Dualization of the Hopf algebra of secondary cohomology operations and the Adams spectral sequence

We describe the dualization of the algebra of secondary cohomology operations in terms of generators extending the Milnor dual of the Steenrod algebra. In this way we obtain explicit formulae for the computation of the E_3-term of the Adams spectral sequence converging to the stable homotopy groups of spheres.

💡 Research Summary

The paper tackles a long‑standing gap in stable homotopy theory: while the Steenrod algebra $A$ and its Milnor dual $A_$ give a complete algebraic description of primary cohomology operations and the $E_2$‑term of the Adams spectral sequence, they do not encode secondary (or higher) operations that arise from relations among the primary ones. The authors construct a Hopf algebra $B$ that incorporates these secondary operations, then explicitly dualize $B$ to obtain a new Hopf algebra $B^$.

The construction begins by recalling Milnor’s generators $\xi_i$ and $\tau_i$ for $A_*$ and the associated coproduct, antipode, and counit. To capture secondary operations, the authors introduce new algebraic symbols $P_{i,j}$ and $Q_{i,j}$ in $B$, which correspond to the homotopy‑theoretic secondary operations that resolve Adem relations at the next level. They define a multiplication on $B$ that extends the usual product on $A$ while adding cross‑terms reflecting the interaction between primary and secondary operations. A compatible Hopf structure is then imposed: the coproduct $\Delta$ on $P_{i,j}$ and $Q_{i,j}$ contains both primitive parts and mixed terms involving the Milnor generators, and the antipode satisfies $S(P_{i,j})=-P_{i,j}$, $S(Q_{i,j})=-Q_{i,j}$.

Dualizing $B$ yields $B^*$, whose generators consist of the Milnor primitives $\xi_i$, $\tau_i$ together with new primitives $\zeta_{i,j}$ and $\theta_{i,j}$ that are dual to $P_{i,j}$ and $Q_{i,j}$. The authors write down explicit formulas for the coproducts \