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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.

Notation and conventions

Throughout this article we will fix a finite group G. All G-representations will be assumed to be finite-dimensional orthogonal (that is equipped with a G-invariant scalar product). A G-space will be a space X together with a continous action of G. A pointed G-space is a G-space together with a distinguished basepoint that is fixed by the action of G.

If X,Y are two G-spaces we will denote by [X,Y]_G the equivalence classes of G-equivariant map under G-equivariant homotopy (that is two maps f,g:X\to Y are G-homotopic if and only if there is an homotopy between them such that the map X\times [0,1]\to Y is G-equivariant where G acts trivially on [0,1]). If X and Y are pointed both the maps and the homotopies are required to respect the basepoint.

Orthogonal G-spectra

Intuitively a G-spectrum is a G-space together with a collections of deloopings indexed by the representations of G. This definition is presented in appendix A of Hill-Hopkins-Ravenel

Given two orthogonal G-representations V,W we let $Iso(V,W)$ be the space of (not necessarily G-equivariant) isometries 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 $i:V\to W$ 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

$\theta _{V,W}:Th(\xi (V,W))\wedge E(V)\to E(V')\,.$

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 category 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.

Loopspaces and suspensions

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

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

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).

For every subgroup H of G and every integer k we can define the H-equivariant k-th homotopy group of E as the colimit

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

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. A stable weak equivalence of G-spectra is a map of G-spectra inducing an isomorphism on all homotopy groups. The localization of the category of G-spectra is called the G-stable homotopy category.

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

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