# Reference for an automorphism in a paper of Toda

In Selick's very pretty paper "Odd primary torsion in $\pi_*(S^3)$" he makes use of an automorphism which was established by Toda in his paper "On the double suspension $E^2$".

Unfortunately, Selick just refers to [10], but the paper is about 40 pages and dense reading; after skimming it several times, I fear that the automorphism is there either implicitly, or with different notation.

Can someone provide a more precise reference? A page or theorem number would be ideal.

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If so, he does not make use of an "automorphism". What he wanted to say is the difference between the map $$(H'\circ\Omega\gamma)_* : H_*(\Omega S^{2p+1}\{p\}) \longrightarrow H_*(\Omega S^{2p+1}_{(p)})$$ and the map $(\Omega i)_*$ is given by a nonzero multiple of an element of $\mathbb{F}_p$ on the generator $y_{2p}$.
You don't need to read Toda's calculations. Generators of these homology Hopf algebras are given in Selick's paper. All you need to do is the image of $y_{2p}$ is nontrivial. This can be done by using the behavior of the mod $p$ Hopf invariant $H : \Omega S^3 \to \Omega S^{2p+1}$ on homology.
By definition, $H$ is designed to be an isomorphism on homology in degree $2p$.