I have spend some time with the geometric approach of framed cobordisms to compute homotopy classes, due to Pontryagin. He computed $\pi_{n+1}(S^n)$ and $\pi_{n+2}(S^n)$. After surveying the literature (not too deeply) I was under the impression that the computation of $\pi_{n+3}(S^n)\cong \mathbb{Z}/24\mathbb{Z}$ for $n\rightarrow \infty$ with similar methods is due to Rohlin in the following paper:
MR0046043 (13,674d) Reviewed
Rohlin, V. A.
Classification of mappings of an (n+3)-dimensional sphere into an n-dimensional one. (Russian)
Doklady Akad. Nauk SSSR (N.S.) 81, (1951). 19–22.
56.0X
It came a bit of a surprise to me that in the review of this paper on Mathscinet, Hilton states that the results in this paper are incorrect. Does the error only concern the unstable groups? Is it fair to cite this paper for the first computation of the third stable homotopy group of spheres, or should I cite papers by Barrat-Paechter, Massey-Whitehead and Serre? As I understand it these methods are much more algebraic and further removed from the applications that I have in mind.
Best Answer
The error is that Rokhlin claimed that $\pi_6(S^3)=\mathbb{Z}/6$, but Hilton, in his review, points out that the paper instead shows that $\pi_6(S^3)/\pi_5(S^2) = \mathbb{Z}/6$. The error lies in a prior calculation (reviewed here) that Rokhlin claimed showed $\eta^3=0$, but in fact this element is 2-torsion.
Rokhlin corrects his mistake and calculates the stable homotopy group $\pi_3^s$ in
The review states that this result "agrees with, and were anticipated by, results of Massey, G. W. Whitehead, Barratt, Paechter and Serre." Serre's CR note Sur les groupes d'Eilenberg-MacLane. C. R. Acad. Sci. Paris 234, (1952). 1243–1245 (BnF) found the correct $\pi_6(S^3)$ by homotopical means. Barratt and Paechter found an element of order 4 in $\pi_{3+k}(S^k)$ when $k\geq 2$.
The reference to Massey-Whitehead is a result presented at the 1951 Summer Meeting of the AMS at Minneapolis; all we have is the abstract in the Bulletin of the AMS 57, no. 6
If one wants to analyse 'dates received' to establish priority, then by all means.