Topological Models for Arithmetic

In this paper, we use topological models to compute the `–adic topological K-theory of certain arithmetic rings A. Our technique is to exploit class field theory to show that the etale topological type of A is equivalent in an appropriate sense to something relatively simple. Calculating the K-theory of this simple “topological model” provides an explicit determination of the `–adic topological K-theory of A and, by means of a comparison map, gives information about the algebraic K-theory of A. For example, we are sometimes able to compute the mod ` cohomology of certain “unstable” topological K-theory spaces and verify that it injects into the cohomology of the corresponding unstable algebraic K-theory spaces. This gives an explicit lower bound for H∗(GL(n,A),Z/`).