Suppose that we are given a CW complex X in terms of the cells and the gluing maps. My understanding is that computing the cup product of the singular cohomology ring from this information is a non-trivial task. I know of two basic strategies that one might take:

1) If the X is homotopy equivalent to a closed oriented manifold, then we can translate from cup product into intersection product and the problem becomes easier to visualize.

2) If X is not too complicated, then we can try to find a simple presentation of X as a finite simplicial complex and compute the cup product explicitly for all the cochains.

My question is: what are other techniques/tricks that can be used to find the cup product?

Surely there must be some general approaches beyond the naive ones I mentioned. Feel free to strengthen the hypotheses or consider specific situations, as I don't expect there to be one trick which works for everything.

`Techniques of geometric topology'', Chapter 1 and S. Buoncristiano, C. P. Rourke, B. J. Sanderson`

A geometric approach to homology theory'', Chapter 2. $\endgroup$