Triple linking and rational homology cobordism

TL;DR

This paper addresses how the Freedman-Krushkal triple torsion linking form $\lambda_3$ behaves on Lagrangians for the classical torsion linking form $\lambda_2$ when a rational homology 3-sphere $M$ bounds a rational homology 4-ball $W$. The main result proves that if $H_2(W)=0$, then $\lambda_3$ vanishes on the kernel $L=\ker(H_1(M)\to H_1(W))$, by expressing $\lambda_3$ via a rational Matsumoto-type pairing on closed surfaces within $W$ and showing a divisibility by $t^3$ that forces the value to lie in $\mathbb{Q}/\mathbb{Z}$ as zero. An explicit example demonstrates how $\lambda_3$ can be nonzero on a particular Lagrangian, highlighting obstructions to bounding with $H_2(W)=0$ and the subtleties of choosing Lagrangians. The paper concludes with open questions about relaxing hypotheses, the nature of deeply nullhomologous classes, and the structure of the algebraic versus topological rational homology cobordism group, guiding future work in topological rational homology cobordism. These results deepen understanding of higher-order linking phenomena and their implications for the cobordism classification of rational homology spheres.

Abstract

If a rational homology 3-sphere $M$ bounds a rational homology 4-ball $W$, then the kernel of the inclusion-induced homomorphism $H_1(M;\mathbb{Z})\to H_1(W;\mathbb{Z})$ is a Lagrangian for the $\mathbb{Q}/\mathbb{Z}$-valued torsion linking form $λ_2$ on $H_1(M;\mathbb{Z})$. In this short paper, we prove that the Freedman-Krushkal triple torsion linking form $λ_3$ (arXiv:2506.11941v3) vanishes on this Lagrangian under the assumption that $H_2(W;\mathbb{Z})=0$. We then pose several questions about topological rational homology cobordism.

Figures (1)

• Figure 1: A Kirby diagram for the manifold $M_{0}$, as seen in FreedmanKrushkal, Figure 4.

Theorems & Definitions (6)

• Theorem 1.1
• Definition 2.1
• Definition 2.2
• Theorem 3.1
• proof
• Conjecture 4.3: KimLivingston