Moduli spaces of compact RCD(0,N)-structures

TL;DR

This paper develops foundational topological results for moduli spaces of non-smooth metric measure spaces with non-negative Ricci curvature, focusing on RCD(0,N)-structures, their convergence, and topological properties.

Contribution

It establishes the relationship between convergence of RCD(0,N)-structures and their universal covers, constructs key maps reflecting structure splitting, and provides examples with non-trivial rational homotopy groups.

Findings

Convergence of RCD(0,N)-structures relates to equivariant convergence on universal covers.

Constructed continuous Albanese and soul maps for these structures.

Provided examples of moduli spaces with non-trivial rational homotopy groups.

Abstract

The goal of the paper is to set the foundations and prove some topological results about moduli spaces of non-smooth metric measure structures with non-negative Ricci curvature in a synthetic sense (via optimal transport) on a compact topological space; more precisely, we study moduli spaces of RCD(0,N)-structures. First, we relate the convergence of RCD(0,N)-structures on a space to the associated lifts' equivariant convergence on the universal cover. Then we construct the Albanese and soul maps, which reflect how structures on the universal cover split, and we prove their continuity. Finally, we construct examples of moduli spaces of RCD(0,N)-structures that have non-trivial rational homotopy groups.