Classifying sufficiently connected PSC manifolds in 4 and 5 dimensions

Abstract

We show that if N is a closed manifold of dimension n D 4 (resp. n D 5) with 2 .N / D 0 (resp. 2 .N / D 3 .N / D 0) that admits a metric of positive scalar curvature, then a finite cover y N of N is homotopy equivalent to S n or connected sums of S n 1 S 1 . 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.

Nonnegative scalar curvature on manifolds with at least two ends

“…For instance, it has recently been shown [6, 7, 16] that closed oriented aspherical manifolds of dimension $$\leqslant 5$$ are $$\mathrm{NPSC^+}$$. Moreover, a closed oriented manifold of dimension $$\leqslant 3$$ which does not admit psc necessarily admits a nonzero degree map to an aspherical $$\mathrm{NPSC^+}$$ manifold.…”

Rigidity and Non-Rigidity of $\mathbb{H}^n/\mathbb{Z}^{n-2}$ with Scalar Curvature Bounded from Below

“…It is well known that T n ∈ C deg for n ≤ 7; also note that the second item in Definition 1.9 is redundant when dim Σ ≤ 5, according to [13].…”