Localisations and completions of nilpotent $G$-spaces

TL;DR

This paper develops a theory for nilpotent G-spaces and their localisations, extending classical concepts to equivariant settings with flexible prime localisations and unbased spaces, using Bousfield localisation techniques.

Contribution

It introduces a comprehensive framework for localising and completing nilpotent G-spaces in the equivariant context, including non-based and non-connected spaces, with prime localisation flexibility.

Findings

The theory applies to fixed point spaces with different prime localisations.

It extends localisation techniques to unbased G-spaces.

The approach generalizes classical non-equivariant results to equivariant settings.

Abstract

We develop the theory of nilpotent $G$-spaces and their localisations, for $G$ a compact Lie group, via reduction to the non-equivariant case using Bousfield localisation. One point of interest in the equivariant setting is that we can choose to localise or complete at different sets of primes at different fixed point spaces - and the theory works out just as well provided that you invert more primes at $K \leq G$ than at $H \leq G$, whenever $K$ is subconjugate to $H$ in $G$. We also develop the theory in an unbased context, allowing us to extend the theory to $G$-spaces which are not $G$-connected.