ALGEBRAIC COCYCLES ON NORMAL, QUASI-PROJECTIVE VARIETIES

This paper extends to quasi-projective varieties earlier work by the a uthor and H. Blaine Lawson concerning spaces of algebraic cocycles on projective varieties. The topological monoid C-r (Y) (U) of effective cocycles on a normal, quasi-projective variety U with values in a proj ective variety Y consists of algebraic cycles on U x Y equi-dimensiona l of relative dimension r over U. A careful choice of topology enables the establishment of various good properties: the definition is essen tially algebraic, the group completion Z(r)(Y) (U) has 'sensible' homo topy groups, the construction is contravariant with respect to U, cova riant with respect to Y, and there is a natural 'duality map' to the t opological group of cycles on U x Y. The fundamental theorem presented here is the extension of Friedlander-Lawson duality to this context: the duality map Z(r) (Y) (U) --> Z(r+m)(U x Y) is a homotopy equivalen ce provided that both U and Y are smooth (where m = dim U). Various ap plication are given, especially the determination of the homotopy type s of certain topological groups of algebraic morphisms.