Modular coinvariants and the mod \(p\) homology of \(QS^{k}\) (Q2797821)

Let \(G\) be a fibrant ring spectrum. Then the spaces \(\{G_k, k \geq 0\}\) appearing in \(G\) carry multiplicative maps \(\mu : G_k \times G_l \rightarrow G_{k+l}.\) In homology, this gives to the graded coalgebra \(\{ H_*(G_k) \}\) the structure of a ring object (in the category of graded coalgebras).NEWLINENEWLINEMoreover, each piece \(H^*(G_k)\) has an action of both the modulo \(p\) Steenrod algebra and of the modulo \(p\) Dyer-Lashof algebra (as \(G_k\) is an infinite loop space).NEWLINENEWLINEIn this paper, the author determines explicitly this very rich structure, in the particular case when \(G\) is the sphere spectrum, when \(p\) is an odd prime number. (This is the content of Theorems 4.4, 4.7, 4.13 and 5.2). The case when \(p=2\) was previously worked out by \textit{P. R. Turner} [Math. Z. 224, No. 2, 209--228 (1997; Zbl 0890.55011)] and \textit{P. J. Eccles} et al. [ibid. 224, No. 2, 229--233 (1997; Zbl 0890.55010)]. The present paper follows the strategy of their proof. As ring spectra are exactly \(S\)-algebras, the study of this case is fundamental in the study of such structures, and for the study of infinite loop spaces in particular.NEWLINENEWLINEDyer and Lashof provided a map NEWLINE\[NEWLINE\mathbb{Z} \times B\Sigma_{\infty} \rightarrow QS^0,NEWLINE\]NEWLINE where \(QS^0 := \Omega^{infty} \Sigma^{\infty} S^0\) is the first space appearing in the sphere spectrum, and \(B\Sigma_{\infty}\) is the classifying space of the infinite symetric group, which induces an isomorphism in homology. The proof now studies the homology of \(B\Sigma_{\infty}\) through the morphisms NEWLINE\[NEWLINE\mathbb{V}_n \hookrightarrow \Sigma_{p^n} \hookrightarrow \Sigma_{\infty}. NEWLINE\]NEWLINE This is done via invariant theory. One major step of the proof is to provide an explicit additive base for the image of \(B\mathbb{V}_n \rightarrow B\Sigma_{p^n}\) in homology (Theorem 3.6).