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nLab > Latest Changes: slice theorem

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2021

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

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 1st 2021

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

    v1, current

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

    added this pointer:

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2021

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

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 12th 2021

    added a general abstract definition of slices as those HH-subspace inclusions whose GG-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.

    diff, v5, current

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 12th 2021

    Under Def 2.1

    Then a slice in a GG-action UU…

    That should be ’HH-slice’?

    We hadn’t yet heard the term on this page. It’s the same as ’slice through GG-orbits modulo HH’ 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 ℝ 1\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.

    diff, v6, current

    • CommentRowNumber8.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 12th 2021

    Added Cauchy surfaces as an example.

    diff, v7, current

    • 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.

    diff, v8, current

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

    Okay, I have added to the statement of the theorem the condition that XX 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.

    diff, v9, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeApr 13th 2021

    added pointer to:

    This has a proof that for SS an HH-slice, then G×S→G⋅SG \times S \to G \cdot S is an open map.

    I had been looking for this statement, since it implies that for SS 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.

    diff, v10, current

    • 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.

    diff, v11, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2021

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

    diff, v12, current

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeNov 8th 2021

    Made explicit (here) the trivial but important example that GG-slices through points whose stabilizer is the entire equivariance group GG are given by the entire GG-space.

    Used this to complete the following example (here) of slices through points in the canonical O(n+1)O(n+1)-action on ℝ n+1\mathbb{R}^{n+1}.

    diff, v14, current

  15. 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

    diff, v16, current

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 15th 2024

    Thanks. Right, that’s also what it says in the references that I cited.

    I have now added pointer to the article by Williams that you point out, together with a remark (here). But I’d need to study this article more closely, maybe you could add more commentary.

    diff, v17, current

    • CommentRowNumber17.
    • CommentAuthorDavidRoberts
    • CommentTimeJun 16th 2024

    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

  18. Just now I updated the reference for slice theorem in the case of proper action. The original proof of Palais is very long (divides itself into 8 cases). The proof provided in GGK is much shorter (half a page).

    Chenchang Zhu

    diff, v18, current

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