Sasaki-Einstein 7-manifolds and Orlik's conjecture

TL;DR

This paper computes homology groups of specific 7-manifolds with Sasaki-Einstein metrics, discovering numerous new examples and classifying their topological types, including rational homology spheres and connected sums of spheres.

Contribution

It introduces new examples of Sasaki-Einstein 7-manifolds, classifies their topological types, and explores their geometric structures, expanding understanding of Sasaki-Einstein geometry in higher dimensions.

Findings

52 new Sasaki-Einstein rational homology 7-spheres

124 new Sasaki-Einstein 2-connected 7-manifolds

Sasaki-Einstein metrics on connected sums of $S^{3} imes S^{4}$ for 22 values of k

Abstract

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.