nLab
proper action

Contents

Context

Topology

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

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

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Representation theory

Contents

Definition

The action of a topological group GG on a topological space XX is called a proper action if the continuous function

G×X X×X (g,x) (gx,x) \array{ G \times X &\longrightarrow& X \times X \\ (g,x) &\mapsto& (g\cdot x, x) }

is a proper continuous function.

Examples

Lie groups actions on smooth manifolds

Proposition

Let XX be a smooth manifold and let GG be a compact Lie group. Then every smooth action of GG on XX is proper.

(e.g. Lee 12, Cor. 7.2)

For more see at equivariant differential topology.

References

For instance

See also

Last revised on October 21, 2020 at 11:29:08. See the history of this page for a list of all contributions to it.