The A∞-structures and differentials of the Adams spectral sequence

Using operad methods and functional homology operations, we obtain inductive formulae for the differentials of the Adams spectral sequence of stable homotopy groups of spheres. The Adams spectral sequence was invented by Adams [1] almost fifty years ago for the calculation of stable homotopy groups of topological spaces (in particular, those of spheres). The calculation of the differentials of the Adams spectral sequence of homotopy groups of spheres is one of the most difficult problems of modern algebraic topology. Here we consider an approach to the solution of this problem based on the use of the A∞-structures introduced by Stasheff [2], operad methods [3]–[6], and functional homology operations [7]–[9]. We apply our results to the Arf invariant problem [10], [11]. § 1. The Adams and Bousfield–Kan spectral sequences Let us recall that the E term of the Adams spectral sequence of stable homotopy groups of a topological space Y with coefficients in Z/2 is the complex F (K, Y∗) : Y∗ → K⊗ Y∗ → · · · → K⊗n ⊗ Y∗ → · · · , where Y∗ is the homology of Y and K is the Milnor coalgebra (dual to the Steenrod algebra), which is the algebra of polynomials in ξi of dimension 2 i − 1. The comultiplication ∇ : K → K ⊗K is defined on the generators ξi by the formula