Coloured metalibrary: a recomputation
RUEN

The coloured metalibrary for the 3-torsion of $\pi_n(S^2)$: our recomputation

We tried to reproduce the “rapacious spectral sequence” from the first lecture of the course “Coloured metalibraries. The mind of space” (R. Mikhailov, 2 September 2016): to compute the $E^2$ page of the coloured metalibrary and to apply to it the rule “eat yourself as fast as possible, constrained only by colour”.

Results

  1. The coloured $E^2$ page of the metalibrary is computed up to $\pi_{150}$. The colour turned out to be the number of Bocksteins (the $Q$-weight) in the unstable Ext; this was checked cell by cell against a direct computation in the lambda algebra up to $\pi_{75}$ and selectively up to $\pi_{85}$ (method and checks).
  2. The colours of the sandwiches and the Toda elements run 3 4 4 5 5 6 …, as stated in the lecture (classes).
  3. The rapacious rule with colour $+1$ agrees with Toda's table for all $n\le48$, including $\pi_{44}=0$. The boundary is exactly the stems $k<46$, the ones Toda determined in 1967.
  4. The rule reproduces the predictions of the lecture: $\pi_{48}=0$, and there is 3-torsion for all $49\le n\le149$; in $\pi_{71}$ there are two copies of $\mathbb Z/3$ under one of the orders of eating (comparison).
  5. Beyond $\pi_{48}$ the rule disagrees with Toda's 2003 table ($\pi_{49}$, $\pi_{50}$, $\pi_{51}$, $\pi_{59}$, $\pi_{60}$, $\pi_{63}$, $\pi_{64}$, $\pi_{66}$, $\pi_{67}$, $\pi_{68}$, $\pi_{69}$, $\pi_{70}$, $\pi_{73}$, $\pi_{74}$, $\pi_{75}$, $\pi_{76}$, $\pi_{77}$, $\pi_{79}$, $\pi_{80}$), leaving more classes in every case. Toda has one copy in $\pi_{71}$ and zeros in $\pi_{51},\pi_{60},\pi_{67},\pi_{75},\pi_{76},\pi_{80}$ (our reading). For $\pi_{49}$–$\pi_{51}$, $\pi_{59}$, $\pi_{60}$ this is proved in a way that can be checked without trusting our program (verifiability). Almost all of the extra classes are removed by colour-preserving $d_5$'s, but not by all the allowed ones (differentials).
  6. The rule does not determine the order of eating when a class can be both a target and a source; $\pi_{71}$, for instance, depends on it (differentials).

Sources