Good objects in the equivariant world

TL;DR

The paper extends localization theory to the category of based $G$-spaces for a compact Lie group, establishing the equivariant localization functor $L_f^G$ with a universal property and a fundamental loop-compatibility $L_f^G \Omega X \simeq_G \Omega L_{\Sigma f}^G X$. It develops the equivariant Segal/Gamma-machine via $\Gamma_G$-spaces and Shimakawa's construction, enabling passage to almost $\Omega$-$G$-spectra, and it formulates equivariant cohomology in the Bredon framework using Mackey functors. A central contribution is the notion of $L^G$-good objects, with a concrete obstruction proving that finite, connected $C_{p^n}$-CW complexes $A$ are $L^{C_{p^n}}$-good only when certain Bredon cohomology groups $H^r_{C_{p^n}}(A; \underline{\mathbb{Q}})$ vanish outside degrees $0$ and $k$; this yields a rational classification in the equivariant setting. The results have implications for rational equivariant homotopy theory, providing criteria for when mapping spaces remain in a restricted rational-type form and offering a framework for analyzing localization in equivariant contexts.

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 $\underline{\Q}$.

Theorems & Definitions (37)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Definition 1.6
• Theorem 1.7
• Theorem 1.8
• Remark 1.9
• Definition 2.1