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New Topological Restrictions For Spaces With Nonnegative Ricci Curvature

Alessandro Cucinotta, Mattia Magnabosco, Daniele Semola

TL;DR

The paper addresses topology of complete manifolds with nonnegative Ricci curvature and extends the framework to $ ext{RCD}(0,n)$ spaces, establishing a Betti-number rigidity theorem and a vanishing simplicial-volume theorem. It develops a novel program built on equivariant blow-downs, splitting theorems for $ ext{RCD}$ spaces under $ ext{R}^k$-actions, and the construction of equivariant harmonic functions with almost linear growth to obtain a measured warped-product description of blow-downs. These tools culminate in a synthetic proof of the 3-manifold classification under nonnegative Ricci curvature and yield broad topological constraints, including the vanishing of simplicial volume in all dimensions under the $ ext{Ric} obreak ext{-} obreak 0$ condition and the extension to $ ext{RCD}(0,n)$ spaces. The results unify and extend classical results (Anderson, Gromov, Liu) within the synthetic setting, providing a versatile framework for rigidity, volume, and topology in collapsed and noncollapsed regimes with potential applications to higher-dimensional topology and Ricci-flow-like analyses. Overall, the work advances the understanding of how curvature bounds control topology in both smooth and synthetic geometric contexts.

Abstract

We obtain new topological restrictions for complete Riemannian manifolds with nonnegative Ricci curvature and RCD(0,n) spaces. Our main results are a Betti number rigidity theorem which answers a question open since work of M.-T. Anderson in 1990, and a vanishing theorem for the simplicial volume generalizing a theorem of M. Gromov from 1982. Combining such results we obtain a new proof of the classification of noncompact 3-manifolds with nonnegative Ricci curvature, originally due to G. Liu in 2011, which extends to the synthetic setting.

New Topological Restrictions For Spaces With Nonnegative Ricci Curvature

TL;DR

The paper addresses topology of complete manifolds with nonnegative Ricci curvature and extends the framework to spaces, establishing a Betti-number rigidity theorem and a vanishing simplicial-volume theorem. It develops a novel program built on equivariant blow-downs, splitting theorems for spaces under -actions, and the construction of equivariant harmonic functions with almost linear growth to obtain a measured warped-product description of blow-downs. These tools culminate in a synthetic proof of the 3-manifold classification under nonnegative Ricci curvature and yield broad topological constraints, including the vanishing of simplicial volume in all dimensions under the condition and the extension to spaces. The results unify and extend classical results (Anderson, Gromov, Liu) within the synthetic setting, providing a versatile framework for rigidity, volume, and topology in collapsed and noncollapsed regimes with potential applications to higher-dimensional topology and Ricci-flow-like analyses. Overall, the work advances the understanding of how curvature bounds control topology in both smooth and synthetic geometric contexts.

Abstract

We obtain new topological restrictions for complete Riemannian manifolds with nonnegative Ricci curvature and RCD(0,n) spaces. Our main results are a Betti number rigidity theorem which answers a question open since work of M.-T. Anderson in 1990, and a vanishing theorem for the simplicial volume generalizing a theorem of M. Gromov from 1982. Combining such results we obtain a new proof of the classification of noncompact 3-manifolds with nonnegative Ricci curvature, originally due to G. Liu in 2011, which extends to the synthetic setting.
Paper Structure (20 sections, 82 theorems, 308 equations)

This paper contains 20 sections, 82 theorems, 308 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Discussion of the key new ideas
  4. Preliminaries
  5. Equivariant Gromov Hausdorff convergence
  6. Some properties of RCD spaces
  7. First Betti number rigidity
  8. Splitting of the blow-downs
  9. Proof of Theorem \ref{['T|splittingkesimo']}, base step: measured warped product structure
  10. Proof of Theorem \ref{['T|splittingkesimo']}, base step: splitting
  11. Proof of Theorem \ref{['T|splittingkesimo']}, inductive step
  12. Proof of Corollary \ref{['cor:blowdownXbarunique']}
  13. Equivariant harmonic functions with almost linear growth
  14. Proof of Theorem \ref{['thm:Bettirigid']}
  15. A vanishing theorem for the simplicial volume
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(M,g)$ be a complete Riemannian $3$-manifold with $\mathop{\mathrm{Ric}}\nolimits\ge 0$. If $\pi_1(M)$ contains a subgroup isomorphic to $\mathbb{Z}$, then the universal cover $(\overline{M},\overline{g})$ of $(M,g)$ splits a line isometrically.

Theorems & Definitions (163)

  • Theorem 1.1: Liu3d
  • Theorem 1.2
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 1.3
  • Theorem 1.4: GromovVolbCoho
  • Remark 5
  • Theorem 1.5
  • ...and 153 more