nLab tubular neighbourhood

Definition

Definition

For $i : X \hookrightarrow Y$ an embedding of manifolds, a tubular neighbourhood of $X$ in $Y$ is

• a real vector bundle $E \to X$;

• an extension of $i$ to an isomorphism

  $$\hat i \;\colon\; E \to U_{i(X)}$$

  with an open neighbourhood of $X$ in $Y$.

Remark

The derivative of $\hat i$ provides an isomorphism of $E$ with the normal bundle $\nu_{X/Y}$ of $X$ in $Y$.

Properties

General

Proposition

(tubular neighbourhood theorem)

Every embedding of smooth manifolds does admit a tubular neighbourhood, def. .

For instance (DaSilva, theorem 3.1).

Moreover, tubular neighbourhoods are unique up to homotopy in a suitable sense:

Definition

For an embedding $i : X \to Y$, write $Tub(i)$ for the topological space whose underlying set is the set of tubular neighbourhoods of $i$ and whose topology is the subspace topology of $Hom(N_i X, Y)$ equipped with the C-infinity topology.

Proposition

If $X$ and $Y$ are compact manifolds, then $Tub(i)$ is contractible for all embeddings $i : X \to Y$.

This appears as (Godin, prop. 31).

Equivariant version

These statements generalize to equivariant differential topology:

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.

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

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.

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

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.

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

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 $G$-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 $G$-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 $G$-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.

Pullbacks of tubular neighbourhoods

(…) propagating flow (…) (Godin).

Related concepts

• tubular neighbourhood of a mapping space

• key application: Pontrjagin-Thom collapse map

References

• René Thom, p. 21 of: Quelques propriétés globales des variétés différentiables, Comment. Math. Helv. 28, (1954). 17-86 (doi:10.1007/BF02566923, dml:139072, digiz:GDZPPN002056259, pdf)

Basics on tubular neighbourhoods are reviewed for instance in

• Ana Cannas da Silva, section 3 of Prerequisites from differential geometry (pdf)

• Stanley Kochman, section 1.2 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996

Discussion in the generality of equivariant differential topology includes

• Glen Bredon, Chapter VI.2 of Introduction to compact transformation groups, Academic Press 1972 (pdf)

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

• 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)

• Wolfgang Ziller, Group actions, 2013 (pdf)

• Marius Crainic, Ivan Struchiner, On the linearization theorem for proper Lie groupoids, Annales scientifiques de l’École Normale Supérieure, Série 4, Volume 46 (2013) no. 5, p. 723-746 (numdam:ASENS_2013_4_46_5_723_0 doi:10.24033/asens.2200)

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

The homotopical uniqueness of tubular neighbourhoods is discussed in

• Veronique Godin, Higher string topology operations (2007)(arXiv:0711.4859)

For an analogue in homotopical algebraic geometry see

• Marc Levine, Motivic tubular neighborhoods, pdf

see also

• wikipedia tubular neighborhood