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By Y. Watatani

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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.

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