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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).

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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.

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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.

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  • $\begingroup$ Is the quotient topology on $\mathrm{Met}(M)/\mathrm{Diff}(M)$ metrizable? $\endgroup$ – Igor Belegradek May 22 '18 at 15:02
  • $\begingroup$ Note that the problem is not local. I think the space is locally the quotient of a slice by its (compact Lie group) stabilizer which gives local metrizability. To get metrizability one also has to show the space is paracompact which I have no idea how to approach. $\endgroup$ – Igor Belegradek May 23 '18 at 1:25
  • $\begingroup$ Actually, Ebin in his thesis (Proposition 148) constructs a $\mathrm{Diff}(M)$-invariant metric in $\mathrm{Met}(M)$, and it descends to a metric on the moduli space. I overlooked it because the result does not appear in the paper based on the thesis. $\endgroup$ – Igor Belegradek May 24 '18 at 0:49
  • $\begingroup$ @Igor Belegradek: See also the generalization of the Ebin metric in [Martin Bauer, Philipp Harms, Peter W. Michor: Sobolev metrics on the manifold of all Riemannian metrics. Journal of Differential Geometry 94, 2 (2013), 187-208] mat.univie.ac.at/~michor/rie-met2.pdf $\endgroup$ – Peter Michor May 25 '18 at 7:22
  • $\begingroup$ In the linked paper I do not see any discussion of the distance function or why it gives a metric space structure on the moduli space. Clark's paper arxiv.org/pdf/0904.0174.pdf seems more relevant except that it also does not exalain why the $L^2$-metric gives a metric space structure on the moduli space; is this even true? $\endgroup$ – Igor Belegradek May 25 '18 at 12:44

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