Unstable Adams operations on p-local compact groups

TL;DR

This paper introduces and constructs unstable Adams operations for p-local compact groups, expanding the algebraic tools available for their study within p-local homotopy theory.

Contribution

It defines unstable Adams operations in the context of p-local compact groups and proves their existence under mild conditions, linking p-adic units to these operations.

Findings

Existence of unstable Adams operations for p-local compact groups.

Construction of an injective homomorphism from p-adic units to Adams operations.

Applicability to a broad class of p-local compact groups.

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 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 G