# nLab global equivariant stable homotopy theory

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Representation theory

representation theory

geometric representation theory

cohomology

# Contents

## Idea

What is called global equivariant stable homotopy theory is a variant of equivariant stable homotopy theory where spectra are equipped with $G$-infinity-actions “for all compact Lie groups $G$ at once”.

Often this is referred to just as “global stable homotopy theory” or even just “global homotopy theory”. But there is also unstable global equivariant homotopy theory.

More precisely, given an orthogonal spectrum $X$, then every representation $\rho \colon G \to O(n)$ of a compact Lie group on the Cartesian space $\mathbb{R}^n$ by orthogonal group actions induces a $G$-equivariant spectrum and hence a notion of $G$-equivariant homotopy groups.

One says that a morphism of orthogonal spectra is a global equivariant equivalence if it induces isomorphisms on all $G$- equivariant homotopy groups, for all $G$, this way. (This definition appears for instance as (Schwede 13, def. 2.9), there referred to just as “global equivalence”. See also at equivariant Whitehead theorem.)

The global equivariant stable homotopy category $\mathcal{GH}$ is the (simplicial) localization of the category of orthogonal spectra at these global equivariant equivalences (this is a stable (infinity,1)-category/triangulated category.)

Since a global equivariant equivalence is in particular an ordinary weak homotopy equivalence of spectra, there is a canonical functor

$U \;\colon\; \mathcal{GH} \longrightarrow \mathcal{SH}$

from the global equivariant to the ordinary stable homotopy category.

This functor has a (derived) left adjoint and right adjoint, which are both full and faithful functors, hence which exhibit discrete object/codiscrete object structure

$\array{ \mathcal{GH} &\stackrel{\overset{L}{\leftarrow}}{\stackrel{\overset{U}{\longrightarrow}}{\underset{R}{\leftarrow}}}& \mathcal{SH} }$

on stable/triangulated categories (Schwede 13, theorem IV 5.2)

Global Borel-type equivariant cohomology is in the image of the right adjoint $R$ (Schwede 13, example IV 5.12)

## Properties

### Relation to plain stable homotopy theory

The forgetful functor from global stable homotopy theory to plain stable homotopy theory exhibits a recollement.

(…)

## References

A comprehensive textbook account is in

Survey (with emphasis on global equivariant bordism homology theory):

Original articles are

• L. Gaunce Lewis, Jr., Peter May, M. Steinberger, chapter II of Equivariant stable homotopy theory. Lecture Notes in Mathematics, 1213, Springer-Verlag, 1986

• John Greenlees, Peter May, section 5 of Localization and completion theorems for $MU$-module spectra Ann. of Math. (2) 146 (1997), 509-544.

Discussion specifically in terms of equivariant orthogonal spectra is in

Discussion for collections of finite subgroups includes

The example of algebraic K-theory: