Euclidean 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



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

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


Relation to representation spheres

The one-point compactification of a Euclidean GG-space is the corresponding representation sphere:

( V) cptS V. \big( \mathbb{R}^V \big)^{cpt} \;\simeq\; S^V \,.

Relation to 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 GG-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

Equivariant configurations in Euclidean GG-spaces



Discussion of equivariant configuration spaces of points in Euclidean GG-spaces: