nLab equivariant homotopy group

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Contents

Context

Stable Homotopy theory

Representation theory

Contents

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 XX a pointed topological G-space and HGH \subset G a closed subgroup, the nnth unstable HH-equivariant homotopy group of XX is simply the ordinary nn-th homotopy group of the HH-fixed point space X HX^H:

π n H(X)π n(X H). \pi_n^H(X) \coloneqq \pi_n(X^H) \,.

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

π n H(X)[G/H +S n,X] G. \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 EE a G-spectrum, and nn \in \mathbb{Z}, write

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

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

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

π n H(X)π 0Map(Σ S n,map G(G/H +,X))π 0Map(Σ ,S n,X H)π n(X H), \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 HX^H is the HH-fixed point spectrum of XX (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 G k(*)[ϵ Σ k𝕊,E] G[Σ k𝕊,F GE]π k(F GE) 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

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

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

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

π V H(X)π 0Map * H(S V,X); \pi^H_V(X) \coloneqq \pi_0 \mathrm{Map}^H_*(S^V,X);

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

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

π V H(X)π 0Map * H(𝕊 V,X). \pi^H_V(X) \coloneqq \pi_0 \mathrm{Map}^H_*(\mathbb{S}^V,X).

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

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

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

π π G:Sp GAb RO(G). \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 GG-spectra

Consider genuine G-spectra modeled on a G-universe UU.

For a finite based G-CW complex XX and base topological G-space YY, write

{X,Y} G=[Σ G X,Σ G Y]lim VU[Σ VX,Σ VY] G \{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 GG-homotopy classes of maps between suspensions Σ VXS VX\Sigma^V X \coloneqq S^V \wedge X, where VV runs through the indexing spaces in the universe and S VS^V denotes its representation sphere.

(May 96, IX.2 def. 2.1)

The equivariant stable homotopy groups of XX are

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

(May 96, IX.2 remark 2.4)

And for subgroups HGH \subset G

π V H(Σ G X){G/H +S V,X} 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 GG-equivariant maps

Let GG be a finite group. For XX a GG-equivariant spectrum modeled as an orthogonal spectrum with GG-action, then for kk \in \mathbb{N} the kkth equivariant homotopy group of XX is the colimit

π k G(X)lim n[S nρ G,(Ω kX)(nρ G)] H, \pi_k^G(X) \coloneqq \underset{\longrightarrow_{\mathrlap{n}}}{\lim} [S^{n \rho_G}, (\Omega^k X)(n \rho_G)]_H \,,

where

(e.g. Schwede 15, section 3)

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

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

Examples

Of equivariant suspension spectra

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

π k G(Σ G X)lim n[S nρ g,(Ω kX)S nρ G] G \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 𝕊=Σ G S 0\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 GG:

π 0 G(𝕊)[HG]π 0 W GH(Σ + E(W GH))[conjugacyclassesofsubgroups] \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 2G = C_2, the RO(G)-graded stable equivariant homotopy groups π s,c(𝕊):=π c+(sc)σ(𝕊)\pi_{s,c}(\mathbb{S}) := \pi_{c + (s-c)\sigma}(\mathbb{S}) of the sphere have been determined in the following ranges:

Properties

Relation to Mackey functors

As HH-varies over the subgroups of a GG-equivariant spectrum EE, the HH-equivariant homotopy groups organize into a contravariant additive functor from the full subcategory of the equivariant stable homotopy category (called a Mackey functor)

π̲ (E):G/Hπ H(X). \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.

References

Last revised on April 24, 2024 at 18:39:51. See the history of this page for a list of all contributions to it.