Universal covers of non-negatively curved manifolds and formality

TL;DR

This paper proves that the formality of the universal cover of a closed non-negatively curved manifold implies the manifold's formality, extending previous results and removing orientability constraints.

Contribution

It establishes a new link between the formality of universal covers and the original manifolds, generalizing prior results by removing orientability assumptions.

Findings

Universal cover formality implies manifold formality.

Closed manifolds with non-negative Ricci curvature and large first Betti number are formal.

The method applies to some non-closed manifolds.

Abstract

We show that if the universal cover of a closed smooth manifold admitting a metric with non-negative Ricci curvature is formal, then the manifold itself is formal. We reprove a result of Fiorenza-Kawai-L\^e-Schwachh\"ofer, that closed orientable manifolds with a non-negative Ricci curvature metric and sufficiently large first Betti number are formal. Our method allows us to remove the orientability hypothesis; we further address some cases of non-closed manifolds.