The Curtis-Wellington spectral sequence through cohomology

TL;DR

This paper explores the Curtis-Wellington spectral sequence using cohomology, revealing a filtration linked to Dickson algebras and providing initial calculations that relate to the image of J.

Contribution

It introduces a cohomology-based approach to the Curtis-Wellington spectral sequence and identifies a new filtration structure with potential implications for the image of J.

Findings

Identified a width filtration with simple Dickson algebra quotients

Performed initial calculations of the spectral sequence towers

Conjectured the image of J is captured by the lowest filtration

Abstract

We study stable homotopy through unstable methods applied to its representing infinite loop space, as pioneered by Curtis and Wellington. Using cohomology instead of homology, we find a width filtration whose subquotients are simple quotients of Dickson algebras. We make initial calculations and determine towers in the resulting width spectral sequence. We also make calculations related to the image of $J$ and conjecture that it is captured exactly by the lowest filtration in the width spectral sequence.