Secondary Algebras Associated to Ring Spectra

Homotopy groups of a connective ring spectrum R form an Ngraded algebra π∗R which is commutative if R is commutative. We describe a secondary algebra π∗,∗R which enriches the structure of the algebra π∗R in a new unexpected way. The algebra π∗,∗R encodes secondary homotopy operations in π∗R, such as Toda brackets, and the first Postnikov invariant of R as a ring spectrum. Moreover, π∗,∗R represents a cohomology class in the third Mac Lane cohomology of the algebra π∗R. If R is commutative then π∗,∗R has an E∞-structure and encodes the cup-one squares in π∗R.