The weak conjecture on spherical classes
type: Item
instance of: scholarly article
title: The weak conjecture on spherical classes
zbMATH Open document ID: 0934.55009
DOI: 10.1007/PL00004750
published in: Mathematische Zeitschrift
publication date: 1999-09-21T00:00:00Z
review text: This paper is a contribution to the study of the stable Hurewicz map, \(H\colon \pi^S_\ast(S^0) \cong \pi_\ast(Q_0S^0) \to H_\ast(Q_0S^0)\), here understood with mod 2 coefficients. It was conjectured that only the Hopf invariant one and Kervaire invariant one elements are detected by this mapping. The target \(H_\ast(Q_0S^0)\) is well understood in terms of the Araki-Kudo-Dyer-Lashof operations, and in work of I.~Madsen, in terms of invariant theory and the Dickson invariants. \textit{W. M. Singer} has taken this theme very far in his analysis of the \(\Lambda\)-algebra in terms of invariants [Trans. Am. Math. Soc. 280, 673-693 (1983; Zbl 0533.55013)] and an algebraic transfer homomorphism [Math. Z. 202, No.~4, 493-523 (1989; Zbl 0687.55014)]. In this note the author formulates a weak conjecture about spherical classes in terms of Singer's transfer, that there are no spherical classes for \(Q_0S^0\) detected by the algebraic transfer except Hopf and Kervaire invariant one elements. The main theorem is an equivalence of this conjecture with the vanishing of the homomorphism induced by inclusion in positive dimensions: \[ j_k\colon \mathbf{F}_2 \otimes_{{\mathcal A}} (\text\textbf{F}_2[x_1, \ldots, x_k]^{Gl_k}) \to (\text\textbf{F}_2 \otimes_{{\mathcal A}} \text\textbf{F}_2[x_1, \ldots, x_k])^{Gl_k}. \] Here \({\mathcal A}\) denotes the mod~2 Steenrod algebra and \(\mathbf{F}_2[x_1, \ldots, x_k]\) is isomorphic to \(H^\ast(B(\mathbf{F}_2)^k; \text\textbf{F}_2)\) on which \(Gl_k\) acts. The analysis is based on the author's work on the action of the Steenrod algebra on modular invariants [Proc. Am. Math. Soc. 113, No.~4, 1097-1104 (1991; Zbl 0736.55014)].
Mathematics Subject Classification ID: 55P47
Mathematics Subject Classification ID: 55Q45
Mathematics Subject Classification ID: 55S10
Mathematics Subject Classification ID: 55T15
zbMATH DE Number: 1336872
zbMATH Keywords: stable Hurewicz homomorphism
zbMATH Keywords: Dickson invariants
zbMATH Keywords: Steenrod algebra
zbMATH Keywords: Lambda algebra
Wikidata QID: Q123140277
author: Nguyen Hu'u Viet Hu'ng
reviewed by: John McCleary
MaRDI profile type: Publication
full work available at URL: pl00004750
OpenAlex ID: W2063902645
Recommended article: The generalized algebraic conjecture on spherical classes
Recommended article: A classification of spherical conjugacy classes
Recommended article: A classification of irreducible weakly spherical homogeneous spaces
Recommended article: Spherical varieties and Wahl's conjecture
Recommended article: Weakly symmetric spaces and spherical varieties
Recommended article: Spherical classes in some finite loop spaces of spheres
Recommended article: Spherical classes and the lambda algebra
Recommended article: Spherical conjugacy classes and involutions in the Weyl group
Recommended article: On spherical twisted conjugacy classes.
Recommended article: On the Sphere Conjecture of Birkhoff
description: scientific article; zbMATH DE number 1336872