nLab equivariant stable homotopy theory



Stable Homotopy theory

Representation theory



Equivariant stable homotopy theory over some topological group GG 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 GG-spectra. Notably a genuine GG-spectrum has homotopy groups graded not by the group of integers, but by the representation ring of GG (usually called RO(G)-grading).

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

cohomology in the presence of ∞-group GG ∞-action:

Borel equivariant cohomologyAAAAAA\phantom{AAA}\leftarrow\phantom{AAA}general (Bredon) equivariant cohomologyAAAAAA\phantom{AAA}\rightarrow\phantom{AAA}non-equivariant cohomology with homotopy fixed point coefficients
AAH(X G,A)AA\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}trivial action on coefficients AAAA[X,A] GAA\phantom{AA}[X,A]^G\phantom{AA}trivial action on domain space XXAAH(X,A G)AA\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 GG-action that is the direct sum of countably many copies of a given set of (finite dimensional? -DMR) representations of GG, at least containing the trivial representation on \mathbb{R} (so that UU contains at least a copy of \mathbb{R}^\infty).

Each such subspace of UU (representation contained in UU? -DMR) is called an indexing space (RO(G)-grading). For VWV \subset W indexing spaces, write WVW-V for the orthogonal complement of VV in WW. Write S VS^V for the one-point compactification of VV; and for XX any (pointed) topological space write Ω V:=[S V,X]\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 GG that fixes the base point. A morphism of GG-spaces is a continuous map that fixes the basepoints and is GG-equivariant.

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

A GG-spectrum EE (indexed on the chosen universe UU) is

  • for each indexing space VUV \subset U a GG-space EVE V;

  • for each pair VWV \subset W of indexing spaces a GG-equivariant homeomorphism

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

A morphism f:EEf : E \to E' of GG-spectra is for each indexing space VV a morphism of GG-spaces f V:EVEVf_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 ff of GG-spectra is a weak equivalence of GG-spectra if for every indexing space VV the component f Vf_V is a weak equivalence of GG-spaces (as defined above).

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

In terms of orthogonal spectra with GG-action

… (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(𝒞)Corr 1(𝒞)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:Corr 1 eff(𝒞)Spectra S \;\colon\; Corr_1^{eff}(\mathcal{C}) \longrightarrow Spectra

For GG a finite group and 𝒞=GSet\mathcal{C}= G Set its category of permutation representations, we have that SS is a genuine GG-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 𝒞\mathcal{C} an abelian category this definition reduces (Barwick 14) Mackey functors as originally defined in (Dress 71).


(equivariant) cohomologyrepresenting
equivariant cohomology
of the point *\ast
of classifying space BGB G
ordinary cohomology
HZBorel equivariance
H G (*)H (BG,)H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})
complex K-theory
KUrepresentation ring
KU G(*)R (G)KU_G(\ast) \simeq R_{\mathbb{C}}(G)
Atiyah-Segal completion theorem
R(G)KU G(*)compl.KU G(*)^KU(BG)R(G) \simeq KU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {KU_G(\ast)} \simeq KU(B G)
complex cobordism cohomology
MUMU G(*)MU_G(\ast)completion theorem for complex cobordism cohomology
MU G(*)compl.MU G(*)^MU(BG)MU_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {MU_G(\ast)} \simeq MU(B G)
algebraic K-theory
K𝔽 pK \mathbb{F}_prepresentation ring
(K𝔽 p) G(*)R p(G)(K \mathbb{F}_p)_G(\ast) \simeq R_p(G)
Rector completion theorem
R 𝔽 p(G)K(𝔽 p) G(*)compl.(K𝔽 p) G(*)^Rector 73K𝔽 p(BG)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{<a href="">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)
stable cohomotopy
K𝔽 1Segal 74K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq} SBurnside ring
𝕊 G(*)A(G)\mathbb{S}_G(\ast) \simeq A(G)
Segal-Carlsson completion theorem
A(G)Segal 71𝕊 G(*)compl.𝕊 G(*)^Carlsson 84𝕊(BG)A(G) \overset{\text{<a href="">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="">Carlsson 84</a>}}{\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

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 GG-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 GG-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

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 April 18, 2024 at 18:14:01. See the history of this page for a list of all contributions to it.