l-adic and Z/l∞-algebraic and topological K-theory

Algebraic K-Theory Eventually Surjects onto Topological K-Theory

Background. For X a smooth scheme over ~, K.t(X; 2g/{") is isomorphic to the usual rood(" topological K-theory of X as a complex manifold ([6]). In this case, Theorem 1 essentially answers a question of Fulton, [9], w 5. For many such X the mapp: Ko(X)~K~ot(X) cannot be a surjection, as non-trivial Hodge conditions on the Chern classes prevent some topological bundles from being stably algebrai...

Topological K-theory of Algebraic K-theory Spectra

One of the central problems of algebraic K-theory is to compute the K-groups KX of a scheme X. Since these groups are, by definition, the homotopy groups of a spectrum KX, it makes sense to analyze the homotopy-type of the spectrum, rather than just the disembodied homotopy groups. In addition to facilitating the computation of the K-groups themselves, knowledge of the spectrum KX can be applie...

Bott Periodicity in Topological, Algebraic and Hermitian K-theory

This paper is devoted to classical Bott periodicity, its history and more recent extensions in algebraic and Hermitian K-theory. However, it does not aim at completeness. For instance, the variants of Bott periodicity related to bivariant K-theory are described by Cuntz in this handbook. As another example, we don’t emphasize here the relation between motivic homotopy theory and Bott periodicit...

Algebraic v. Topological K-theory: a Friendly Match

The Localization Sequence for the Algebraic K-theory of Topological K-theory

We prove a conjecture of Rognes that establishes a localization cofiber sequence of spectra K(Z) → K(ku) → K(KU) → ΣK(Z) for the algebraic K-theory of topological K-theory. We deduce the existence of this sequence as a consequence of a devissage theorem identifying the K-theory of the Waldhausen category of Postnikov towers of modules over a connective A∞ ring spectrum R with the Quillen K-theo...