Algebraic K-theory of topological K-theory
John Rognes
We view rings as special cases of (structured) ring spectra, by way
of their Eilenberg - MacLane spectra. This extended scope makes room
for further constructions in the arithmetic of ring spectra than are
available within the arithmetic of rings, e.g. in classical number theory.
The algebraic K-theory of ring spectra captures some of this extended
number theory, and is a field wide open for computational exploration.
In a joint work with Christian Ausoni (now at ETH, Zurich), we have
computed the algebraic K-theory of (the p-completed Adams summand of
connective) topological K-theory, viewed as a ring spectrum. Here
topological K-theory is represented by complex vector bundles, and the
ring operations are induced by Whitney sum and tensor product of bundles.
The calculation sheds light on the interaction of algebraic K-theory
with the chromatic periodicity phenomena in stable homotopy theory.
Last modified: Thu Nov 11 16:11:15 MET 1999