On the cohomology of stable two stage Postnikov systems

Abstract

We study the cohomology of certain fibre spaces. The spaces are the total spaces of stable two stage Postnikov systems. We study their cohomology as Hopf algebras over the Steenrod algebra. The first theorem determines the cohomology as a Hopf algebra over the ground field, the algebi a structure being known previously. The second theorem relates the action of the Steenrod algebra to the Hopf algebra structure and other available structures. The work is in the direction of explicit computations of these structures but is not quite complete with regard to the action of the Steenrod algebra. The ideas of Massey and Peterson [7], Mem. Amer. Math. Soc. No. 74, are used extensively, and mod 2 cohomology is used throughout. Introduction. Let F -* E -* B be a two stage Postinkov system with stable kinvariant. Under several special assumptions (listed in ?3) we study the mod 2 cohomology of QE over the Steenrod algebra A. We shall suppress further mention of the coefficient group, since mod 2 cohomology is used exclusively. A complete description of H*(QE) as an algebra over Z2 is available in [3], [11], [6] and [7]. These papers also contain certain information about the other structures. What has been missing are methods for explicitly calculating coproducts for the Hopf algebra structure and the action of the Steenrod algebra. Theorem 3.2, combined with results of [6] and [7], solves the first of these problems and gives partial information about the second. We use the work of [6] and [7] as our starting point. Our main contribution is Theorem 3.2. It gives coproducts for a certain subset of H *(QE). The fundamental sequence of the two stage Postnikov system gives one a systematic means of extending the information of Theorem 3.2, to obtain the, coproduct for IH*(QE). The fundamental sequence also provides a systematic means of using the Hstructure to obtain information about the A-structure. It would be instructive to have the connection between the H-structure and A-structure expressed in a functorial manner. The paper is organized as follows. In ?1 terminology and definitions are given. In ?2 we study special cases. The proof of Theorem 3.2 appears in ?3. Essentially the proof is a reduction of general cases to one of the special cases of ?2. In ?4 we use the H-structure to gain Received by the editors November 10, 1969. AMS 1969 subject classifications. Primary 5550; Secondary 5534.