Talk:Homotopy groups of spheres/table

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Experiment with table format

Contents

[edit] Clean and colored

π1 π2 π3 π4 π5 π6 π7 π8 π9 π10 π11 π12 π13 π14 π15
S1 Z 0 0 0 0 0 0 0 0 0 0 0 0 0 0
S2 0 Z Z Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2 Z22 Z12×Z2 Z84×Z22 Z22
S3 0 0 Z Z2 Z2 Z12 Z2 Z2 Z3 Z15 Z2 Z22 Z12×Z2 Z84×Z22 Z22
S4 0 0 0 Z Z2 Z2 Z×Z12 Z22 Z22 Z24×Z3 Z15 Z2 Z23 Z120×Z12×Z2 Z84×Z25
S5 0 0 0 0 Z Z2 Z2 Z24 Z2 Z2 Z2 Z30 Z2 Z23 Z72×Z2
S6 0 0 0 0 0 Z Z2 Z2 Z24 0 Z Z2 Z60 Z24×Z2 Z23
S7 0 0 0 0 0 0 Z Z2 Z2 Z24 0 0 Z2 Z120 Z23
S8 0 0 0 0 0 0 0 Z Z2 Z2 Z24 0 0 Z2 Z×Z120
S9 0 0 0 0 0 0 0 0 Z Z2 Z2 Z24 0 0 Z2

[edit] Complete piped

n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 n = 11 n = 12 n ≥ k + 2
k = 0 Z
k = 1 Z 2
k = 2 2 2 2
k = 3 2 12 Z + 12 24
k = 4 12 2 22 2 0
k = 5 2 2 22 2 Z 0
k = 6 2 3 24 + 3 2 2 2 2
k = 7 3 15 15 30 60 120 Z + 120 240
k = 8 15 2 2 2 8 + 6 23 24 23 22
k = 9 2 22 23 23 23 24 25 24 Z + 23 23
k = 10 22 12 + 2 40 + 4 + 2 + 32 18 + 8 18 + 8 24 + 2 82 + 2 + 32 24 + 2 12 + 2 22 + 3 2 + 3
k = 11 12 + 2 84 + 22 84 + 25 504 + 22 504 + 4 504 + 2 504 + 2 504 + 2 504 504 Z + 504 504
k = 12 84 + 22 22 26 23 240 0 0 0 4 + 3 2 22 See
below
k = 13 22 6 8 + 22 + 32 22 + 3 6 6 22 + 3 6 6 22 + 3 22 + 3
k = 14 6 30 840 + 9 + 22 22 + 3 12 + 2 24 + 4 60 + 48 + 8 16 + 4 16 + 2 16 + 2 24 + 16
k = 15 30 30 30 15 + 22 10 + 4 + 32 120 + 23 120 + 25 240 + 23 240 + 22 30 + 16 30 + 16
k = 16 30 22 + 3 23 + 32 22 504 + 22 24 27 24 30 + 16 2 2
k = 17 22 + 3 12 + 22 8 + 42 + 22 + 32 4 + 22 24 24 25 + 3 24 23 23 24
k = 18 12 + 22 12 + 22 40 + 4 + 25 + 32 24 + 22 8 + 22 + 32 24 + 2 82 + 42 + 9 8 + 2 + 3 8 + 22 + 3 8 + 4 + 2 32 + 30 + 42
k = 19 12 + 22 132 + 2 132 + 25 66 + 8 264 + 32 264 + 2 264 + 2 264 + 2 22 + 32 264 + 23 264 + 25


n = 13 n = 14 n = 15 n = 16 n = 17 n = 18 n = 19 n = 20 n ≥ k + 2
k = 12 2 0
k = 13 6 Z + 3 3
k = 14 16 + 2 8 + 2 4 + 2 22
k = 15 32 + 30 32 + 30 32 + 30 Z + 32 + 30 32 + 30
k = 16 2 8 + 6 23 24 23 22
k = 17 24 24 25 26 25 Z + 24 24
k = 18 82 + 2 82 + 2 83 + 2 + 3 83 + 2 + 3 82 + 2 8 + 6 8 + 22 8 + 2
k = 19 264 + 23 66 + 8 + 4 264 + 22 8 + 22 + 3 + 11 264 + 22 66 + 8 66 + 8 Z + 66 + 8 132 + 8

[edit] Complete and uncluttered


n = 2 n = 3 n = 4 n = 5 n = 6 n = 7 n = 8 n = 9 n = 10 n = 11 n = 12 n ≥ k + 2
k = 0 Z
k = 1 Z 2
k = 2 2 2 2
k = 3 2 12 Z + 12 24
k = 4 12 2 22 2 0
k = 5 2 2 22 2 Z 0
k = 6 2 3 24 + 3 2 2 2 2
k = 7 3 15 15 30 60 120 Z + 120 240
k = 8 15 2 2 2 8 + 6 23 24 23 22
k = 9 2 22 23 23 23 24 25 24 Z + 23 23
k = 10 22 12 + 2 40 + 4 + 2 + 32 18 + 8 18 + 8 24 + 2 82 + 2 + 32 24 + 2 12 + 2 22 + 3 2 + 3
k = 11 12 + 2 84 + 22 84 + 25 504 + 22 504 + 4 504 + 2 504 + 2 504 + 2 504 504 Z + 504 504
k = 12 84 + 22 22 26 23 240 0 0 0 4 + 3 2 22 See
below
k = 13 22 6 8 + 22 + 32 22 + 3 6 6 22 + 3 6 6 22 + 3 22 + 3
k = 14 6 30 840 + 9 + 22 22 + 3 12 + 2 24 + 4 60 + 48 + 8 16 + 4 16 + 2 16 + 2 24 + 16
k = 15 30 30 30 15 + 22 10 + 4 + 32 120 + 23 120 + 25 240 + 23 240 + 22 30 + 16 30 + 16
k = 16 30 22 + 3 23 + 32 22 504 + 22 24 27 24 30 + 16 2 2
k = 17 22 + 3 12 + 22 8 + 42 + 22 + 32 4 + 22 24 24 25 + 3 24 23 23 24
k = 18 12 + 22 12 + 22 40 + 4 + 25 + 32 24 + 22 8 + 22 + 32 24 + 2 82 + 42 + 9 8 + 2 + 3 8 + 22 + 3 8 + 4 + 2 32 + 30 + 42
k = 19 12 + 22 132 + 2 132 + 25 66 + 8 264 + 32 264 + 2 264 + 2 264 + 2 22 + 32 264 + 23 264 + 25


n = 13 n = 14 n = 15 n = 16 n = 17 n = 18 n = 19 n = 20 n ≥ k + 2
k = 12 2 0
k = 13 6 Z + 3 3
k = 14 16 + 2 8 + 2 4 + 2 22
k = 15 32 + 30 32 + 30 32 + 30 Z + 32 + 30 32 + 30
k = 16 2 8 + 6 23 24 23 22
k = 17 24 24 25 26 25 Z + 24 24
k = 18 82 + 2 82 + 2 83 + 2 + 3 83 + 2 + 3 82 + 2 8 + 6 8 + 22 8 + 2
k = 19 264 + 23 66 + 8 + 4 264 + 22 8 + 22 + 3 + 11 264 + 22 66 + 8 66 + 8 Z + 66 + 8 132 + 8

[edit] Current

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

[edit] Chopped and colored

π1 π2 π3 π4 π5 π6 π7
S1 Z 0 0 0 0 0 0
S2 0 Z Z Z/(2) Z/(2) Z/(12) Z/(2)
S3 0 0 Z Z/(2) Z/(2) Z/(12) Z/(2)
S4 0 0 0 Z Z/(2) Z/(2) Z×Z/(12)
S5 0 0 0 0 Z Z/(2) Z/(2)
S6 0 0 0 0 0 Z Z/(2)
S7 0 0 0 0 0 0 Z
πn+0 πn+1 πn+2 πn+3 πn+4 πn+5 πn+6
S1 Z 0 0 0 0 0 0
S2 Z Z Z/(2) Z/(2) Z/(12) Z/(2) Z/(2)
S3 Z Z/(2) Z/(2) Z/(12) Z/(2) Z/(2) Z/(3)
S4 Z Z/(2) Z/(2) Z×Z/(12) Z/(2)2 Z/(2)2 Z/(24)×Z/(3)
S5 Z Z/(2) Z/(2) Z/(24) Z/(2) Z/(2) Z/(2)
S6 Z Z/(2) Z/(2) Z/(24) 0 Z Z/(2)
S7 Z Z/(2) Z/(2) Z/(24) 0 0 Z/(2)

[edit] See also