Homotopy Groups of Spheres - Table of Homotopy Groups

Table of Homotopy Groups

Tables of homotopy groups of spheres are most conveniently organized by showing πn+k(Sn).

The following table shows many of the groups πn+k(Sn). (These tables are based on the table of homotopy groups of spheres in Toda (1962).) The stable homotopy groups are highlighted in blue, the unstable ones in red. Each homotopy group is the product of the cyclic groups of the orders given in the table, using the following conventions:

  • The entry "⋅" denotes the trivial group.
  • Where the entry is an integer, m, the homotopy group is the cyclic group of that order (generally written Zm).
  • Where the entry is ∞, the homotopy group is the infinite cyclic group, Z.
  • Where entry is a product, the homotopy group is the cartesian product (equivalently, direct sum) of the cyclic groups of those orders. Powers indicate repeated products. (Note that when a and b have no common factor, Za×Zb is isomorphic to Zab.)

Example: π19(S10) = π9+10(S10) = Z×Z2×Z2×Z2, which is denoted by ∞⋅23 in the table.

Sn → S0 S1 S2 S3 S4 S5 S6 S7 S8 S9 S10 S11 S12 S≥13
π<n(Sn) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
π0+n(Sn) 2 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞
π1+n(Sn) ⋅ ⋅ ∞ 2 2 2 2 2 2 2 2 2 2 2
π2+n(Sn) ⋅ ⋅ 2 2 2 2 2 2 2 2 2 2 2 2
π3+n(Sn) ⋅ ⋅ 2 12 ∞⋅12 24 24 24 24 24 24 24 24 24
π4+n(Sn) ⋅ ⋅ 12 2 22 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
π5+n(Sn) ⋅ ⋅ 2 2 22 2 ∞ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
π6+n(Sn) ⋅ ⋅ 2 3 24⋅3 2 2 2 2 2 2 2 2 2
π7+n(Sn) ⋅ ⋅ 3 15 15 30 60 120 ∞⋅120 240 240 240 240 240
π8+n(Sn) ⋅ ⋅ 15 2 2 2 24⋅2 23 24 23 22 22 22 22
π9+n(Sn) ⋅ ⋅ 2 22 23 23 23 24 25 24 ∞⋅23 23 23 23
π10+n(Sn) ⋅ ⋅ 22 12⋅2 120⋅12⋅2 72⋅2 72⋅2 24⋅2 242⋅2 24⋅2 12⋅2 6⋅2 6 6
π11+n(Sn) ⋅ ⋅ 12⋅2 84⋅22 84⋅25 504⋅22 504⋅4 504⋅2 504⋅2 504⋅2 504 504 ∞⋅504 504
π12+n(Sn) ⋅ ⋅ 84⋅22 22 26 23 240 ⋅ ⋅ ⋅ 12 2 22 See
below
π13+n(Sn) ⋅ ⋅ 22 6 24⋅6⋅2 6⋅2 6 6 6⋅2 6 6 6⋅2 6⋅2
π14+n(Sn) ⋅ ⋅ 6 30 2520⋅6⋅2 6⋅2 12⋅2 24⋅4 240⋅24⋅4 16⋅4 16⋅2 16⋅2 48⋅4⋅2
π15+n(Sn) ⋅ ⋅ 30 30 30 30⋅2 60⋅6 120⋅23 120⋅25 240⋅23 240⋅22 240⋅2 240⋅2
π16+n(Sn) ⋅ ⋅ 30 6⋅2 62⋅2 22 504⋅22 24 27 24 240⋅2 2 2
π17+n(Sn) ⋅ ⋅ 6⋅2 12⋅22 24⋅12⋅4⋅22 4⋅22 24 24 6⋅24 24 23 23 24
π18+n(Sn) ⋅ ⋅ 12⋅22 12⋅22 120⋅12⋅25 24⋅22 24⋅6⋅2 24⋅2 504⋅24⋅2 24⋅2 24⋅22 8⋅4⋅2 480⋅42⋅2
π19+n(Sn) ⋅ ⋅ 12⋅22 132⋅2 132⋅25 264⋅2 1056⋅8 264⋅2 264⋅2 264⋅2 264⋅6 264⋅23 264⋅25
Sn → S13 S14 S15 S16 S17 S18 S19 S20 S≥21
π12+n(Sn) 2 ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
π13+n(Sn) 6 ∞⋅3 3 3 3 3 3 3 3
π14+n(Sn) 16⋅2 8⋅2 4⋅2 22 22 22 22 22 22
π15+n(Sn) 480⋅2 480⋅2 480⋅2 ∞⋅480⋅2 480⋅2 480⋅2 480⋅2 480⋅2 480⋅2
π16+n(Sn) 2 24⋅2 23 24 23 22 22 22 22
π17+n(Sn) 24 24 25 26 25 ∞⋅24 24 24 24
π18+n(Sn) 82⋅2 82⋅2 82⋅2 24⋅82⋅2 82⋅2 8⋅4⋅2 8⋅22 8⋅2 8⋅2
π19+n(Sn) 264⋅23 264⋅4⋅2 264⋅22 264⋅22 264⋅22 264⋅2 264⋅2 ∞⋅264⋅2 264⋅2

Read more about this topic:  Homotopy Groups Of Spheres

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