The Singer transfer for infinite real projective space
Let \(A\) be the mod. \(2\) Steenrod algebra, let \(M\) be an \(A\)-module (\(M_{*}\) denotes its dual), let \(V_{s}\) be an elementary abelian \(2\)-group of rank \(s\) and let \(\mathrm{GL}_{s}\) be the general linear group over \(\mathbb{F}_{2}\). We denote by \(P(H_{*}V_{s} \otimes M_{*})\) the primitive part of \( H_{*}V_{s} \otimes M_{*}\) under the action of \(A\). \textit{W. M. Singer} [Math. Z. 202, No. 4, 493--523 (1989; Zbl 0687.55014)] defined the algebraic transfer for \(M\) which is a homomorphism \( \mathrm{Tr}_{s}^{M}: \mathbb{F}_{2} \otimes_{\mathrm{GL}_{s}} P(H_{*}V_{s} \otimes M_{*}) \rightarrow \mathrm{Ext}_{A}^{s}(\Sigma^{-s}M, \mathbb{F}_{2})\). The aim of this paper is to study the Singer transfer, \(\mathrm{Tr}_{s}^{\widetilde{H^{*}}(\mathbb{R}P^{\infty})}\), for the infinite real projective space \(\mathbb{R}P^{\infty}\), using the algebraic Kahn-Priddy homomorphism, (see [\textit{D. S. Kahn} and \textit{S. B. Priddy}, Bull. Am. Math. Soc. 78, 981--987 (1972; Zbl 0265.55009)]), and in connection with the Singer transfer, \(\mathrm{Tr}_{s}^{\widetilde{H^{*}}(S^{0})}\), for the sphere \(S^{0}\). The authors prove, in particular, that: \begin{itemize} \item[i)] The image of \(\mathrm{Tr}_{*}^{\widetilde{H^{*}}(\mathbb{R}P^{\infty})}\) is a module over the image of \(\mathrm{Tr}_{*}^{\widetilde{H^{*}}(S^{0})}\). \item[ii)] The Kahn-Priddy homomorphism is an epimorphism from \(\mathrm{Im}(\mathrm{Tr}_{*}^{\widetilde{H^{*}}(\mathbb{R}P^{\infty})})\) onto \(\mathrm{Im}(\mathrm{Tr}_{*}^{\widetilde{H^{*}}(S^{0})})\). \item[iii)] The Singer transfer \(\mathrm{Tr}_{s}^{\widetilde{H^{*}}(\mathbb{R}P^{\infty})}\) is not a monomorphism in every positive homological degree. \end{itemize} The stability of \(\mathrm{Tr}_{s}^{\widetilde{H^{*}}(\mathbb{R}P^{\infty})}\) under the iterated squaring operation and its behavior with respect to elements that are not products of Adams elements are also studied by the authors.
scientific article; zbMATH DE number 7612310
scientific article