topological G-space




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Representation theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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 (often, but crucially not always, taken to be 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.


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 point spaces for all suitable subgroups. By Elmendorf's theorem, this 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).


We discuss some classes of examples of G-spaces.

Euclidean GG-spaces

Let VRO(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 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 VRO(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 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 VRO(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

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

then the GG-action on the Euclidean space 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.

Rezk-global equivariant homotopy theory:

cohesive (∞,1)-toposits (∞,1)-sitebase (∞,1)-toposits (∞,1)-site
global equivariant homotopy theory PSh (Glo)PSh_\infty(Glo)global equivariant indexing category GloGlo∞Grpd PSh (*) \simeq PSh_\infty(\ast)point
sliced over terminal orbispace: PSh (Glo) /𝒩PSh_\infty(Glo)_{/\mathcal{N}}Glo /𝒩Glo_{/\mathcal{N}}orbispaces PSh (Orb)PSh_\infty(Orb)global orbit category
sliced over BG\mathbf{B}G: PSh (Glo) /BGPSh_\infty(Glo)_{/\mathbf{B}G}Glo /BGGlo_{/\mathbf{B}G}GG-equivariant homotopy theory of G-spaces L weGTopPSh (Orb G)L_{we} G Top \simeq PSh_\infty(Orb_G)GG-orbit category Orb /BG=Orb GOrb_{/\mathbf{B}G} = Orb_G

See also


See also the references at equivariant homotopy theory.

Last revised on November 10, 2020 at 06:03:17. See the history of this page for a list of all contributions to it.