An infinite family in 𝜋_{∗}𝑆⁰ derived from Mahowald’s 𝜂ⱼ family

Abstract

Combining the relationship due to D. S. Kahn between ∪ i { \cup _i} operations in homotopy and Steenrod operations in the E 2 {E_2} term of the Adams spectral sequence with Mahowald’s result that h 1 h j {h_1}{h_j} is a permanent cycle for j ⩾ 4 j \geqslant 4 , we show that h 2 h j 2 {h_2}h_j^2 is also a permanent cycle for j ⩾ 5 j \geqslant 5 . This gives another infinite family of nonzero elements in the stable homotopy of spheres. Properties of the ∪ i { \cup _i} homotopy operations further imply that these elements generate Z 2 {Z_2} direct summands.