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equivariant bundle

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Contents

Idea

Generally, for GG some group, a GG-equivariant bundle is a bundle, specifically a fiber bundle (principal bundle, vector bundle, etc.) all whose component spaces (total space EE, base space XX, fiber FF, but also possibly the structure group Γ\Gamma) are equipped with GG-actions, such that all structure morphisms (in particular the projection EpXE \overset{p}{\to} X, but also the Γ\Gamma-action for principal bundles etc.) are GG-equivariant functions.

In short, this should mean (GSS 21, Def. below) that GG-equivariant (fiber-, principal-,…) bundles are (fiber-, principal, …) bundles internal to a category of G-spaces (e.g. topological G-spaces, G-manifolds but also G-sets etc.).

While the existing literature does not state the definition of equivariant bundles via internalization, one sees (Prop. below) that the explicit definition in tom Dieck 69, tom Dieck 87, Sec. I.8 (for the case of principal bundles) is the equivalent external description, including, thereby, an action of the equivariance group GG on the structure group Γ\Gamma – together with the respective compatibility conditions, which equivalently say (as highlighted in tom Dieck 69, Sec. 1.2, also Murayama-Shimakawa 95, below 1.1, see the discussion here) that the joint action is that of the semidirect product group ΓG\Gamma \rtimes G.

Beware that this action of the equivariance group GG on the structure group Γ\Gamma of an equivariant principal bundle is often and traditionally disregarded, i.e. implicitly taken to be the trivial action (e.g. Lashof 82, Lashof-May-Segal 83), which equivalently means that the semidirect product group that acts is reduced to the direct product group Γ×G\Gamma \times G, meaning that the action of the equivariance group commutes with that of the structure group. This special case is the default meaning of equivariant bundle in much of the literature!

(The definition of “generalized equivariant bundles” proposed in Lashof-May 86 and advertized in Lewis-May-Steinberger 86, Sec. IV.1 allows any group extension of GG by Γ\Gamma to act. This reduces to semidirect products ΓG\Gamma \rtimes G for split extensions (see there) where Γ\Gamma is a normal subgroup, and that is the case that May 90, Guillou-May-Merling 17 falls back to).

Much of the literature on equivariant bundles is interested

and/or

Definition

We discuss equivariant groups in/as topological spaces, for definiteness and due to their relevance as models in equivariant homotopy theory. All of the discussion generalizes, say to smooth manifolds or general toposes.

Preliminaries

Definition

(convenient category of topological spaces)
We write

TopologicalSpacesCategories TopologicalSpaces \;\in\; Categories

for any convenient category of topological spaces whose mapping space serves as an internal hom, such as

This means in particular that for X,Y,ATopologicalSpacesX,Y,A \,\in\, TopologicalSpaces, we have a natural bijection

(1)X×YAadjunctsXMaps(Y,A) X \times Y \longrightarrow A \;\;\;\;\;\;\; \overset{adjuncts}{\leftrightarrow} \;\;\;\;\;\;\; X \longrightarrow Maps(Y,A)

between maps (meaning: continuous functions) out of the product topological space with YY and maps into the mapping space.

Definition

(topological GG-spaces)
For GG be a topological group – to be called the equivariance group – we write

GActions(TopologicalSpaces)Categories G Actions(TopologicalSpaces) \;\in\; Categories

for the category whose

Remark

(further conditions on the equivariance group)
For purposes of equivariant homotopy theory one typically assumes the topological equivariance group GG in Def. to be that underlying a compact Lie group, such as a finite group (as that guarantees that G-CW-complexes are well-behaved and that the equivariant Whitehead theorem holds). But for the plain point-set topology of equivariant bundles, this condition is not necessary.

Remark

(topological G-spaces are cartesian monoidal) The category of topological G-spaces (Def. ) is a Cartesian monoidal category: The Cartesian product of two topological G-spaces (X i,ρ i)(X_i, \rho_i) is the underlying product topological space equipped with the diagonal action by GG:

(X 1,ρ 1)×(X 2,ρ 2)=(X 1×X 2,ρ 1,2(g)(x 1,x 2)=(ρ 1(x 1),ρ 2(x 2))). (X_1, \rho_1) \times (X_2, \rho_2) \;=\; \Big( X_1 \times X_2, \, \rho_{1,2}(g)(x_1,x_2) \,=\, \big( \rho_1(x_1), \rho_2(x_2) \big) \Big) \,.

In a Cartesian monoidal category there is a notion of internal group objects:

Equivariant groups

Definition

(equivariant topological groups)
Given an equivariance group GG (Def. ), a GG-equivariant topological group (Γ,α)(\Gamma, \alpha) is a group object internal to topological G-spaces (Def. ):

(Γ,α)Groups(GActions(TopologicalSpaces)). \big( \Gamma, \, \alpha \big) \;\in\; Groups \big( G Actions ( TopologicalSpaces ) \big) \,.

(See Prop. below for the choice of notation used here.)

Remark

Since the forgetful functor from topological G-spaces (Def. ) to underlying topological spaces

GActions(TopologicalSpaces) TopologicalSpaces \array{ G Actions ( TopologicalSpaces ) & \overset{ }{\longrightarrow} & TopologicalSpaces }

preservesCartesian products (explicitly so by Remark ), it preserves group objects and hence sends GG-equivariant topological groups (Def. ) to underlying plain topological groups:

(2)Groups(GActions(TopologicalSpaces)) Groups(TopologicalSpaces) (Γ,α) Γ. \array{ Groups \big( G Actions ( TopologicalSpaces ) \big) & \overset{ }{\longrightarrow} & Groups ( TopologicalSpaces ) \\ \big( \Gamma , \alpha \big) &\mapsto& \Gamma \,. }

Equivariant bundles

We say that a GG-equivariant principal bundle is a principal bundle internal to topological G-spaces.

(Beware that, following tradition in equivariant bundle theory, we do not impose a local trivializability condition at this point, but add this as an extra clause later and then speak explicitly of locall trivial equivariant bundles – for more on this see at Notions of equivariant local triviality.)

Definition

(equivariant principal bundle)
Given

a GG-equivariant (Γ,α)(\Gamma,\alpha)-principal bundle over (X,ρ)(X,\rho) is:

such that the following conditions holds:

Remark

(on the internal definition of principality) Notice that in Def. we do not explicitly demand that the base space is the quotient space of the total space by the group action (it defines an internal pseudo-torsor instead of a torsor). This has several reasons:

  1. The way the definition is stated, it involves only finite limits (no colimits), which means that this notion of equivariant principal bundle will be preserved by all right adjoint functors on the ambient category, such as notably the fixed locus-functors (see there).

  2. Nevertheless, the expected quotient condition is both implied as well as circumvented, as need be:

    Once equivariant local trivializability is imposed below it follows that the underlying continuous function PXP \to X is a locally trivial fiber bundle. There are then two cases:

    1. Either the typical fiber is inhabited. This implies that PXP \to X is an effective epimorphism, and with that property the shear map condition above implies that XX is the quotient of PP by GG.

    2. Or the typical fiber is the empty space. This case, often disregarded in discussion of fiber bundles, is actually an example of Def. , because for the empty bundle the shear map is an isomorphism:

      Γ×× X \Gamma \times \varnothing \overset{\simeq}{\longrightarrow} \varnothing \times_X \varnothing

      (since both domain and codomain are isomorphic to the empty space):

      Empty bundles are principal! (and locally trivial!), in the above sense.

      While this degenerate case is irrelevant for ordinary principal bundles, it is crucial for equivariant principal bundles, since their fixed loci often have empty total space. In fact, if the action α\alpha of GG on the structure group Γ\Gamma is trivial (as is often the case in applications) then it follows at once that any fixed locus of an equivariant principal bundle can have only one of two typical fibers, Γ\Gamma or \varnothing, while the structure group is Γ\Gamma, in both cases.

Properties

As (G,α,Γ)(G,\alpha,\Gamma)-principal bundles

Proposition

(principal bundles internal to GG-spaces are equivalent to tom Dieck-bundles)
The GG-equivariant (Γ,α)(\Gamma,\alpha)-principal bundles in the internal sense of Def. are equivalent to the (G,α,Γ)(G,\alpha,\Gamma)-principal bundles in the sense of tom Dieck 69 (both without any condition of local trivializability, at this point).

Proof

This follows immediately by the fact (this Prop.) that GG-equivariant actions of equivariant groups (G,α)(G,\alpha) are equivalent to plain actions of the semidirect product group Γ αG\Gamma \rtimes_\alpha G.

Restriction to fixed loci

The internal-formulation of GG-equivariant principal; bundles (Def. ) serves to make manifest that they have good behaviour under passage to fixed loci:

Notice that for any subgroup HGH \subset G we have the fixed point functor (here)

(3)() H:TopologicalGSpacesTopologicalN(H)/HSpaces. (-)^H \;\colon\; Topological G Spaces \overset{\;\;\;\;\;}{\longrightarrow} Topological N(H)/H Spaces \,.

Proposition

For (Γ,α)(\Gamma,\alpha) an GG-equivariant topological group (Def. ), PXP \to X a GG-equivariant (Γ,α)(\Gamma,\alpha)-principal bundle (Def. ) and HGH \subset G any subgroup, the HH-fixed locus (3) P HX HP^H \to X^H is canonically an N(H)/HN(H)/H-equivariant (Γ H,α H)(\Gamma^H, \alpha^H)-principal bundle.

Proof

The point is that passage to HH-fixed loci (3) is a right adjoint functor (this Prop.) and therefore preserves all limits, and so in particular preserves Cartesian products and fiber products, hence preserves internal group objects as well as the principality condition.

Over coset spaces

Proposition

(semidirect product coset space bundles are locally trivial) Let HGH \subset G be a closed subgroup, and let H^{\widehat H} be a lift of HH to a closed subgroup of the semidirect product group, as shown on the left here:

(4)H^ Γ αG A (Γ αG)/H^ pr 2 p H G G/H. \array{ {\widehat H} &\subset& \Gamma \rtimes_\alpha G &{\phantom{A}}& \big( \Gamma \rtimes_\alpha G \big) \big/ {\widehat H} \\ {}^{\mathllap{\simeq}} \big\downarrow && \big\downarrow {}^{\mathrlap{ pr_2 }} && \big\downarrow {}^{ \mathrlap{p} } \\ H &\subset& G && G/H \mathrlap{ \,. } }

Then the induced bundle of coset spaces, shown on the right of (4):

  1. is a (Γ,α)(\Gamma,\alpha)-principal bundle (under Prop. )

  2. is locally trivial as an ordinary Γ\Gamma-fiber bundle in TopologicalSpaces as soon as the the coset space coprojection GG/HG \to G/H admits local sections.

This is tom Dieck 69, Lemma in Sec. 2.1

Proof

The first statement is evident.

For the second statement, let σ\sigma denote a local section over an open subset UG/HU \subset G/H:

G |𝒰 G σ (pb) U = U G/H, \array{ && G_{\vert \mathcal{U}} &\longrightarrow& G \\ & {}^{\mathllap{ \sigma }} \nearrow & \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \\ U &=& U &\longrightarrow& G/H \mathrlap{\,,} }

We claim that then

Γ×U (γ,u)[γ,σ(u)] ((Γ αG)/H^) |U U \array{ \Gamma \times U && \underoverset {\simeq} { (\gamma,u) \mapsto [\gamma,\sigma(u)] } { \longrightarrow } && \Big( \big( \Gamma \rtimes_\alpha G \big) / {\widehat H} \Big)_{\vert U} \\ & \searrow && \swarrow \\ && U }

is an isomorphism, of Γ\Gamma-principal bundles over UG/HU \subset G/H.

  • It is clear by construction that the map is

    1. continuous.

    2. Γ\Gamma-equivariant.

  • To see that map is an isomorphism it is sufficient to see that it is so over each uUu \in U:

    • Injectivity here follows from the fact that H^{\widehat H} is a lift of HH, so that the unique element lifting the neutral element is the neutral element.

    • To see that the map is surjective over each uUu \in U:

      That [γ,g](Γ αG)/H^[\gamma, g] \in \big( \Gamma \rtimes_\alpha G \big)/{\widehat H} is in the fiber over uu means that g 1σ(u)HG g^{-1} \sigma(u) \in H \subset G. Then for h^H^\hat h \in \widehat H the unique element with pr 2(h^)=g 1σ(u)pr_2(\hat h) = g^{-1} \sigma(u) we see that [γ,g]=[(γ,g)q]=[,σ(u)][\gamma, g] = \big[ (\gamma,g) \cdot q \big] = [\cdots, \sigma(u)], which is manifestly in the image of our map.

      (Filling in the ellipses here, which is straightforward but unenlightning, actually gives the continuous inverse map.)

In fact every equivariant principal bundle over a coset space is of this form:

Proposition

(equivariant principal bundle over coset spaces are semidirect product coset space bundles) Let HGH \subset G be a closed subgroup. Then every Hausdorff space (Γ,α)(\Gamma,\alpha)-principal bundle (Def. ) over the coset space

PpG/H, P \overset{p}{\longrightarrow} G/H \,,

which is locally trivial as an ordinary Γ\Gamma-principal bundle, is isomorphic

(5)(Γ αG)/H^ [γ,g](γ,g)p P G/H \array{ (\Gamma \rtimes_\alpha G)/{\widehat H} && \underoverset {\simeq} { [\gamma,g] \mapsto (\gamma,g)\cdot p } {\longrightarrow} && P \\ & \searrow && \swarrow \\ && G/H }

to a coset space-bundle (Γ αG)/H^G/H(\Gamma \rtimes_\alpha G)/{\widehat H} \longrightarrow G/H from Prop. , for H^{\widehat H} a lift of HH to a closed subgroup of the semidirect product group:

H^ Γ αG pr 2 H G. \array{ {\widehat H} &\subset& \Gamma \rtimes_\alpha G \\ {}^{\mathllap{\simeq}} \big\downarrow && \big\downarrow {}^{\mathrlap{ pr_2 }} \\ H &\subset& G \mathrlap{ \,. } }

Proof

Regard PP as equipped with the action of the semidirect product group Γ αG\Gamma \rtimes_\alpha G, by Prop. .

Let eP [H]e \in P_{[H]} be any point in the fiber over [H]G/H[H] \in G/H, and take

H^Stab Γ αG(e) {\widehat H} \;\coloneqq\; Stab_{\Gamma \rtimes_\alpha G}(e)

to be its stabilizer subgroup under the semidirect product group-action. We observe that this has the stated properties:

  • H^{\widehat H} is a closed subgroup because stabilizer subgroups are closed subgroups under the Hausdorff-ness assumption;

  • under Γ αGpr 2G\Gamma \rtimes_\alpha G \overset{pr_2}{\longrightarrow} G we have H^H{\widehat H} \to H, since for all h^H^\hat h \in \widehat H:

    [H] =p(h^e) =pr 2(h^)p(e) =pr 2(h^)[H] \begin{aligned} [H] &= p(\hat h \cdot e) \\ & = pr_2(\hat h) \cdot p(e) \\ & = pr_2(\hat h) \cdot [H] \end{aligned}
  • H^H{\widehat H} \to H is

    (by principality).

Moreover, it is clear by construction that (5) is a (Γ αG)(\Gamma \rtimes_\alpha G)-equivariant continuous function over G/HG/H.

Hence to see that (5) is an isomorphism (a homeomorphism of underlying topological spaces) it is sufficient to see that after forgetting the GG-action we have a morphism between ordinary Γ\Gamma-principal bundles over their common base space, because any such is an isomorphism, as is manifest from its restriction to any common local trivialization.

Therefore it is now sufficient to see that the coset bundle (Γ αG)/H^G/H(\Gamma \rtimes_\alpha G)/{\widehat H} \longrightarrow G/H is locally trivial as a Γ\Gamma-principal bundle. But this is the statement of Prop. .

Notions of equivariant local triviality

The literature considers various different notions of local triviality of equivariant bundles. We list them and then (…eventually…) discuss sufficient conditions under which these imply each other.

under construction

Definition

(tom Dieck’s equivariant local triviality condition – tom Dieck 69, Def. 2.3, tom Dieck 87, p. 58)
An equivariant principal bundle (Def. , Prop. )

PpB(Γ,α)PrincipalBundles P \overset{p}{\longrightarrow} B \;\; \in \; (\Gamma,\alpha)PrincipalBundles

is locally trivial if there exists

  1. an index-set II,

  2. an II-indexed set of sub-G-spaces

    U iBGActions(TopologicalSpaces) U_i \;\subset\; B \;\; \in G Actions \big( TopologicalSpaces \big)
  3. an II-indexed set of closed subgroup H iGH_i \subset G;

  4. an II-indexed set of E ip iG/H i(Γ,α)PrincipalBundlesE_i \overset{p_i}{\to} G/H_i \;\in\; (\Gamma,\alpha)PrincipalBundles over their coset spaces;

such that

  1. {U iB} iI\big\{ U_i \hookrightarrow B \big\}_{i \in I} is an open cover in TopologicalSpaces;

  2. for each iIi \in I there is a homomorphism of (Γ,α)PrincipalBundles(\Gamma,\alpha)PrincipalBundles

    (6)E |U i E i p U i G/H \array{ E_{\vert U_i} & \overset{}{\longrightarrow} & E_i \\ \big\downarrow & {}^{{}_{}} & \big\downarrow {}^{\mathrlap{p}} \\ U_i &\longrightarrow& G/H }

    from the restriction of EE to U iU_i and the given equivariant bundle over the coset space (as above).

Definition

(Bierstone’s equivariant local triviality condition – Bierstone 78, Sec. 4, p. 619-620)
An equivariant principal bundle (Def. , Prop. )

PpX(Γ,α)PrincipalBundles P \overset{p}{\longrightarrow} X \;\; \in \; (\Gamma,\alpha)PrincipalBundles

is locally trivial if for each point xXx \in X (with isotropy group/stabilizer group denoted G xStab G(x)XG_x \coloneqq Stab_G(x) \subset X) there exists

  1. an open neighbourhood

    U xXG xActions(TopologicalSpaces) U_x \;\subset\; X \;\; \in \; G_x Actions(TopologicalSpaces)
  2. a G xG_x-equivariant homomorphism of ΓPrincipalBundles\Gamma PrincipalBundles

    P |U x U x×p 1({x}) p |U x pr 1 U x = U x \array{ P_{\vert U_x} & \longrightarrow & U_x \times p^{-1}(\{x\}) \\ {}^{\mathllap{ p_{\vert U_x} }} \big\downarrow && \big\downarrow {}^{ \mathrlap{ pr_1 } } \\ U_x &=& U_x }

    from the restriction of PP to U xU_x to the Cartesian product of U xU_x with the fiber of PP over xx.

Definition

(Lashof’s equivariant local triviality condition – Lashof 82, p. 258, Lashof-May 86, p. 267)
An equivariant principal bundle (Def. , Prop. )

PpX(Γ,α)PrincipalBundles P \overset{p}{\longrightarrow} X \;\; \in \; (\Gamma,\alpha)PrincipalBundles

is locally trivial if for each point xXx \in X (with isotropy group/stabilizer group denoted G xStab G(x)XG_x \coloneqq Stab_G(x) \subset X) there exists

  1. an index-set II;

  2. an II-indexed set of closed subgroups H iGH_i \subset G;

  3. an II-indexed set of H iH_i slices S iXS_i \subset X;

  4. an II-indexed set of Γ iH iActions(TopologicalSpaces)\Gamma_i \;\in\; H_i Actions(TopologicalSpaces) lifting ΓTopologicalSpaces\Gamma \in TopologicalSpaces;

such that

  1. the orbits of the slices {GS iX} iI\big\{ G\cdot S_i \subset X \big\}_{i \in I} form an open cover over XX;

  2. for each iIi \in I there is a GG-equivariant homomorphism of ΓPrincipalBundles\Gamma PrincipalBundles

    (7)P |GS i G× H i(S i×Γ i) GS i = G× HS i \array{ P_{\vert G\cdot S_i} &\longrightarrow& G \times_{H_i} \big( S_i \times \Gamma_i \big) \\ \big\downarrow && \big\downarrow \\ G \cdot S_i &=& G \times_H S_i }

    from the restriction of PP over the orbit of the iith slice to …

(…)

Proposition

(…) Lashof82 \leftrightarrow Bierstone78 (…)

(Lashof 82, Lemmas 1.1, 1.3)

Lemma

(Lashof’s local models are locally trivial as ordinary fiber bundles) If the equivariance group GG is a compact Lie group then for every closed subgroup HGH \subset G and topological H-spaces S,FS, F the canonical projection

G× H(S×F) G× HS \array{ G \times_H \big( S \times F \big) \\ \big\downarrow \\ G \times_H S }

is a locally trivial FF-fiber bundle.

(Lashof 82, Cor. 1.2)

Proof

Since GG is assumed to be a compact Lie group, it admits a bi-invariant Riemannian metric (this Prop.). With respect to this metric, consider a small open normal ϵ\epsilon-neighbourhood to HH at ee in GG

D eD ϵN eHNHexpTH D_e \,\coloneqq\, D_\epsilon N_e H \,\subset \, N H \; \underoverset{\simeq}{\exp}{\longrightarrow} \; T H

i.e. of points that with respect to some choice of tubular neighbourhood of HGH \subset G are a normal distance <ϵ\lt \epsilon from HH.

Then the multiplication action

D e×H ()() D eH (d,h) dh \array{ D_e \times H & \underoverset{ \simeq }{ (-)\cdot(-) }{\longrightarrow} & D_e \cdot H \\ (d,h) &\mapsto& d \cdot h }

is a diffeomorphism, because, by right-invariance of the chosen metric, the operation

D e D h d dh \array{ D_e &\overset{}{\longrightarrow}& D_h \\ d &\mapsto& d \cdot h }

is even an isometry, for each hHh \in H.

It follows

1) by dimension reasons that

DHG D \cdot H \;\subset\; G

is an open neighbourhood of HH in GG, and hence that

(DH)× HSG× HS \big(D \cdot H\big) \times_H S \hookrightarrow G \times_H S

is an open neighbourhood of SG H×SS \subset G_H \times S.

2) that the restriction of the projection to this neighbourhood is isomorphic to the trivial FF-fiber bundle:

D×S×F (D×H)× H(S×F) (DH)× H(S×F) G× H(S×F) (pb) D×S (D×H)× HS m× Hid (DH)× HS G× HS. \array{ D \times S \times F & \simeq & (D \times H) \times_H (S \times F) & \overset{\simeq}{\longrightarrow} & (D \cdot H) \times_H (S \times F) & \longrightarrow & G \times_H ( S \times F ) \\ \big\downarrow && && \big\downarrow & {}^{{}_{(pb)}} & \big\downarrow \\ D \times S &\simeq& (D \times H) \times_H S & \underoverset { m \times_H id } {\simeq} {\longrightarrow} & (D \cdot H) \times_H S &\hookrightarrow& G \times_H S \,. }

Finally, since G× HSG \times_H S is covered by left GG-translates of the open subset (DH)× HS(D \cdot H) \times_H S, and since the same argument applies to each of theses, by left-invariance of the metric, the claim follows.

Proposition

(Lashof’s local trivializability implies tom Dieck’s for α=1\alpha = 1) In the case that α=1\alpha = 1 (in Def. ), equivariant local trivializability in the sense of Lashof (Def. ) implies local trivializability in the sense of tom Dieck (Def. ).

Proof

It is sufficient to see that Lashof’s local model bundles in (7) are examples of tom Dieck’s local model bundles in (6).

So let HGH \subset G be a closed subgroup, SXS \subset X be an HH-slice and

(8)ϕ:HΓ \phi \colon H \to \Gamma

be the homomorphism through which HH acts on Γ\Gamma in Lashof’s model for the equivariant bundle over the orbit of the slice.

G× H(S×Γ) G× HS. \array{ G \times_H (S \times \Gamma) \\ \big\downarrow \\ G \times_H S \,. }

By Prop. and using that α=1\alpha = 1 we obtain from ϕ\phi an equivariant principal bundle over G/HG/H by taking the subgroup H^{\widehat H} in (4) to be the graph of ϕ\phi (8)

H^graph(ϕ)Γ×G. {\widehat H} \;\coloneqq\; graph(\phi) \;\subset\; \Gamma \times G \,.

Observing that the coset space by a subgroup of this form coincindes with the quotient by the diagonal action

(Γ×G)/graph(ϕ)G× HΓ, (\Gamma \times G)/ graph(\phi) \;\simeq\; G \times_H \Gamma \,,

makes it evident that tom Dieck’s condition (6) is satisfied for Lashof’s local model in that the following square is manifestly a pullback:

G× H(S×Γ) G× HΓ = (Γ×G)/(graph(HϕΓ)) G× HS G× HH \array{ G \times_H (S \times \Gamma) & \longrightarrow & G \times_H \Gamma & = & (\Gamma \times G) \big/ ( graph(H \overset{\phi}{\to} \Gamma) ) \\ \big\downarrow && \big\downarrow & \swarrow \\ G \times_H S & \underoverset {} {} {\longrightarrow} & G \times_H H }

(…)

Remark

(sufficient condition for plain local trivialization to have equivariant enhancement) Sufficient conditions for existence of plain local trivialization to imply a GG-equivariant local trivialization, and hence for the inclusion in Prop. to be an actual equivalence of categories, are:

  1. GG is a finite group; or

  2. XX is a locally compact separable metric space of finite dimension and with a finite number of GG-orbit types; or

  3. Γ\Gamma is a locally compact separable metric space such that for every GG-orbit type G/HXG/H \subset X the fixed locus Γ H\Gamma^H is an absolute neighbourhood retract (such as a finite-dimensional topological manifold or a finite-dimensional and locally finite CW-complex, by the discussion there).

Because (Atiyah 66, p. 374) then for every plain local trivialization around any orbit the equivariant Tietze extension theorem implies the existence of an equivariant function to Γ\Gamma on an open neighbourhood of that orbit, which thus constitutes a GG-equivariant local trivialization.


Universal equivariant principal bundles

Definition

(Murayama-Shimakawa groupoid for discrete GG)
For GG a discrete group and (Γ,α)(\Gamma,\alpha) a GG-equivariant topological group (hence a topological group equipped with an action α\alpha of GG by continuous group automorphisms), consider the topological groupoid which is the functor groupoid from the pair groupoid of GG to the delooping groupoid of Γ\Gamma:

(9)ΓGroupoids(TopSpaces)(G×Gpr 2pr 1G,Γ*) \mathcal{B}\Gamma \;\coloneqq\; Groupoids(TopSpaces) \left( G \times G \underoverset{pr_2}{pr_1}{\rightrightarrows} G, \; \Gamma \rightrightarrows \ast \right)

but regarded as an internal groupoid in topological G-spaces

ΓGroupoids(GActions(TopologicalSpaces)) \mathcal{B}\Gamma \;\in\; Groupoids \big( G Actions(TopologicalSpaces) \big)

by equipping it with the GG-action given on functors FF and natural transformations η\eta in (9) by:

(gF)(g 1,g 2)α(g)(F(g 1g,g 2g)),AAA(gη)(g 1)α(g)(η(g 1)). (g \cdot F) (g_1, g_2) \;\coloneqq\; \alpha(g) \big( F(g_1 g, g_2 g) \big) \,, {\phantom{AAA}} (g \cdot \eta) (g_1) \;\coloneqq\; \alpha(g)(\eta(g_1)) \,.

add word on choice of topology on the mapping spaces, see Murayama-Shimakawa 95, p. 1293 (5 of 7)

Remark

This means that for any HGH \subset G the HH-fixed point space of Γ\mathcal{B}\Gamma is the HH-homotopy fixed point-space of BΓB \Gamma (Thomason 83, (3.2)).

Proposition

The fat geometric realization of the nerve of the GG-topological groupoid from Def.

ΓGActions(TopSpaces) \left\Vert \mathcal{B}\Gamma \right\Vert \;\; \in G Actions(TopSpaces)

is a classifying space for GG-equivariant (Γ,α)(\Gamma,\alpha)-principal bundles.

Proof

This is Theorem 3.1 in Murayama-Shimakawa 95, using the remark on the bottom of p. 1294 (6 of 7) that for discrete group GG the construction in Theorem 3.1 may be simplified.

For GG discrete and Γ\Gamma discrete or compact Lie the same statement appears as Guillou, May & Merling 17, Thm. 3.11. Notice that Scholium 3.12 there doubts that Murayama-Shimakawa 95‘s result holds for non-discrete GG.

Proposition

Let HGH \subset G any subgroup, there is an equivalence of topological groupoids between the HH-fixed locus of the GG-groupoid from Def. and

  1. for trivial α\alpha: the functor groupoid between the delooping groupoids of GG and Γ\Gamma:

    (Γ) HGroupoids(TopSpaces)(G*,,Γ*) \big( \mathcal{B}\Gamma \big)^H \;\; \simeq \;\; Groupoids(TopSpaces) \big( G \rightrightarrows \ast, \,, \Gamma \rightrightarrows \ast \big)
  2. for general α\alpha: the action groupoid of the conjugation action of Γγ(γ,e)Γ αG\Gamma \overset{\gamma \mapsto (\gamma,e)}{\hookrightarrow} \Gamma \rtimes_\alpha G on group homomorphisms ϕ:GΓ αG\phi \colon G \to \Gamma \rtimes_\alpha G which are sections (pr 2ϕ=id Gpr_2 \circ \phi = id_G) of pr 2:Γ αGGpr_2 \colon \Gamma \rtimes_\alpha G \to G:

    (Γ) H(Groups(TopSpaces) /G(G,Γ αG))Γ. \big( \mathcal{B}\Gamma \big)^H \;\; \simeq \;\; \Big( Groups(TopSpaces)_{/G} \big( G, \, \Gamma \rtimes_\alpha G \big) \Big) \sslash \Gamma \,.

Proof

The first statement is the evident specialization of the second. It may help to go through the proof first in this special case of trivial α\alpha, as the presence of α\alpha can tend to obscure the simple logic behind it.

First consider the case H=GH = G. Here the point to notice is that GG-invariance of a functor F:G×GΓ F \colon G \times G \to \Gamma means that its value depends only on the difference of its arguments

F(g 1,g 2)=F(g 1g 2 1) F(g_1, g_2) = F'( g_1 g_2^{-1} )

and functoriality then means equivalently that g(F(g),g)g \mapsto (F'(g), g) is a group homomorphism to Γ α\Gamma \rtimes_\alpha.

Similarly, GG-invariance of a naturaltrans formation η:GΓ\eta \colon G \to \Gamma means that its components are fixed by its values on eGe \in G via

η(g 1)=α(g)(η(e)). \eta( g^{-1} ) \;=\; \alpha(g)(\eta(e)) \,.

With this, the naturality square for η\eta commutes precisely if two sections g(F(g),g)g \mapsto (F'(g), g), as above, are related by conjugation with γ eΓ αG\gamma_e \in \Gamma \rtimes_\alpha G.

This proves the equivalence in the case H=GH = G.

Next to see the claim for general HGH \subset G, notice that restriction along (H×HH)(G×GG)(H \times H \rightrightarrows H) \hookrightarrow (G \times G \rightrightarrows G) followed by the above equivalence (now for GG replaced by HH) gives a canonical comparison functor pp. We claim that a homotopy-inverse is given by the evident inclusion functor ii that fills up unspecified data by neutral elements. It is clear that pi=idp \circ i = id and one checks that there is a homotopy (idip)(id \Rightarrow i \circ p ) (which is straightforward if one chooses good diagrammatic notation…).

Remark

To the extent that passage to fixed loci commutes with realization (this would be guaranteed if we could use ordinary geometric realization in Prop. which works as soon as Γ\Gamma is compact Lie), Prop. immediately implies the behaviour of equivariant classifying spaces under fixed loci according to Lashof 82, Thm. 2.17 and Lashof & May 86, Thm. 10.

Related discussion is in Guillou, May & Merling 17, pp. 15.

(…)


Examples

References

Equivariant fiber bundles

On general (topological) equivariant fiber bundles/principal bundles:

Precursor discussion:

The original definition:

With abelian structure group:

More on equivariant principal bundles and their classifying spaces/universal principal bundles:

Review and examples over the 2-sphere:

See also:

Equivariant vector bundles

On equivariant vector bundles:

In the context of equivariant K-theory:

In a context of equivariant differential topology:

As twists for twisted equivariant K-theory

On equivariant pricipal bundles with structure group the projective unitary group, hence providing twists for twisted equivariant K-theory:

For equivariant elliptic cohomology

In a context of equivariant elliptic cohomology:

For gauge theory on orbifolds

The general notion of equivariant bundles from tom Dieck 69 (with action of the semidirect product of the gauge group with the equivariance group) gets a brief mentioning in

in a context of Chern-Simons theory on orbifolds.

Last revised on June 6, 2021 at 10:10:50. See the history of this page for a list of all contributions to it.