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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2019

    need to record some results on equivariant tubular neighbourhoods etc. Didn’t know where to put these, so I thought we’d need a dedicated entry on equivariant differential topology.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 6th 2019

    okay, added statement of existence of GG-invariant tubular neighbourhoods and statement that fixed loci of proper smooth actions are submanifolds.

    v1, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeFeb 7th 2019
    • (edited Feb 7th 2019)

    I am trying to nail down the following statement, which should be completeley standard, but right now I am not sure about some details:

    For GG a compact Lie group acting smoothly and properly on a smooth manifold, there should be a GG-representation V xV_x associated with each point xX Gx \in X^G in the fixed locus, obtained from a choice of linear structure on G-invariant tubular neighbourhoods.

    This should follow for instance with Wasserman 69, Lemma 3.5. Maybe its made explicit in Bredon 72? In Kankaanrinta 07, p. 7 the statement seems to be taken for granted in the absence of boundaries, and the author seems to be concerned with establishing it also for boundaries, but the wording remains a little inconclusive.

    Next, these V xV_x should be independent of the choices made. Now as xx varies, do these representations arrange into a smooth GG-vector bundle over X GX^G? Or what’s good sufficient conditions that they do?

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 7th 2019
    • (edited Feb 7th 2019)

    The GG-representation exists for instance by the local linearisation for proper Lie groupoids. Corollary 3.10 of https://arxiv.org/abs/1101.0180 might also be useful. X GX^G is the smallest-dimensional stratum in the orbit-type stratification. V xV_x will have underlying vector space T xXT_x X, the normal bundle to each orbit=a point, so T(X G)T(X^G) should be the bundle you want.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 7th 2019
    • (edited Feb 7th 2019)

    Thanks! Okay, from the discussion on top of p. 5 in Crainic-Struchiner the essential uniqueness of the linear rep at a fixed point is pretty immediate.

    It is clear that as vector spaces these glue to the normal bundle. But do we need an argument that the fiberwise GG-action on that glues to a global GG-action? Seems pretty clear, but maybe needs an argument.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeFeb 7th 2019
    • (edited Feb 7th 2019)

    Ah, no, of course that’s obvious, from their equation (1).

    Okay, so we have canonical fiberwise linear GG-action on the normal bundle N(X G)N(X^G).

    [ edit: made that a Prop here ]

    Next I’d want a GG-equivariant diffeomorphism from this N(X G)N(X^G) onto a GG-invariant tubular neighbourhood of X GX^G.

    That maybe follows by using theorem 4.1 from arXiv:1101.0180 at all points in X GX^G with one and the same metric on XX?

    • CommentRowNumber7.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 7th 2019

    I think so, you can get a tubular neighbourhood of the zero section of the normal bundle, and with the adapted metric (ie GG-invariant) should be able to use the exponential map to get the tubular nhd. Since the Lie group is compact I don’t think the more general setting, where GG-invariant tubular nhds only exist locally, is a worry.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 8th 2019

    found the required statement in Bredon, added a few more references (thanks to kind help from Markus Pflaum!) and then accordingly expanded the text here, and rearranged slightly.

    Finally, I copied these statements over also to tubular neighbourhood into a new subsection Properties – Equivariant version

    diff, v5, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeFeb 9th 2019
    • (edited Feb 9th 2019)

    Another thought, maybe give me a sanity check:

    I suppose the orthogonal G-representations which are the fibers of the normal bundle on X GX^G never contain a direct summand of a trivial rep. Right? For if they did, that trivial rep summand would identify with a non-empty fixed locus in the tubular neighbourhood around and away from X GX^G, contradicting the definition of X GX^G.

    That in turn should imply, by Schur’s lemma, that any equivariant map from XX to some representation sphere S VS^V, when linearised around X GX^G, should locally split as the Cartesian product of a map X GS (V G)X^G \to S^{(V^G)} and a linear map N xX GVV GN_x X^G \to V-V^G.

    Hm, might we have such a splitting even beyond the linear approximation?

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