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equivariant stable homotopy category

Contents

Context

Stable Homotopy theory

Representation theory

Contents

Idea

The generalization of the stable homotopy category from stable homotopy theory to equivariant stable homotopy theory.

Definition

A homomorphism f:E 1E 2f \colon E_1 \longrightarrow E_2 between two G-spectra, indexed on a G-universe 𝒰\mathcal{U}, is called an equivariant weak homotopy equivalence if the following equivalent conditions hold

  1. For each V𝒰V\in \mathcal{U} the component map f V:ff_V \colon f induces ordinary weak homotopy equivalences (E 1) V H(E 2) V H(E_1)_V^H \to (E_2)_V^H on all fixed point spaces for all closed subgroups HGH \hookrightarrow G.

  2. For each nn \in \mathbb{Z} and each closed subgroup HGH \hookrightarrow G the morphism ff induces an isomorphism of Mackey functors of equivariant homotopy groups π n H(E 1)π n H(E 2)\pi_n^H(E_1) \stackrel{\simeq}{\longrightarrow} \pi_n^H(E_2).

(The equivalence of these conditions is part of the equivariant Whitehead theorem.)

The GG-equivariant stable homotopy category is the homotopy category of G-spectra with respect to these weak equivalences.

Properties

Relation to Mackey functors

The full subcategory G\mathcal{B}_G of the equivariant stable homotopy category on the objects of the form

G/H *Σ S n G/H_* \wedge \Sigma^\infty S^n

is, as an additive category, the domain of Mackey functors, such as the equivariant homotopy group-functors.

References

  • John Greenlees, Peter May, section 2 of Equivariant stable homotopy theory, in I.M. James (ed.), Handbook of Algebraic Topology , pp. 279-325. 1995. (pdf)

  • Anna Marie Bohmann, Basic notions of equivariant stable homotopy theory, (pdf)

Last revised on January 16, 2016 at 06:07:18. See the history of this page for a list of all contributions to it.