# Geometric Graph Manifolds with non-negative scalar curvature

## TL;DR

This paper classifies n-dimensional geometric graph manifolds with nonnegative scalar curvature, revealing their universal cover structure and providing a detailed classification for 3-dimensional cases, including moduli space properties.

## Contribution

It offers a comprehensive classification of geometric graph manifolds with nonnegative scalar curvature, including universal cover splitting and moduli space structure, extending understanding in geometric topology.

## Key findings

- Universal cover splits off a codimension 3 Euclidean factor for n>3
- 3D manifolds are either lens or prism spaces with rigid metrics
- Moduli space has infinitely many components for lens spaces, connected for prism spaces

## Abstract

We classify $n$-dimensional geometric graph manifolds with nonnegative scalar curvature, and first show that if $n>3$, the universal cover splits off a codimension 3 Euclidean factor. We then proceed with the classification of the 3-dimensional case by showing that such a manifold is either a lens space or a prism manifold with a very rigid metric. This allows us to also classify the moduli space of such metrics: it has infinitely many connected components for lens spaces, while it is connected for prism manifolds.