  In this article, we are interested in the question whether any complete
contractible $3$-manifold of positive scalar curvature is homeomorphic to
$\mathbb{R}^{3}$. We study the fundamental group at infinity,
$\pi_{1}^{\infty}$, and its relationship with the existence of complete metrics
of positive scalar curvature. We prove that a complete contractible
$3$-manifold with positive scalar curvature and trivial $\pi^{\infty}_{1}$ is
homeomorphic to $\mathbb{R}^{3}$.
