Table of Stable Homotopy Groups
The stable homotopy groups πk are the product of cyclic groups of the infinite or prime power orders shown in the table. (For largely historical reasons, stable homotopy groups are usually given as products of cyclic groups of prime power order, while tables of unstable homotopy groups often give them as products of the smallest number of cyclic groups.) The main complexity is in the 2-, 3-, and 5-components: for p > 5, the p-components in the range of the table are accounted for by the J-homomorphism and are cyclic of order p if 2(p−1) divides k+1 and 0 otherwise (Fuks 2001). (The 2-components can be found in Kochman (1990), though there were some errors for k≥54 that were corrected by Kochman & Mahowald (1995), and the 3- and 5-components in Ravenel (2003).) The mod 8 behavior of the table comes from Bott periodicity via the J-homomorphism, whose image is underlined.
| n → | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|---|
| π0+nS | ∞ | 2 | 2 | 8⋅3 | ⋅ | ⋅ | 2 | 16⋅3⋅5 |
| π8+nS | 2⋅2 | 2⋅22 | 2⋅3 | 8⋅9⋅7 | ⋅ | 3 | 22 | 32⋅2⋅3⋅5 |
| π16+nS | 2⋅2 | 2⋅23 | 8⋅2 | 8⋅2⋅3⋅11 | 8⋅3 | 22 | 2⋅2 | 16⋅8⋅2⋅9⋅3⋅5⋅7⋅13 |
| π24+nS | 2⋅2 | 2⋅2 | 22⋅3 | 8⋅3 | 2 | 3 | 2⋅3 | 64⋅22⋅3⋅5⋅17 |
| π32+nS | 2⋅23 | 2⋅24 | 4⋅23 | 8⋅22⋅27⋅7⋅19 | 2⋅3 | 22⋅3 | 4⋅2⋅3⋅5 | 16⋅25⋅3⋅3⋅25⋅11 |
| π40+nS | 2⋅4⋅24⋅3 | 2⋅24 | 8⋅22⋅3 | 8⋅3⋅23 | 8 | 16⋅23⋅9⋅5 | 24⋅3 | 32⋅4⋅23⋅9⋅3⋅5⋅7⋅13 |
| π48+nS | 2⋅4⋅23 | 2⋅2⋅3 | 23⋅3 | 8⋅4⋅22⋅3 | 23⋅3 | 24 | 4⋅2 | 16⋅3⋅3⋅5⋅29 |
| π56+nS | 2⋅2 | 2⋅23 | 22 | 8⋅22⋅9⋅7⋅11⋅31 | 4 | ⋅ | 4⋅22⋅3 | 128⋅23⋅3⋅5⋅17 |
Read more about this topic: Homotopy Groups Of Spheres
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