The A-module Structure for the Cohomology of Two Stage Spaces

The purpose of the present paper is to give a general method for determining the structure of the cohomology of stable two-stage space (for definitions see Section 1) as a module over the Steenrod algebra A. For convenience we only consider cohomology with coefficients in the integers in Z 2 , the integers mod 2. The general part of our paper, however, works equally well with arbitrary coefficients. A two-stage space with stable k-invariant is an H-space and the cohomology consequently a Hopf algebra. We also evaluate the diagonal in the cohomology. The idea of the paper is to express the A-module structure of the cohomology of a two-stage system in terms of Massey products in A. This is done in Section 3. A Massey product in A (definition in section 2) < A, B, C >⊂ A, is a coset defined for matrices A, B, and C with entries from A of type (1, s), (s, t) and (t, 1) such that AB = 0 and BC = 0. The indeterminacy is given by {AX + Y C} where X and Y run over all matrices of type (s, t) and (1, t). When dealing with Massey products in A one can usually avoid indeterminacy. This is also the case in the present paper. We shall always be dealing with a particular element from the coset. The calculation of Massey products in general is not done in the present paper. It is, however, in principle possible to calculate any Massey product in A as follows. Kristensen and Madsen [6] worked out the reduced diagonal ψ : A → A ⊗ A of a Massey product. It contains Massey products of lower degrees together with some " primary " terms which were left undecided in Kristensen [4]. These terms, which arise from the primary terms in the Cartan formula for a secondary operation, were then computed in Kristensen [5].