# “Standard arguments” in Mahowald's eta_j paper

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.)

The short answer is that composition in Ext does correspond to composition of the maps, if nothing intervenes. In the case in hand, if $p$ is the projection of $X_j$ onto its top cell, then $f_j$ is represented by an element $a \in Ext^1$ such that $p_*(a) = h_j$, and $g_j$ is represented by $p^*(h_1)$, so that $f_jg_j$ is represented by $ap^*(h_1) = p_*(a)h_1 = h_jh_1$.
Since the paper actually refers to a secondary operation associated with the Adem relation $$Sq^{2^i+1}Sq^1+Sq^2Sq^{2^i}+Sq^4Sq^{2^i-2}+Sq^{2^i}Sq^2=0$$ then a standard argument to show that the composition $S^{2^i}\stackrel{g_i}\to X_i\stackrel{f_i}\to S^0$ is essential is to proceed and prove by contradiction. If the composition is null then one may look at the cohomology of the double mapping cone $S^0\cup_{\overline{f_i}}C(X_i\cup e^{2^i+1})$ where existence of a map $\overline{f_i}:C_{g_i}\to S^0$ follows from the assumption that $f_i\circ g_i$ is null. Now, evaluation of the above Adem relation on the $0$-dimensional class of in the cohomology of the double mapping cone gives the desired contradiction. This is a standard argument prior to the use of Adams Spectral Sequence arguments.