  We show that if $N$ is a closed manifold of dimension $n=4$ (resp. $n=5$)
with $\pi_2(N) = 0$ (resp. $\pi_2(N)=\pi_3(N)=0$) that admits a metric of
positive scalar curvature, then a finite cover $\hat N$ of $N$ is homotopy
equivalent to $S^n$ or connected sums of $S^{n-1}\times S^1$. Our approach
combines recent advances in the study of positive scalar curvature with a novel
argument of Alpert--Balitskiy--Guth.
  Additionally, we prove a more general mapping version of this result. In
particular, this implies that if $N$ is a closed manifold of dimensions $4$ or
$5$, and $N$ admits a map of nonzero degree to a closed aspherical manifold,
then $N$ does not admit any Riemannian metric with positive scalar curvature.
