nLab
rational equivariant stable homotopy theory

Contents

Context

Stable Homotopy theory

Representation theory

Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

Rational equivariant stable homotopy theory is the study of equivariant spectra just on the level of their rationalization, hence concerning only their non-torsion homotopy groups. This is the equivariant and stable version of rational homotopy theory.

A key general statement of the theory is that rationally the homotopy theory of GG-equivariant spectra is equivalently given by homological algebra of Mackey functors (even for non-finite GG). At the level of equivalences of homotopy categories this was established by John Greenlees, at the level of zig-zags of Quillen equivalences of model categories this was established Greenlees and by his students, David Barnes and Magdalena Kedziorek.

Properties

Greenlees-May splitting into equivariant Eilenberg-MacLane spectra

Let GG be a finite group. For XX a G-spectrum, write π (X)[G]\pi_\bullet(X) \in \mathcal{M}[G] for its Mackey functor, the one which sends G/HG/H to the HH-equivariant homotopy groups of XX.

Every rational GG-equivariant spectrum EE is the direct sum of the Eilenberg-MacLane spectra (Mackey functors) on its equivariant homotopy groups:

E nΣ nHπ n(E). E \simeq \prod_n \Sigma^n H\pi_n(E) \,.

(Greenlees-May 95, theorem A.1, Greenlees, theorem 5.1)

For X,YX,Y two GG-spectra, there is a canonical morphism

[X,Y] GnHom [G](π n(X),π n(Y)). [X,Y]_G \longrightarrow \underset{n}{\prod} Hom_{\mathcal{M}[G]}(\pi_n(X),\pi_n(Y)) \,.

When YY is rational, then this is an isomorphism (Greenlees-May 95, theorem A.4).

Rational tom Dieck splitting

for the moment see at tom Dieck splitting

Examples

Just as for the plain sphere spectrum, the equivariant homotopy groups of the equivariant sphere spectrum in ordinary integer degrees nn are all torsion, except at n=0n = 0:

π n H(𝕊)={ forn=0 0 otherwise \pi_n^H(\mathbb{S})\otimes \mathbb{Q} = \left\{ \array{ \cdots & for \; n = 0 \\ 0 & otherwise } \right.

(Greenlees-May 95, prop. A.3)

But in some RO(G)-degrees there may appear further non-torsion groups, see at equivariant sphere spectrum the section Examples.

References

General:

For G=O(2)G = O(2) or SO(2)SO(2) and so also for G=G = dihedral group and cyclic group:

For G=(S 1) × nG = (S^1)^{\times_n} a torus:

For G=SO(3)G = SO(3) and hence also for the finite groups of ADE type:

  • John Greenlees, Rational SO(3)-Equivariant Cohomology Theories, in Homotopy methods in algebraic topology (Boulder, CO, 1999), Contemp. Math. 271, Amer. Math. Soc. (2001) 99 (web)

  • Magdalena Kedziorek, Algebraic models for rational G-spectra, 2014 (web)

Last revised on September 17, 2018 at 08:36:15. See the history of this page for a list of all contributions to it.