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equivariant suspension spectrum

Contents

Context

Stable Homotopy theory

Representation theory

Contents

Idea

The generalization of the concept of suspension spectrum from stable homotopy theory to GG-equivariant stable homotopy theory.

Definition

As a spectrum indexed on a GG-universe

Let XX be a pointed topological G-space. For a representation VV in the G-universe UU, write S VS^V for its representation sphere.

As a G-spectrum indexed on a G-universe:

  1. the suspension GG-pre-spectrum is Π X:VS VX\Pi^\infty X \colon V \mapsto S^V \wedge X;

  2. the suspension GG-spectrum is Σ X:VQ(S VX)\Sigma^\infty X \colon V \mapsto Q(S^V \wedge X)

where Q()=VUΩ VΣ VXQ (-) = \underset{V \in U}{\cup} \Omega^V \Sigma^V X.

(Greenlees-May 95)

As an orthogonal spectrum

The equivariant suspension spectrum Σ G X\Sigma^\infty_G X of a pointed topological G-space XX is the G-spectrum which, modeled as an orthogonal spectrum with GG-action, is in degree nn given by the smash product

(Σ G X) nXS n (\Sigma^\infty_G X)_n \coloneqq X \wedge S^n

of XX with the n-sphere, equipped with the canonical action of the orthogonal group O(n)O(n) just on the S nS^n-factor and equipped with the given action of GG on just XX.

(e.g. Schwede 15, example 2.11)

Properties

Monoidalness

In generalization to the strong monoidal-structure on the ordinary suspension spectrum functor with respect to the symmetric monoidal smash product of spectra (see there) also the equivariant suspension spectrum functor ought to constitute a monoidal (infinity,1)-functor from GG-equivariant homotopy theory to GG-equivariant stable homotopy theory.

This follows from general properties of stabilization when regarding equivariant stable homotopy theory as the result of inverting smash product with all representation spheres, via Robalo 12, last clause of Prop. 4.1 with last clause of Prop. 4.10 (1), generalized to sets of objects as in Hoyois 15, section 6.1, see also Hoyois 15, Def. 6.1.

Alternatively, under the equivalence of genuine G-spectra with spectral Mackey functors on the Burnside category, it follows as in Nardin 12, Remark A.12.

RO(G)RO(G)-degrees

For VV an orthogonal linear GG-representation then the value of the equivariant suspension spectrum in that RO(G)-degree is the smash product of XX with the corresponding representation sphere.

(Σ G X)(V)XS V (\Sigma^\infty_G X)(V) \simeq X \wedge S^V

Relation between genuine and Bredon-equivariant suspension

For UU a complete GG-universe and U GU^G its fixed point universe, then the inclusion i:U GUi \colon U^G \longrightarrow U induces an adjunction

Spectra(GTop)i *i *GSpectra Spectra(G Top) \stackrel{\overset{i^\ast}{\longleftarrow}}{\underset{i_\ast}{\longrightarrow}}G Spectra

between naive G-spectra and genuine G-spectra. The genuine GG-suspension spectrum is the naive GG-suspension spectrum followed by ii:

i *Σ U G Σ U . i_\ast \circ \Sigma^\infty_{U^G } \simeq \Sigma^\infty_U \,.

(Greenlees-May 95, p. 16)

Examples

The GG-equivariant sphere spectrum is

𝕊=Σ G S 0 \mathbb{S} = \Sigma^\infty_G S^0

for S 0S^0 regarded as equipped with the (necessarily) trivial GG-action. It follows that for VV an orthogonal linear GG-representation then in RO(G)-degree VV the equivariant sphere spectrum is the corresponding representation sphere 𝕊(V)S V\mathbb{S}(V) \simeq S^V.

Equivariant homotopy groups and fixed point spectra

The equivariant homotopy groups and the fixed point spectra of equivariant suspension spectra Σ G X\Sigma^\infty_G X decompose into the naive fixed points of the GG-action on XX. This is the tom Dieck splitting, see there for details.

References

Discussion in terms of spectral Mackey functors on the Burnside category:

Last revised on January 4, 2019 at 17:59:45. See the history of this page for a list of all contributions to it.