Function spaces and continuous algebraic pairings for varieties

Given a quasi-projective complex variety X and a projective variety Y, one may endow the set of morphisms, Mor(X, Y), from X to Y with the natural str ucture of a topological space. We introduce a convenient technique (namely, the notion of a functor on the category of 'smooth curves') for studying t hese function complexes and for forming continuous pairings of such. Buildi ng on this technique, we establish several results, including (1) the exist ence of cap and join product pairings in topological cycle theory; (2) the agreement of cup product and intersection product for topological cycle the ory; (3) the agreement of the motivic cohomology cup product with morphic c ohomology cup product; and (4) the Whitney sum formula for the Chern classe s in morphic cohomology of vector bundles.