Husemoller-Witt decompositions and actions of the Steenrod algebra

Abstract

Recently, there has been renewed interest in the homology of connective covers of the classifying spaces BU and BO, and their associated Thom spectra-see e.g. [4,6,9,10,15]. There are now numerous families of generators as well as structural results on the action of the Steenrod algebra. However, these two areas have not been well related since the methods used have tended to emphasise one goal rather than the other. In this paper we show that there are in fact canonical Hopf algebra decompositions for the sub-Hopf algebras of the homology of BU, and BO constructed by S. Kochman in [9], generalising those of [8]. Furthermore, these are clearly and consistently related to the Steenrod algebra action, and provide canonical sets of algebra generators. They should thus allow calculations of the type exemplified in [6] to be carried out in all cases, although of course the complexity of the answer increases rapidly! A by-product of our approach is that we can easily obtain results on these homologies as Hopf algebras, such as selfduality and a computation of endomorphism groups over the Steenrod algebra. We feel that the methods will also give interesting information in the case of some other familiar spaces even if their homology is not self dual (or bipolynomial); we intend to return to this in a sequel.