# The Goodwillie Tower and the EHP Sequence

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*Mark Behrens*

The author studies 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. He relates the Goodwillie filtration to the
\(P\) map, and the Goodwillie differentials to the \(H\) map.
Furthermore, he studies an iterated Atiyah-Hirzebruch spectral sequence
approach to the homotopy of the layers of the Goodwillie tower of the identity
on spheres. He shows that differentials in these spectral sequences give rise
to differentials in the EHP spectral sequence. He uses his theory to recompute
the \(2\)-primary unstable stems through the Toda range (up to the
\(19\)-stem). He also studies 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.

#### Table of Contents

# Table of Contents

## The Goodwillie Tower and the EHP Sequence

- Introduction vii8 free
- Chapter 1. Dyer-Lashof operations and the identity functor 114 free
- Chapter 2. The Goodwillie tower of the EHP sequence 1326
- Chapter 3. Goodwillie filtration and the P map 2538
- Chapter 4. Goodwillie differentials and Hopf invariants 3346
- 4.1. Hopf invariants and the transfinite EHPSS 3346
- 4.2. Stable Hopf invariants and metastable homotopy 3548
- 4.3. Goodwillie d1 differentials and stable Hopf invariants 3649
- 4.4. Higher Goodwillie differentials and unstable Hopf invariants 3750
- 4.5. Propagating differentials with the P and E maps 3851
- 4.6. Calculus form of the Whitehead conjecture 4053
- 4.7. Exotic Goodwillie differentials 4255

- Chapter 5. EHPSS differentials 4558
- Chapter 6. Calculations in the 2-primary Toda range 5366
- 6.1. AHSS calculations 5366
- 6.2. Calculation of the GSS for S1 5568
- 6.3. GSS calculations 5568
- 6.4. Calculation of the EHPSS 5568
- 6.5. Tables of computations 5669
- 6.5.1. The AHSS for k(L(1)) 5770
- 6.5.2. The AHSS for k(L(2)) 5972
- 6.5.3. The AHSS for k(L(3)) 6174
- 6.5.4. The EHPSS 6174
- 6.5.5. The GSS for n+1(S1) 6477
- 6.5.6. The GSS for n+2(S2) 6679
- 6.5.7. The GSS for n+3(S3) 6881
- 6.5.8. The GSS for n+4(S4) 7083
- 6.5.9. The GSS for n+5(S5) 7285
- 6.5.10. The GSS for n+6(S6) 7487

- Appendix A. Transfinite spectral sequences associated to towers 7790
- Bibliography 89102