Bochner-type theorems for distributional category

TL;DR

This paper establishes new bounds on topological invariants of manifolds using distributional category under geometric conditions like non-negative Ricci curvature, and explores the implications of equality cases for manifold structure.

Contribution

It introduces bounds on invariants such as Betti number and fundamental group rank via distributional category, extending Bochner-type theorems to this setting.

Findings

Distributional category bounds the first Betti number and macroscopic dimension.

Equality cases impose geometric and topological constraints on manifolds.

Refined bounds are obtained for cohomologically symplectic manifolds.

Abstract

We show that in the presence of a geometric condition such as non-negative Ricci curvature, the distributional category of a manifold may be used to bound invariants, such as the first Betti number and macroscopic dimension, from above. Moreover, \`a la Bochner, when the bound is an equality, special constraints are imposed on the manifold. We show that the distributional category of a space also bounds the rank of the Gottlieb group, with equality imposing constraints on the fundamental group. These bounds are refined in the setting of cohomologically symplectic manifolds, enabling us to get specific computations for the distributional category and LS-category.