### Semi-topological K-theory using function complexes, by Eric M. Friedlander and Mark E. Walker

The semi-topological K-theory (written K_{*}^{semi}(X)) of a
quasi-projective complex algebraic variety X is based on the notion of
algebraic vector bundles modulo algebraic equivalence. This theory is given
as the homotopy groups of a certain infinite loop space which comes equipped
with maps both to the infinite loop space giving the topological K-theory of
(the analytic space associated to) X and from the infinite loop space giving
the algebraic K-theory of X. The composition of these maps is the natural map
from the algebraic K-theory of X to the topological K-theory of X.

We give an explicit description of K_{0}^{semi}(X) in terms
of K_{0}^{algebraic}(X), a description of
K_{q}^{semi}(X) in terms of K_{0}^{semi}(X)
for projective varieties X, a Poincare duality theorem for projective
varieties, a projective bundle formula for projective varieties, and
computations of the semi-topological K-theory of a product of projective
spaces and of any complete smooth curve.

For X a smooth quasi-projective variety, there are natural Chern class
maps from the semi-topological K-groups of X to its morphic cohomology
groups compatible with similarly defined Chern class maps from
algebraic K-theory to motivic cohomology and compatible with the
classical Chern class maps from topological K-theory to the singular
cohomology of X.

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

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