TITLE: Calculating cup products using cellular cohomology
QUESTION [39 upvotes]: Most algebraic topology books (for instance, Hatcher) contain a recipe for computing cup products in singular or simplicial homology.  In other words, given two explicit singular or simplicial cocycles, they contain a recipe for computing an explicit cocycle representing the cup product of the cocycles in question.
Is there a similar recipe in cellular cohomology?  In other words, if I have a very explicit CW complex and two explicit cellular cocycles, then is there a recipe for computing a cellular cocycle representing their cup product?
Of course, one answer is to subdivide everything up into a simplicial complex, but that is messy (and not always possible).  Is there a better way?
I'm especially interested in the special case of 2-dimensional CW complexes, where the only interesting cup products are between elements of $H^1$.

REPLY [3 votes]: A recipe is given in Whitney's 1937 paper "On Products in a Complex" (Annals of Mathematics, vol. 39, no 2). You can find a copy here.
In section 12 he gives a very explicit construction of the cap product for low-dimensional, regular complexes from which the cup product follows immediately (via Eqn 5.3 in the same paper).