The Hopf ring for Bockstein-nil homology of \(QS^n\)
In this article, the author describes the Hopf ring structure of the mod \(p\) homology of \(QS^n=\Omega^\infty\Sigma^\infty S^n\) for odd primes \(p\). The author [C. R. Acad. Sci., Paris, Sér. I 320, No. 9, 1119--1122 (1995; Zbl 0840.55008)] gave a proof for the prime 2 case which uses the Quillen isomorphism in the prime 2 case. The author makes use of the ideas required in the proof of the Quillen isomorphism in the odd prime case to describe the mod \(p\) homology of \(QS^n\) for odd primes \(p\). Let \(C(G)\) denote the category whose objects are elementary abelian \(p\)-subgroups of a group \(G\) and whose morphisms are inclusions induced by conjugations in \(G\). \textit{J. H. Gunawardena}, \textit{J. Lannes} and \textit{S. Zarati} [Lond. Math. Soc. Lect. Note Ser. 139, 61--68 (1989; Zbl 0693.55012)] proved that the Quillen map \( H^\ast(B\Sigma_n, \mathbb{Z}/2)) \to \varprojlim_{C(\Sigma_n)} H^\ast( BE, \mathbb{Z}/2)\) is an isomorphism for \(p=2\). In the odd prime case, it is no longer an isomorphism and \textit{Lê Minh Hà} and \textit{K. Lesh} [J. Pure Appl. Algebra 190, No. 1--3, 137--153 (2004; Zbl 1058.20042)] proved that on the Bockstein-nil part the Quillen map \( H^{\bullet} (B\Sigma_n, \mathbb{Z}/p) \to \varprojlim_{C(\Sigma_n)} H^\bullet (BE, \mathbb{Z}/p) \) is an isomorphism. In this article, the author defines the Bockstein-nil quotient of homology in \(p\) coefficients for an odd prime \(p\). Let \(C X= \coprod_n E\Sigma_n \times_{\Sigma_n} X^n/\sim\). The author first describes certain sub objects of \(H_\ast(X, \mathbb{Z}/p)\) where \(X\) is either \(CS^n \) or \(QS^n\) in terms of generators and relations. The author then computes the Hopf ring structure of the Bockstein-nil part and the Bockstein quotient of \(H^\ast(X, \mathbb{Z}/p)\) and \(H_\ast(X, \mathbb{Z}/p)\) for an odd prime \(p\) .
scientific article; zbMATH DE number 5999963
scientific article