Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

nLab > Latest Changes: equivariant differential topology

Bottom of Page

1 to 14 of 14

  1.  
    • 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 x∈X 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 G→S (V G)X^G \to S^{(V^G)} and a linear map N xX G→V−V GN_x X^G \to V-V^G.

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

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 3rd 2020

    added ISBN to

    diff, v7, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeOct 21st 2020

    I have added to the clauses of the proposition on GG-fixed submanifolds (here) the parenthetical remark that the assumptions are met for any smooth action if the group is compact

    diff, v8, current

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMar 17th 2021

    added pointer to:

    diff, v12, current

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeApr 9th 2021
    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 25th 2021

1 to 14 of 14

  1.