Notes on sources

Here is what we noticed in other people's tables and programs while checking our data against them. Unlike the other pages, this one is about the 2-primary part of the homotopy groups of spheres.

N. Strickland's Mathematica package (Toda.m)

N. Strickland, a package on the unstable homotopy groups of spheres up to the 19th stem: strickland1.org/toda (files Toda.m, LambdaAlgebra.m, LambdaData.m, TodaChecks.m). The data come mainly from Toda's book Composition Methods in Homotopy Groups of Spheres (1962). The package stores the groups $\pi_{n+k}(S^n)$ modulo odd torsion, generators, orders and 962 relations among compositions, the suspension $E$, the Hopf invariant $H$ and the Whitehead product $P$. Its lambda algebra is mod 2 only.

We have no Mathematica, so we read the package's data with our own program and ran consistency checks on it: a port of the author's checks (TodaChecks.m) and our own, on degrees, orders and lambda representatives. An independent mod-2 lambda algebra agreed with LambdaData.m completely. The rest turned up as errors in the data:

1. Toda.m, line 1623: the rule Sigma[Sigma[sigmasecond]] -> 2 Sigma[sigma[8]] equates elements of different groups, $\pi_{15}(S^8)$ and $\pi_{16}(S^9)$; and the rule does fire.
2. Line 2126: o[Sigma[sigmaprime],eta[15],epsilon[16]] -> Sigma[zetaprime]: the right-hand side should be $\Sigma^2\zeta'$. This is the only violation of $E\circ P=0$ that the checks find.
3. Lines 1947–1960: a typo in a function name (AddRelations), because of which three relations are not added.
4. Line 1144: AddOrderRule is called with four arguments instead of five, so the element $w_n$ for odd $n$ does not get order 2.
5. Line 1450: SetOrder[o[nu[6],kappa[9]],4,"Toda, Theorem 12.7"]: the order of $\nu_6\circ\kappa_9$ is 2, not 4 (see below). Because of this the package gives $\pi_{23}(S^6)\cong\mathbb Z_4\oplus(\mathbb Z_2)^3$ instead of $(\mathbb Z_2)^4$.
6. Line 1563: SetOrder[o[omega[17],nu[32]],2,...]: a typo, it should be nu[33]; so the order of $\omega_{17}\circ\nu_{33}$ is not set, and a spurious summand $\mathbb Z$ appears in $\pi_{36}(S^{17})$.
7. The order of $\nu_7\circ\kappa_{10}$ is not set (it is 2: $\pi_{24}(S^7)\cong(\mathbb Z_2)^4$, Toda, Theorem 12.7), hence a spurious $\mathbb Z$ in $\pi_{24}(S^7)$.
8. Minor errors in the examples of the documentation toda.pdf.

Besides these, the checks show gaps: the images under $E$, $H$, $P$ in 33, 14 and 29 groups respectively do not reduce to generators. The author himself warns of such incompleteness on the package's page; no contradictions were found behind it.

$\pi_{23}(S^6)\cong(\mathbb Z_2)^4$: the order of $\nu_6\kappa_9$ is 2

Comparing the package's groups with Toda's table as reprinted by Ravenel (Complex Cobordism and Stable Homotopy Groups of Spheres, 3rd ed., Table A3.4, p. 390) gave 377 matches out of 380 cells ($2\le n\le21$, $k\le19$). All three discrepancies are errors of the package (items 5–7 above). On $\pi_{23}(S^6)$:

• Toda's table in Ravenel and its copy in the PKU table: $(\mathbb Z_2)^4$ (both are reprints of Toda's book);
• Miyauchi, Mukai, Ohara, arXiv:1705.00273, §3, take an extension of $\nu_6\kappa_9$ to the Moore space $M^{24}$, which exists only if $2\nu_6\kappa_9=0$;
• arXiv:2107.00154, proof of Prop. 3.4: $\{P(E\theta)+\nu_6\kappa_9,\ \eta_6\mu_7\sigma_{16}\}\cong(\mathbb Z_2)^2$; since $P(E\theta)$ has order 2 (the package itself has this too, line 1438), $2\nu_6\kappa_9=0$;
• the same from the relations: $2\kappa_7=\bar\nu_7\nu_{15}^2$ and $\nu_6\bar\nu_9=P(\nu_{13}^2)$ give $2\nu_6\kappa_9=P(\nu_{13}^4)=0$, since $\pi^s_{12}=0$ (our derivation).

This is stated explicitly by Y. Hirato, H. Kachi, J. Mukai, 21-st and 22-nd homotopy groups of the n-th rotation group, J. Fac. Sci. Shinshu Univ. 41 (2006), p. 12 (open access): $\pi^6_{23}=\{P(\Sigma\theta),\nu_6\kappa_9,\bar\mu_6,\eta_6\mu_7\sigma_{16}\}\cong(\mathbb Z_2)^4$. It is the neighbouring $\nu_5\kappa_8$ that has order 4 (ibid., p. 11: $2\nu_5\kappa_8\ne0$), which is probably where the error comes from. We have not found the text of Theorem 12.7 of Toda's book itself in open access.