Good objects in the equivariant world

TL;DR

This paper investigates equivariant localization in G-spaces, establishing a commutation rule with the loop functor and classifying certain good objects via Bredon cohomology with rational coefficients.

Contribution

It introduces a new classification of good objects in equivariant topology and proves a commutation rule between localization and loop functors.

Findings

Established a commutation rule for localization and loop functors.

Classified good objects in G-spaces using Bredon cohomology.

Provided new insights into equivariant localization theory.

Abstract

This article explores equivariant localization in the category of $G$-spaces, where $G$ is a compact Lie group. We establish a commutation rule for the localization functor and the equivariant loop functor. Additionally, we introduce and classify certain good objects in this category up to their Bredon cohomology with coefficients in the constant rational Mackey functor $\underline{\Q}$.