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The Hatcher-Waldhausen map is a spectrum map. (English) Zbl 0803.19002

We prove that the map \(G/O\to \Omega W h^{\text{Diff}}(*)\) which was constructed by Waldhausen and proved to be a rational equivalence by Bökstedt, is in fact an infinite loop map if we use a multiplicative infinite loop space structure on the target. Here \(G/O\) is the homotopy fiber of the \(j\)-homomorphism \(BO\to BG\), and \(Wh^{\text{Diff}} (*)\) is the differentiable Whitehead space of a point, which is a double delooping of the stable concordance (pseudoisotopy) space of a point.
As an application we investigate the obstruction to improving Bökstedt’s two-primary results on splitting the étale \(K\)-theory space \(JK(\mathbb{Z})\) off from \(K(\mathbb{Z})\), to the unlooped space or spectrum level. In particular, if the usual map \(Q(S^ 0)\to K(\mathbb{Z})\) from stable homotopy to the algebraic \(K\)-theory of the integers factors through the image of \(J\)-spectrum as a spectrum map, such a spectrum level splitting exists. Note that S. A. Mitchell has proved the analogous space level factorization.

MSC:

19D10 Algebraic \(K\)-theory of spaces
19D50 Computations of higher \(K\)-theory of rings
55P42 Stable homotopy theory, spectra
19L20 \(J\)-homomorphism, Adams operations
55S25 \(K\)-theory operations and generalized cohomology operations in algebraic topology
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References:

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