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Odd-dimensional manifolds with infinitely many different geometries of positive Ricci curvature

Anand Dessai

TL;DR

The paper proves that in every odd dimension $n\ge 5$ there exist closed $n$-manifolds with infinitely many geometries of positive Ricci curvature, by constructing families of manifolds as boundaries of generalized plumbings and distinguishing the induced metrics via index-theoretic invariants. The core methods combine topological plumbing (generalized and equivariant) with geometric constructions that yield $scal>0$ interiors and $Ric>0$ boundaries, then apply Perelman-type gluing and eta/APS index-theory to detect distinct components in the moduli space. The work distinguishes geometries using both basic index differences in dimension $4k+3$ and eta-invariants in dimension $4k+1$, including equivariant and Spin$^c$-settings, yielding numerous new classes of examples (e.g., sphere bundles and disk/plumbing constructions). The results significantly expand the known landscape of moduli spaces of Ricci-positive metrics, highlighting rich topology in the space of such metrics beyond merely their existence. Collectively, the paper provides a robust framework for generating infinite families of geometries in high dimensions and demonstrates the power of combining geometric analysis with topological plumbing and index theory for moduli-space questions.

Abstract

In every odd dimension $n\geq 5$ we exhibit large classes of closed $n$-dimensional manifolds which admit infinitely many different geometries of positive Ricci curvature, i.e., manifolds for which their moduli space of metrics of positive Ricci curvature has infinitely many connected components.

Odd-dimensional manifolds with infinitely many different geometries of positive Ricci curvature

TL;DR

The paper proves that in every odd dimension there exist closed -manifolds with infinitely many geometries of positive Ricci curvature, by constructing families of manifolds as boundaries of generalized plumbings and distinguishing the induced metrics via index-theoretic invariants. The core methods combine topological plumbing (generalized and equivariant) with geometric constructions that yield interiors and boundaries, then apply Perelman-type gluing and eta/APS index-theory to detect distinct components in the moduli space. The work distinguishes geometries using both basic index differences in dimension and eta-invariants in dimension , including equivariant and Spin-settings, yielding numerous new classes of examples (e.g., sphere bundles and disk/plumbing constructions). The results significantly expand the known landscape of moduli spaces of Ricci-positive metrics, highlighting rich topology in the space of such metrics beyond merely their existence. Collectively, the paper provides a robust framework for generating infinite families of geometries in high dimensions and demonstrates the power of combining geometric analysis with topological plumbing and index theory for moduli-space questions.

Abstract

In every odd dimension we exhibit large classes of closed -dimensional manifolds which admit infinitely many different geometries of positive Ricci curvature, i.e., manifolds for which their moduli space of metrics of positive Ricci curvature has infinitely many connected components.

Paper Structure

This paper contains 32 sections, 19 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Moduli spaces and geometries of positive Ricci curvature
  3. Topological Plumbing
  4. Plumbing of disk bundles
  5. Generalized plumbing
  6. Geometric plumbing
  7. Preliminaries
  8. Generalized geometric plumbing
  9. Equivariant plumbing
  10. Index theory for $Spin^c$-manifolds
  11. The group $Spin^c(k)$
  12. $Spin^c$-manifolds
  13. Dirac operators
  14. The APS index theorem
  15. Eta invariants on connected components of $\mathcal{R}_{scal>0}$
  16. ...and 17 more sections

Key Result

Lemma 3.1

Let $E_1 \square \ldots \square E_{2l}$ be the plumbing of disk bundles over spheres according to a straight line, $B_1=S^q,\ldots , B_{2l}=S^p$. We identify $E_{2i+1}$ with $D^q\times D^p\cup_{\Phi _{2i+1}} D^q\times D^p$ and $E_{2i}$ with $D^p\times D^q\cup_{\Phi _{2i}} D^p\times D^q$. Then the bo ∎

Theorems & Definitions (48)

  • Definition 2.1
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Theorem 4.1: P97
  • Remark 4.2
  • Proposition 4.3
  • Proposition 4.4
  • ...and 38 more