proper action




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

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


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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.



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)


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.


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.


The original definition are due to:

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Further discussion

Last revised on April 13, 2021 at 06:33:55. See the history of this page for a list of all contributions to it.