The Hopf ring for \(bo\) and its connective covers
If \(F_*(\enskip)\) is a multiplicative homology theory and \(E_*\) is a ring spectrum then the homology groups \(F_*(E_*)\) have a coproduct, conjugation, counit and a pair of products which give them the structure of a Hopf ring, i.e., a ring object in the category of coalgebras. The study of such objects was initiated by \textit{D. C. Ravenel} and \textit{W. S. Wilson} [J. Pure Appl. Algebra 9, 241--280 (1977; Zbl 0373.57020)] and their computation is of interest to topologists because of their connection with the unstable operations between \(E^* (\enskip)\) and \(F^*(\enskip)\). The paper under review gives a complete computation of this Hopf ring for \(E_* = bo\), the connective real \(K\)-theory spectrum, and \(F_* = \overline{H}_* = H_*(\enskip ; {\mathbb Z}/2)\), ordinary mod \(2\) homology. In an earlier paper the author and \textit{N. Strickland} [J. Pure Appl. Algebra 166, 247--265 (2002; Zbl 0987.55007)] computed the corresponding Hopf ring for nonconnective real \(K\)-theory, \(\overline{H}_*(KO_*)\). The case of integral ordinary homology, \(H_*(\overline{H}_*)\), is classical. The author shows that \(\overline{H}_*(bo_*)\) injects into the product \(H_* (KO) \otimes H_* (\overline{H}_*)\) and she describes the image. The result is a very large graded tensor product of families of polynomial and exterior algebras coming from these two factors. The method of computation is repeated use of the bar spectral sequence. The paper concludes with the computation of \(\overline{H}_*(bo \langle n \rangle)\), the homology of the connective covers of \(bo\), as Hopf modules over \(\overline{H}_* (bo_*)\).
scientific article; zbMATH DE number 5146871