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naive G-spectrum

Contents

Context

Stable Homotopy theory

Representation theory

Contents

Idea

For GG a topological group, a naive GG-spectrum is a spectrum object in the category of G-spaces (e.g. Carlsson 92, def. 3.1). The stable homotopy theory of spectra with GG-action is part of the subject of equivariant stable homotopy theory.

There one typically considers a richer concept of G-spectra. In that context a spectrum with GG-action is a G-spectrum for the “trivial universe” and is called a naive G-spectrum. Naive G-spectra are obtained by stabilizing G-spaces with respect to the sphere S 1S^1, while (genuine) G-spectra are obtained by inverting all representation spheres.

Properties

Relation to GG-spectra

For UU a universe of GG-representations in the sense of genuine G-spectra, there is the inclusion

i:U GU i \;\colon\; U^G \longrightarrow U

of the fixed points. A genuine G-spectrum modeled on U GU^G is a spectrum with G-action (“naive G-spectrum”). The induced adjunction

(i *i *):GSpectraSpectra(PSh (Orb G)) (i_\ast \dashv i^\ast) \;\colon\; G Spectra \longrightarrow Spectra(PSh_\infty(Orb_G))

has unit and counit which are equivalences on the underlying bare spectra.

(e.g. Carlsson 92, p. 14, Greenlees May, p. 16)

Homotopy fixed points and homotopy quotients

Write

() G:Spectra(GTop)Spectra. (-)^G \;\colon\; Spectra(G Top) \longrightarrow Spectra \,.

for the fixed point (invariants) functor and

()/Q:Spectra(GTop)Spectra (-)/Q \;\colon\; Spectra(G Top) \longrightarrow Spectra

for the quotient (coinvariants) functor.

Write EGE G, as usual, for a contractible topological G-space whose GG-action is free. Write EG +E G_+ for this regarded as a pointed GG-space with a point freely adjoined.

Then there are two functor

EG +(),[EG +,]:Spectra(GTop)Spectra(GTop) EG_+ \wedge (-) , [EG_+,-] \;\colon\; Spectra(G Top) \to Spectra(G Top)

given by forming smash product with EG +E G_+ and forming the internal hom out of EG +E G_+.

Then the homotopy fixed point functor is

() hg[EG +,] G:Spectra(GTop)Spectra (-)^{h g} \coloneqq [E G_+, -]^G \;\colon\; Spectra(G Top) \longrightarrow Spectra

and the homotopy quotient functor is

()//G() hG(EG +())/G:Spectra(GTop)Spectra (-)//G \coloneqq (-)_{h G} \coloneqq (E G_+ \wedge (-))/G \;\colon\; Spectra(G Top) \longrightarrow Spectra

Relation to equivariant cohomology

Spectra with GG-action represent \mathbb{Z}-graded equivariant cohomology on G-spaces.

For XX a G-space and EE a spectrum with GG-action, then the corresponding cohomology is

E (X)π (Hom(X,E) G), E^\bullet(X) \coloneqq \pi_\bullet(Hom(X,E)^G) \,,

where on the right we have the homotopy fixed points of the mapping spectrum, which inherits a conjugation action by GG from the GG-action on XX and EE.

More abstractly, in terms of the tangent cohesive (∞,1)-topos TPSh (Orb) /BGT PSh_\infty(Orb)_{/\mathbf{B}G} of the slice (∞,1)-topos of orbispaces over BG\mathbf{B}G, this means that

E(X)=BG[X,E] E(X) = \underset{\mathbf{B}G}{\prod} [X,E]

is the dependent product over BG\mathbf{B}G of the intrinsic cohomology of the tangent slice topos. See at ∞-action for more on this.

Notice here Elmendorf's theorem which identifies G-spaces with (∞,1)-presheaves over the orbit category Orb GOrb_G. It is via this equivalence that spectra with GG-action represent equivariant cohomology in the form of Bredon cohomology.

Hence exhibiting a spectrum EE with GG-action as a spectrum-valued presheaf on the orbit category means to assign to any coset space G/HG/H of GG the HH-homotopy fixed points of EE:

E//G:G/HE H. E//G \;\colon\; G/H \mapsto E^{H} \,.

References

Created on December 19, 2017 at 07:27:58. See the history of this page for a list of all contributions to it.