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In algebraic topology, a branch of mathematics, a G-spectrum for a finite group G is an object representing an equivariant generalized cohomology theory. There are different models for G-spectra, but they all determine the same homotopy theory.

Orthogonal G-spectra

There are various versions of orthogonal G-spectra. The version presented here is the one used in the solution of the Kervaire invariant one problem. Intuitively an orthogonal G-spectrum is a G-space together with a collections of deloopings indexed by the representations of G.

Given two orthogonal G-representations V,W we let $Iso(V,W)$ be the space of (not necessarily G-equivariant) isometric embeddings from V to W. This is a G-space with the natural conjugation action. There is a G-equivariant vector bundle $\xi (V,W)$ over $Iso(V,W)$ whose fiber over $i:V\hookrightarrow W$ is the orthogonal complement of the image of i in W. There is a natural map of vector bundles $\xi (V,W)\times \xi (W,U)\to \xi (V,U)$ lying above the composition map $Iso(V,W)\times Iso(W,U)\to Iso(V,U)$. This data allows us to construct a category $J_{G}$ enriched in G-spaces, whose objects are G-representations and such that $Map_{J_{G}}(V,W)$ is the Thom spaces $Th(\xi (V,W))$ of the vector bundle $\xi (V,W)\to Iso(V,W)$.

An orthogonal G-spectrum E is an enriched functor from $J_{G}$ to pointed G-spaces. Concretely it is the datum of

• For every finite-dimensional orthogonal G-representation V a pointed G-space $E_{V}$;
• For every pair of G-representations V,W a G-equivariant map

$Th(\xi (V,W))\wedge E_{V}\longrightarrow E_{W}\,.$

These maps are required to satisfy the obvious associativity and unitality constraints. The Thom space $Th(\xi (V,W))$ should be thought of as the "union" of $S^{W-iV}$ across all possible isometric embeddings i.

The category of orthogonal G-spectra has a natural symmetric monoidal structure given by the Day convolution using the direct sum monoidal structure in the source and the smash product of pointed G-spaces in the target. This product is called the smash product of G-spectra.

Homotopy groups

In analogy with the homotopy groups of a spectrum, we can define for every integer n the n-th homotopy Mackey functor of a G-spectrum E. It is the Mackey functor whose value on a G-set U is the colimit

$\pi _{k}(E)(U)=\mathrm {colim} _{V\supseteq -k}[U_{+}\wedge S^{V+k},X_{V}]_{G}.$

where V runs through all H-representations equipped with a subrepresentation isomorphic to the direct sum of -k copies of the trivial representation (this condition is empty when k≥0) and V+k is either the direct sum of V with k copies of the trivial representation (if k≥0) or the orthogonal complement of the aforementioned subrepresentation. Specializing to the case of an orbit U=G/H we can define the H-equivariant k-th homotopy group of E as

$\pi _{k}^{H}(E)=\pi _{k}(E)(G/H)=\mathrm {colim} _{V\supseteq -k}[S^{V+k},X_{V}]_{H}$

A stable equivalence of G-spectra is a map of G-spectra inducing an isomorphism on all homotopy Mackey functors. The localization of the category of G-spectra at the stable equivalences is called the G-stable homotopy category.

There is also a variant of the homotopy groups using G-representations. For every virtual G-representation V the V-th homotopy group of a G-spectrum E is

$\pi _{V}(E)=\mathrm {colim} _{W\supseteq -V}[S^{W\oplus V},E_{W}]_{G}\,.$

Care needs to be taken that, unlike the homotopy Mackey functors, the RO(G)-indexed homotopy groups in general do not detect stable equivalences (that is the analogue of the Whitehead theorem is false). The RO(G)-graded group of $\pi _{V}(E)$ for all representations is denoted $\pi _{\star }(E)$.

Similarly to the nonequivariant case we can define for every spectrum E the corresponding homology and cohomology theory

$E_{*}X=\pi _{*}(E\wedge X),\qquad E^{*}X=\pi _{*}F(X,E)\,.$

Examples

• For every pointed G-space X we can define its suspension spectrum as

$(\Sigma ^{\infty }X)(V)=S^{V}\wedge X$

In particular the suspension spectrum of the zero sphere with trivial action is called the sphere spectrum and denoted by $\mathbb {S}$, as in the nonequivariant case. The suspension spectrum functor has a right adjoint, denoted $\Omega ^{\infty }$ given by.

$\mathrm {hocolim} _{V}\Omega ^{V}E_{V}\,.$

• For every V=[V₀-V₁] virtual representation of G, we can define a representation sphere $\mathbb {S} ^{V}$ as

$\mathbb {S} ^{V}(W)=\mathrm {colim} _{U}\Omega ^{V_{1}\oplus U}S^{V_{0}\oplus U\oplus W}$

where we write S^W to denote the one-point compactification of the representation W.

• For every Mackey functor M there is a G-spectrum HM such that

$\pi _{i}HM={\begin{cases}M&{\textrm {if}}\ \ i=0\\0&{\textrm {otherwise}}\end{cases}}$

called the Eilenberg-MacLane G-spectrum of M. It represents equivariant cohomology with coefficients in M.

• There exists a C₂-spectrum KR called real K-theory spectrum representing KR-theory. It satisfies a form of Bott periodicity given by $S^{\rho }\wedge KR\cong KR$ where ρ is the regular representation of C₂.
• Every orthogonal G-spectrum E has a canonical representation as

$E=\mathrm {hocolim} _{V}\mathbb {S} ^{-V}\wedge \Sigma ^{\infty }E_{V}$

G-spectra as infinite loop spaces

For any G-spectrum E we can define the underlying space

$\Omega ^{\infty }E=\mathrm {hocolim} _{V}\Omega ^{V}E(V)$

where the homotopy colimit is computed over the topological category of G-representations. More generally for every G-representation W we can write

$\Omega ^{\infty -W}E=\mathrm {hocolim} _{V}\Omega ^{V}E(V\oplus W)$

so that

$\Omega ^{\infty }E\cong \Omega ^{W}(\Omega ^{\infty -W}E)\,.$

Hence an orthogonal G-spectrum should be thought of as a G-space $\Omega ^{\infty }E$ together a family of deloopings for every G-representation (similarly to how a spectrum consists of a space together with a family of iterated deloopings).

Fixed points

If E is an orthogonal G-spectrum, its Lewis-May fixed points are the spectrum E^G whose n-space is (E^G)_n = E(\mathbb{R}^n)^G and with the maps...

G-spectra as spectral Mackey functors

An important observation is that the suspension spectra of finite G-sets span a subcategory of the G-equivariant stable homotopy category which is equivalent to the Burnside category. So the homotopy groups of a G-spectrum can be assemble to for an homotopy Mackey functor. In fact this is true even if we consider the topological subcategory provided we

Theorem (Guillou-May, Schwede-Shipley): The category of orthogonal G-spectra is equivalent to the category of additive topological functors from the Burnside category to spectra.

Under this point of view a lot of the constructions we have described above are evident. For example if E is a spectral Mackey functor its Lewis-May fixed points $E^{H}$ are simply the value of the spectral Mackey functor on the G-set G/H. The geometric fixed points instead are obtained as a left Kan extension ...

The norm

Let H be a subgroup of G. If E is an orthogonal H-spectrum we can define the norm as the G-spectrum

$N_{H}^{G}E(V)=\mathrm {hocolim} _{V}\Omega ^{\mathrm {ind} _{H}^{G}V}\Sigma ^{\infty }N_{H}^{G}(X(V))$

Borel equivariant G-spectra

Sometimes the name G-spectrum is used to mean a spectrum with an action of the group G. These objects form in fact a full subcategory of the category of G-spectra, corresponding to those E such that the map $E^{H}\to E^{hH}$ is an equivalence for every subgroup H.

References

• Hill, Micheal A.; Hopkins, Micheal J.; Ravenel, Douglas C. (2016). "On the nonexistence of elements of Kervaire invariant one". Annals of Mathematics. 184 (1): 1–262. doi:10.4007/annals.2016.184.1.1.
• Lewis, L. Gaunce; May, Peter; Steinberger, Mark (1986). Equivariant Stable Homotopy Theory. Springer Lecture Notes in Mathematics.