Two-primary algebraic K-theory of pointed spaces, by John Rognes

We compute the mod 2 cohomology of Waldhausen's algebraic K-theory spectrum A(*) of the category of finite pointed spaces, as a module over the Steenrod algebra. This also computes the mod 2 cohomology of the smooth Whitehead spectrum of a point, denoted Wh^{DIFF}(*). Using an Adams spectral sequence we compute the 2-primary homotopy groups of these spectra in dimensions * <= 18, and up to extensions in dimensions 19 <= * <= 21.

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 <>