Introduction All spaces and spectra in this book are implicitly localized at the prime 2. The purpose of this book is to describe the relationship between two machines for computing the 2-primary unstable homotopy groups of spheres. The first of these machines is the EHP sequence. James [Jam55] constructed fiber sequences Ω2S2n+1 P Sn E ΩSn+1 H ΩS2n+1. Applying π∗ to the resulting filtered space (0.0.1) · · · ΩnSn Ωn+1Sn+1 · · · QS0 yields the EHP spectral sequence (EHPSS) E1 m,t = πt+m+1S2m+1 πs. t By truncating the filtration (0.0.1) we also get truncated EHPSS’s Em,t(n) 1 = πt+m+1S2m+1, m n, 0, otherwise πt+n(Sn). The Curtis algorithm (see, for instance, [Rav86, Sec. 1.5]) gives an inductive tech- nique for using the various truncated EHPSS’s to compute the 2-primary unstable stems, provided one is able to compute differentials in the EHPSS. Unfortunately, as the differentials come from the P map, they are closely related to Whitehead products, and therefore are difficult to compute. The second computational machine is the Goodwillie tower of the identity. Let Top∗ denote the category of pointed topological spaces. Goodwillie calculus associates to a reduced finitary homotopy functor F : Top∗ Top∗ (i.e. a functor which preserves weak equivalences and filtered homotopy colim- its, and satisfies F (∗) ∗) a tower Pi(F ) of i-excisive approximations [Goo90], [Goo92], [Goo03]. D3(F ) D2(F ) D1(F ) · · · P3(F ) P2(F ) P1(F ) The fibers of the tower take the form of infinite loop spaces Di(F ) = Ω∞Di(F ), where Di(F )(X) (∂i(F ) X∧i)hΣ i vii
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