nLab equivariant tubular neighbourhood




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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


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The notion of equivariant tubular neighbourhood (often called “invariant”!) is the generalization of the notion of tubular neighbourhood from differential topology to equivariant differential topology.



(GG-equivariant tubular neighbourhood)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

For ΣX GX\Sigma \subset X^G \subset X a closed smooth submanifold inside the fixed locus, a GG-equivariant tubular neighbourhood 𝒩(ΣX)\mathcal{N}(\Sigma \subset X) of Σ\Sigma in XX is

  1. a smooth vector bundle EΣE \to \Sigma equipped with a fiber-wise linear GG-action;

  2. an equivariant diffeomorphism EXE \overset{}{\longrightarrow} X onto an open neighbourhood of Σ\Sigma in XX which takes the zero section identically to Σ\Sigma.




(GG-action on normal bundle to fixed locus)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

Then linearization of the GG-action aroujnd the fixed locus X GXX^G \subset X equips the normal bundle N X(X G)N_X\left( X^G\right) with smooth and fiber-wise linear GG-action.

(e.g. Crainic-Struchiner 13, Example 1.7)


(existence of GG-equivariant tubular neighbourhoods)

Let XX be a smooth manifold, GG a Lie group and ρ:G×XX\rho \;\colon\; G \times X \to X a proper action by diffeomorphisms.

If ΣιX\Sigma \overset{\iota}{\hookrightarrow} X is a closed smooth submanifold inside the GG-fixed locus


  1. Σ\Sigma admits a GG-equivariant tubular neighbourhood ΣUX\Sigma \subset U \subset X (Def. );

  2. any two choices of such GG-equivariant tubular neighbourhoods are GG-equivariantly isotopic;

  3. there always exists an GG-equivariant tubular neighbourhood parametrized specifically by the normal bundle N(ΣX)N(\Sigma \subset X) of SigmaSigma in XX, equipped with its induced GG-action from Def. , and such that the GG-equivariant diffeomorphism is given by the exponential map

    exp ϵ:N(ΣX)𝒩(ΣX) \exp_\epsilon \;\colon\; N(\Sigma \subset X) \overset{\simeq}{\longrightarrow} \mathcal{N}(\Sigma \subset X)

with respect to a GG-invariant Riemannian metric (which exists according to Prop. ):

The existence of the GG-equivariant tubular neighbourhoods is for instance in Bredon 72 VI Theorem 2.2, Kankaanrinta 07, theorem 4.4. The uniqueness up to equivariant isotopy is in Kankaanrinta 07, theorem 4.4, theorem 4.6. The fact that one may always use the normal bundle appears at the end of the proof of Bredon 72 VI Theorem 2.2, and as a special case of a more general statement about invariant tubular neighbourhoods in Lie groupoids it follows from Pflaum-Posthuma-Tang 11, Theorem 4.1 by applying the construction there to each point in Σ\Sigma for one and the same choice of background metric. See also for instance Pflaum-Wilkin 17, Example 2.5.


Created on April 9, 2021 at 07:31:57. See the history of this page for a list of all contributions to it.