nLab seventh stable homotopy group of spheres

Idea

The seventh stable homotopy group of spheres (the seventh stable stem) is the cyclic group of order 240 (cf. Toda 1961 p. 186):

(1)$$\array{ \pi_7^s &\simeq& \mathbb{Z}/240 \\ [h_{\mathbb{O}}] &\leftrightarrow& [1] }$$

where the generator $[1] \in \mathbb{Z}/240$ is represented by (the suspensions of) the octonionic Hopf fibration $S^{15} \overset{h_{\mathbb{O}}}{\longrightarrow} S^{8}$ (cf. Prop. below).

Properties

Generator

Proposition

A generator of $\pi_7^s$, is represented by the suspension of the quaternionic Hopf fibration.

One would hope this could be cited directly from classical literature. But Toda 1962 makes the statement only for the 2-primary projection, Adams 1966 leaves it rather implicit, and also the review in Ravenel 2004 Thm. 1.13 still does not quite say it. Here is how to stitch this information together:

Proof

From Toda 1962 p. 186 we know that $\pi_7^s \simeq \mathbb{Z}_{/240}$. With this, Adams 1966 Thm. 1.6 implies that the J-homomorphism is surjective onto $\pi_7^s$. (Use here that, in Adams’s notation, $s = 2$, and then that the Bernoulli number $B_{2s} = B_4 = -1/30$ so that $B_{2s}/4s = -1/240$.) Finally, Adams 1966 §10 observes (stated more explicitly in Ravenel 2004 Thm. 1.13), that the image of the J-homomorphism in degree 7 is generated by the (suspensions $\sigma$ of) the octonionic Hopf fibration.

See also Randal-Williams 2025 p. 1991 (20), ftn 5.

Related concepts

• first stable homotopy group of spheres

• second stable homotopy group of spheres

• third stable homotopy group of spheres

References

The identification of the group as such is due to:

• Hirosi Toda; p. 186 of: Composition Methods in Homotopy Groups of Spheres, Annals of Mathematics Studies 49, Princeton University Press (1962) [doi:10.1515/9781400882625, jstor:j.ctt1bgzb5t]

That the suspension of the octonionic Hopf fibration is a generator (which Toda 1962 discusses only under 2-primary projection):

• John Adams; Thm. 1.6 in: On the groups $J(X)$ IV, Topology 5 1 (1966) 21–71 [doi:10.1016/0040-9383(66)90004-8, pdf]

• Doug Ravenel; Thm 1.1.13 in: Complex cobordism and stable homotopy groups of spheres, Academic Press Orland (1986) reprinted as: AMS Chelsea Publishing 347 (2004) [ISBN:978-0-8218-2967-7, webpage, pdf]

• Oscar Randal-Williams; p. 1991 (20) of: Monodromy and mapping class groups of 3-dimensional hypersurfaces, Mathematische Annalen 391 (2025) 1965–2003 [doi:10.1007/s00208-024-02951-4, arXiv:2308.06397]