The Singer transfer for infinite real projective space
type: Item
instance of: scholarly article
title: The Singer transfer for infinite real projective space
published in: Forum Mathematicum
publication date: 2022-11-04T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 55T15
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55P47
zbMATH DE Number: 7612310
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: Adams spectral sequences
zbMATH Keywords: Singer transfer
zbMATH Keywords: lambda algebra
zbMATH Keywords: invariant theory
zbMATH Keywords: Dickson algebra
author: Nguyen Hu'u Viet Hu'ng
author: Luu X. Truong
reviewed by: Saïd Zarati
MaRDI profile type: Publication
full work available at URL: forum-2021-0245
OpenAlex ID: W4293443532
cites work: On the structure and applications of the Steenrod algebra
cites work: On the non-existence of elements of Hopf invariant one
cites work: Q5180579
cites work: Q4278105
cites work: The mod-\(p\) lower central series and the Adams spectral sequence
cites work: On the behavior of the algebraic transfer
cites work: Determination of \(\mathrm{Ext}^{5,*}_{\mathcal A}(\mathbb Z/2,\mathbb Z/2)\)
cites work: The Adams spectral sequence of the real projective spaces
cites work: An infinite family in the cohomology of the Steenrod algebra
cites work: Spherical classes and the algebraic transfer
cites work: The weak conjecture on spherical classes
cites work: Spherical classes and the lambda algebra
cites work: On triviality of Dickson invariants in the homology of the Steenrod algebra
cites work: The cohomology of the Steenrod algebra and representations of the general linear groups
cites work: The \(\mathcal{A}\)-decomposability of the Singer construction
cites work: The image of Singer's fourth transfer
cites work: The algebraic transfer for the real projective space
cites work: The generalized algebraic conjecture on spherical classes
cites work: Sur les foncteurs dérivés de la déstabilisation
cites work: Algebraic Kahn-Priddy theorem
cites work: \(Ext_A^{4,\ast}(\mathbb Z/2,\mathbb Z/2)\) and \(Ext_A^{5,\ast}(\mathbb Z/2,\mathbb Z/2)\)
cites work: Q4218772
cites work: The factorization of cyclic reduced powers by secondary cohomology operations
cites work: Q5653373
cites work: The iterated transfer analogue of the new doomsday conjecture
cites work: Q4102036
cites work: Algebraic transfer and action of linear groups on mod 2 divided powers
cites work: Invariant Theory and the Lambda Algebra
cites work: The transfer in homological algebra
cites work: Cohomology Operations (AM-50)
cites work: On the cohomology of the Steenrod algebra
cites work: On the cohomology of the mod-2 Steenrod algebra and the non-existence of elements of Hopf invariant one
cites work: Classifying spaces, Steenrod operations and algebraic closure
cites work: Steenrod squares of polynomials and the Peterson conjecture
cites work: Problems in the Steenrod Algebra
DOI: 10.1515/FORUM-2021-0245
description: scientific article; zbMATH DE number 7612310