By Mark Behrens

The writer reports the interplay among the EHP series and the Goodwillie tower of the id evaluated at spheres on the major $2$. either provide upward thrust to spectral sequences (the EHP spectral series and the Goodwillie spectral series, respectively) which compute the volatile homotopy teams of spheres. He relates the Goodwillie filtration to the $P$ map, and the Goodwillie differentials to the $H$ map. in addition, he stories an iterated Atiyah-Hirzebruch spectral series method of the homotopy of the layers of the Goodwillie tower of the identification on spheres. He exhibits that differentials in those spectral sequences supply upward push to differentials within the EHP spectral series. He makes use of his concept to recompute the $2$-primary volatile stems during the Toda variety (up to the $19$-stem). He additionally reports the homological habit of the interplay among the EHP series and the Goodwillie tower of the identification. This homological research includes the advent of Dyer-Lashof-like operations linked to M. Ching's operad constitution at the derivatives of the id. those operations act at the mod $2$ reliable homology of the Goodwillie layers of any functor from areas to areas

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**Extra resources for The Goodwillie tower and the EHP sequence**

**Sample text**

Proof. 1 to the TGSS’s of the ﬁber sequence H P E Ω2 S m+1 −→ Ω2 S 2m+1 − → Sm − → ΩS m+1 . By assumption, α is a permanent cycle in the TGSS for S m+1 . 1, we are in Case (5). Thus there exists a lift α ∈ πt+m+1 (S m+1 ) of α so that m dS (α) = β[J, m] and either β[J] detects H(α) or it is the target of a longer diﬀerential in the TGSS for S 2m+1 . However, any longer diﬀerentials in the spectral sequence have source in a zero group. We conclude that β[J, m] ∈ GHI(α). 4. 5. 1 to give a plethora of dr -diﬀerentials throughout the GSS.

STABLE HOPF INVARIANTS AND METASTABLE HOMOTOPY 35 For 0 = α ∈ πts we deﬁne the generalized Hopf invariant to be the coset GHI(α) ⊂ πt+|J| (S J +m ) of elements which detect α in the TEHPSS. If γ[J, m] ∈ GHI(α) then α is born on S m+1 , with Hopf invariant which is detected by γ[J] in the TGSS for S 2m+1 . 2. Stable Hopf invariants and metastable homotopy For X ∈ Top∗ , let ∧2 JH : QX → QXhΣ 2 denote the James-Hopf map. It is adjoint to the map ∧2 Σ∞ QX → Σ∞ XhΣ 2 coming from the Snaith splitting.

Proof. 1 to the TGSS’s of the ﬁber sequence H P E Ω2 S m+1 −→ Ω2 S 2m+1 − → Sm − → ΩS m+1 . By assumption, α is a permanent cycle in the TGSS for S m+1 . 1, we are in Case (5). Thus there exists a lift α ∈ πt+m+1 (S m+1 ) of α so that m dS (α) = β[J, m] and either β[J] detects H(α) or it is the target of a longer diﬀerential in the TGSS for S 2m+1 . However, any longer diﬀerentials in the spectral sequence have source in a zero group. We conclude that β[J, m] ∈ GHI(α). 4. 5. 1 to give a plethora of dr -diﬀerentials throughout the GSS.