nLab equivariant suspension spectrum

Idea

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

Definition

As a spectrum indexed on a $G$-universe

Let $X$ be a pointed topological G-space. For a representation $V$ in the G-universe $U$, write $S^V$ for its representation sphere.

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

1. the suspension $G$-pre-spectrum is $\Pi^\infty X \colon V \mapsto S^V \wedge X$;

2. the suspension $G$-spectrum is $\Sigma^\infty X \colon V \mapsto Q(S^V \wedge X)$

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

(Greenlees-May 95)

As an orthogonal spectrum

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

$$(\Sigma^\infty_G X)_n \coloneqq X \wedge S^n$$

of $X$ with the n-sphere, equipped with the canonical action of the orthogonal group $O(n)$ just on the $S^n$-factor and equipped with the given action of $G$ on just $X$.

(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 $G$-equivariant homotopy theory to $G$-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.

$RO(G)$-degrees

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

$$(\Sigma^\infty_G X)(V) \simeq X \wedge S^V$$

Relation between genuine and Bredon-equivariant suspension

For $U$ a complete $G$-universe and $U^G$ its fixed point universe, then the inclusion $i \colon U^G \longrightarrow U$ induces an adjunction

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

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

$$i_\ast \circ \Sigma^\infty_{U^G } \simeq \Sigma^\infty_U \,.$$

(Greenlees-May 95, p. 16)

Examples

The $G$-equivariant sphere spectrum is

$$\mathbb{S} = \Sigma^\infty_G S^0$$

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

Equivariant homotopy groups and fixed point spectra

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

References

• Andrew Blumberg, Example 2.5.9 in Equivariant homotopy theory, 2017 (pdf, GitHub)

• John Greenlees, Peter May, Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

• Stefan Schwede, Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)

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

• Denis Nardin, section A.2 of Stability and distributivity over orbital ∞-categories, 2012 (hdl.handle.net/1721.1/112895, pdf)