Some relations between the topological and geometric filtration for smooth projective varieties

In the first part of this paper, we show that the assertion "T_pH_k(X,Q)=G_pH_k(X,Q)" (which is called the Friedlander-Mazur conjecture) is a birationally invariant statement for smooth projective varieties X when p=dim(X)-2 and when p=1. We also establish the Friedlander-Mazur conjecture in certain dimensions. More precisely, for a smooth projective variety X, we show that the topological filtration T_pH_{2p+1}(X,Q) coincides with the geometric filtration G_pH_{2p+1}(X,Q) for all p. (Friedlander and Mazur had previously shown that T_pH_{2p}(X,Q})=G_pH_{2p}(X,Q)). As a corollary, we conclude that for a smooth projective threefold X, T_pH_k(X,Q)=G_pH_k(X,Q) for all k\geq 2p\geq 0 except for the case p=1,k=4. Finally, we show that the topological and geometric filtrations always coincide if Suslin's conjecture holds.