Linking forms revisited

TL;DR

This paper rigorously analyzes the properties of linking forms on rational homology spheres, demonstrating their symmetry and providing a formula for their computation via Heegaard splittings.

Contribution

It offers clear, rigorous proofs of known properties of linking forms and introduces a practical method to compute them using Heegaard splittings.

Findings

Linking forms are (anti-) symmetric.

Linking form can be computed from Heegaard splitting.

Provides rigorous proofs for known properties.

Abstract

We show that the $\mathbb{Q}/\mathbb{Z}$-valued linking forms on rational homology spheres are (anti-) symmetric and we compute the linking form of a 3-dimensional rational homology sphere in terms of a Heegaard splitting. Both results have been known to a larger or lesser degree, but it is difficult to find rigorous down-to-earth proofs in the literature.