Topological $K$-theory of the integers at the prime 2

Abstract

Recent results of Voevodsky and others have effectively led to the proof of the Lichtenbaum-Quillen conjectures at the prime 2, and consequently made it possible to determine the 2-homotopy type of the $K$-theory spectra for various number rings. The basic case is that of $ BGL({\Bbb Z})$; in this note we use these results to determine the 2-local (topological) $K$-theory of the space $BGL({\Bbb Z})$, which can be described as a completed tensor product of two quite simple components; one corresponds to a real `image of $J$' space, the other to $BBSO$.