# nLab equivariant homotopy group

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 homotopy group from homotopy theory and stable homotopy theory to equivariant homotopy theory and equivariant stable homotopy theory.

## Definition

### Abstractly

For $X$ a pointed topological G-space and $H \subset G$ a closed subgroup, the $n$th unstable $H$-equivariant homotopy group of $X$ is simply the ordinary $n$-th homotopy group of the $H$-fixed point space $X^H$:

$\pi_n^H(X) \coloneqq \pi_n(X^H) \,.$

With $G/H$ denoting the quotient space, this is equivalently the $G$-homotopy classes of $G$-equivariant continuous functions from the smash product $S^n \wedge G/H_+$ to $X$:

$\pi_n^H(X) \simeq [G/H_+ \wedge S^n, X]^G \,.$

In this form the definition directly generalizes to G-spectra and hence to stable equivariant homotopy groups: for $E$ a G-spectrum, then

$\pi_n^H(X) \simeq [G/H_+ \wedge \Sigma^\infty S^n, X]^G \,.$

where now $\Sigma^\infty S^n \simeq \Sigma^n \mathbb{S}$ is the suspension spectrum of the n-sphere and $[-,-]^G$ now denotes the hom functor in the equivariant stable homotopy category.

### Via genuine $G$-spectra

Consider genuine G-spectra modeled on a G-universe $U$.

For a finite based G-CW complex $X$ and base topological G-space $Y$, write

$\{X,Y\}_G = [\Sigma^\infty_G X, \Sigma^\infty_G Y] \coloneqq \underset{\longrightarrow}{\lim}_{V \subset U} [\Sigma^V X, \Sigma^V Y]_G$

for the colimit over $G$-homotopy classes of maps between suspensions $\Sigma^V X \coloneqq S^V \wedge X$, where $V$ runs through the indexing spaces in the universe and $S^V$ denotes its representation sphere.

The equivariant stable homotopy groups of $X$ are

$\pi_V^G(\Sigma^\infty_G X) \coloneqq \{S^V,X\}_G \,.$

And for subgroups $H \subset G$

$\pi_V^H(\Sigma^\infty_G X) \coloneqq \{G/H_+ \wedge S^V,X\}_G$

### Via orthogonal spectra and $G$-equivariant maps

Let $G$ be a finite group. For $X$ a $G$-equivariant spectrum modeled as an orthogonal spectrum with $G$-action, then for $k \in \mathbb{N}$ the $k$th equivariant homotopy group of $X$ is the colimit

$\pi_k^G(X) \coloneqq \underset{\longrightarrow_{\mathrlap{n}}}{\lim} [S^{n \rho_G}, (\Omega^k X)(n \rho_G)]_H \,,$

where

• $\rho_G$ denotes the fundamental representation of $G$

• $n \rho_G = (\rho_G)^{\oplus_n}$;

• $S^{n \rho_G}$ is its representation sphere;

• $X(n \rho_G)$ is the value of $X$ in RO(G)-degree $n\rho_G$;

• $[-,-]_G$ is the set of homotopy classes of $G$-equivariant maps of pointed topological G-spaces.

(e.g. Schwede 15, section 3)

More generally for $H \hookrightarrow G$ a subgroup then one writes $\pi_\bullet^H(X)$ for the $H$-equivariant subgroups of $X$ with $X$ regarded now as an $H$-equivariant spectrum, via restriction of the action.

(e.g. Schwede 15, p. 16)

### Via fixed point spectra

Equivalently, the $k$th $G$-equivariant homotopy group of a $G$-equivariant spectrum $E$ is the plain $k$th stable homotopy group of its fixed point spectrum $F^G E$

(1)$\pi_k^G(E) \simeq \pi_k(F^G E) \,.$

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

### Via equivariant cohomology of the point

The identification (1) in turn is the equivariant cohomology of the point,

$E^{-k}_G(\ast) \;\coloneqq\; \left[ \epsilon^\sharp \Sigma^k \mathbb{S} , E\right]_G \;\simeq\; \left[ \Sigma^k \mathbb{S}, F^G E \right] \;\simeq\; \pi_k(F^G E)$

due to the base change adjunction

$G Spectra \underoverset { \underset{ F^G }{\longrightarrow} } {\overset{ \epsilon^\sharp }{\longleftarrow}} { \phantom{AA} \bot \phantom{AA} } Spectra$

## Examples

### Of equivariant suspension spectra

For $X$ a pointed topological G-space, then by the discussion there) the formula for the equivariant homotopy groups of its equivariant suspension spectrum $\Sigma^\infty_G X$ reduces to

$\pi_k^G(\Sigma^\infty_G X) \coloneqq \underset{\longrightarrow_n}{\lim} [S^{n \rho_g}, (\Omega^k X)\wedge S^{n \rho_G}]_G$

which in turn decomposes as a direct sum of ordinary homotopy groups of Weyl group-homotopy quotients of naive fixed point spaces – see at tom Dieck splitting.

### Of the equivariant sphere spectrum

For the equivariant sphere spectrum $\mathbb{S} = \Sigma^\infty_G S^0$ the tom Dieck splitting gives that its 0th equivariant homotopy group is the free abelian group on the set of conjugacy classes of subgroups of $G$:

$\pi_0^G(\mathbb{S}) \simeq \underset{[H \subset G]}{\oplus} \pi_0^{W_G H}(\Sigma_+^\infty E (W_G H)) \simeq \mathbb{Z}[conjugacy\;classes\;of\;subgroups]$

(e.g. Schwede 15, p. 64)

## Properties

### Relation to Mackey functors

As $H$-varies over the subgroups of a $G$-equivariant spectrum $E$, the $H$-equivariant homotopy groups organize into a contravariant additive functor from the full subcategory of the equivariant stable homotopy category (called a Mackey functor)

$\underline{\pi}_\bullet(E) \colon G/H \mapsto \pi^H_\bullet(X) \,.$

(…)