links that arise as
Thom-Sebastiani sums of chain type singularities and cycle type singularities.
Among these links, we found 52 new examples of Sasaki-Einstein rational
homology 7-spheres and 124 new examples of Sasaki-Einstein 2-connected
7-manifolds homeomorphic to connected sums of $S^{3} \times S^{4}.$
Furthermore, we found that manifolds of the form $k \#\left(S^{3} \times
S^{4}\right)$ admit Sasaki-Einstein metrics for 22 different values of $k.$ We
also describe the diffeomorphism type of certain families of homotopy 9-spheres
admitting positive Ricci curvature. These manifolds are branched covers of
$S^{11}$ branched over Sasaki-Einstein rational homology 7-spheres.
