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.)



"Standard arguments" in Mahowald's eta_j paper
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.)


