nLab equivariant tubular neighbourhood

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

($G$-equivariant 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 $G$-equivariant 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 $\Sigma$.

Properties

Existence

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.

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

Proposition

(existence of $G$-equivariant 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 $G$-equivariant tubular neighbourhood $\Sigma \subset U \subset X$ (Def. );

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

3. there always exists an $G$-equivariant 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 $G$-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 $\Sigma$ for one and the same choice of background metric. See also for instance Pflaum-Wilkin 17, Example 2.5.

References

• Glen Bredon, Introduction to compact transformation groups, Academic Press 1972 (pdf, ISBN:9780080873596)

• Marja Kankaanrinta, Equivariant collaring, tubular neighbourhood and gluing theorems for proper Lie group actions, Algebr. Geom. Topol. Volume 7, Number 1 (2007), 1-27 (euclid:agt/1513796653)

• Markus Pflaum, Hessel Posthuma, X. Tang, Geometry of orbit spaces of proper Lie groupoids, Journal für die reine und angewandte Mathematik (Crelles Journal) 2014.694 (arXiv:1101.0180, doi:10.1515/crelle-2012-0092)

• Markus Pflaum, Graeme Wilkin, Equivariant control data and neighborhood deformation retractions, Methods and Applications of Analysis, 2019 (arXiv:1706.09539)