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I am trying to proove that $\pi_{n+1}(S^n) \cong \mathbb{Z}_2$ using the Pontryagin-Thom construction and the special case $n=1$ of the $J$-homomorphism $$ J_1:\pi_1(SO(n))\rightarrow \pi_{n+1}(S^n). $$ but I can't find any reference of this specific case of the $J$-homomorphism being an isomorphism.

Where can I find a proof?

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  • $\begingroup$ You can check chapter 19 of Switzer's book Algebraic Topology, around page 480 in my version. $\endgroup$ – Tyrone Jun 24 '18 at 9:26

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