The cohomology of finite H-spaces as U(M) algebras: I

Abstract

By an H-space (X, μ) we will mean a topological space X having the homotopy type of a connected CW complex of finite type together with a basepoint preserving map μ: x × X → X with two sided homotopy unit. Let p be a prime and let /p be the integers reduced modp. Given an H-space (X, μ) then H*(X;/p) is a commutative associative Hopf algebra over the Steenrod algebra A*(p) and H*(X;/p) is iso- morphic, as an algebra, to a tensor product [⊗ , where each algebra At is generated by a single element ai (see Theorem 7.11 of (24)). The decomposition Ai is called a Borel decomposition and the elements {ai} are called the Borel generators of the decomposition. The decomposition ⊗ Ai and the resulting generators {at} are far from unique. Many choices are possible. Since A*(p) acts on H*(X;/p) an obvious restriction would be to choose the Borel decomposition to be compatible with this action. We would like the /p module generated by the Borel generators and their iterated pth powers to be invariant under the action of A * (p). More precisely we would like H*(X;/p) to be the enveloping algebra U(M) of an unstable Steenrod module M (see § 2). If H*(X;/p) admits such a choice then it is called a U(M) algebra. The fact that H*(X; /p) is a U(M) algebra has applications in homotopy theory. In particular there exist unstable Adams spectral sequences which can be used to calculate the homotopy groups of X (see (21) and (7)). However, the question of U(M) structures for modp cohomology seems of most interest simply as a classification device for finite H-spaces.