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

TL;DR

This paper introduces a novel approach using the Goodwillie tower spectral sequence to prove nonrealization results in algebraic topology, strengthening previous theorems about module structures over the Steenrod algebra.

Contribution

It replaces the traditional Eilenberg--Moore spectral sequence with the Goodwillie tower spectral sequence for more effective nonrealization proofs.

Findings

Established new nonrealization results for certain Steenrod algebra modules.

Developed foundational properties of the Goodwillie tower spectral sequence.

Provided a simplified method for proving nonrealization theorems.

Abstract

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.