  The goal of this paper is to investigate the topological structure of open
simply-connected 3-manifolds whose scalar curvature has a slow decay at
infinity. In particular, we show that the Whitehead manifold does not admit a
complete metric, whose scalar curvature decays slowly, and in fact that any
contractible complete 3-manifolds with such a metric is homeomorphic to
$\mathbb{R}^{3}$. Furthermore, using this result, we prove that any open
simply-connected 3-manifold $M$ with $\pi_{2}(M)=\mathbb{Z}$ and a complete
metric as above, is homeomorphic to $\mathbb{S}^{2}\times \mathbb{R}$.
