Essential extensions, the nilpotent filtration and the Arone-Goodwillie tower

TL;DR

This paper uses spectral sequences and the nilpotent filtration to compare the first non-trivial layers of the mod 2 cohomology of connected spaces, introducing new unstable module theory tools.

Contribution

It introduces a generalized class of almost unstable modules and applies spectral sequence techniques to relate nilpotent filtrations in cohomology.

Findings

The first two layers of the nilpotent filtration are comparable for certain spaces.

A non-vanishing result for extension groups in unstable modules is established.

The work advances understanding of the structure of cohomology in algebraic topology.

Abstract

The spectral sequence associated to the Arone-Goodwillie tower for the n-fold loop space functor is used to show that the first two non-trivial layers of the nilpotent filtration of the reduced mod 2 cohomology of a (sufficiently connected) space with nilpotent cohomology are comparable. This relies upon the theory of unstable modules over the mod 2 Steenrod algebra, together with properties of a generalized class of almost unstable modules which is introduced here. An essential ingredient of the proof is a non-vanishing result for certain extension groups in the category of unstable modules localized away from nilpotents.