Since handle decompositions and Morse functions are intimately related, I'm imagining that a given explicit handle decomposition allows for an explicit description of the cellular complex and thus of (co)homology, which is directly isomorphic to Morse (co)homology. More precisely, the $k$-th grade of the complex is generated by the $k$-handles.

It is known how to compute the cup product $\smile\,\colon H^i(M) \otimes H^j(M) \to H^{i+j}(M)$ for the Morse cohomology $H^*$ of a smooth manifold $M$. One has to chose two different Morse functions and study certain Y-shaped Morse flow lines. The cap product should be related.

Is there a way to translate this to handle decompositions, i.e. an algorithm to compute the cap product $\frown\,\colon H^i(M) \otimes H_j(M) \to H_{j-i}(M)$ for given handle decompositions? In particular, how do you compute it for Kirby diagrams?

Edit: As far as I understand, the reason why this isn't trivial is because the diagonal map $\Delta\colon M \to M \times M$ is used in the cup product: The first part of the cup product is a special case of the Künneth isomorphism $H^*(M) \otimes H^*(M) \cong H^*(M \times M)$, and then we pull back along the diagonal, $\Delta^*\colon H^*(M \times M) \to H^*(M)$. But the diagonal is not a cellular map, so it doesn't have a straightforward description in terms of handles.