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 obtained from the Goodwillie tower for . Much of the paper develops basic properties of this spectral sequence.
"Topological nonrealization results via the Goodwillie tower approach to iterated loopspace homology." Algebr. Geom. Topol. 8 (4) 2109 - 2129, 2008. https://doi.org/10.2140/agt.2008.8.2109