geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
The generalization of the concept of suspension spectrum from stable homotopy theory to $G$-equivariant stable homotopy theory.
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:
the suspension $G$-pre-spectrum is $\Pi^\infty X \colon V \mapsto S^V \wedge X$;
the suspension $G$-spectrum is $\Sigma^\infty X \colon V \mapsto Q(S^V \wedge X)$
where $Q (-) = \underset{V \in U}{\cup} \Omega^V \Sigma^V X$.
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
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)
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.
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
between naive G-spectra and genuine G-spectra. The genuine $G$-suspension spectrum is the naive $G$-suspension spectrum followed by $i$:
The $G$-equivariant sphere spectrum is
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 $\mathbb{S}(V) \simeq S^V$.
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.
Andrew Blumberg, Ezample 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)
Last revised on December 4, 2018 at 00:45:49. See the history of this page for a list of all contributions to it.