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

Ran Levi and Assaf Libman

Algebraic & Geometric Topology 17 (2017) 355–418

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.

$p$–local compact groups, unstable Adams operations
Mathematical Subject Classification 2010
Primary: 55R35
Secondary: 20D20
Received: 20 December 2015
Revised: 3 June 2016
Accepted: 17 June 2016
Published: 26 January 2017
Ran Levi
Institute of Mathematics
University of Aberdeen
Fraser Noble Building 138
AB24 3UE
United Kingdom
Assaf Libman