Semi-topological K-theory of real varieties, by Eric M. Friedlander and Mark E. Walker

The semi-topological K-theory of real varieties is an oriented multiplicative (generalized) cohomology theory which extends the authors' earlier theory for complex algebraic varieties. Motivation comes from consideration of algebraic equivalence of vector bundles (sharpened to real semi-topological equivalence), consideration of Z/2-equivariant mapping spaces of morphisms of algebraic varieties to Grassmannian varieties, and consideration of the algebraic K-theory of real varieties. The authors verify that the semi-topological K-theory of a real variety X interpolates between the algebraic K-theory of X and Atiyah's Real K-theory of the associated Real space of complex points. The resulting natural maps of spectra, from algebraic K-theory to semi-topological K-theory to Atiyah's real K-theory, satisfy numerous good properties: The map from algebraic K-theory to semi-topological K-theory is a mod-n equivalence for any projective real variety and positive integer n. The map from semi-topological K-theory to Atiyah's Real K-theory is an equivalence for smooth projective real curves and real flag varieties. Furthermore, the triple of maps fits in a commutative diagram of spectra, mapping via total Segre classes to a triple of cohomology theories. The authors also establish results for the semi-topological K-theory of real varieties, such as Nisnevich excision and a type of localization result, which were previously unknown even for complex varieties.

Eric M. Friedlander <>
Mark E. Walker <>