Mathematical Research Letters

Volume 29 (2022)

Number 6

Negative Sasakian structures on simply-connected $5$-manifolds

Pages: 1827 – 1857



Vicente Muñoz (Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Spain)

Matthias Schütt (Institut für Algebraische Geometrie, Leibniz Universität, Hannover, Germany; and Riemann Center for Geometry and Physics, Leibniz Universität, Hannover, Germany)

Aleksy Tralle (Faculty of Mathematics and Computer Science, University of Warmia and Mazury, Olsztyn, Poland)


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].

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Received 30 November 2020

Accepted 4 July 2021

Published 4 May 2023