The equivariant \(J\)-homomorphism for finite groups at certain primes
type: Item
instance of: scholarly article
title: The equivariant \(J\)-homomorphism for finite groups at certain primes
zbMATH Open document ID: 1177.19004
author: Christopher P. French
published in: Algebraic \& Geometric Topology
publication date: 2009-10-06T00:00:00Z
review text: In order to compute the group \(J(X)\), \textit{J. F. Adams} [Topology 3, 137-171 (1965; Zbl 0137.16801)] introduced two groups \(J'(X)\) and \(J''(X)\) equipped with canonical epimorphisms \(J''(X) \to J(X) \to J'(X)\) such that the composite of these epimorphisms induces an identity \(J'(X)=J''(X)\), which of course depends on the solution to the Adams conjecture. The purpose of this paper is to consider this identity in the \(G\)-equivariant setting where \(G\) is a finite group. The above identity is rephrased in the \(J\)-theory diagram of \textit{J. P. May} [``\(E_\infty\) ring spaces and \(E_\infty\) ring spectra'', Lect. Notes Math. 577 (1977; Zbl 0345.55007)]. The central columns of this diagram form a square, which is called here the Adams-May square: \[ \begin{tikzcd} BO \rar["\psi^k-1"]\dar["\sigma^k" '] & B\mathrm{Spin}\dar["\rho^k"] \\ BO_\otimes \rar["\psi^k/1" '] & B\mathrm{Spin}_\otimes \end{tikzcd} \] where \(\sigma^k\) denotes a map which arises from using a space level solution to the Adams conjecture and \(\rho^k\) is a geometric interpretation of the Adams-Bott cannibalistic class. It is asserted there that after localizing at a prime \(p\), for a suitable choice of \(k\), this square becomes a homotopy pullback, which implies immediately that \(J'(X)=J''(X)\). In a previous paper [\textit{C. French}, Homology Homotopy Appl. 5, No.~1, 161--212 (2003; Zbl 1032.55016)] the author constructed an equivariant version of this Adams-May square, and proved that when \(G\) is a \(p\)-group with \(p\neq 2\), for suitable \(k\), after restricting to \(G\)-connected covers and completing at the prime \(p\), this equivariant Adams-May square becomes a homotopy pullback. In this paper, the author considers the case when \(G\) is a finite group such that none of the prime divisors of \(|G|\) are congruent to 1 modulo \(p\) (a prime), so that \(G\) is not a \(p\)-group. Here one restricts oneself to the case when \(p\) is odd for the sake of brevity. The autho…
reviewed by: Haruo Minami
Mathematics Subject Classification ID: 19L20
Mathematics Subject Classification ID: 19L47
Mathematics Subject Classification ID: 55R91
Mathematics Subject Classification ID: 55Q50
zbMATH DE Number: 5610805
zbMATH Keywords: \(J\)-homomorphism
zbMATH Keywords: Adams operations
zbMATH Keywords: \(J\)-theory diagram
zbMATH Keywords: Adams-May square
zbMATH Keywords: Adams-Bott map
zbMATH Keywords: equivariant \(K\)-theory
zbMATH Keywords: equivariant Adams splitting
zbMATH Keywords: equivariant classifying spaces
zbMATH Keywords: equivariant fiber spaces and bundles
MaRDI profile type: Publication
full work available at URL: agt.2009.9.1885
OpenAlex ID: W1999401215
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DOI: 10.2140/AGT.2009.9.1885
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description: scientific article; zbMATH DE number 5610805