In the context of topology, a topological GG-space (traditionally just GG-space, for short, if the context is clear) is a topological space equipped with an action of a topological group GG – the equivariance group (usually taken to be the topological group underlying a compact Lie group, such as a finite group).

The canonical homomorphisms of topological GG-spaces are GG-equivariant continuous functions, and the canonical choice of homotopies between these are GG-equivariant continuous homotopies (for trivial GG-action on the interval).

A GG-equivariant version of the Whitehead theorem says that on G-CW complexes these GG-equivariant homotopy equivalences are equivalently those maps that induce weak homotopy equivalences on all fixed point spaces for all subgroups of GG (compact subgroups, if GG is allowed to be a Lie group).

By Elmendorf's theorem, this, in turn, is equivalent to the (∞,1)-presheaves over the orbit category of GG. See below at In topological spaces – Homotopy theory.

See (Henriques-Gepner 07) for expression in terms of topological groupoids/orbispaces.

In the context of stable homotopy theory the stabilization of GG-spaces is given by spectra with G-action; these lead to equivariant stable homotopy theory. See there for more details. (But beware that in this context one considers the richer concept of G-spectra, which have a forgetful functor to spectra with G-action but better homotopy theoretic properties. ) The union of this as GG is allowed to vary is the global equivariant stable homotopy theory.

Properties

Change of equivariance groups

In the following TopSpTopSp is a cartesian closed convenient category of topological spaces, such as that of compactly generated topological spaces.

We discuss how any homomorphism of topological groups induces an adjoint triple of functors between the corresponding TopologicalGSpacesTopological G Spaces (e.g. May 96, Sec. I.1, DHLPS 19, p.8, see also at induced representation for a formulation in homotopy type theory)

Throughout, let G 1,G 2∈G_1, G_2 \in TopologicalGroups and consider a continuous homomorphism of topological groups

(1)ϕ:G 1⟶G 2. \phi \;\colon\; G_1 \longrightarrow G_2 \,.

Pullback action

Definition

(pullback action)
Given a topological group homomorphism ϕ\phi (1), write

TopologicalG 1Spaces⟵ϕ *TopologicalG 2Spaces Topological G_1 Spaces \overset{ \;\;\; \phi^\ast \;\;\; }{\longleftarrow} Topological G_2 Spaces

for the functor which takes a topological G2-space (X,ρ)(X,\rho) to the same underlying topological space ρ\rho, equipped with the G 1G_1-action ρ(ϕ(−)) \rho(\phi(-)).

Induced action

Definition

(induced action)
Given a topological group homomorphism ϕ\phi (1), the induced action functor

TopologicalG 1Actions⟶G 2× G 1(−)TopologicalG 2Actions Topological G_1 Actions \overset{ G_2 \times_{G_1} (-) }{\longrightarrow} Topological G_2 Actions

sends X∈TopologicalG 1SpacesX \in Topological G_1 Spaces to the quotient space of its Cartesian product with G 2G_2 by the diagonal action of G 1G_1 (which on G 2G_2 is by inverse right multiplication through ϕ\phi):

G 2× G 1X≔(G 2×X)/((g 2,x)∼(g 2⋅ϕ(g 1) −1,g 1⋅x)|g 1∈G 1) G_2 \times_{G_1} X \;\coloneqq\; \big( G_2 \times X \big) \big/ \big( (g_2, x) \sim (g_2 \cdot \phi(g_1)^{-1}, g_1 \cdot x) \,\vert\, g_1 \in G_1 \big)

and equipped with the G 2G_2-action given by left multiplication on the G 2G_2-factor.

Proposition

(induced action is left adjoint to pullback action)
Given a topological group homomorphism ϕ\phi (1), the induced action functor (Def. ) is left adjoint to the pullback action functor (Def. ):

G 1Spaces⊥⟵ϕ *⟶G 2× G 1(−)G 2Spaces. G_1 Spaces \underoverset {\underset{\phi^\ast}{\longleftarrow}} {\overset{G_2 \times_{G_1} (-)}{\longrightarrow}} {\;\;\; \bot \;\;\; } G_2 Spaces \,.

Proof

For XX a G 1G_1-space and YY a G 2G_2-space, consider a G 2G_2-equivariant function out of the induced action

G 2× G 1X⟶fY. G_2 \times_{G_1} X \overset{\;\;\; f \;\;\;}{\longrightarrow} Y \,.

Since the G 2G_2-orbit of

X=G 1× G 1X↪G 2× G 1X X = G_1 \times_{G_1} X \hookrightarrow G_2 \times_{G_1} X

is the entire domain space on the left, by G 2G_2 equivariance, this function is completely determined by its restriction along this inclusion to a G 1G_1-equivariant function f˜:X→Y\tilde f \;\colon\; X \to Y, hence to a homomorphism f˜:X→ϕ *Y\tilde f \;\colon\; X \to \phi^\ast Y. This correspondence

f↔f˜ f \leftrightarrow \tilde f

is clearly a natural bijection and hence establishes the hom-isomorphisms characterizing the adjunction.

Coinduced action

Definition

(coinduced action)

Given a topological group homomorphism ϕ\phi (1), the coinduced action functor

TopologicalG 1Actions⟶Maps(G 2,−) G 1TopologicalG 2Actions Topological G_1 Actions \overset{ Maps(G_2,-)^{G_1} }{\longrightarrow} Topological G_2 Actions

sends X∈TopologicalG 1SpacesX \in Topological G_1 Spaces to the G 1G_1-fixed locus in the mapping space between topological spaces equipped with G 1G_1-actions (on G 2G_2 the ϕ\phi-induced left multiplication action) and itself equipped with the G 2G_2-action given by

(2)G 2×Maps(G 2,X) G 1 ⟶ Maps(G 2,X) G 1 (g 2,h) ↦ h((−)⋅g 2). \array{ G_2 \times Maps(G_2,X)^{G_1} & \overset{}{\longrightarrow} & Maps(G_2,X)^{G_1} \\ (g_2, h) &\mapsto& h\big( (-) \cdot g_2 \big) \,. }

Proposition

(coinduced action is right adjoint to pullback action)

Given a topological group homomorphism ϕ\phi (1), the pullback action functor (Def. ) is the left adjoint and the coinduced action functor (Def. ) is the right adjoint in a pair of adjoint functors

G 1Spaces⊥⟶Maps(G 2,−) G 1⟵ϕ *G 2Spaces G_1 Spaces \underoverset { \underset{ Maps \big( G_2, - \big)^{G_1} }{\longrightarrow} } { \overset{ \phi^\ast }{ \longleftarrow } } {\bot} G_2 Spaces

Proof

To see the defining hom-isomorphism, consider a G 1G_1-equivariant continuous function

ϕ *X⟶fY. \phi^\ast X \overset{ \;\;\; f \;\;\; }{ \longrightarrow } Y \,.

From this we obtain the following function

X ⟶f˜ Maps(G 2,Y) G 1 x ↦ (g 2↦f(g 2⋅x)), \array{ X & \overset{ \tilde f }{ \longrightarrow } & Maps \big( G_2, Y \big)^{G_1} \\ x &\mapsto& \big( g_2 \mapsto f( g_2 \cdot x ) \big) \,, }

where e∈G 2e \in G_2 denotes the neutral element.

This is manifestly:

well-defined, due to the G 1G_1-equivariance of ff;
continuous, being built from composition of continuous map;
G 2G_2-equivariant with respect to the action (2).

Conversely, given a G 2G_2-equivariant continuous function X⟶f˜Maps(G 2,Y) G 1X \overset{\tilde f}{\longrightarrow} Maps\big(G_2, Y\big)^{G_1}, we obtain the following function

ϕ *X ⟶ Y x ↦ f˜(x)(e). \array{ \phi^\ast X &\overset{}{\longrightarrow}& Y \\ x &\mapsto& \tilde f(x)(e) \,. }

This is:

continuous, being the composition of continuous functions;
G 1G_1-equivariant due to the equivariance properties of f˜\tilde f:

ϕ(g 1)⋅x ↦f˜(ϕ(g 1)⋅x)(e) =f˜(x)(e⋅ϕ(g 1)) =f˜(x)(ϕ(g 1)⋅e) =g 1⋅(f˜(x)(e)) \begin{aligned} \phi(g_1) \cdot x & \mapsto \tilde f \big( \phi(g_1)\cdot x \big) (e) \\ & = \tilde f ( x ) \big( e \cdot \phi(g_1) \big) \\ & = \tilde f ( x ) \big( \phi(g_1) \cdot e \big) \\ & = g_1 \cdot \big( \tilde f ( x ) ( e ) \big) \end{aligned}

Finally, it is clear that these transformations f↔f˜f \leftrightarrow \tilde f are natural, hence it only remains to see that they are bijective:

Plugging in the above constructions we find indeed:

f˜˜:x↦f(e⋅x)=f(x) \widetilde {\tilde f} \;\colon\; x \mapsto f(e \cdot x) = f(x)

and

f˜˜˜:x↦(g 2↦f˜(g 2⋅x)(e)⏟=f˜(x)(e⋅g 2)=f˜(x)(g 2)). \widetilde {\widetilde {\tilde f}} \;\colon\; x \mapsto \big( g_2 \mapsto \underset{ {= \tilde f(x)(e \cdot g_2)} \atop {= \tilde f(x)(g_2)} }{ \underbrace{ \tilde f(g_2\cdot x)(e) } } \big) \,.

Fixed loci with residual Weyl group action

Combining these change-of-equivariance grouo adjunctions (from above) to “pull-push” through the correspondence

we obtain the fixed locus-functor in the form in which it appears in Elmendorf's theorem, namely with the residual Weyl group-action on the fixed loci:

Limits and colimits

Recalling that the ambient category is assumed to be a cartesian closed convenient category of topological spaces, such as that of compactly generated topological spaces:

(e.g. Schwede 2018, p. 736-737)

Proof

By this Prop. the given convenient category of topological spaces is regular. Since regularity is purely a condition on limits and colimits (this Def.) it transfers along any forgetful functor which creates limits and colimits. Therefore the statement follows by Prop. .

(Bykov-Flores 15, Prop. 4.1)

Equivariant Tietze extension theorem

See at equivariant Tietze extension theorem

Model structure and homotopy theory

The standard homotopy theory on GG-spaces used in equivariant homotopy theory considers weak equivalences which are weak homotopy equivalence on all (ordinary) fixed loci for all suitable subgroups. This is presented by the fine model structure on topological G-spaces, which, by Elmendorf's theorem, is equivalent to (∞,1)-presheaves over the orbit category of GG.

On the other hand there is also the standard homotopy theory of infinity-actions, presented by the Borel model structure, in this context also called the “coarse” or “naive” equivariant model structure (Guillou).

Examples

We discuss some classes of examples of G-spaces.

Euclidean GG-spaces

Let V∈RO(G)V \in RO(G) be an orthogonal linear representation of a finite group GG on a real vector space VV. Then the underlying Euclidean space ℝ V\mathbb{R}^V inherits the structure of a G-space

We may call this the Euclidean G-space associated with the linear representation VV.

Representation spheres

Let V∈RO(G)V \in RO(G) be an orthogonal linear representation of a finite group GG on a real vector space VV. Then the one-point compactification of the underlying Euclidean space ℝ V\mathbb{R}^V inherits the structure of a G-space with the point at infinity a fixed point. This is called the VV-representation sphere

Representation tori

Let V∈RO(G)V \in RO(G) be an orthogonal linear representation of a finite group GG on a real vector space VV.

If GG is the point group of a crystallographic group inside the Euclidean group

N⋊G↪Iso(ℝ V) N \rtimes G \hookrightarrow Iso(\mathbb{R}^V)

then the GG-action on the Euclidean space ℝ V\mathbb{R}^V descends to the quotient by the action of the translational normal subgroup lattice NN (this Prop.). The resulting GG-space is an n-torus with GG-action, which might be called the representation torus of VV

graphics grabbed from SS 19

Projective GG-space

Let GG be a finite group (or maybe a compact Lie group) and let VV be a GG-linear representation over some topological ground field kk.

Then the corresponding projective G-space is the quotient space of the complement of the origin in (the Euclidean space underlying) VV by the given action of the group of units of kk (from the kk-vector space-structure on VV):

kP(V):=(V∖{0})/k × k P(V) \;:=\; \big( V \setminus \{0\} \big) / k^\times

and equipped with the residual GG-action on VV (which passes to the quotient space since it commutes with the kk-action, by linearity).

G-CW complexes

See at G-CW complex.

References

(For basics see also the references at group actions.)

Early appearance of the notion (as “transformation groups”):

Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872) Mathematische Annalen volume 43, pages 63–100 1893 (doi:10.1007/BF01446615)

English translation by M. W. Haskell:

A comparative review of recent researches in geometry, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (euclid:1183407629 retyped pdf)