A048648 Order of n-th stable homotopy group of spheres.

2, 2, 24, 1, 1, 2, 240, 4, 8, 6, 504, 1, 3, 4, 960, 4, 16, 16, 528, 24, 4, 4, 3144960, 4, 4, 12, 24, 2, 3, 6, 65280, 16, 32, 32, 114912, 6, 12, 120, 1267200, 384, 32, 96, 552, 8, 5760, 48, 12579840, 64, 12, 24, 384, 24, 16, 8, 20880, 2, 8, 4, 687456, 4, 1

Offset: 1

Author

Stephen A. Silver

Keywords

• nonn
• nice

Comments

Proved by Serre to be finite for all positive n.

The best current reference is Isaksen-Wang-Xu, Table 1. - Charles Rezk, Aug 22 2020

Examples

			Pi_1^S = Pi_4(S^3) = Z/2Z, so a(1) = |Z/2Z| = 2.
		

References

• D. B. Fuks, "Spheres, homotopy groups of the", Encyclopaedia of Mathematics, Vol. 8.
• S. O. Kochman, Stable homotopy groups of spheres. A computer-assisted approach. Lecture Notes in Mathematics, 1423. Springer-Verlag, Berlin, 1990. 330 pp. ISBN: 3-540-52468-1. [Math. Rev. 91j:55016]
• Douglas C. Ravenel, Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, 2003.
• Hirosi Toda, Composition Methods in Homotopy Groups of Spheres, Princeton University Press, 1962.

Links

• Andrey Zabolotskiy, Table of n, a(n) for n = 1..83 (terms 1..81 from Charles Rezk, terms 82..83 using data from Isaksen, Wang & Xu (2023))
• Kevin Hartnett, An Old Conjecture Falls, Making Spheres a Lot More Complicated, Quanta Magazine (2023)
• A. Hatcher, Stable Homotopy Groups of Spheres
• Daniel C. Isaksen, Guozhen Wang and Zhouli Xu, Stable homotopy groups of spheres: from dimension 0 to 90, Publications mathématiques de l'IHÉS, 137 (2023), 107-243; arXiv:2001.04511 [math.AT], 2020-2023.
• S. O. Kochman and M. E. Mahowald, On the computation of stable stems, The Cech Centennial (Boston, MA, 1993), 299-316, Contemp. Math., 181, Amer. Math. Soc., Providence, RI, 1995. [Math. Rev. 96j:55018]
• John W. Milnor, Differential Topology Forty-six Years Later, Notices Amer. Math. Soc. 58 (2011), 804-809.
• Robert Scharein's program sphere-link.c linked from the Linked Spheres page [has incorrect a(23) and a(29)-a(33)]
• Wikipedia, Homotopy groups of spheres

Crossrefs

Cf. A001676.

Formula

a(n) = |Pi_n^S| = |Pi_{k+n}(S^k)| for k > n+1.

Extensions

More terms from Alex Fink (finka(AT)math.ucalgary.ca), Aug 10 2006

a(23) and a(29)-a(33) corrected by Charles Rezk, Aug 22 2020

More terms from Charles Rezk, Aug 25 2020