nLab
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

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

Last revised on December 4, 2018 at 00:45:49. See the history of this page for a list of all contributions to it.