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# Homology, Homotopy and Applications

## Volume 2 (2000)

### Number 1

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

Pages: 119 – 126

DOI: http://dx.doi.org/10.4310/HHA.2000.v2.n1.a9

#### Author

#### 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 $BGL(\mathbb{Z})$; 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