  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].
