Semi-topological \(K\)-theory using function complexes
Recall that for a quasi-projective variety \(X\) over a field \(k\), an element of \(K_{0}(X)\) is defined to be algebraically equivalent to zero provided that it lies in the image of \(t_{0}^{*} - t_{1}^{*} : K_{0}(X \times T) \longrightarrow K_{0}(X)\) for some smooth, connected \(k\)-variety \(T\) with \(k\)-rational points \(t_{0}, t_{1}\). The authors denote by \(K_{0}^{\text{semi}}(X)\) the quotient of \(K_{0}(X)\) by the subgroup of elements algebraically equivalent to zero. Now suppose that \(X\) is a weakly normal, quasi-projective complex variety. Using techniques from the theory of spaces which are ``algebras'' over \(E_{\infty}\)-operads (the \(E_{\infty}\)-operad in question here is a sort of complex analytic version of the classical linear isometries operad) the authors construct an infinite loopspace \({\mathcal K}^{\text{semi}}(X)\) having \(K_{0}^{\text{semi}}(X)\) as its space of components and sitting, as an infinite loopspace between the infinite loospace of Quillen algebraic K-theory \({\mathcal K}(X)\) and that of topological K-theory of the underlying analytic space \({\mathcal K}_{\text{top}}(X^{\text{an}})\). Just as \({\mathcal K}_{\text{top}}(X^{\text{an}})\) is related to Betti cohomology and \({\mathcal K}(X)\) is related to motivic cohomology via the Atiyah-Hirzebruch and Bloch-Lichtenbaum-Suslin spectral sequences, respectively, the authors conjecture that \({\mathcal K}^{\text{semi}}(X)\) is related to morphic cohomology. They make a number of conjectures about the relationships between these three types of K-theory and prove several of the standard K-theory properties (e.g. Mayer-Vietoris, projective bundle theorem, etc.) for \({\mathcal K}^{\text{semi}}(X)\). This paper is one of a series by the authors on this topic and in a footnote they announce that they have been able, using work of Voevodsky-Suslin, to verify all their conjectures [\textit{E. M. Friedlander} and \textit{M. E. Walker}, Am. J. Math. 123, No.~5, 779-810 (2001; Zbl 1018.19001)].
scientific article; zbMATH DE number 1750417