Sasaki–Einstein 7-manifolds and Orlik’s conjecture

Abstract

We study the homology groups of certain 2-connected 7-manifolds admitting quasiregular Sasaki-Einstein metrics, among them, we found 52 new examples of Sasaki-Einstein rational homology 7-spheres, extending the list given by Boyer, Galicki and Nakamaye in [6]. As a consequence, we exhibit new families of positive Sasakian homotopy 9-spheres given as cyclic branched covers, determine their diffeomorphism types and find out which elements do not admit extremal Sasaki metrics. We also improve previous results given by Boyer [12] showing new examples of Sasaki-Einstein 2-connected 7-manifolds homeomorphic to connected sums of S 3 × S 4 . Actually we show that manifolds of the form #k S 3 × S 4 admit Sasaki-Einstein metrics for 22 different values of k. All these links arise as Thom-Sebastiani sums of chain type singularities and cycle type singularities where Orlik's conjecture holds due to a recent result by Hertling and Mase [19].