# nLab equivariant suspension spectrum

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

The generalization of the concept of suspension spectrum from stable homotopy theory to $G$-equivariant stable homotopy theory.

## Definition

### As a spectrum indexed on a $G$-universe

Let $X$ be a pointed topological G-space. For a representation $V$ in the G-universe $U$, write $S^V$ for its representation sphere.

As a G-spectrum indexed on a G-universe:

1. the suspension $G$-pre-spectrum is $\Pi^\infty X \colon V \mapsto S^V \wedge X$;

2. the suspension $G$-spectrum is $\Sigma^\infty X \colon V \mapsto Q(S^V \wedge X)$

where $Q (-) = \underset{V \in U}{\cup} \Omega^V \Sigma^V X$.

### As an orthogonal spectrum

The equivariant suspension spectrum $\Sigma^\infty_G X$ of a pointed topological G-space $X$ is the G-spectrum which, modeled as an orthogonal spectrum with $G$-action, is in degree $n$ given by the smash product

$(\Sigma^\infty_G X)_n \coloneqq X \wedge S^n$

of $X$ with the n-sphere, equipped with the canonical action of the orthogonal group $O(n)$ just on the $S^n$-factor and equipped with the given action of $G$ on just $X$.

(e.g. Schwede 15, example 2.11)

## Properties

### $RO(G)$-degrees

For $V$ an orthogonal linear $G$-representation then the value of the equivariant suspension spectrum in that RO(G)-degree is the smash product of $X$ with the corresponding representation sphere.

$(\Sigma^\infty_G X)(V) \simeq X \wedge S^V$

### Relation between genuine and Bredon-equivariant suspension

For $U$ a complete $G$-universe and $U^G$ its fixed point universe, then the inclusion $i \colon U^G \longrightarrow U$ induces an adjunction

$Spectra(G Top) \stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}G Spectra$

between naive G-spectra and genuine G-spectra. The genuine $G$-suspension spectrum is the naive $G$-suspension spectrum followed by $i$:

$i_\ast \circ \Sigma^\infty_{U^G } \simeq \Sigma^\infty_U \,.$

## Examples

The $G$-equivariant sphere spectrum is

$\mathbb{S} = \Sigma^\infty_G S^0$

for $S^0$ regarded as equipped with the (necessarily) trivial $G$-action. It follows that for $V$ an orthogonal linear $G$-representation then in RO(G)-degree $V$ the equivariant sphere spectrum is the corresponding representation sphere $\mathbb{S}(V) \simeq S^V$.

### Equivariant homotopy groups and fixed point spectra

The equivariant homotopy groups and the fixed point spectra of equivariant suspension spectra $\Sigma^\infty_G X$ decompose into the naive fixed points of the $G$-action on $X$. This is the tom Dieck splitting, see there for details.

## References

Last revised on December 4, 2018 at 00:45:49. See the history of this page for a list of all contributions to it.