Groups of unstable Adams operations on $p$–local compact groups

Abstract

A $p$–local compact group is an algebraic object modelled on the homotopy theory associated with $p$–completed classifying spaces of compact Lie groups and $p$–compact groups. In particular $p$–local compact groups give a unified framework in which one may study $p$–completed classifying spaces from an algebraic and homotopy theoretic point of view. Like connected compact Lie groups and $p$–compact groups, $p$–local compact groups admit unstable Adams operations: self equivalences that are characterised by their cohomological effect. Unstable Adams operations on $p$–local compact groups were constructed in a previous paper by F Junod, R Levi, and A Libman. In the present paper we study groups of unstable operations from a geometric and algebraic point of view. We give a precise description of the relationship between algebraic and geometric operations, and show that under some conditions, unstable Adams operations are determined by their degree. We also examine a particularly well behaved subgroup of unstable Adams operations.