nLab equivariant differential topology

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) – the equivariance group.

Properties

Fixed submanifolds

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 (e.g. any smooth action if $G$ is compact, by this Prop.).

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.

Invariant Riemannian metrices

Proposition

(existence of $G$-invariant Riemannian metrics on G-manifolds)

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 $X$ which is 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)

Equivariant tubular neighbourhoods

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 $\Sigma$.

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.

Applications

• equivariant Hopf degree theorem

• equivariant Pontrjagin theorem

Related concepts

• equivariant triangulation, equivariant triangulation theorem

• equivariant bundle

• equivariant good open cover

• equivariant extension theorem

• equivariant homotopy theory

• equivariant stable homotopy theory

• equivariant rational homotopy theory

• equivariant cohomology

• equivariant cobordism theory

References

• Arthur Wasserman, section 3 of: Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (doi:10.1016/0040-9383(69)90005-6pdf)

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

• Sören Illman, Equivariant algebraic topology, Princeton University 1972 (pdf)

• Tammo tom Dieck, Section I.5 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

• 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, 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)

See also:

• Richard Lashof, Melvin Rothenberg, Section 1 of: $G$-Smoothing theory, p. 211-266 in: Algebraic and Geometric Topology, Proceedings of Symposia in Pure Mathematics, Vol. XXXII, Part 1, American Mathematical Society 1978 (doi:10.1090/pspum/032.1, pdf)

In relation to Chern-Weil theory and equivariant de Rham cohomology:

• Eckhard Meinrenken, Group actions on manifolds, Lecture Notes 2003 (pdf, pdf)