Secondary compositions and the Adams spectral sequence

Abstract

In each term of the Adams spectral sequence [1, 2] Massey products [10] can be formed and in the E2-term it has been found convenient to describe specific elements by means of these products. In Theorem (1.1) of this paper a description is given of the action of the differential d r on the Massey product in E r. In homotopy an analogous construction to the Massey product is the Toda bracket [15] and Theorem (1.2) discusses the convergence of Massey products in the Adams spectral sequence to Toda brackets in homotopy. It states that, under favorable conditions, a Massey product that is formed from permanent cycles will itself contain a permanent cycle that is realized in homotopy by an element of the appropriate Toda bracket. The statement of both Theorems (1.1) and (1.2) will be found in w 1 and the proofs are given in w167 5, 6. The intervening sections contain various geometric and algebraic preliminaries. A previous account of Theorem (1.2) was given in [12] and I am grateful to Professor J.F.Adams for suggesting the problem to which this theorem provides a partial solution. In 1966, Ivanovskil announced, at the Moscow congress, various results concerning the Adams spectral sequence and there appears to be some overlap between these results and Theorem (1.2). More recently, in Chicago, Lawrence has developed an account of the higher Massey product behaviour in the Adams spectral sequence and his results generalize the main theorems of this paper.