Algebraic \(K\)-theory of the two-adic integers
In [\textit{J. Rognes}, J. Pure Appl. Algebra 134, No. 3, 219-286 (1999; Zbl 0929.19003); see the paper above] the author calculated the mod two homotopy of the algebraic K-theory of the \(2\)-adic integers \(\widehat{\mathbb{Z}}_2\). In this paper the author uses this to determine the homotopy type and homotopy groups of the two-completed algebraic K-theory spectrum \(K(\widehat{\mathbb{Z}}_2)_2^\wedge\). For \(*>0\) the homotopy groups are given by \[ \pi_*(K(\widehat{\mathbb{Z}}_2)_2^\wedge)=\begin{cases} \widehat{\mathbb{Z}}_2\oplus\mathbb{Z}/2^{v_2(k)+3}& \text{for \(*=4k-1\)}\\ \mathbb{Z}/2^{v_2(k)+3}& \text{for \(*=4k\)}\\ \widehat{\mathbb{Z}}_2\oplus\mathbb{Z}/2& \text{for \(*=4k+1\)}\\ \mathbb{Z}/2& \text{for \(*=4k+2\)}\end{cases} \] where \(v_2(k)\) is the two-adic valuation of \(k\). These homotopy groups are the direct sum of the homotopy groups of \(\text{Im } J {\mathbb{C}}_2^\wedge\) (the complex image of \(J\)), i.e., the fiber of \(\mathbb{Z}\times BU@>{\psi^3-1}>>BU\), \(B \text{ Im } J {\mathbb{C}}_2^\wedge\) and \(BBU_2^\wedge\), but as opposed to the odd-primary case the homotopy type is more complicated. More precisely, \(K(\widehat{\mathbb{Z}}_2)_2^\wedge\) fits into fiber sequences built out of these spectra, but the sequences do not split. The extension problems are analyzed and identified, determining the infinite loop space structure of \(K(\widehat{\mathbb{Z}}_2)_2^\wedge\) up to homotopy equivalence.
scientific article; zbMATH DE number 1308293
scientific article