# nLab rational equivariant stable homotopy theory

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Representation theory

representation theory

geometric representation theory

## Theorems

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## Idea

Rational equivariant stable homotopy theory is the study of equivariant spectra just on the level of their rationalization, hence concerning only their non-torsion homotopy groups. This is the equivariant and stable version of rational homotopy theory.

A key general statement of the theory is that rationally the homotopy theory of $G$-equivariant spectra is equivalently given by homological algebra of Mackey functors (even for non-finite $G$). At the level of equivalences of homotopy categories this was established by John Greenlees, at the level of zig-zags of Quillen equivalences of model categories this was established Greenlees and by his students, David Barnes and Magdalena Kedziorek.

## Properties

### Greenlees-May splitting into equivariant Eilenberg-MacLane spectra

Let $G$ be a finite group. For $X$ a G-spectrum, write $\pi_\bullet(X) \in \mathcal{M}[G]$ for its Mackey functor, the one which sends $G/H$ to the $H$-equivariant homotopy groups of $X$.

Every rational $G$-equivariant spectrum $E$ is the direct sum of the Eilenberg-MacLane spectra (Mackey functors) on its equivariant homotopy groups:

$E \simeq \prod_n \Sigma^n H\pi_n(E) \,.$

For $X,Y$ two $G$-spectra, there is a canonical morphism

$[X,Y]_G \longrightarrow \underset{n}{\prod} Hom_{\mathcal{M}[G]}(\pi_n(X),\pi_n(Y)) \,.$

When $Y$ is rational, then this is an isomorphism (Greenlees-May 95, theorem A.4).

### Rational tom Dieck splitting

for the moment see at tom Dieck splitting

## Examples

Just as for the plain sphere spectrum, the equivariant homotopy groups of the equivariant sphere spectrum in ordinary integer degrees $n$ are all torsion, except at $n = 0$:

$\pi_n^H(\mathbb{S})\otimes \mathbb{Q} = \left\{ \array{ \cdots & for \; n = 0 \\ 0 & otherwise } \right.$

But in some RO(G)-degrees there may appear further non-torsion groups, see at equivariant sphere spectrum the section Examples.

General:

For $G = O(2)$ or $SO(2)$ and so also for $G =$ dihedral group and cyclic group:

• John Greenlees, Rational $O(2)$-equivariant cohomology theories. In Stable and unstable homotopy (Toronto, ON, 1996), volume 19 of Fields

Inst. Commun., pages 103–110. Amer. Math. Soc., Providence, RI, 1998. (web)

• John Greenlees, Rational $S^1$-equivariant stable homotopy theory, Memoirs of the AMS, 1999

• Brooke Shipley, An algebraic model for rational $S^1$-equivariant stable homotopy theory (pdf)

• David Barnes, Classifying Dihedral $O(2)$-Equivariant Spectra (arXiv:0804.3357)

• David Barnes, Rational $Z_p$-Equivariant Spectra, Algebr. Geom. Topol. 11 (2011) 2107-2135 (arXiv:1011.5785)

• David Barnes, Rational $O(2)$-Equivariant Spectra (arXiv:1201.6610)

• David Barnes, A monoidal algebraic model for rational $SO(2)$-spectra (arXiv:1412.1700)

• David Barnes, John Greenlees, Magdalena Kedziorek, Brooke Shipley, Rational $SO(2)$-Equivariant Spectra (arXiv:1511.03291)

For $G = (S^1)^{\times_n}$ a torus:

For $G = SO(3)$ and hence also for the finite groups of ADE type:

• John Greenlees, Rational SO(3)-Equivariant Cohomology Theories, in Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math. 271, Amer. Math. Soc. (2001) 99 (web)

• Magdalena Kedziorek, Algebraic models for rational G-spectra, 2014 (web)