Negative Sasakian structures on simply-connected 5-manifolds

TL;DR

This paper investigates the existence of negative Sasakian structures on simply connected 5-manifolds, showing that such structures exist on all rational homology spheres with positive Sasakian structures and on connected sums of S^2×S^3.

Contribution

It proves that all simply connected rational homology spheres with positive Sasakian structures also admit negative ones, and that connected sums of S^2×S^3 admit negative quasi-regular Sasakian structures.

Findings

Any simply connected rational homology sphere with positive Sasakian structure admits a negative one.

Connected sums of S^2×S^3 admit negative quasi-regular Sasakian structures.

Complete classification of negative Sasakian structures on certain 5-manifolds.

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 [BG], 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 [BG].