Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$?

Question

Who proved that $\pi_{n+1}(S^n) \cong \mathbb{Z}/2\mathbb{Z}$ for $n\geq 3$? I can't find any reference about who did it first.

Answer 1

You can use the Freudenthal’s Suspension theorem to get that:

$\pi_{n+1}(\mathbb{S}^n)\cong\pi_n(\mathbb{S}^{n-1})$

And so you must only prove that $\pi_4(\mathbb{S}^3)\cong \frac{\mathbb{Z}}{2\mathbb{Z}}$

Answer 2

It goes back to 1938:

H. Freudenthal, Über die Klassen der Sphärenabbildungen I. Große Dimensionen, Comp. Math. 5 (1938), 299 - 314

http://www.numdam.org/item?id=CM_1938__5__299_0

Freudenthal proves $\pi_4(S^3) = \mathbb{Z}_2$ and the suspension theorem. He introduces the notation $(d^e) = \pi_e(S^d)$ and explicitly states "$(d^{d+1})$ besteht für $d \ge 3$ aus genau zwei Elementen" - in English "$(d^{d+1})$ consists for $d \ge 3$ of exactly two elements".

See also http://math.mit.edu/~guozhen/homotopy%20groups.pdf.

By the way, in his paper Freudenthal states modestly: "Our results are - in contrast to that what else has come to be known - of very general nature. The special conclusions that one can draw immediately from them are not very numerous.

Another remark concerning the year of publication: Some sources say 1937, other 1938. The paper belongs to volume 5 of Comp. Math. which definitely contains papers from 1938 (e.g. H. Hopf, Eine Charakterisierung der Bettischen Gruppen von Polyedern durch stetige Abbildungen, received February 8, 1938). Freudenthal's paper was received September 4, 1937. I guess volume 5 has various issues ranging from 1937 to 1938.

Answer 3

I believe Hopf first calculated $\pi_3S^2\cong \mathbb{Z}$ in his paper

H. Hopf: Über die Abbildungen der dreidimensionalen Sphäre auf die
Kugelfläche, Math. Ann., 104 (1931), 637-665.

The calculation was performed by computing what is now know as the Hopf invariant of the eponymous Hopf map $\eta:S^3\rightarrow S^2$, which he defined in the paper. My German is not good enough for me to be familiar with his exact statement, however, so it may be that he simply showed that $\pi_3S^3$ contained a $\mathbb{Z}$ summand, although I believe that the full result was known to him.

In any case the result was revisited by Hurewicz in 1935

W. Hurewicz: Beiträge zur Topologie der Deformationen, Proc. Akad. Wetensch. Amsterdam; I: Höherdimensionalen Homotopiegruppen, 38(1935), 112-119

in a paper which used the long exact sequence of the fibration $S^1\rightarrow S^3\xrightarrow{\eta}S^2$ to conclude that $\pi_3S^2\cong\pi_3S^3\cong\mathbb{Z}$.

Later on, in 1937, Freudenthal introduced his famous suspension theorem

H. Freudenthal: Über die Klassen von Sphärenabbildungen, Comp. Math., 5 (1937), 299-314

and obtained enough information to show that the suspension $\Sigma:\pi_3S^2\rightarrow \pi_4S^3$ was surjective with kernel consisting of those maps of even Hopf invariant. Thus he concluded that $\pi_4S^3\cong\mathbb{Z}_2$.

Other results of the same paper showed him that that the suspension $\Sigma:\pi_{n+1}S^n\rightarrow\pi_{n+2}S^{n+1}$ was an isomorphism for $n\geq 3$, and he concluded the result

$$\pi_{n+1}S^n\cong\mathbb{Z}_2,\qquad n\geq 3.$$