nLab equivariant homotopy group

Idea

The generalization of the concept of homotopy group from homotopy theory and stable homotopy theory to equivariant homotopy theory and equivariant stable homotopy theory.

Definition

Abstractly

For $X$ a pointed topological G-space and $H \subset G$ a closed subgroup, the $n$th unstable $H$-equivariant homotopy group of $X$ is simply the ordinary $n$-th homotopy group of the $H$-fixed point space $X^H$:

$$\pi_n^H(X) \coloneqq \pi_n(X^H) \,.$$

With $G/H$ denoting the quotient space, this is equivalently the $G$-homotopy classes of $G$-equivariant continuous functions from the smash product $S^n \wedge G/H_+$ to $X$:

$$\pi_n^H(X) \simeq [G/H_+ \wedge S^n, X]^G \,.$$

In this form the definition directly generalizes to G-spectra and hence to stable equivariant homotopy groups: for $E$ a G-spectrum, and $n \in \mathbb{Z}$, write

$$\pi_n^H(X) \simeq [G/H_+ \wedge S^n, X]^G \,.$$

where now $S^n$ is the $n$-fold suspension of the suspension spectrum of the n-sphere and $[-,-]^G$ now denotes the hom functor in the equivariant stable homotopy category..

In particular, the (∞,1)-category of $G$-spectra is stable, and hence it is cotensored over the (∞,1)-category of spectra; using this structure, we produce a chain of natural equivalences

$$\pi_n^H(X) \simeq \pi_0 \mathrm{Map}(\Sigma^\infty S^n, \mathrm{map}^G(G/H_+, X)) \simeq \pi_0 \mathrm{Map}(\Sigma^\infty, S^n, X^H) \simeq \pi_n(X^H),$$

where $X^H$ is the $H$-fixed point spectrum of $X$ (e.g. Schwede 15, prop. 7.2).

Via equivariant cohomology of the point

The identification via fixed point spectra in turn is the equivariant cohomology of the point,

$$E^{-k}_G(\ast) \;\coloneqq\; \left[ \epsilon^\sharp \Sigma^k \mathbb{S} , E\right]_G \;\simeq\; \left[ \Sigma^k \mathbb{S}, F^G E \right] \;\simeq\; \pi_k(F^G E)$$

due to the base change adjunction

$$G Spectra \underoverset { \underset{ F^G }{\longrightarrow} } {\overset{ \epsilon^\sharp }{\longleftarrow}} { \phantom{AA} \bot \phantom{AA} } Spectra$$

With $\mathrm{RO}(G)$-grading

If $V$ is a real orthogonal $G$-representation and $H \subset G$ is a subgroup, then we define the functor $\pi^H_V(-):\mathcal{S}_{G,*} \rightarrow \mathrm{Set}$ by

$$\pi^H_V(X) \coloneqq \pi_0 \mathrm{Map}^H_*(S^V,X);$$

for fixed $V$ these are functorial in $H$, and the resulting structure is called a $V$-Mackey functor by Lewis.

Similarly, if $\mathrm{RO}(G)$ denotes the real orthogonal representation ring and $V \in \mathrm{RO}(G)$ is a real orthogonal virtual representation of $G$, then we define the functor $\pi^H_V(-):\mathrm{Sp}_{G} \rightarrow \mathrm{Ab}$ by

$$\pi^H_V(X) \coloneqq \pi_0 \mathrm{Map}^H_*(\mathbb{S}^V,X).$$

These together bear the structure of an $\mathrm{RO}(G)$-graded Mackey functor, i.e. they take the form of a functor

$$\pi_{\star}^{\bullet}:\mathrm{Sp}_{G} \rightarrow \mathcal{M}_G(\mathrm{Ab})^{\mathrm{RO}(G)}.$$

Often one fixes $H = G$, in which case one simply writes

$$\pi_{\star} \coloneqq \pi_{\star}^{G}:\mathrm{Sp}_G \rightarrow \mathrm{Ab}^{\mathrm{RO}(G)}.$$

See more at C_2-equivariant homotopy groups of spheres.

Via genuine $G$-spectra

Consider genuine G-spectra modeled on a G-universe $U$.

For a finite based G-CW complex $X$ and base topological G-space $Y$, write

$$\{X,Y\}_G = [\Sigma^\infty_G X, \Sigma^\infty_G Y] \coloneqq \underset{\longrightarrow}{\lim}_{V \subset U} [\Sigma^V X, \Sigma^V Y]_G$$

for the colimit over $G$-homotopy classes of maps between suspensions $\Sigma^V X \coloneqq S^V \wedge X$, where $V$ runs through the indexing spaces in the universe and $S^V$ denotes its representation sphere.

(May 96, IX.2 def. 2.1)

The equivariant stable homotopy groups of $X$ are

$$\pi_V^G(\Sigma^\infty_G X) \coloneqq \{S^V,X\}_G \,.$$

(May 96, IX.2 remark 2.4)

And for subgroups $H \subset G$

$$\pi_V^H(\Sigma^\infty_G X) \coloneqq \{G/H_+ \wedge S^V,X\}_G$$

(Greenlees-May 95, p. 11)

Via orthogonal spectra and $G$-equivariant maps

Let $G$ be a finite group. For $X$ a $G$-equivariant spectrum modeled as an orthogonal spectrum with $G$-action, then for $k \in \mathbb{N}$ the $k$th equivariant homotopy group of $X$ is the colimit

$$\pi_k^G(X) \coloneqq \underset{\longrightarrow_{\mathrlap{n}}}{\lim} [S^{n \rho_G}, (\Omega^k X)(n \rho_G)]_H \,,$$

where

• $\rho_G$ denotes the fundamental representation of $G$

• $n \rho_G = (\rho_G)^{\oplus_n}$;

• $S^{n \rho_G}$ is its representation sphere;

• $X(n \rho_G)$ is the value of $X$ in RO(G)-degree $n\rho_G$;

• $[-,-]_G$ is the set of homotopy classes of $G$-equivariant maps of pointed topological G-spaces.

(e.g. Schwede 15, section 3)

More generally for $H \hookrightarrow G$ a subgroup then one writes $\pi_\bullet^H(X)$ for the $H$-equivariant subgroups of $X$ with $X$ regarded now as an $H$-equivariant spectrum, via restriction of the action.

(e.g. Schwede 15, p. 16)

Examples

Of equivariant suspension spectra

For $X$ a pointed topological G-space, then by the discussion there) the formula for the equivariant homotopy groups of its equivariant suspension spectrum $\Sigma^\infty_G X$ reduces to

$$\pi_k^G(\Sigma^\infty_G X) \coloneqq \underset{\longrightarrow_n}{\lim} [S^{n \rho_g}, (\Omega^k X)\wedge S^{n \rho_G}]_G$$

which in turn decomposes as a direct sum of ordinary homotopy groups of Weyl group-homotopy quotients of naive fixed point spaces – see at tom Dieck splitting.

Of the equivariant sphere spectrum

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

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

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

In the case $G = C_2$, the RO(G)-graded stable equivariant homotopy groups $\pi_{s,c}(\mathbb{S}) := \pi_{c + (s-c)\sigma}(\mathbb{S})$ of the sphere have been determined in the following ranges:

• stems $s \leq 25$ and coweights $-1 \leq c \leq 9$ (Guillou-Isaksen 24);

• $s \leq 7$ and coweights $-9 \leq c \leq -2$ (Guillou-Isaksen 24).

Properties

Relation to Mackey functors

As $H$-varies over the subgroups of a $G$-equivariant spectrum $E$, the $H$-equivariant homotopy groups organize into a contravariant additive functor from the full subcategory of the equivariant stable homotopy category (called a Mackey functor)

$$\underline{\pi}_\bullet(E) \colon G/H \mapsto \pi^H_\bullet(X) \,.$$

(e.g. Schwede 15, p. 16 and section 4)

(…)

Equivariant Whitehead theorem

See at equivariant Whitehead theorem.

Related concepts

• equivariant weak homotopy equivalence

• equivariant Whitehead theorem

• homotopy group

• stable homotopy group

References

• Gaunce Lewis, The equivariant Hurewicz map (1992) (TAMS)

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

• Peter May, section IX.4 of Equivariant homotopy and cohomology theory CBMS Regional Conference Series in Mathematics, vol. 91, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1996. With contributions by M. Cole, G. Comeza˜na, S. Costenoble, A. D. Elmenddorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J. Piacenza, G. Triantafillou, and S. Waner. (pdf)

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

• Bertrand Guillou, Daniel Isaksen, $C_2$-Equivariant Stable Stemsm (arXiv:2404.14627)