Comparison by n: the rule, Toda, the lecture
Each row is the group $\pi_n(S^2)=\pi_n(S^3)$ ($n\ge4$), its 3-primary part. A number is the rank of the 3-torsion.
- E²: the unstable $E_2$ for $S^3$ at $p=3$ without the tower over $\pi_3$, that is, the homology of the metalibrary; the colours are the $Q$-weight of Ext (method), $n\le150$.
- rapacious: what remains after the rapacious rule with colour $+1$. The two columns are two ways of filling the chains, “low” and “high” (differentials). The result is final up to $n=149$.
- Toda 2003: our reading of Toda's 2003 table ($n\le82$); Toda 1962: his book ($n\le22$).
The rapacious rule (“low”) agrees with Toda for all $n\le48$. These are exactly the stems $k=n-3<46$ that Toda determined in 1967. Disagreements for $n\le82$: $\pi_{49}$, $\pi_{50}$, $\pi_{51}$, $\pi_{59}$, $\pi_{60}$, $\pi_{63}$, $\pi_{64}$, $\pi_{66}$, $\pi_{67}$, $\pi_{68}$, $\pi_{69}$, $\pi_{70}$, $\pi_{73}$, $\pi_{74}$, $\pi_{75}$, $\pi_{76}$, $\pi_{77}$, $\pi_{79}$, $\pi_{80}$; in every case the rule leaves more classes than Toda. For $49\le n\le149$ the rule gives no zeros in either variant.
| n | E² | rapacious low | rapacious high | Toda 2003 | Toda 1962 | family | lecture | remark |
|---|---|---|---|---|---|---|---|---|
| 4 | 0 | 0 | 0 | — | 0 | |||
| 5 | 0 | 0 | 0 | — | 0 | |||
| 6 | 1 | 1 | 1 | 1 | 1 | Toda element | ||
| 7 | 0 | 0 | 0 | 0 | 0 | |||
| 8 | 0 | 0 | 0 | 0 | 0 | |||
| 9 | 1 | 1 | 1 | 1 | 1 | sandwich | ||
| 10 | 1 | 1 | 1 | 1 | 1 | Toda element | ||
| 11 | 0 | 0 | 0 | 0 | 0 | |||
| 12 | 0 | 0 | 0 | 0 | 0 | |||
| 13 | 1 | 1 | 1 | 1 | 1 | sandwich | ||
| 14 | 1 | 1 | 1 | 1 | 1 | Toda element | ||
| 15 | 0 | 0 | 0 | 0 | 0 | |||
| 16 | 1 | 1 | 1 | 1 | 1 | |||
| 17 | 1 | 1 | 1 | 1 | 1 | sandwich | ||
| 18 | 1 | 1 | 1 | 1 | 1 | Toda element | ||
| 19 | 1 | 1 | 1 | 1 | 1 | |||
| 20 | 1 | 1 | 1 | 1 | 1 | |||
| 21 | 1 | 1 | 1 | 1 | 1 | sandwich | ||
| 22 | 1 | 1 | 1 | 1 | 1 | Toda element | ||
| 23 | 1 | 0 | 0 | 0 | ||||
| 24 | 2 | 0 | 0 | 0 | ||||
| 25 | 2 | 1 | 1 | 1 | sandwich | |||
| 26 | 2 | 2 | 2 | 2 | Toda element | |||
| 27 | 2 | 1 | 1 | 1 | ||||
| 28 | 2 | 0 | 0 | 0 | ||||
| 29 | 3 | 2 | 2 | 2 | sandwich | |||
| 30 | 2 | 2 | 2 | 2 | Toda element | |||
| 31 | 1 | 0 | 0 | 0 | ||||
| 32 | 2 | 0 | 0 | 0 | ||||
| 33 | 3 | 1 | 1 | 1 | sandwich | |||
| 34 | 3 | 1 | 1 | 1 | Toda element | |||
| 35 | 3 | 1 | 1 | 1 | ||||
| 36 | 4 | 2 | 2 | 2 | ||||
| 37 | 4 | 2 | 2 | 2 | sandwich | |||
| 38 | 3 | 1 | 1 | 1 | Toda element | |||
| 39 | 3 | 1 | 1 | 1 | ||||
| 40 | 3 | 1 | 1 | 1 | ||||
| 41 | 3 | 1 | 2 | 1 | sandwich | |||
| 42 | 4 | 2 | 2 | 2 | Toda element | |||
| 43 | 4 | 1 | 0 | 1 | ||||
| 44 | 4 | 0 | 0 | 0 | no 3-torsion | |||
| 45 | 5 | 2 | 2 | 2 | sandwich | |||
| 46 | 5 | 3 | 3 | 3 | Toda element | |||
| 47 | 4 | 1 | 1 | 1 | ||||
| 48 | 4 | 0 | 0 | 0 | ||||
| 49 | 5 | 2 | 2 | 1 | sandwich | conjecture: present | rule ≠ Toda | |
| 50 | 5 | 3 | 3 | 1 | Toda element | conjecture: present | rule ≠ Toda | |
| 51 | 4 | 1 | 1 | 0 | conjecture: present | rule ≠ Toda; Toda: 0 for n ≥ 49 | ||
| 52 | 6 | 2 | 1 | 2 | conjecture: present | |||
| 53 | 7 | 3 | 2 | 3 | sandwich | conjecture: present | ||
| 54 | 6 | 2 | 2 | 2 | Toda element | conjecture: present | ||
| 55 | 5 | 1 | 1 | 1 | conjecture: present | |||
| 56 | 6 | 2 | 2 | 2 | conjecture: present | |||
| 57 | 6 | 2 | 2 | 2 | sandwich | conjecture: present | ||
| 58 | 6 | 2 | 2 | 2 | Toda element | conjecture: present | ||
| 59 | 6 | 2 | 2 | 1 | conjecture: present | rule ≠ Toda | ||
| 60 | 6 | 1 | 1 | 0 | conjecture: present | rule ≠ Toda; Toda: 0 for n ≥ 49 | ||
| 61 | 7 | 2 | 3 | 2 | sandwich | conjecture: present | ||
| 62 | 7 | 3 | 3 | 3 | Toda element | conjecture: present | ||
| 63 | 8 | 3 | 2 | 2 | conjecture: present | rule ≠ Toda | ||
| 64 | 8 | 2 | 2 | 1 | conjecture: present | rule ≠ Toda | ||
| 65 | 7 | 2 | 2 | 2 | sandwich | conjecture: present | ||
| 66 | 7 | 3 | 3 | 2 | Toda element | conjecture: present | rule ≠ Toda | |
| 67 | 7 | 2 | 2 | 0 | conjecture: present | rule ≠ Toda; Toda: 0 for n ≥ 49 | ||
| 68 | 8 | 2 | 2 | 1 | conjecture: present | rule ≠ Toda | ||
| 69 | 8 | 3 | 3 | 2 | sandwich | conjecture: present | rule ≠ Toda | |
| 70 | 7 | 2 | 2 | 1 | Toda element | conjecture: present | rule ≠ Toda | |
| 71 | 8 | 1 | 2 | 1 | predicted $(\mathbb Z/3)^2$; conjecture: present | Toda: 1, not 2 | ||
| 72 | 10 | 1 | 1 | 1 | conjecture: present | |||
| 73 | 11 | 3 | 2 | 1 | sandwich | conjecture: present | rule ≠ Toda | |
| 74 | 10 | 3 | 4 | 1 | Toda element | conjecture: present | rule ≠ Toda | |
| 75 | 10 | 2 | 2 | 0 | conjecture: present | rule ≠ Toda; Toda: 0 for n ≥ 49 | ||
| 76 | 12 | 4 | 3 | 0 | conjecture: present | rule ≠ Toda; Toda: 0 for n ≥ 49 | ||
| 77 | 11 | 3 | 4 | 1 | sandwich | conjecture: present | rule ≠ Toda | |
| 78 | 10 | 2 | 3 | 2 | Toda element | conjecture: present | ||
| 79 | 11 | 3 | 2 | 1 | conjecture: present | rule ≠ Toda | ||
| 80 | 11 | 2 | 1 | 0 | conjecture: present | rule ≠ Toda; Toda: 0 for n ≥ 49 | ||
| 81 | 12 | 2 | 3 | 2 | sandwich | conjecture: present | ||
| 82 | 14 | 3 | 3 | 3 | Toda element | conjecture: present | ||
| 83 | 16 | 3 | 3 | — | conjecture: present | |||
| 84 | 16 | 3 | 5 | — | conjecture: present | |||
| 85 | 14 | 3 | 4 | — | sandwich | conjecture: present | ||
| 86 | 15 | 5 | 4 | — | Toda element | conjecture: present | ||
| 87 | 15 | 3 | 3 | — | conjecture: present | |||
| 88 | 14 | 1 | 2 | — | conjecture: present | |||
| 89 | 14 | 4 | 4 | — | sandwich | conjecture: present | ||
| 90 | 14 | 4 | 3 | — | Toda element | conjecture: present | ||
| 91 | 16 | 2 | 2 | — | conjecture: present | |||
| 92 | 18 | 1 | 2 | — | conjecture: present | |||
| 93 | 20 | 4 | 5 | — | sandwich | conjecture: present | ||
| 94 | 20 | 6 | 7 | — | Toda element | conjecture: present | ||
| 95 | 18 | 4 | 3 | — | conjecture: present | |||
| 96 | 18 | 4 | 4 | — | conjecture: present | |||
| 97 | 18 | 4 | 6 | — | sandwich | conjecture: present | ||
| 98 | 18 | 4 | 5 | — | Toda element | conjecture: present | ||
| 99 | 20 | 6 | 5 | — | conjecture: present | |||
| 100 | 20 | 5 | 3 | — | conjecture: present | |||
| 101 | 20 | 3 | 4 | — | sandwich | conjecture: present | ||
| 102 | 23 | 5 | 5 | — | Toda element | conjecture: present | ||
| 103 | 25 | 5 | 4 | — | conjecture: present | |||
| 104 | 24 | 3 | 4 | — | conjecture: present | |||
| 105 | 23 | 4 | 4 | — | sandwich | conjecture: present | ||
| 106 | 24 | 5 | 5 | — | Toda element | conjecture: present | ||
| 107 | 25 | 3 | 4 | — | conjecture: present | |||
| 108 | 25 | 2 | 3 | — | conjecture: present | |||
| 109 | 25 | 5 | 6 | — | sandwich | conjecture: present | ||
| 110 | 26 | 6 | 5 | — | Toda element | conjecture: present | ||
| 111 | 27 | 5 | 4 | — | conjecture: present | |||
| 112 | 27 | 5 | 5 | — | conjecture: present | |||
| 113 | 28 | 5 | 6 | — | sandwich | conjecture: present | ||
| 114 | 29 | 4 | 5 | — | Toda element | conjecture: present | ||
| 115 | 29 | 4 | 4 | — | conjecture: present | |||
| 116 | 29 | 4 | 5 | — | conjecture: present | |||
| 117 | 29 | 4 | 5 | — | sandwich | conjecture: present | ||
| 118 | 29 | 5 | 5 | — | Toda element | conjecture: present | ||
| 119 | 28 | 4 | 5 | — | conjecture: present | |||
| 120 | 29 | 3 | 3 | — | conjecture: present | |||
| 121 | 30 | 4 | 4 | — | sandwich | conjecture: present | ||
| 122 | 30 | 6 | 7 | — | Toda element | conjecture: present | ||
| 123 | 31 | 5 | 5 | — | conjecture: present | |||
| 124 | 31 | 4 | 5 | — | conjecture: present | |||
| 125 | 31 | 6 | 6 | — | sandwich | conjecture: present | ||
| 126 | 30 | 6 | 6 | — | Toda element | conjecture: present | ||
| 127 | 29 | 3 | 4 | — | conjecture: present | |||
| 128 | 31 | 3 | 3 | — | conjecture: present | |||
| 129 | 33 | 6 | 6 | — | sandwich | conjecture: present | ||
| 130 | 34 | 5 | 5 | — | Toda element | conjecture: present | ||
| 131 | 36 | 5 | 6 | — | conjecture: present | |||
| 132 | 37 | 6 | 8 | — | conjecture: present | |||
| 133 | 38 | 5 | 6 | — | sandwich | conjecture: present | ||
| 134 | 38 | 6 | 6 | — | Toda element | conjecture: present | ||
| 135 | 36 | 5 | 4 | — | conjecture: present | |||
| 136 | 36 | 4 | 5 | — | conjecture: present | |||
| 137 | 37 | 5 | 7 | — | sandwich | conjecture: present | ||
| 138 | 38 | 6 | 5 | — | Toda element | conjecture: present | ||
| 139 | 39 | 5 | 4 | — | conjecture: present | |||
| 140 | 41 | 3 | 4 | — | conjecture: present | |||
| 141 | 44 | 7 | 9 | — | sandwich | conjecture: present | ||
| 142 | 44 | 9 | 10 | — | Toda element | conjecture: present | ||
| 143 | 43 | 6 | 5 | — | conjecture: present | |||
| 144 | 43 | 4 | 6 | — | conjecture: present | |||
| 145 | 44 | 6 | 8 | — | sandwich | conjecture: present | ||
| 146 | 44 | 8 | 7 | — | Toda element | conjecture: present | ||
| 147 | 43 | 6 | 5 | — | conjecture: present | |||
| 148 | 45 | 6 | 5 | — | conjecture: present | |||
| 149 | 47 | 8 | 5 | — | sandwich | conjecture: present | ||
| 150 | 48 | — | — | — | Toda element | conjecture: present |