nLab
equivariant differential topology

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

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 subject of equivariant differential topology is the enhancement of results of differential topology from plain manifolds/topological spaces to those equipped with actions of some group (G-spaces).

Properties

Proposition

(fixed loci of smooth proper actions are submanifolds)

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 the GG-fixed locus X GXX^G \hookrightarrow X is a smooth submanifold.

If in addition XX is equipped with a Riemannian metric and GG acts by isometries then the submanifold X GX^G is a totally geodesic submanifold.

(e.g. Ziller 13, theorem 3.5.2, see also this MO discussion)

Proof

Let xX GXx \in X^G \subset X be any fixed point. Since this is in particular a closed invariant submanifold, Prop. applies and shows that an open neighbourhood of xx in XX is GG-equivariantly diffeomorphic to a linear representation VRO(G)V \in RO(G). The fixed locus V GVV^G \subset V of that is hence diffeomorphic to an open neighbourhood of xx in Σ\Sigma.

Remark

Without the assumption of proper action in Prop. the conclusion generally fails. See this MO comment for a counter-example.

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-invariant Riemannian metric)

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

Then there exists a Riemannian metric on XX with its invariant with respect to the GG-action, hence such that all elements of GG act by isometries.

(Bredon 72, VI Theorem 2.1, see also Ziller 13, Theorem 3.0.2)

Definition

(GG-invariant 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-invariant 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.

Proposition

(existence of GG-invariant 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-invariant tubular neighbourhood ΣUX\Sigma \subset U \subset X (Def. );

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

  3. there always exists an GG-invariant 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-invariant 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

Last revised on February 8, 2019 at 08:38:30. See the history of this page for a list of all contributions to it.