#### Volume 9, issue 4 (2009)

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The equivariant $J$–homomorphism for finite groups at certain primes

### Christopher P French

Algebraic & Geometric Topology 9 (2009) 1885–1949
##### Abstract

Suppose $G$ is a finite group and $p$ a prime, such that none of the prime divisors of $G$ are congruent to $1$ modulo $p$. We prove an equivariant analogue of Adams’ result that ${J}^{\prime }={J}^{\prime \prime }$. We use this to show that the $G$–connected cover of ${Q}_{G}{S}^{0}$, when completed at $p$, splits up to homotopy as a product, where one of the factors of the splitting contains the image of the classical equivariant $J$–homomorphism on equivariant homotopy groups.

##### Keywords
$J$–homomorphism, Adams operations, equivariant $K$–theory, equivariant fiber spaces and bundles
##### Mathematical Subject Classification 2000
Primary: 19L20, 19L47, 55R91
##### Publication
Received: 7 May 2007
Revised: 17 July 2009
Accepted: 3 August 2009
Published: 3 October 2009
##### Authors
 Christopher P French Department of Mathematics and Statistics Grinnell College Grinnell, IA 50112 United States http://www.math.grin.edu/~frenchc/