Price Inequalities and Betti Number Growth on Manifolds without Conjugate Points

TL;DR

This paper establishes Price inequalities for harmonic forms on manifolds without conjugate points, analyzes Betti number growth in coverings, and proves vanishing results for $L^{2}$-Betti numbers, advancing understanding of geometric and topological properties of such manifolds.

Contribution

It introduces new Price inequalities tailored for manifolds without conjugate points and applies them to study Betti number asymptotics and $L^{2}$-Betti number vanishing.

Findings

Derived Price inequalities for harmonic forms.

Analyzed asymptotic Betti number growth in coverings.

Proved vanishing of $L^{2}$-Betti numbers for certain manifolds.

Abstract

We derive Price inequalities for harmonic forms on manifolds without conjugate points and with a negative Ricci upper bound. The techniques employed in the proof work particularly well for manifolds of non-positive sectional curvature, and in this case we prove a strengthened Price inequality. We employ these inequalities to study the asymptotic behavior of the Betti numbers of coverings of Riemannian manifolds without conjugate points. Finally, we give a vanishing result for $L^{2}$-Betti numbers of closed manifolds without conjugate points.