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.