  We prove a strengthened version of a theorem of Lionel Schwartz that says
that certain modules over the Steenrod algebra cannot be the mod 2 cohomology
of a space.
  What is most interesting is our method, which replaces his iterated use of
the Eilenberg--Moore spectral sequence by a single use of the spectral sequence
converging to the mod 2 cohomology of Omega^nX obtained from the Goodwillie
tower for the suspension spectrum of Omega^nX. Much of the paper develops basic
properties of this spectral sequence.
