# Who computed the third stable homotopy group?

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)
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.

• Was it really Pontryagin who was the first to compute π_n(S^n)? – Dmitri Pavlov Jun 27 '17 at 11:19
• @DmitriPavlov: Good question. I guess not, probably it was Hopf in "Abbildungsklassen $n$-dimensionaler Mannigfaltigkeiten". – Thomas Rot Jun 27 '17 at 12:52
• And for the two-dimensional case it was probably Brouwer. – Thomas Rot Jun 27 '17 at 16:00

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

Rohlin, V. A. MR0052101
New results in the theory of four-dimensional manifolds. (Russian)

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.

• Note that Massey-Whitehead only found the order of $\pi_3^s$, not that fact it is cyclic. So Rokhlin it is! – David Roberts Jun 26 '17 at 22:54
• This clears up the history to me. I was most interested in the geometric viewpoint, and it does seem it is due to Rohlin. Thank you very much for your effort. – Thomas Rot Jun 26 '17 at 23:07

The mistake is corrected in [Rohlin, V. A. New results in the theory of four-dimensional manifolds. (Russian) Doklady Akad. Nauk SSSR (N.S.) 84, (1952). 221–224, MR0052101].

This and other matters are discussed in the monograph

[À la recherche de la topologie perdue. (French) [Remembrance of topology past] I. Du côté de chez Rohlin. II. Le côté de Casson. [I. Rokhlin's way. II. Casson's way] Edited by Lucien Guillou and Alexis Marin. Progress in Mathematics, 62. Birkhäuser Boston, Inc., Boston, MA, 1986, MR0900243]

where Rohlin's four papers are reproduced with comments. This book was also translated into Russian (I own a copy) but it seems not into English.

• French will be easier than Russian. Thank you for both references. – Thomas Rot Jun 26 '17 at 23:07
• @ThomasRot, "French will be easier than Russian." -- я не думаю. – Wlod AA Jun 26 '17 at 23:40
• @WlodAA "@ThomasRot, "French will be easier than Russian." -- я не думаю." Ваш пробег может варьироваться – David Roberts Jun 27 '17 at 0:09
• Мне нравится google translate. – Thomas Rot Jun 27 '17 at 8:24