Interpretation
It is possible to view the cup product as induced from the following composition:
in terms of the chain complexes of and, where the first map is the Künneth map and the second is the map induced by the diagonal .
This composition passes to the quotient to give a well-defined map in terms of cohomology, this is the cup product. This approach explains the existence of a cup product for cohomology but not for homology: induces a map but would also induce a map, which goes the wrong way round to allow us to define a product. This is however of use in defining the cap product.
Bilinearity follows from this presentation of cup product, i.e. and
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