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

The triple-cup product form $μ$ 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 $μ$ from a Heegaard diagram of $M$. Then, we show that the triple-cup product form $μ$ can be recovered as a reduction of Turaev's homotopy intersection form $η$ of the Heegaard surface.