Husemoller-Witt decompositions and actions of the Steenrod algebra. (English) Zbl 0558.55012
We generalize D. Husemoller’s Hopf algebra decomposition of \(H_*(BU;{\mathbb{F}}_ p)\) to a family of sub-Hopf algebras over the Steenrod algebra, first constructed in full generality by S. O. Kochman, and generalizing \(H_*(BSU)\). The algebra is based on the existence of the ring of p-Witt vectors, and using their algebraic properties we are able to deduce results on the endomorphisms over the Steenrod algebra of these Hopf algebras. We give explicit choices of generators and a method for calculating the Steenrod action on them; we also describe the related sub-algebras of \(H_*(MU;{\mathbb{F}}_ p)\) over the Steenrod algebra.
MSC:
| 55R40 | Homology of classifying spaces and characteristic classes in algebraic topology |
| 55R45 | Homology and homotopy of \(B\mathrm{O}\) and \(B\mathrm{U}\); Bott periodicity |
| 55S10 | Steenrod algebra |
| 55T05 | General theory of spectral sequences in algebraic topology |
| 55R50 | Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory |
Keywords:
sub-Hopf algebras over the Steenrod algebra; ring of p-Witt vectors; homology of the classifying space of the infinite unitary group; complex K-theory; Steenrod actionReferences:
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