nLab proper action

Idea

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

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

Definition

Consider

• a topological group $G$,

• a locally compact Hausdorff space $X$,

• a group action of $G$ on $X$ (making $X$ a topological G-space):

  $$\array{ G \times X &\overset{\rho}{\longrightarrow}& X \\ (g,x) &\mapsto& g \cdot x \mathrlap{\,.} }$$

The action $\rho \colon (g,x) \mapsto g \cdot x$ is called proper (Palais 61, Def. 1.2.2) if one of the following equivalent conditions hold (their equivalence relies on $X$ 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:

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

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

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

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

   is compact.

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

3. (Palais properness) Every point $x \in X$ has a neighbourhood $U_x$ such that every point $y \in U_x$ has a neighbourhood $V_y$ such that

   $$(U_x \vert V_y) \;\coloneqq\; \big\{ g \in G \,\vert\, g \cdot U_x \,\cap\, V_y \neq \varnothing \big\} \;\subset\; G$$

   has compact closure.

   (Palais 61, Def. 1.2.2)

Examples

Lie group actions on smooth manifolds

Proposition

Let $X$ be a smooth manifold and let $G$ be a compact Lie group. Then every continuous action of $G$ on $X$ is proper.

(e.g. Lee 12, Cor. 21.6)

For more see at equivariant differential topology.

Related concepts

• properly discontinuous action

• free action, semi-free action, transitive action, regular action

• equivariant differential topology

References

The original definition are due to:

• Richard Palais, On the Existence of Slices for Actions of Non-Compact Lie Groups, Annals of Mathematics

  Second Series, Vol. 73, No. 2 (Mar., 1961), pp. 295-323 (jstor:1970335, doi:10.2307/1970335, pdf)

  (in the context of proving the slice theorem)

• Nicolas Bourbaki, Éléments de mathématique, Topologie générale, Chap. 3: Groupes topologiques, 1960

Textbook accounts:

• John M. Lee, Section 12.3 of: Introduction to topological manifolds. Graduate Texts in Mathematics 202 (2000), Springer. ISBN: 0-387-98759-2, 0-387-95026-5.

  Second edition: Springer, 2011. ISBN: 978-1-4419-7939-1 (doi:10.1007/978-1-4419-7940-7, errata pdf)

• John Lee, around p. 147 of: Introduction to Smooth Manifolds, Springer 2012 (doi:10.1007/978-1-4419-9982-5, pdf)

• Marja Kankaanrinta, Def. 2.1 in Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions, Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 (euclid:agt/1513796653)

See also

• J. Lee, MO comment, Oct 2014

Further discussion

• Sini Karppinen, The existence of slices in $G$-spaces, when $G$ is a Lie group, Helsinki 2016 (hdl:10138/190707)

  (in the context of the slice theorem)

• Sergey Antonyan, Universal proper G-spaces, Topology and its Applications Volume 117, Issue 1, 15 January 2002, Pages 23-43 (doi:10.1016/S0166-8641(00)00115-2)