In “A new infinite family in $ _{2}\pi^S_*$ ” (1976), Mark Mahowald constructs elements $ \eta_j \in \pi_{2^j}(S^0)$ for $ j \neq 2$ which come from permanent cycles in the Adams Spectral Sequence that are generated by $ h_1h_j \in Ext_A^{2, 2^j}(\mathbb{Z}_2, \mathbb{Z}_2)$ . Let $ H^*$ denote reduced mod-2 cohomology and for $ Y$ a CW-complex let $ Y_\ell$ denote the $ \ell$ -skeleton. Mahowald actually constructs a certain map from a stable complex $ f_j: X_j \to S^0$ where $ X_j$ has dimension $ 2^j-1$ , as well as a map $ g_j: S^{2^j} \to X_j$ , so that $ X_j/(X_j)_{2^j-2} \simeq S^{2^j-1}$ , the composition of $ g_j$ with the quotient $ X_j \to X_j/(X_j)_{2^j-2}$ is the Hopf map, $ H^{< 2^j – 2^{j-3}}(X_j) = 0$ , and $ Sq^{2^j}$ is nonzero in the mapping cone of $ f_j$ . Then he defines $ \eta_j$ be the composition $ f_j \circ g_j$ and concludes that by “standard arguments”, $ h_1h_j$ is a permanent cycle, etc.

What are these standard arguments? I know that a good way to show that maps are nonzero in the $ \pi^S_*$ is to show that they are detected by a primary or secondary stable cohomology operation. I know that secondary cohomology operations come from relations in the Steenrod algebra; I know that relations in the Steenrod algebra give rise to the second column in the Adams Spectral Sequence. Unfortunately, I can’t quite put things together to see why, for example $ f_j \circ g_j$ “represents $ h_1h_j$ ” (as Mahowald says)! In particular, I have no idea why the product on the ASS should be related to composition of maps of complexes.

(Why I am asking this question: I am not much of a homotopy theorist, but for some reason I had to read a later paper of Mahowald’s that was based on observations of this one in which he shows that certain Eilenberg-Maclane spectra are Thom spectra. This paper seemed interesting.)