nLab equivariant differential topology




topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


Basic concepts

Universal constructions

Extra stuff, structure, properties


Basic statements


Analysis Theorems

topological homotopy theory

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Representation theory



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.


Fixed submanifolds


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

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)


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.


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

Invariant Riemannian metrices


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

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 which is 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)

Equivariant tubular neighbourhoods


(GG-equivariant 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-equivariant 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.


(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)


(existence of GG-equivariant 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


  1. Σ\Sigma admits a GG-equivariant tubular neighbourhood ΣUX\Sigma \subset U \subset X (Def. );

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

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



See also:

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

Last revised on September 25, 2022 at 12:43:35. See the history of this page for a list of all contributions to it.