Moduli spaces of Ricci positive metrics in dimension five

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.

27 pages; v2: main theorem expanded, theorem added on the classification of 5-manifolds with fundamental group ${\mathbb{Z}}_2$ admitting free $S^1$ actions

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