Two-primary algebraic K-theory of pointed spaces

We compute the mod 2 cohomology of Waldhausenʹs 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 WhDiff(∗). 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→ΩWhDiff(∗) does not admit a four-fold delooping, and there is a 2-complete spectrum map M:WhDiff(∗)→Σg/o⊕ which is precisely 9-connected. Here g/o⊕ is a spectrum whose underlying space has the 2-complete homotopy type of G/O.