nLab equivariant stable homotopy theory

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 …

In terms of orthogonal spectra with $G$-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 $Corr_1^{eff}(\mathcal{C}) \hookrightarrow Corr_1(\mathcal{C})$ 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 sphere spectrum, equivariant stable cohomotopy

• equivariant bordism homology theory, equivariant cobordism cohomology theory

• equivariant connective topological K-theory

(equivariant) cohomology	representing spectrum	equivariant cohomology of the point $\ast$	cohomology of classifying space $B G$
(equivariant) ordinary cohomology	HZ	Borel equivariance $H^\bullet_G(\ast) \simeq H^\bullet(B G, \mathbb{Z})$
(equivariant) complex K-theory	KU	representation 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 $R_{\mathbb{F}_p}(G) \simeq K (\mathbb{F}_p)_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {(K \mathbb{F}_p)_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Rector+completion+theorem">Rector 73</a>}}{\simeq} \!\!\!\!\!\! K \mathbb{F}_p(B G)$
(equivariant) stable cohomotopy	$K \mathbb{F}_1 \overset{\text{<a href="stable cohomotopy#StableCohomotopyIsAlgebraicKTheoryOverFieldWithOneElement">Segal 74</a>}}{\simeq}$ S	Burnside ring $\mathbb{S}_G(\ast) \simeq A(G)$	Segal-Carlsson completion theorem $A(G) \overset{\text{<a href="https://ncatlab.org/nlab/show/Burnside+ring+is+equivariant+stable+cohomotopy+of+the+point">Segal 71</a>}}{\simeq} \mathbb{S}_G(\ast) \overset{ \text{compl.} }{\longrightarrow} \widehat {\mathbb{S}_G(\ast)} \!\! \overset{\text{<a href="https://ncatlab.org/nlab/show/Segal-Carlsson+completion+theorem">Carlsson 84</a>}}{\simeq} \!\!\!\!\!\! \mathbb{S}(B G)$

• KR cohomology theory

• equivariant complex cobordism cohomology theory

• equivariant Morava K-theory

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.)

Related concepts

• equivariant differential topology

• global equivariant homotopy theory

• rational equivariant stable homotopy theory

• equivariant motivic homotopy theory

• equivariant spectrum,

  equivariant sphere spectrum, equivariant suspension spectrum, equivariant homotopy group, equivariant stable homotopy category, tom Dieck splitting, slice spectral sequence

• G-spectra, naive G-spectra, spectra with G-action

  G-spaces

• Burnside category, Burnside ring

  Mackey functor

• Parametrized Higher Category Theory and Higher Algebra

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:

• L. Gaunce Lewis, Peter May, Mark Steinberger (with contributions by J.E. McClure), Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics 1213 (1986) [pdf, doi:10.1007/BFb0075778]

• Tammo tom Dieck, Section II.6 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

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

• Michael Mandell, Peter May, Equivariant orthogonal spectra and S-modules, Mem. Amer. Math. Soc. 159 (2002), no. 755, x+108. MR 2003i:55012 (pdf, K-theory archive)

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

• Michael Hill, Michael Hopkins, Douglas Ravenel, On the non-existence of elements of Kervaire invariant one, Annals of Mathematics 184 1 (2016)[doi:10.4007/annals.2016.184.1.1, arXiv:0908.3724, talk slides]

• Michael Hill, Michael Hopkins, Douglas Ravenel, Equivariant Stable Homotopy Theory and the Kervaire Invariant Problem, New Mathematical Monographs, Cambridge University Press (2021) [doi:10.1017/9781108917278]

Lecture notes are in

• Andrew Blumberg, The Burnside category, 2017 (pdf, GitHub)

Further introductions and surveys include the following

• Gunnar Carlsson, A survey of equivariant stable homotopy theory,Topology, Vol 31, No. 1, pp. 1-27, 1992 (pdf)

• Anna Marie Bohmann, Basic notions of equivariant stable homotopy theory (pdf)

• John Greenlees, Peter May, Equivariant stable homotopy theory, in Ioan James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

• Brooke Shipley, An introduction to equivariant homotopy theory (pdf)

• Talbot workshop 2016

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

• Stefan Schwede, Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)

An alternative perspective on this is in

• Mark Hovey, David White, An alternative approach to equivariant stable homotopy theory (arXiv:1312.3846)

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

• Halvard Fausk, Equivariant homotopy theory for pro-spectra (arXiv:math/0609635)

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

• Andrew Blumberg, Continuous functors as a model for the equivariant stable homotopy category (arXiv:math.AT/0505512)

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

• Stefan Schwede, Brooke Shipley, Classification of stable model categories (pdf)

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:

• Clark Barwick, Spectral Mackey functors and equivariant algebraic K-theory (I), Adv. Math., 304:646–727 (arXiv:1404.0108)

• Clark Barwick, Saul Glasman, Denis Nardin, Spectral Mackey functors and equivariant algebraic K-theory (II), (arxiv:1505.03098)

A universal property characterizing equivariant stable homotopy theory:

• Tim Campion, Free duals and a new universal property for stable equivariant homotopy theory [arXiv:2302.04207]