nLab equivariant stable homotopy theory

Context

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.)

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

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: