On the topology of moduli spaces of non-negatively curved Riemannian metrics

TL;DR

This paper investigates the topological properties of spaces of Riemannian metrics with non-negative curvature, revealing new examples with complex rational homotopy and infinitely many components in high dimensions.

Contribution

It constructs the first examples of manifolds with non-trivial rational homotopy, homology, and cohomology in their metric moduli spaces, and shows these spaces can have infinitely many components.

Findings

Existence of manifolds with non-trivial rational homotopy in their metric spaces.

Presence of infinitely many path components in high-dimensional metric spaces.

Analogous results for both Ricci and sectional curvature conditions.

Abstract

We study spaces and moduli spaces of Riemannian metrics with non-negative Ricci or non-negative sectional curvature on closed and open manifolds. We construct, in particular, the first classes of manifolds for which these moduli spaces have non-trivial rational homotopy, homology and cohomology groups. We also show that in every dimension at least seven (respectively, at least eight) there exist closed (respectively, open) manifolds for which the space and moduli space of Riemannian metrics with non-negative sectional curvature has infinitely many path components. A completely analogous statement holds for spaces and moduli spaces of non-negative Ricci curvature metrics.