nLab rational equivariant stable homotopy theory

Contents

Context

Representation theory

representation theory

geometric representation theory

Theorems

Homological algebra

homological algebra

Introduction

diagram chasing

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 $G$-equivariant spectra is equivalently given by homological algebra of Mackey functors (even for non-finite $G$). 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 $G$ be a finite group. For $X$ a G-spectrum, write $\pi_\bullet(X) \in \mathcal{M}[G]$ for its Mackey functor, the one which sends $G/H$ to the $H$-equivariant homotopy groups of $X$.

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

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

For $X,Y$ two $G$-spectra, there is a canonical morphism

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

When $Y$ 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 $n$ are all torsion, except at $n = 0$:

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

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

General:

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

• John Greenlees, Rational $O(2)$-equivariant cohomology theories. In Stable and unstable homotopy (Toronto, ON, 1996), volume 19 of Fields

Inst. Commun., pages 103–110. Amer. Math. Soc., Providence, RI, 1998. (web)

• John Greenlees, Rational $S^1$-equivariant stable homotopy theory, Memoirs of the AMS, 1999

• Brooke Shipley, An algebraic model for rational $S^1$-equivariant stable homotopy theory (pdf)

• David Barnes, Classifying Dihedral $O(2)$-Equivariant Spectra (arXiv:0804.3357)

• David Barnes, Rational $Z_p$-Equivariant Spectra, Algebr. Geom. Topol. 11 (2011) 2107-2135 (arXiv:1011.5785)

• David Barnes, Rational $O(2)$-Equivariant Spectra (arXiv:1201.6610)

• David Barnes, A monoidal algebraic model for rational $SO(2)$-spectra (arXiv:1412.1700)

• David Barnes, John Greenlees, Magdalena Kedziorek, Brooke Shipley, Rational $SO(2)$-Equivariant Spectra (arXiv:1511.03291)

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

For $G = SO(3)$ 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)