Positive scalar curvature and strongly inessential manifolds

TL;DR

The paper proves that most closed manifolds with positive scalar curvature and abelian fundamental groups have finite covers that can be deformed into lower-dimensional skeletons, except for certain 4-manifolds with spin universal covers.

Contribution

It establishes a new inessentiality property for a broad class of manifolds with positive scalar curvature and abelian fundamental groups, excluding specific 4-manifolds.

Findings

Finite covers of such manifolds are strongly inessential.

The result applies to all dimensions except certain 4-manifolds with spin universal covers.

Provides a new link between scalar curvature and topological inessentiality.

Abstract

We prove that a closed $n$-manifold $M$ with positive scalar curvature and abelian fundamental group admits a finite covering $M'$ which is strongly inessential. The latter means that a classifying map $u:M'\to K(\pi_1(M'),1)$ can be deformed to the $(n-2)$-skeleton. This is proven for all $n$-manifolds with the exception of 4-manifolds with spin universal coverings.