Secondary Steenrod Operations in Cohomology of Infinite-Dimensional Projective Spaces

The important role of the Steenrod operations Sq in the description of the cohomology of topological spaces, the calculation of various spectral sequences, and the study of homotopy properties of topological spaces and their continuous maps is well known [1, 2]. Projective spaces, in particular RP∞ and CP∞ , play a special role in the construction of Steenrod operations and their computation. In the present paper, we consider secondary Steenrod operations and give formulas for computing these operations in the cohomology of the spaces RP∞ and CP∞ . Recall that the Steenrod operations Sq : H(X) → H(X) arise in the cohomology H∗(X) of a topological space X (with coefficients in Z/2) from the action of an E∞-operad on the singular cochain complex C∗(X) (see [1, 3]). An operad in the category of chain complexes is a family E = {E(j)} of chain complexes E(j) , j ≥ 1 , on which the permutation groups Σj act, together with maps γ : E(k)⊗ E(j1)⊗ · · · ⊗E(jk) → E(j1 + · · ·+ jk),