Two-primary algebraic K-theory of pointed spaces
John Rognes
Department of Mathematics
University of Oslo
P.O. Box 1053, Blindern
Norway
rognes@math.uio.no
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.