# nLab fixed point spectrum

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Given a $G$-equivariant spectrum, its fixed point spectrum $F^G X$ is the plain spectrum which is the value of the derived functor of the naive $G$-fixed point functor on $X$. This constitutes a suitably derived functor

$F^G \;\colon\; G Spectra \longrightarrow Spectra$

More generally, for $N \subset G$ a normal subgroup, there is the $N$-fixed point spectrum regarded as $G/N$-spectrum, equivariant under the remaining quotient group $G/N$:

$F^N \;\colon\; G Spectra \longrightarrow G/N Spectra$

These are derived right adjoint to the operation of regarding a $G/N$-spectrum as a $G$-spectrum, via the projection $G \to G/N$

## Properties

### Relation to equivariant homotopy groups

The plain homotopy groups of the $G$-fixed point spectrum of $X$ are the equivariant homotopy groups of the $G$-spectrum $X$:

$\pi_k(F^G X) \simeq \pi_k^G(X)$

(e.g. Schwede 15, prop. 7.2)

## Examples

### For equivariant suspension spectra

For $\Sigma^\infty_G X$ an equivariant suspension spectrum the fixed point spectrum is given by the tom Dieck splitting formula

$F^G(\Sigma^\infty_G X) \simeq \underset{[H\subset G]}{\oplus} \Sigma^\infty( E W(X)_+ \wedge_{W H} X^H ) \,.$

(e.g. Schwede 15, example 7.7)

## References

Last revised on December 30, 2018 at 13:27:21. See the history of this page for a list of all contributions to it.