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Mathematical Research Letters
Volume 29 (2022)
Number 6
Negative Sasakian structures on simply-connected $5$-manifolds
Pages: 1827 – 1857
DOI: https://dx.doi.org/10.4310/MRL.2022.v29.n6.a9
Authors
Abstract
We study several questions on the existence of negative Sasakian structures on simply connected rational homology spheres and on Smale–Barden manifolds of the form $\#_k (S^2 \times S^3)$. First, we prove that any simply connected rational homology sphere admitting positive Sasakian structures also admits a negative one. This result answers the question, posed by Boyer and Galicki in their book [3], of determining which simply connected rational homology spheres admit both negative and positive Sasakian structures. Second, we prove that the connected sum $\#_k (S^2 \times S^3)$ admits negative quasi-regular Sasakian structures for any $k$. This yields a complete answer to another question posed in [3].
Received 30 November 2020
Accepted 4 July 2021
Published 4 May 2023