As applications we show that the linearization map L : A(*) -> K(Z) induces the zero homomorphism in mod 2 spectrum cohomology in positive dimensions, the space level Hatcher-Waldhausen map hw : G/O -> Omega Wh^{DIFF}(*) does not admit a four-fold delooping, and there is a 2-complete spectrum map M : Wh^{DIFF}(*) \to Sigma g/o_{oplus} which is precisely 9-connected. Here g/o_{oplus} is a spectrum whose underlying space has the 2-complete homotopy type of G/O.

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John Rognes <rognes@math.uio.no>