Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

We show that if N is a closed manifold of dimension n D 4 (resp. n D 5) with π₂ (N) D 0 (resp. π₂ (N) D π₃ (N) D 0) that admits a metric of positive scalar curvature, then a finite cover yN of N is homotopy equivalent to Sⁿ or connected sums of S^{n 1} X S¹. Our approach combines recent advances in the study of positive scalar curvature with a novel argument of Alpert, Balitskiy and 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.