The algebra of tertiary cohomology operations

The Adams spectral sequence computes homotopy classes of maps between spectra starting from cohomological information. The $E_2$ term is given by Ext groups over the Steenrod algebra, the algebra of primary cohomology operations. The $E_3$ term can be described as a secondary Ext group, over the "algebra" of secondary cohomology operations, a structure which is more complicated than an algebra. H.J. Baues showed that this algebraic structure can be replaced by a DG-algebra over the ring $Z/p^{^2}$. 

This was used with M. Jibladze to compute the Adams differential $d_2$.

In an ongoing program with H.J. Baues, we are aiming to prove an analogous structural result for tertiary cohomology operations: that they can be encoded by a DG-algebra. In this talk, I will describe the project, some known results, and some recent developments.