Semi-topological K-homology and Thomason’s Theorem

In this paper, we introduce the “semi-topological K-homology” of complex varieties, a theory related to semi-topological K-theory much as connective topological K-homology is related to connective topological K-theory. Our main theorem is that the semi-topological K-homology of a smooth, quasiprojective complex variety Y coincides with the connective topological Khomology of the associated analytic space Y an. From this result, we deduce a pair of results relating semi-topological K-theory with connective topological K-theory. In particular, we prove that the “Bott inverted” semi-topological K-theory of a smooth, projective complex variety X coincides with the topological K-theory of Xan. In combination with a result of Friedlander and the author [12, 3.8], this gives a new proof, in the special case of smooth, projective complex varieties, of Thomason’s celebrated theorem that “Bott inverted” algebraic K-theory with Z/n coefficients coincides with topological K-theory with Z/n coefficients.