nLab
equivariant homotopy group

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, then

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

where now Σ S nΣ n𝕊\Sigma^\infty S^n \simeq \Sigma^n \mathbb{S} is the suspension spectrum of the n-sphere and [,] G[-,-]^G now denotes the hom functor in the equivariant stable homotopy category.

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)

Via fixed point spectra

Equivalently, the kkth GG-equivariant homotopy group of a GG-equivariant spectrum EE is the plain kkth stable homotopy group of its fixed point spectrum F GEF^G E

(1)π k G(E)π k(F GE). \pi_k^G(E) \simeq \pi_k(F^G E) \,.

(e.g. Schwede 15, prop. 7.2)

Via equivariant cohomology of the point

The identification (1) 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

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)

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 stable Whitehead theorem

The equivariant version of the stable Whitehead theorem holds:

a mpa of GG-spectra f:EFf \colon E \longrightarrow F is a weak equivalence (e.g. an RO(G)RO(G)-degree-wise weak homotopy equivalence of topological G-spaces in the model via indexing on all representations) precisely it if induces isomorphisms π (f):π (E)π (F)\pi_\bullet(f) \colon \pi_\bullet(E) \longrightarrow \pi_{\bullet}(F) on all equivariant homotopy group Mackey functors.

(Greenlees-May, theorem 2.3)

References

  • 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)

Last revised on September 22, 2018 at 05:09:02. See the history of this page for a list of all contributions to it.