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The Goodwillie Tower and the EHP Sequence
Mark Behrens, Massachusetts Institute of Technology, Cambridge, MA
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Memoirs of the American Mathematical Society
2012; 90 pp; softcover
Volume: 218
ISBN-10: 0-8218-6902-7
ISBN-13: 978-0-8218-6902-4
List Price: US$67
Individual Members: US$40.20
Institutional Members: US$53.60
Order Code: MEMO/218/1026
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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

  • Introduction
  • Dyer-Lashof operations and the identity functor
  • The Goodwillie tower of the EHP sequence
  • Goodwillie filtration and the \(P\) map
  • Goodwillie differentials and Hopf invariants
  • EHPSS differentials
  • Calculations in the \(2\)-primary Toda range
  • Appendix A. Transfinite spectral sequences associated to towers
  • Bibliography
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