Negative Sasakian structures on simply-connected \(5\)-manifolds (Q6043370)

The following questions appear in the well-known seminal book on Sasakian geometry by \textit{C. P. Boyer} and \textit{K. Galicki} [Sasakian geometry. Oxford: Oxford University Press (2008; Zbl 1155.53002)]: Question 1. Which simply connected rational homology \(5\)-spheres admit negative Sasakian structures? Question 2. Which simply connected rational homology spheres admit both positive and negative Sasakian structures? The authors contribute to answer the first question, and give a complete answer to the second question by proving the following: Theorem. All positive Sasakian simply connected rational homology spheres also admit negative Sasakian structures.