Negative Sasakian structures on simply-connected 5-manifolds

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 )x 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 x S-3) admits negative quasi -regular Sasakian structures for any k. This yields a complete an-swer to another question posed in [3].