User:DenisNardin/sandbox

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.
 * $$\mathrm{hocolim}_V \Omega^VE_V\,.$$


 * 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 SW to denote the one-point compactification of the representation W.
 * For every Mackey functor M there is a G-spectrum HM such that
 * $$\pi_iHM=\begin{cases} M &\textrm{ if }\ \ i=0\\ 0 &\textrm{otherwise}\end{cases}$$

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
 * $$\Omega^\infty E=\mathrm{hocolim}_V \Omega^VE(V)$$

where the homotopy colimit is computed over the topological category of G-representations. More generally for every G-representation W we can write
 * $$\Omega^{\infty-W} E = \mathrm{hocolim}_V \Omega^VE(V\oplus W)$$

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
 * $$N^G_HE(V)=\mathrm{hocolim}_V \Omega^{\mathrm{ind}_H^GV} \Sigma^\infty N^G_H(X(V))$$

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.