Action that is Bourbaki proper but not Palais proper

Question

I'm working with different definitions of proper action (Cartan, Bourbaki and Palais) and the relation between them. All the spaces I'm working with are $T_{3.5}$, the definitions are:

If $U$ and $V$ are subsets of a $G$-space $X$ then we say that $U$ is thin relative to $V$ if $\{g \in G \; : \; gU \cap V \neq \emptyset \}$ has compact closure in $G$. If $U$ is thin relative to itself then we say that $U$ is thin. A subset $S$ of a $G$-space $X$ is small if every point of $X$ has a neighborhood wich is thin relative to $S$.

Bourbaki proper. Let $X$ be a $G$-space. We say that the action is Bourbaki proper (B-proper) if the function $\delta: G \times X \to X \times X$, $\delta(g,x) = (x,gx)$, is perfect (i. e. closed and the fibers of points $\delta^{-1}((x,x))$ are compact). If $G$ is locally compact, the action on $X$ is Bourbaki proper if and only if for every $x, y \in X$ there are neighborhoods $V_x$ and $V_y$ in $X$ of $x, y$, that are thin relative.

Palais proper. Let $X$ be a $G$-space with $G$ locally compact. We say that the action is Palais proper (P-proper) if each point of $X$ has a small neighborhood.

I need and example of an action that is B-proper but not P-proper . Some facts that could help are that P-proper implies B-proper and if the action is P-proper then the orbit space $X/G$ is $T_{3.5}$, like $X$, but if the action is B-proper then $X/G$ is only $T_2$.

Also if $X$ is locally compact then the two definitions are equivalent. Then, to build the example we need a $G$-space with $X$ not locally compact but that satisfy the axiom $T_{3.5}$ (or $T_{3}$) whose orbit space is $T_2$ but not $T_{3.5}$ $(T_{3})$. Some examples of space that are $T_{3.5}$ but not locally compact (where I have been looking for the example) are the Moore plane (Niemytski plane), Sorgenfrey line and infinite dimensional vector spaces (I have been trying with Hilbert cube).

Answer 1

Let $M$ and $N$ be smooth finite dimensional manifolds with $M$ compact, and let $\dim(N)\ge \dim(M)$. Let $\text{Imm}(M,N)$ be the space of smooth immersions $N\to N$, which is a smooth manifold modelled on spaces $\Gamma(f^*TN)$ of smooth sections alonf immersions. Consider the regular Frechet Lie group $\text{Diff}(M)$ of all diffeomorphisms of $M$. See the Wikipedia article here for background.

In the paper here you find a more general version of the following result (there are some misprints but no mistake in the paper):

• The action $\text{Imm}(M,N)\times \text{Diff}(M)\to \text{Imm}(M,N)$ by composition from the right is smooth. The orbit space is Hausdorff in the quotient topology.

• On the open subspace $\text{Imm}_{free}(M,N)$ of free immersions the action of $\text{Diff}(M)$ is free (this is the definition of free) and is the action of smooth principal bundle.

• The isotropy group in $\text{Diff}(M)$ of a (non-free) immersion $f$ is always a finite group which acts strictly discontinuously on $M$, so that $f$ factors to a smooth manifold finitely covered by the isotropy group. Thus the orbit space $\text{Imm}(M,N)/\text{Diff}(M)$ is an infinite dimensional orbifold.

So this is a B-proper action, which is not P-proper since the group $\text{Diff}(M)$ is not locally compact - admittedly a trivial reason.

Answer 2

Here is another example: Consider a compact finite dimensional smooth manifold $M$, and let $\text{Met}(M)$ be the space of all smooth Riemannian metrics on $M$, an open subset in the Frechet space $\Gamma(S^2T^*M)$. Then the regular Frechet Lie group $\text{Diff}(M)$ of all diffeomorphisms of $M$ acts smoothly on $\text{Met}(M)$ by pullback. The quotient space $\text{Met}(M)/\text{Diff}(M)$ is Hausdorff in the quotient topology. The action is free on generic Riemannian metrics, and is the isometry group of a metric $g$ in general, which is a compact Lie group in general, since $M$ is compact. Thus this action is B-proper, but not P-proper, since the group $\text{Diff}(M)$ is not locally compact.