Moduli spaces of flat Riemannian metrics on 3- and 4-dimensional closed manifolds

TL;DR

This paper explores the topology of moduli spaces of flat Riemannian metrics on all closed 3-manifolds and some 4-manifolds, providing algebraic descriptions and topological insights.

Contribution

It offers a comprehensive topological analysis of moduli spaces for flat metrics on 3-manifolds and algebraic descriptions for certain 4-manifolds with specific holonomy.

Findings

Topology of moduli spaces for all 3-dimensional closed manifolds described.

Algebraic descriptions of moduli spaces for 4-dimensional flat manifolds with single-generator holonomy.

Topological properties of these moduli spaces studied in specific cases.

Abstract

We describe the topology of the moduli spaces of flat metrics for all the 3-dimensional closed manifolds. We give an algebraic description of the moduli spaces for the 4-dimensional closed flat manifolds with a single generator in their holonomy and, in some cases, also study their topology.