Moduli spaces of Ricci positive metrics in dimension five

TL;DR

This paper uses eta invariants of spin^c Dirac operators to distinguish connected components in the moduli spaces of Ricci positive metrics on five-dimensional manifolds, revealing infinitely many components for certain manifolds.

Contribution

It introduces a method to distinguish moduli space components using eta invariants and constructs infinitely many non-diffeomorphic 5-manifolds with complex moduli space structures.

Findings

Identified infinitely many components in moduli spaces of Ricci positive metrics.

Classified 5-manifolds with fundamental group Z_2 admitting free S^1 actions.

Constructed examples of manifolds with complex moduli space topology.

Abstract

We use the $\eta$ invariants of spin$^c$ Dirac operators to distinguish connected components of moduli spaces of Riemannian metrics with positive Ricci curvature. We then find infinitely many non-diffeomorphic five dimensional manifolds for which these moduli spaces each have infinitely many components. The manifolds are total spaces of principal $S^1$ bundles over $\#^a\mathbb{C}P^2\#^b\overline{\mathbb{C}P^2}$ and the metrics are lifted from Ricci positive metrics on the bases. Along the way we classify 5-manifolds with fundamental group $\mathbb{Z}_2$ admitting free $S^1$ actions with simply connected quotients.