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Rational homology ribbon cobordism is a partial order

Stefan Friedl, Filip Misev, Raphael Zentner

TL;DR

The paper proves that ribbon $\mathbb{Q}$-homology cobordism defines a partial order on irreducible closed oriented 3-manifolds, with a stronger conclusion for aspherical manifolds that mutual comparability implies an orientation-preserving homeomorphism. Building on Agol's knot-concordance techniques, the authors analyze inclusions of fundamental groups through ribbon cobordisms, construct maps between manifolds with prescribed $\pi_1$-behaviour, and use classifying-space and Whitehead arguments to deduce degree and homeomorphism results. They handle finite fundamental-group cases via lens-space classifications and Huber's lens-space results, ensuring the overall partial-order statement holds in the irreducible setting. A key component is the adaptation of Agol’s representation-variety method to show that self-ribbon cobordisms force isomorphisms on $\pi_1$, underpinning the rigidity statements. The work thus extends the rigidity paradigm from knot theory to 3-manifold topology, providing a structured, robust ordering among irreducible 3-manifolds.

Abstract

We show that ribbon rational homology cobordism is a partial order within the class of irreducible 3-manifolds. This makes essential use of the methods recently employed by Ian Agol to show that ribbon knot concordance is a partial order.

Rational homology ribbon cobordism is a partial order

TL;DR

The paper proves that ribbon -homology cobordism defines a partial order on irreducible closed oriented 3-manifolds, with a stronger conclusion for aspherical manifolds that mutual comparability implies an orientation-preserving homeomorphism. Building on Agol's knot-concordance techniques, the authors analyze inclusions of fundamental groups through ribbon cobordisms, construct maps between manifolds with prescribed -behaviour, and use classifying-space and Whitehead arguments to deduce degree and homeomorphism results. They handle finite fundamental-group cases via lens-space classifications and Huber's lens-space results, ensuring the overall partial-order statement holds in the irreducible setting. A key component is the adaptation of Agol’s representation-variety method to show that self-ribbon cobordisms force isomorphisms on , underpinning the rigidity statements. The work thus extends the rigidity paradigm from knot theory to 3-manifold topology, providing a structured, robust ordering among irreducible 3-manifolds.

Abstract

We show that ribbon rational homology cobordism is a partial order within the class of irreducible 3-manifolds. This makes essential use of the methods recently employed by Ian Agol to show that ribbon knot concordance is a partial order.
Paper Structure (5 sections, 9 theorems, 4 equations)

This paper contains 5 sections, 9 theorems, 4 equations.

Table of Contents

  1. Introduction
  2. Preparations
  3. Proof of Theorem \ref{['thm: main theorem-2']}
  4. Proof of Theorem \ref{['thm: main theorem']}
  5. Sketch of proof of Theorem \ref{['thm: ribbon to itself']}

Key Result

Theorem 1.2

The preorder $\geq$ is a partial order on the set of homeomorphism classes of irreducible, closed, connected, oriented 3-manifolds. In particular, if $Y_0 \geq Y_1$ and $Y_1 \geq Y_0$ then $Y_0$ and $Y_1$ are homeomorphic.

Theorems & Definitions (18)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Corollary 1.6
  • Remark
  • Proposition 2.1: Daemi, Lidman, Vela-Vick, Wong
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 8 more