Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology

Abstract

We prove a strengthened version of a theorem of Lionel Schwartz [Invent. Math. 134 (1998) 211–227] 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 $H^{\ast }\left({\Omega }^{n}X;\mathbb{Z}∕2\right)$ obtained from the Goodwillie tower for ${\Sigma }^{\infty }{\Omega }^{n}X$. Much of the paper develops basic properties of this spectral sequence.