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

Idea

The continuous group action of a topological group GG is called “proper” if it behaves like the action of a compact topological group, without GG necessarily being compact.

In particular, the isotropy groups of a proper action are compact.

Definition

Consider

The action ρ:(g,x)gx\rho \colon (g,x) \mapsto g \cdot x is called proper (Palais 61, Def. 1.2.2) if one of the following equivalent conditions hold (their equivalence relies on XX being locally compact and Hausdorff, Palais 61, Thm. 1.2.9, Karppinen 16, Rem. 5.2.4):

  1. (Bourbaki properness) The shear map is a proper continuous function:

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

    (Bourbaki 60, Ch. III, Sec. 4.4, review in Lee 00, p. 266)

  2. (Borel properness) For every compact subspace KXK \subset X the subset

    (K|K){gG|gKK}G (K \vert K) \;\coloneqq\; \big\{ g \in G \,\vert\, g \cdot K \,\cap\, K \neq \varnothing \big\} \;\subset\; G

    is compact.

    (Palais 61, Thm 1.2.9 (5), attributed there to Borel)

  3. (Palais properness) Every point xXx \in X has a neighbourhood U xU_x such that every point yU xy \in U_x has a neighbourhood V yV_y such that

    (U x|U y){gG|gV xU y}G (U_x \vert U_y) \;\coloneqq\; \big\{ g \in G \,\vert\, g \cdot V_x \,\cap\, U_y \neq \varnothing \big\} \;\subset\; G

    has compact closure.

    (Palais 61, Def. 1.2.2)

Examples

Lie group actions on smooth manifolds

How is the following not a trivial consequence of the fact that compact groups have proper action? Needs clarification.

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

The original definition are due to:

Textbook accounts:

See also

Further discussion

Last revised on September 24, 2021 at 11:27:46. See the history of this page for a list of all contributions to it.