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added this pointer:
Under Def 2.1
Then a slice in a $G$-action $U$…
That should be ’$H$-slice’?
We hadn’t yet heard the term on this page. It’s the same as ’slice through $G$-orbits modulo $H$’ earlier?
Is there a way to see Cauchy surface as such a thing?
I see, okay I made it say “H-slice” in Def. 2.1.
And yes, if time evolution on some manifold is already given as an $\mathbb{R}^1$-action with timelike flow lines, then slices for this action (“1-slices”) are Cauchy surfaces!
I won’t edit further right now, since I am just on my phone at the moment, but we could add this as an example.
Okay, I have added to the statement of the theorem the condition that $X$ be locally compact, and then added a Remark (here) that Palais61 goes to some trouble to generalize away from this assumption by carefully adjusting the definition of proper action.
But I won’t go down that road now, will assume local compactness and keep fingers crossed that this won’t bite me later.
added pointer to:
This has a proof that for $S$ an $H$-slice, then $G \times S \to G \cdot S$ is an open map.
I had been looking for this statement, since it implies that for $S$ a slice through some point, also its intersection with any open neighbourhood of the point is still a slice through the point. This is used without comment in Lashof’s “Equivariant bundles and I fail to see how it doesn’t require an argument. Such as Antonyan’s.
Made explicit (here) the trivial but important example that $G$-slices through points whose stabilizer is the entire equivariance group $G$ are given by the entire $G$-space.
Used this to complete the following example (here) of slices through points in the canonical $O(n+1)$-action on $\mathbb{R}^{n+1}$.
I corrected the statement regarding the existence of slices for compact group actions: One has to assume that the group is a Lie group, otherwise the theorem is false even for group actions on compact metric spaces (as proven by Kolmogorov). See
R. F. Williams, A useful functor and three famous examples in topology. Trans. Amer. Math. Soc. 106 (1963) 319–329.
Michael Kapovich
There’s a version of this slice theorem for proper Lie groupoids. I once made some notes around this at https://ncatlab.org/davidroberts/show/cocompact+proper+Lie+groupoids
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