#### Volume 12, issue 1 (2012)

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Unstable Adams operations on $p$–local compact groups

### Fabien Junod, Ran Levi and Assaf Libman

Algebraic & Geometric Topology 12 (2012) 49–74
##### Abstract

A $p$–local compact group is an algebraic object modelled on the $p$–local homotopy theory of classifying spaces of compact Lie groups and $p$–compact groups. In the study of these objects unstable Adams operations are of fundamental importance. In this paper we define unstable Adams operations within the theory of $p$–local compact groups and show that such operations exist under rather mild conditions. More precisely, we prove that for a given $p$–local compact group $\mathsc{G}$ and a sufficiently large positive integer $m$, there exists an injective group homomorphism from the group of $p$–adic units which are congruent to 1 modulo ${p}^{m}$ to the group of unstable Adams operations on $\mathsc{G}$.

##### Keywords
p-local compact group, unstable Adams operation, classifying space
##### Mathematical Subject Classification 2010
Primary: 55R35
Secondary: 55R40, 20D20