The kernel of the loop suspension map

Abstract

Let X be a 1-connected H space such that the Hopf algebra H*(X; Zp) has finite type. In this paper we characterize elements of the kernel of the loop map a" QH(X; Z,) PH-(KX; Z,) both in terms of restricted types of Massey products and, of more interest, in terms of elementary stable cohomology operations. Basically the main result, Theorem B, states that if ax 0, then either x kt(U) or x kJk,(v), where flk is the pkth order Bockstein, t and J are particularly simple primary operations, k is a specific secondary cohomology operation, and u and v are indecomposable cohomology classes of H*(X; Z). One of the applications is a characterization of differentials in certain spectral sequences in terms of these stable cohomology operations.