The Goodwillie tower and the EHP sequence

TL;DR

This paper explores the relationship between the EHP spectral sequence and the Goodwillie tower of the identity functor at spheres, providing new insights into unstable homotopy groups and spectral sequence differentials at the prime 2.

Contribution

It establishes a connection between the Goodwillie filtration and the P map, relates differentials in spectral sequences, and re-computes unstable stems up to the 19-stem using these methods.

Findings

Linked Goodwillie differentials to EHP spectral sequence differentials

Recomputed 2-primary unstable stems up to the 19-stem

Introduced Dyer-Lashof-like operations on homology of Goodwillie layers

Abstract

We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime 2. Both give rise to spectral sequences (the EHP spectral sequence and the Goodwillie spectral sequence, respectively) which compute the unstable homotopy groups of spheres. We relate the Goodwillie filtration to the P map, and the Goodwillie differentials to the H map. Furthermore, we study an iterated Atiyah-Hirzebruch spectral sequence approach to the homotopy of the layers of the Goodwillie tower of the identity on spheres. We show that differentials in these spectral sequences give rise to differentials in the EHP spectral sequence. We use our theory to re-compute the 2-primary unstable stems through the Toda range (up to the 19-stem). We also study the homological behavior of the interaction between the EHP sequence and the Goodwillie tower of the identity. This homological analysis involves the introduction of Dyer-Lashof-like operations associated to M. Ching's operad structure on the derivatives of the identity. These operations act on the mod 2 stable homology of the Goodwillie layers of any functor from spaces to spaces.