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 $\mathbb{R}^{n+1}$.

]]>added the statement of the slice theorem for compact groups acting on completely regular spaces (here)

]]>I have added a new section “Examples and Applications – Principal bundles” (here), with proof that free proper actions of Lie groups on locally compact Hausdorff spaces have quotient coprojections which are principal bundles.

]]>added pointer to:

- Sergey Antonyan,
*Characterizing slices for proper actions of locally compact groups*, Topology and its Applications Volume 239, 15 April 2018, Pages 152-159 (arXiv:1702.08093, doi:10.1016/j.topol.2018.02.026)

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.

]]>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 a warning that Palais’s definition of “proper action” is not quite the usual one, unless some extra conditions are met.

Will straighten this out tomorrow. Have to call it quits now.

]]>Added Cauchy surfaces as an example.

]]>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.

]]>Under Def 2.1

Then a

slicein 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?

]]>added a general abstract definition of slices as those $H$-subspace inclusions whose $G$-induced action adjunct is an isomorphism.

(I have never seen an author admit this, but that’s evidently the right abstract definition.)

Also added an Idea-section.

]]>added something closer to the traditional form of the definition of a “slice”

]]>added this pointer:

- Richard Palais,
*Slices and equivariant embeddings*, chapter VIII in: Armand Borel (ed.),*Seminar on Transformation Groups*, Annals of Mathematics Studies 46, Princeton University Press 1960 (jstor:j.ctt1bd6jxd)

now some minimum content (and all or most original references) in place.

]]>starting something. There is nothing to be seen yet, but I need to save.

]]>