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Moduli spaces of Ricci positive metrics in dimension five

McFeely Jackson Goodman

Abstract

We use the $η$ 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.

Moduli spaces of Ricci positive metrics in dimension five

Abstract

We use the invariants of spin 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 bundles over and the metrics are lifted from Ricci positive metrics on the bases. Along the way we classify 5-manifolds with fundamental group admitting free actions with simply connected quotients.

Paper Structure

This paper contains 7 sections, 28 theorems, 134 equations.

Table of Contents

  1. $S^1$ actions on 5 manifolds with $\pi_1=\mathbb{Z}_2$
  2. $\eta$ Invariant
  3. $\eta$ invariant in dimension $4n+1$ with free $S^1$ actions
  4. metric and connection
  5. Metric
  6. Connection
  7. Curvature

Key Result

Theorem A

Let $B^4=\#^a\mathbb{C}P^2\#^b\overline{\mathbb{C}P^2}$, $a+b\geq2$, and let $S^1\to M^5\to B^4$ be a principal bundle with first Chern class 2d, where $d\in H^2(B^4,\mathbb{Z})$ is primitive and $w_2(TB^4)=d$ mod 2. Then $\mathfrak{M}_{\text{Ric}>0}(M^5)$ has infinitely many path components.

Theorems & Definitions (47)

  • Theorem A
  • Theorem B
  • Corollary
  • Theorem 1.2
  • Lemma 1.3
  • Lemma 1.5
  • Lemma 1.6
  • proof
  • Lemma 1.7
  • proof
  • ...and 37 more