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

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 $BGL(\mathbb{Z})$; in this note we use these results to determine the 2-local (topological) $K$-theory of the space $BGL(\mathbb{Z})$, 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 $BBSO$.

Keywords

$K$-theory, general linear group of integers, Rothenberg-Steenrod spectral sequence, Bousfield localization

2010 Mathematics Subject Classification

19Dxx, 55N15, 55P15

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Published 1 January 2000