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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.)

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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 $ap^*(h_1) = p_*(a)h_1 = h_jh_1$.

For more detail about Mahowald's maps, and some simplification in their construction, see my paper with David Hunter: [David Hunter and Nicholas Kuhn, Mahowaldean families of elements in stable homotopy groups revisted, Math. Proc. Camb. Phil Soc. 127 (1999), 237-251]. I think we did a good job in breaking the construction down into understandable bits. (Unfortunately our paper is just too old to be on the Arxiv.)

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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.

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