Triple linking and rational homology cobordism

TL;DR

This paper investigates the properties of linking forms on rational homology 3-spheres bounding rational homology 4-balls, proving the vanishing of a specific triple linking form under certain conditions and exploring implications for homology cobordism.

Contribution

It proves the vanishing of the Freedman-Krushkal triple torsion linking form on a Lagrangian in certain rational homology cobordisms, extending understanding of linking forms in 4-manifold topology.

Findings

The kernel of the inclusion-induced homomorphism forms a Lagrangian for the torsion linking form.

The triple linking form vanishes on this Lagrangian when $H_2(W;\mathbb{Z})=0$.

Poses new questions about topological rational homology cobordism.

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 $\lambda_2$ on $H_1(M;\mathbb{Z})$. In this short paper, we prove that the Freedman-Krushkal triple torsion linking form $\lambda_3$ (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.