# How well-defined is $\bar\kappa$ in the stable $20$-stem?

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)?

• Would it work better if we tried to detect it in the Adams-Novikov spectral sequence by $\beta_4$? – Dylan Wilson Jan 7 at 15:36
• @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 ..." – John Rognes Jan 7 at 16:39