Semi-topological \(K\)-homology and Thomason's theorem
Let \(X\) be a smooth, projective variety, and denote by \(X^{\text{an}}\) the underlying analytic space. The semi-topological \(K\)-theory of \(X\) (written \(K^{\text{semi}}\)) may be regarded as a theory between \(K\)-theory and topological \(K\)-theory. The author proves various important results comparing these theories. The main results are: there is a natural isomorphism of graded rings \[ K_{*}^{\text{semi}}(X)[1/\beta]\cong K^{-*}_{\text{top}}(X^{\text{an}}), \] where \(\beta\in K_{2}(X)\) is the Bott element. From the above the author infers that there are natural isomophisms \[ K_*^{\text{alg}}(X;\mathbb{Z}/n)[1/\beta]\cong K^{-*}_{\text{top}}(X^{\text{an}};\mathbb{Z}/n). \] The author develops all the machinery of bivariant versions of semi-topological \(K\)-theory and connective topological \(K\)-theory and a natural map between these. One important ingredient is the development of operations in semi-topological \(K\)-theory which are compatible with those of topological \(K\)-theory.
scientific article; zbMATH DE number 1876947