Mark Walker
In recent work with Eric Friedlander and Christian Haesemeyer, we have exploited the weight filtration on the singular homology of a complex variety (due to Deligne and Gillet-Soule) in order to better understand the relationship between Lawson homology and singular homology. This technique leads to a proof that Lawson homology and singular homology are isomorphic in a range of degrees for a certain class of complex varieties. Using our recently developed "semi-topological spectral sequence", such results lead to a similar comparison of semi-topological K-theory and topological K-theory. In the case of a smooth projective complex variety X, this result amounts to the assertion that the space of all continuous maps from X to Grassmann varieties is well approximated (homotopically speaking) by the subspace of algebraic morphisms.