The 𝐻-space squaring map on Ω³𝑆⁴ⁿ⁺¹ factors through the double suspension

Richter, William

doi:10.1090/S0002-9939-1995-1273520-6

The $H$-space squaring map on $\Omega ^ 3S^ {4n+1}$ factors through the double suspension
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Proc. Amer. Math. Soc. 123 (1995), 3889-3900 Request permission

Abstract:

We compute the first EHP spectral sequence differential followed by the double suspension. We show that $2{\pi _ \ast }({S^{4n + 1}}) \subset \operatorname {Im} ({E^2})$, which refines the exponent for ${\pi _ \ast }({S^{2n + 1}})$ of James and Selick. The proof follows an odd primary program of Gray and Harper, and uses Barratt’s theory of unsuspended Hopf invariants and Boardman and Steer’s geometric Hopf invariants.

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