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Theorem 1.1 immediately implies that 2π∗(S ) ⊂ Im(E), which is suggested by but does not follow from Selick’s exponent theorem. With Barratt, Cohen, Gray and Mahowald [B-C-G&], we gave simple proofs of weaker compression theorems, and deduced that the E2 term of the EHP spectral sequence π∗(S ) ⇒ π ∗ is a Z/2 module, and that 4 is the order of the identity on ΩW (n), where W (n) is the fiber of the double suspension E : S2n−1 −→ S. Our proof follows a program of Gray’s [Gr3, pp. 106–107], completed at odd primes by Harper [Ha]. Roughly, we show that a colifting β : ΩS −→ ΩS determined by sequence ΩS ΩE −→ ΩS ΩH2 −−→ ΩS 1+Ω 2(−1)q → ΩS is homotopic to (−1)q−1ΩE2 · H2 : ΩS −→ ΩS, by computing the H-deviation of the colifting β (cf. [Gr2]). We compute the H-deviation of β by dualizing Barratt and Toda’s theorem involving the Hopf invariant of a Toda bracket (Theorem 3.1),

The $H$-space squaring map on $\Omega\sp 3S\sp {4n+1}$ factors through the double suspension