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 K0semi(X) in terms of K0algebraic(X), a description of Kqsemi(X) in terms of K0semi(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 <>
Mark E. Walker <>