# nLab equivariant differential topology

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### Representation theory

representation theory

geometric 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 $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.

Then the $G$-fixed locus $X^G \hookrightarrow X$ is a smooth submanifold.

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

###### Proof

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

###### Remark

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

###### Proposition/Definition

($G$-action on normal bundle to fixed locus)

Let $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.

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

###### Proposition

(existence of $G$-invariant Riemannian metric)

Let $X$ be a smooth manifold, $G$ a compact Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.

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

###### Definition

($G$-invariant tubular neighbourhood)

Let $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.

For $\Sigma \subset X^G \subset X$ a closed smooth submanifold inside the fixed locus, a $G$-invariant tubular neighbourhood $\mathcal{N}(\Sigma \subset X)$ of $\Sigma$ in $X$ is

1. a smooth vector bundle $E \to \Sigma$ equipped with a fiber-wise linear $G$-action;

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

###### Proposition

(existence of $G$-invariant tubular neighbourhoods)

Let $X$ be a smooth manifold, $G$ a Lie group and $\rho \;\colon\; G \times X \to X$ a proper action by diffeomorphisms.

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

then

1. $\Sigma$ admits a $G$-invariant tubular neighbourhood $\Sigma \subset U \subset X$ (Def. );

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

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

$\exp_\epsilon \;\colon\; N(\Sigma \subset X) \overset{\simeq}{\longrightarrow} \mathcal{N}(\Sigma \subset X)$

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

The existence of the $G$-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 $\Sigma$ 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.