nLab
homotopy groups of spheres

Idea

The homotopy groups of spheres …

The stable homotopy groups of the sphere spectrum …

Tables

The first stable homotopy groups of the sphere spectrum $𝕊$

$k=$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	$\cdots$
${\pi }_k(𝕊)=$	$\mathbb{Z}$	${\mathbb{Z}}_2$	${\mathbb{Z}}_2$	${\mathbb{Z}}_{24}$	$0$	$0$	${\mathbb{Z}}_2$	${\mathbb{Z}}_{240}$	$({\mathbb{Z}}_2{)}^2$	$({\mathbb{Z}}_2{)}^3$	${\mathbb{Z}}_6$	${\mathbb{Z}}_{504}$	$0$	${\mathbb{Z}}_3$	$({\mathbb{Z}}_2{)}^2$	${\mathbb{Z}}_{480}\oplus {\mathbb{Z}}_2$

References

• Alex Writght, Homotopy groups of spheres: A very basic introduction (pdf)

• Doug Ravenal, Complex cobordism and stable homotopy groups of spheres (web)

• Wikipedia, Homotopy groups of spheres

• MO, Computational complexity of computing homotopy groups of spheres

For formalization in homotopy type theory see at

• UF-IAS-2012, HomotopyGroupsOfSpheres