Let $Y$ be an orientable, smooth 3-manifold and let $X=Y\times S^{1}$. My question is that: when does $X$ admits a Riemannian metric with positive scalar curvature? 


An obvious case is when $Y$ itself admits a psc metric. Are there any other case?
 It was proved by Gromov and Lawson that any 3-manifold which contains a $K(\pi,1)$ prime factor does not admits a psc metric (another result in the paper implies that $Y$ should not be hyperbolic if $Y\times S^{1}$ admits psc), which, together with the Geometrization conjecture classifies all the 3-manifold admits psc metric. Does their proof works for $Y\times S^{1}$?


(I am most interested in the special case that $Y$ is an irreducible integer homology sphere) 



For a 3-manifold $Y$, when does $Y\times S^{1}$ admits a Riemannian metric with positive scalar curvature?
It is a theorem of Schoen and Yau (see [Jonathan Rosenberg's survey](http://www.math.umd.edu/~jmr/psc2006.pdf), theorem 1.10) that no closed aspherical 4-manifold is psc. That means that the three-manifold should be non-aspherical, which narrows it down to a very small list (especially if you insist on integer homology spheres).


