nLab tom Dieck splitting

Idea

The objects of equivariant stable homotopy theory – genuine G-spectra – are very rich: Already the fixed point spectra of equivariant suspension spectra contain considerably more information than just the suspension spectra of the plain underlying fixed point spaces – the latter are just the geometric fixed point spectra.

The tom Dieck splitting (tom Dieck 75, Lewis-May-Steinberger 86, V.11) gives an explicit description of all the wedge summands appearing in fixed point spectra of equivariant suspension spectra. These wedge sums start out with the geometric fixed point spectra and then have one summand for each conjugacy class of subgroups $H \subset G$, given by the plain suspension spectra of the homotopy quotient of the $H$-fixed point spaces by the corresponding Weyl group-action.

Induced from this wedge sum splitting formula for the spectra themselves is a corresponding direct sum-formula of the equivariant stable homotopy groups in terms of plain stable homotopy groups.

The richness of this splitting, hence of G-spectra, is witnessed by its simplest non-trivial example, which is the equivariant stable homotopy groups of the equivariant sphere spectrum: This yields the abelian group underlying the Burnside ring, which is freely generated from the conjugacy classes of subgroups of $G$ (see below).

Statement

For $H\subset G$ a subgroup, write

• $[H \subset G]$ for its conjugacy class;

• $W_G H \coloneqq (N_G H)/H$ for its Weyl group, the quotient group of the normalizer subgroup (of $H$ in $G$) by $H$;

• $E W_G H$ for the universal principal bundle of the Weyl group.

Of fixed point spectra of equivariant suspension spectra

The fixed point spectrum of an equivariant suspension spectrum is given by the wedge sum formula

$$F^G(\Sigma^\infty_G X) \simeq \underset{[H\subset G]}{\bigvee} \Sigma^\infty( E (W_G H)_+ \wedge_{W_G H} X^H )$$

where $\Sigma^\infty$ is the plain suspension spectrum construction.

(e.g. Guillou-May 12, theorem 5.3, Schwede 15, example 7.7)

In particular, since $W_G G = 1$ and $W_G 1 = G$, the extremal summands for $H = G$ and $H = 1$ are just the suspension spectrum of the plain fixed point space $X^G$ and of the homotopy quotient $X\sslash G$ (equivalently the Borel construction $X \sslash G \simeq E G_+ \times_G X$ ) of the full space, respectively:

$$F^G(\Sigma^\infty_G X) \simeq \Sigma^\infty( X^G ) \vee \left( \underset{{[H\subset G]} \atop {1 \neq H \neq G}}{\bigvee} \Sigma^\infty( E (W_G H)_+ \wedge_{W_G H} X^H ) \right) \vee \Sigma^\infty( E G_+ \wedge_{G} X ) \,.$$

Here the first summand is the geometric fixed point spectrum inside the full fixed point spectrum

$$\Phi^G(\Sigma^\infty_G X) \;\simeq\; \Sigma^\infty( X^G ) \hookrightarrow F^G(\Sigma^\infty_G X)$$

(Schwede 15, Example 7.7)

For equivariant homotopy groups

It follows that for $X$ a pointed topological G-space, its equivariant homotopy groups are

$$\begin{aligned} \pi_\bullet^G(\Sigma^\infty X) & \simeq \underset{[H \subset G]}{\bigoplus} \pi_\bullet^{W_G H}(\Sigma^\infty (E (W_G H)_+ \wedge X^H)) \\ &\simeq \pi_\bullet(\Sigma^\infty X^G) \oplus \underset{{[H \subset G]} \atop {1 \neq H \neq G}}{\bigoplus} \pi_\bullet^{W_G H}(\Sigma^\infty (E (W_G H)_+ \wedge X^H)) \oplus \pi_\bullet^{G}(\Sigma^\infty (E G_+ \wedge X)) \end{aligned}$$

where the direct sum is over conjugacy classes of subgroups $H$ of $G$.

(e.g. Schwede 15, theorem 6.12)

For rational equivariant homotopy theory

For $G$ finite and for rational equivariant stable homotopy theory this becomes (Greenlees, 6.2)

$$G RationalSpectra \simeq \underset{[H \subset G] }{\prod} \mathbb{Q}(W_G H) Mod$$

where the product is over conjugacy classes of subgroups $H$ of $G$ and $W_G H$ denotes the Weyl group of $H$ in $G$

Examples

For the equivariant sphere spectrum

For the equivariant sphere spectrum $\mathbb{S} = \Sigma^\infty_G S^0$ the tom Dieck splitting says that its 0th equivariant homotopy group is the free abelian group on the set of conjugacy classes of subgroups of $G$:

$$\pi_0^G(\mathbb{S}) \simeq \underset{[H \subset G]}{\oplus} \pi_0^{W H}(\Sigma_+^\infty E W H) \simeq \mathbb{Z}\big[\text{conjugacy classes of subgroups}\big]$$

(e.g. Schwede 15, p. 64)

This is the group underlying the Burnside ring.

Related concepts

• equivariant Whitehead theorem

• Elmendorf's theorem

• G-CW approximation

References

The theorem at the level of stable homotopy groups is due to

• Tammo tom Dieck, Satz 2 of Orbittypen und äquivariante Homologie. II., Arch. Math. (Basel) 26 (1975), no. 6, 650–662

The refinement to spectra is achieved in section V.11 of

• L. Gaunce Lewis, Peter May, and Mark Steinberger (with contributions by J.E. McClure), Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics Vol.1213. 1986 (pdf)

An alternative proof is in

• Bert Guillou, Peter May, section 6.2 of Permutative $G$-categories in equivariant infinite loop space theory, Algebr. Geom. Topol. 17 (2017) 3259-3339 (arXiv:1207.3459)

Detailed lecture notes are in

• Stefan Schwede, section 6 of Lectures on Equivariant Stable Homotopy Theory, 2015 (pdf)

• Andrew Blumberg, section 2.7 of The Burnside category, 2017 (pdf, GitHub)

A brief mentioning appears in this survey of rational equivariant stable homotopy theory:

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