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.

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    For those seeking more details about the cup product in Morse cohomology (as I was) there are some here math.stackexchange.com/questions/887571/… – j.c. Apr 11 at 13:55
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    There is a reasonably easy way to compute the intersection product on homology, which is close to what you're asking (via Poincaré duality). I'm pretty sure it's explained in some detail in Gompf and Stipsicz's book. – Marco Golla Apr 16 at 9:32
  • @MarcoGolla, that's encouraging! I've read quite a few parts of that book in close detail, but the only thing relating homology and handles I recall is page 111. As far as I understand, it doesn't discuss the issue at heart, which seems to be that the diagonal map is not a cellular map. I'm not so familiar with part III of the book though, so there might be something. Can you be more specific where I might find more information? – Manuel Bärenz Apr 16 at 9:39
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    @ManuelBärenz I was thinking of Proposition 4.5.11. – Marco Golla Apr 16 at 9:53
  • @MarcoGolla, good point! So that explains the middle part of the cap product for 4-manifolds. I wonder whether there is a general description along those lines. – Manuel Bärenz Apr 16 at 10:21

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