# nLab equivariant stable homotopy theory

Contents

### Context

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Representation theory

representation theory

geometric representation theory

# Contents

## Idea

Equivariant stable homotopy theory over some topological group $G$ is the stable homotopy theory of G-spectra. This includes the naive G-spectra which constitute the actual stabilization of equivariant homotopy theory, but is more general, one speaks of genuine $G$-spectra. Notably a genuine $G$-spectrum has homotopy groups graded not by the group of integers, but by the representation ring of $G$ (usually called RO(G)-grading).

The concept of cohomology in equivariant stable homotopy theory is equivariant cohomology:

cohomology in the presence of ∞-group $G$ ∞-action:

Borel equivariant cohomology$\phantom{AAA}\leftarrow\phantom{AAA}$general (Bredon) equivariant cohomology$\phantom{AAA}\rightarrow\phantom{AAA}$non-equivariant cohomology with homotopy fixed point coefficients
$\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}$trivial action on coefficients $A$$\phantom{AA}[X,A]^G\phantom{AA}$trivial action on domain space $X$$\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}$

## Basic definitions

### In terms of looping by representation spheres

The definition of G-spectrum is typically given in generalization of the definition of coordinate-free spectrum.

A G-universe in this context is (e.g. Greenlees-May, p. 10) an infinite dimensional real inner product space equipped with a linear $G$-action that is the direct sum of countably many copies of a given set of (finite dimensional? -DMR) representations of $G$, at least containing the trivial representation on $\mathbb{R}$ (so that $U$ contains at least a copy of $\mathbb{R}^\infty$).

Each such subspace of $U$ (representation contained in $U$? -DMR) is called an indexing space (RO(G)-grading). For $V \subset W$ indexing spaces, write $W-V$ for the orthogonal complement of $V$ in $W$. Write $S^V$ for the one-point compactification of $V$; and for $X$ any (pointed) topological space write $\Omega^V := [S^V,X]$ for the corresponding (based) sphere space.

A G-space in the following means a pointed topological space equipped with a continuous action of the topological group $G$ that fixes the base point. A morphism of $G$-spaces is a continuous map that fixes the basepoints and is $G$-equivariant.

A weak equivalence of $G$-spaces is a morphism that induces isomorphism on all $H$-fixed homotopy groups (…)

A $G$-spectrum $E$ (indexed on the chosen universe $U$) is

• for each indexing space $V \subset U$ a $G$-space $E V$;

• for each pair $V \subset W$ of indexing spaces a $G$-equivariant homeomorphism

$E V \stackrel{\simeq}{\to} \Omega^{W-V} E W \,.$

A morphism $f : E \to E'$ of $G$-spectra is for each indexing space $V$ a morphism of $G$-spaces $f_V : E V \to E' V$, such that this makes the obvious diagrams commute.

This becomes a category with weak equivalences by setting:

a morphism $f$ of $G$-spectra is a weak equivalence of $G$-spectra if for every indexing space $V$ the component $f_V$ is a weak equivalence of $G$-spaces (as defined above).

This may be expressed directly in terms of the notion of homotopy group of a $G$-spectrum: this is …

… (Schwede 15)…

### In terms of Mackey-functors

A Mackey functor with values in spectra (“spectral Mackey functor”) is an (∞,1)-functor on a suitable (∞,1)-category of correspondences $Corr_1^{eff}(\mathcal{C}) \hookrightarrow Corr_1(\mathcal{C})$ which sends coproducts to smash product. (This is similar to the concept of sheaf with transfer.)

$S \;\colon\; Corr_1^{eff}(\mathcal{C}) \longrightarrow Spectra$

For $G$ a finite group and $\mathcal{C}= G Set$ its category of permutation representations, we have that $S$ is a genuine $G$-equivariant spectrum (Guillou-May 11). So in this case the homotopy theory of spectral Mackey functors is a presentation for equivariant stable homotopy theory (Guillou-May 11, Barwick 14).

For $\mathcal{C}$ an abelian category this definition reduces (Barwick 14) Mackey functors as originally defined in (Dress 71).

## Examples

(equivariant) cohomologyrepresenting
spectrum
equivariant cohomology
of the point $\ast$
cohomology
of classifying space $B G$
(equivariant)
ordinary cohomology
HZBorel equivariance
$H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant)
complex K-theory
KUrepresentation ring
$KU_G(\ast) \simeq R_{\mathbb{C}}(G)$
Atiyah-Segal completion theorem
$R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)$
(equivariant)
complex cobordism cohomology
MU$MU_G(\ast)$completion theorem for complex cobordism cohomology
$MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)$
(equivariant)
algebraic K-theory
$K \mathbb{F}_p$representation ring
$(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)$
Rector completion theorem
$R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{Rector 73}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant)
stable cohomotopy
$K \mathbb{F}_1 \overset{\text{Segal 74}}{\simeq}$ SBurnside ring
$\mathbb{S}_G(\ast) \simeq A(G)$
Segal-Carlsson completion theorem
$A(G) \overset{\text{Segal 71}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{Carlsson 84}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

## Equivariant cohomology

The notion of cohomology relevant in equivariant stable homotopy theory is the flavor of equivariant cohomology (see there for details) called Bredon cohomology. (See also at orbifold cohomology.)

## References

Original articles include

• Graeme Segal, Equivariant stable homotopy theory, Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 2, pp. 59–63. Gauthier-Villars, Paris, 1971. (pdf)

Textbook accounts:

and a more modern version taking into account the theory of symmetric monoidal categories of spectra is in

Solving the Arf-Kervaire invariant problem with methods of equivariant stable homotopy theory (and reviewing these):

Lecture notes are in

Further introductions and surveys include the following

Lecture notes on G-spectra modeled as orthogonal spectra with $G$-actions are

An alternative perspective on this is in

Generalization from equivariance under compact Lie groups to compact topological groups (Hausdorff) and in particular to profinite groups and pro-homotopy theory is in

The May recognition theorem for G-spaces and genuine G-spectra is discussed in

• Costenoble and Warner, Fixed set systems of equivariant infinite loop spaces Transactions of the American mathematical society, volume 326, Number 2 (1991) (JSTOR)

Characterization of G-spectra via excisive functors on G-spaces is in

The characterization of $G$-equivariant functors in terms of topological Mackey functors is discussed in example 3.4 (i) of

A construction of equivariant stable homotopy theory in terms of spectral Mackey functors is due to

• Bert Guillou, Peter May, Models of $G$-spectra as presheaves of spectra, (arXiv:1110.3571)

Permutative $G$-categories in equivariant infinite loop space theory (arXiv:1207.3459)

see at spectral Mackey functor for more references.

A fully (∞,1)-category theoretic formulation:

A universal property characterizing equivariant stable homotopy theory:

Last revised on March 2, 2023 at 16:46:08. See the history of this page for a list of all contributions to it.