nLab rational equivariant stable homotopy theory

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:

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

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

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 $n = 0$:

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

(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.

Related concepts

• rational homotopy theory

• rational equivariant homotopy theory

  • rational topological G-space

  • model structure on equivariant dgc-algebras

• rational equivariant stable homotopy theory

• rational stable homotopy theory

• rational parameterized stable homotopy theory

References

General:

• John Greenlees, Peter May, appendix A of Generalized Tate cohomology, Mem. Amer. Math. Soc. 113 (1995) no 543 (pdf)

• John Greenlees, p. 3 of Triangulated categories of rational equivariant cohomology theories (pdf, pdf)

• David Barnes, Rational Equivariant Spectra (arXiv:0802.0954)

• David Barnes, Classifying Rational $G$-Spectra for Finite $G$ (arXiv:0812.0317)

• David Barnes, Magdalena Kedziorek, An introduction to algebraic models for rational G-spectra (arXiv:2004.01566)

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:

• John Greenlees, Brooke Shipley, An algebraic model for rational torus-equivariant spectra (arXiv:1101.2511)

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, GBooks)

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

• Magdalena Kedziorek, An algebraic model for rational $SO(3)$–spectra, Algebraic & Geometric Topology 17 (2017) 3095–3136 (arXiv:1611.08415, doi:10.2140/agt.2017.17.3095)

Discussion of commutative ring-structures with an eye towards rational equivariant K-theory:

• Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, Magdalena Kędziorek, Clover May, Naive-commutative structure on rational equivariant K-theory for abelian groups (arXiv:2002.01556)

• Anna Marie Bohmann, Christy Hazel, Jocelyne Ishak, Magdalena Kędziorek, Clover May, Genuine-commutative ring structure on rational equivariant K-theory for finite abelian groups (arXiv:2104.01079)