Sasaki-Einstein 7-manifolds and Orlik's conjecture

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 $S^3\times S^4\mathrm{.}$ Furthermore, we found that manifolds of the form $k\mathrm{\#}(S^3\times S^4)$ admit Sasaki-Einstein metrics for 22 different values of $k\mathrm{.}$ 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.