An infinite family in $\pi _\ast S^{0}$ derived from Mahowald’s $\eta _{j}$ family

Abstract:

Combining the relationship due to D. S. Kahn between ${ \cup _i}$ operations in homotopy and Steenrod operations in the ${E_2}$ term of the Adams spectral sequence with Mahowald’s result that ${h_1}{h_j}$ is a permanent cycle for $j \geqslant 4$, we show that ${h_2}h_j^2$ is also a permanent cycle for $j \geqslant 5$. This gives another infinite family of nonzero elements in the stable homotopy of spheres. Properties of the ${ \cup _i}$ homotopy operations further imply that these elements generate ${Z_2}$ direct summands.