Groups of unstable Adams operations on p-local compact groups

TL;DR

This paper studies groups of unstable Adams operations on p-local compact groups, providing a detailed algebraic and geometric analysis, and showing conditions under which these operations are determined by their degree.

Contribution

It offers a precise description of the relationship between algebraic and geometric unstable Adams operations and identifies conditions where these are determined by their degree.

Findings

Unstable Adams operations are characterized by their cohomological effect.

Under certain conditions, these operations are determined by their degree.

A well-behaved subgroup of operations is examined.

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 and the authors. In the current 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 operations.