### Rational isomorphisms between K-theories and cohomology theories, by Eric M. Friedlander and Mark E. Walker

The well known isomorphism relating the rational algebraic K-theory
groups and the rational motivic cohomology groups of a smooth variety
over a field of characteristic 0 is shown to be realized by a map
(the "Segre map") of infinite loop spaces. Moreover, the associated
Chern character map on rational homotopy groups is shown to be a ring
isomorphism. A technique is introduced which establishes a useful
general criterion for a natural transformation of functors on
quasi-projective complex varieties to induce a homotopy equivalence of
semi-topological singular complexes. Since semi-topological
K-theory and morphic cohomology can be formulated as the
semi-topological singular complexes associated to K-theory and
motivic cohomology, this criterion provides a rational isomorphism
between the semi-topological K-theory groups and the morphic
cohomology groups of a smooth complex variety. Consequences include a
Riemann-Roch theorem for the Chern character on semi-topological
K-theory and an interpretation of the "topological filtration" on
singular cohomology groups in K-theoretic terms.

Eric M. Friedlander <eric@math.northwestern.edu>

Mark E. Walker <mwalker@math.unl.edu>