## 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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