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Ribbon cobordisms as a partial order

Marius Huber

TL;DR

The paper proves that ribbon rational homology cobordisms induce a partial order on aspherical 3-manifolds by extending Agol's knot-level partial order to the 3-manifold setting. It adapts the analysis of maps between fundamental groups to the realm of representation varieties $R_n(X)$ into $SO(n)$, leveraging residual finiteness to show inclusions are isomorphisms and yield orientation-preserving homotopy equivalences. In the aspherical case, these homotopy equivalences lift to homeomorphisms (by the Borel conjecture in dimension three), thereby proving that if there exist ribbon cobordisms in either direction between aspherical or lens spaces, the manifolds are homeomorphic. This confirms the antisymmetry aspect of the conjectured preorder for aspherical 3-manifolds and ties the result to the broader Daemi–Lidman–Vela-Vick–Wong program, with independent contemporary confirmations by FMZ.

Abstract

We show that the notion of ribbon rational homology cobordism yields a partial order on the set of aspherical $3$-manifolds, thus supporting a conjecture formulated by Daemi, Lidman, Vela-Vick and Wong. Our proof is built on Agol's recent proof of the fact that ribbon concordance yields a partial order on the set of knots in the 3-sphere.

Ribbon cobordisms as a partial order

TL;DR

The paper proves that ribbon rational homology cobordisms induce a partial order on aspherical 3-manifolds by extending Agol's knot-level partial order to the 3-manifold setting. It adapts the analysis of maps between fundamental groups to the realm of representation varieties into , leveraging residual finiteness to show inclusions are isomorphisms and yield orientation-preserving homotopy equivalences. In the aspherical case, these homotopy equivalences lift to homeomorphisms (by the Borel conjecture in dimension three), thereby proving that if there exist ribbon cobordisms in either direction between aspherical or lens spaces, the manifolds are homeomorphic. This confirms the antisymmetry aspect of the conjectured preorder for aspherical 3-manifolds and ties the result to the broader Daemi–Lidman–Vela-Vick–Wong program, with independent contemporary confirmations by FMZ.

Abstract

We show that the notion of ribbon rational homology cobordism yields a partial order on the set of aspherical -manifolds, thus supporting a conjecture formulated by Daemi, Lidman, Vela-Vick and Wong. Our proof is built on Agol's recent proof of the fact that ribbon concordance yields a partial order on the set of knots in the 3-sphere.
Paper Structure (2 sections, 4 theorems, 1 equation)

This paper contains 2 sections, 4 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Proofs of results

Key Result

Theorem 1.2

Let $Y_{1}$ and $Y_{2}$ be compact, oriented, $3$-manifolds, possibly with boundary, such that there exists a ribbon cobordism $W_{i}$ from $Y_{i}$ to $Y_{j}$, $\{i,j\}=\{1,2\}$. If $Y_{i}$ is either aspherical or a lens space, $i=1,2$, then $Y_{1}\cong Y_{2}$.

Theorems & Definitions (9)

  • Conjecture 1.1: Daemi-Lidman-Vela-Vick-Wong DLVVW
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • proof : Proof of Theorem \ref{['mainthm']}
  • proof : Proof of Theorem \ref{['mainthm_2']}
  • proof : Proof of Theorem \ref{['thm_conj']}