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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeApr 1st 2021

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

• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeApr 1st 2021

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

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeApr 2nd 2021
• (edited Apr 2nd 2021)

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeApr 9th 2021

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

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeApr 12th 2021

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.)

• CommentRowNumber6.
• CommentAuthorDavid_Corfield
• CommentTimeApr 12th 2021

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?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeApr 12th 2021
• (edited Apr 12th 2021)

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.

• CommentRowNumber8.
• CommentAuthorDavid_Corfield
• CommentTimeApr 12th 2021

Added Cauchy surfaces as an example.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 12th 2021

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.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeApr 13th 2021
• (edited Apr 13th 2021)

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.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeApr 13th 2021

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.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeSep 22nd 2021

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.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeSep 22nd 2021

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

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeNov 8th 2021

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