Triple-cup product forms of 3-manifolds and Heegaard diagrams

TL;DR

This paper provides an explicit formula to compute the triple-cup product form of 3-manifolds from Heegaard diagrams and relates it to Turaev's homotopy intersection form.

Contribution

It introduces a new explicit method to derive the triple-cup product form from Heegaard diagrams and connects it to Turaev's intersection form.

Findings

Derived an explicit formula for the triple-cup product form from Heegaard diagrams.

Showed that the triple-cup product form can be obtained as a reduction of Turaev's homotopy intersection form.

Enhanced understanding of the algebraic invariants of 3-manifolds.

Abstract

The triple-cup product form $\mu$ is a classical invariant of $3$-manifolds, determining the cohomology ring up to torsion. Given a closed, connected, oriented $3$-manifold $M$, we describe an explicit formula for computing $\mu$ from a Heegaard diagram of $M$. Then, we show that the triple-cup product form $\mu$ can be recovered as a reduction of Turaev's homotopy intersection form $\eta$ of the Heegaard surface.