nLab
equivariant tubular neighbourhood

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

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}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Representation theory

Contents

Idea

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.

Definition

Definition

(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.

Properties

Existence

Proposition/Definition

(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)

Proposition

(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

then

  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.

References

Created on April 9, 2021 at 03:31:57. See the history of this page for a list of all contributions to it.