The $2$-completed stable $20$-stem $\pi_{20}(S)_2$ is cyclic of order $8$.

Mimura and Toda (1963, Lemma 15.4) mr=157384 show the existence of a class $\bar\kappa_7 \in \pi_{27}(S^7)$ whose stable class $\bar\kappa$ generates $\pi_{20}(S)_2$. Equivalently, $\bar\kappa$ is detected by the class $g \in E_2^{4,24} = E_\infty^{4,24}$ in the mod $2$ Adams spectral sequence.

Note that these descriptions only determine $\bar\kappa$ up to an odd multiple, i.e., up to a factor in $\{1,3,5,7\}$.

Kochman (1990, page 104 and Lemma 5.3.8(e)) mr=1052407 finds a generator $C[20] = d^{12}(\nu^3 M_1^3 \bar M_2)$ for $\pi_{20}(S)_2$, which is an element of the four-fold Toda bracket $\langle \nu, \eta, 2, A[14] \rangle$, where I believe $A[14] = \kappa$.

Bauer (2008, equation (8.1)) mr=2508200 sets $\bar\kappa$ equal to the four-fold Toda bracket $\langle \kappa, 2, \eta, \nu \rangle$.

According to Kochman's Theorem 2.3.1(b) the indeterminacy of each of these four-fold brackets is $\nu \pi_{17}(S) \cong \mathbb{Z}/2$, generated by $4 \bar\kappa$. Hence these descriptions only determine $\bar\kappa$ up to a factor in $\{1,5\}$.

Is there a way to uniquely specify an element $\bar\kappa$ that generates $\pi_{20}(S)$ (modulo the odd torsion)?

  • $\begingroup$ Would it work better if we tried to detect it in the Adams-Novikov spectral sequence by $\beta_4$? $\endgroup$ – Dylan Wilson Jan 7 at 15:36
  • $\begingroup$ @DylanWilson: No, the Adams-Novikov coset $\{\beta_4\}$ has the same four elements as the Adams coset $\{g\}$. In other words, the homotopy classes detected by $\beta_4$ are the same as those detected by $g$. See e.g. Figure A3.2 in Ravenel's "Complex Cobordism ..." $\endgroup$ – John Rognes Jan 7 at 16:39

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