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ISSN 1947-6221(e) ISSN 0065-9266(p)

The Goodwillie tower and the EHP sequence

Author: Mark Behrens
Journal: Memoirs of the AMS
MSC (2010): Primary 55Q40; Secondary 55Q15, 55Q25, 55S12
Posted: November 21, 2011
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Abstract: We study the interaction between the EHP sequence and the Goodwillie tower of the identity evaluated at spheres at the prime . 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 map, and the Goodwillie differentials to the 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 recompute the -primary unstable stems through the Toda range (up to the -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 stable homology of the Goodwillie layers of any functor from spaces to spaces.

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