Timeline for The homotopy class of the loop $(S^1,\ast)\to (\Omega^2(S^2),\operatorname{id}_{S^2})$ that rotates the sphere

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May 29, 2021 at 21:29	comment	added	Connor Malin		The argument I had in mind only worked for the stabilization, i.e. $\pi_1(\Omega^3 S^3)$. In that case, it is just factorizing the second Stiefel-Whitney class $[S^2,BSO(2)] \rightarrow [S^2,B \Omega^3 S^3] \rightarrow H^2(S^2,\mathbb{Z}/2)$ and noticing that this is isomorphic to a factorization $\mathbb{Z} \rightarrow \mathbb{Z}/2 \rightarrow \mathbb{Z}/2$ where the entire composition is surjective.
May 29, 2021 at 21:04	history	edited	LSpice	CC BY-SA 4.0	`\operatorname`
May 28, 2021 at 14:59	comment	added	Chris Schommer-Pries		This is not too hard to see from the point of view of the Pontryagin-Thom construction. You adjoin the map into a map $S^3 \to S^2$, and then look at the inverse image of a generic point (not the base point or the fixed points of the rotation). You also keep track of the framing from $\mathbb{R}^2$ at that point. The inverse image is a framed unknot whose framing "twists" around once, and so has self linking number (i.e. Hopf invariant) equal to one.
May 27, 2021 at 16:17	comment	added	Connor Malin		In this case, if you already know $\pi_3(S^2) =\mathbb{Z}$ to characterize the image of the J-homomorphism really just requires knowledge of the Euler class of a vector bundle and of a spherical fibration. These are pretty easy to define.
May 27, 2021 at 14:59	history	asked	J.K.T.	CC BY-SA 4.0