nLab equivariant bundle

Idea

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

In short, this should mean (SS 2021, Sec. 2.1, Def. below) that $G$-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 $G$ on the structure group $\Gamma$ – 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 $\Gamma \rtimes G$.

Beware that this action of the equivariance group $G$ on the structure group $\Gamma$ 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 $\Gamma \times G$, 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 $G$ by $\Gamma$ to act. This reduces to semidirect products $\Gamma \rtimes G$ for split extensions (see there) where $\Gamma$ 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

• in equivariant vector bundles regarded as representing cocycles for equivariant K-theory (Segal 68, Sec 1, Wasserman 69, Sec. 2,HJJS 08, Sec. 13);

and/or

• as twists for twisted equivariant cohomology theories, and here most of the existing literature focuses on twisted equivariant K-theory and specifically on its equivariant degree-3 twist by equivariant projective bundles (e.g. Barcenas-Espinoza-Joachim-Uribe 12, Uribe & Lück 2014, Sec. 15).

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

$$TopologicalSpaces \;\in\; Categories$$

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

• D-topological spaces

• compactly generated topological spaces.

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

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

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

Definition

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

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

for the category whose

• objects$(X,\rho)$ are topological spaces $X$ equipped with continuous $G$-actions

  $$G \times X \overset{}{\longrightarrow} X \;\;\;\;\;\; \leftrightarrow \;\;\;\;\;\; G \overset{\rho}{\longrightarrow} Aut(X) \subset Maps(X,X) \,;$$

• morphisms are $G$-equivariant continuous functions between them (“maps”); i.e. for the category of “G-spaces”, often denote “$G Spaces$” or even $G Sp$ or similar.

Remark

(further conditions on the equivariance group)
For purposes of equivariant homotopy theory one typically assumes the topological equivariance group $G$ 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 $(X_i, \rho_i)$ is the underlying product topological space equipped with the diagonal action by $G$:

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

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

Equivariant groups

We follow the terminology of SS 2021, Def. 2.1.2:

Definition

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

$$\big( \Gamma, \, \alpha \big) \;\in\; Groups \big( G Actions ( TopologicalSpaces ) \big) \,.$$

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

Remark

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

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

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

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

Equivariant bundles

We say that a $G$-equivariant principal bundle is a principal bundle internal to topological G-spaces (SS 2021, Def. 2.1.3).

(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 locally trivial equivariant bundles – for more on this see at Notions of equivariant local triviality.)

Definition

(equivariant principal bundle)
Given

• an equivariance group $G \in Groups(TopologicalSpaces)$;

• an equivariant topological group (Def. )

  $(\Gamma, \alpha) \;\in\; Groups\big( G Actions(TopologicalSpaces) \big)$

  to be called the $G$-equivariant structure group;

a $G$-equivariant $(\Gamma,\alpha)$-principal bundle over $(X,\rho)$ is:

• a internal bundle, namely a morphism $P \overset{p}{\to} X$ in topological G-spaces;

• the structure of a $(\Gamma,\alpha)$-action object on the total space $P$:

such that the following conditions holds:

• (fiberwise action) the action is over $X$, in that the following diagram commutes:

• (principality) the shear map

  $$\array{ \Gamma \times P &\longrightarrow& P \times_X P \\ (\gamma, p) &\mapsto& (\rho(\gamma)(p), p) }$$

  is an isomorphism, hence this commuting diagram is moreover a pullback square (in topological G-spaces):

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 $P \to X$ is a locally trivial fiber bundle. There are then two cases:

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

   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:

      $$\Gamma \times \varnothing \overset{\simeq}{\longrightarrow} \varnothing \times_X \varnothing$$

      (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 $\alpha$ of $G$ on the structure group $\Gamma$ 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, $\Gamma$ or $\varnothing$, while the structure group is $\Gamma$, in both cases.

Properties

As $(G,\alpha,\Gamma)$-principal bundles

Proposition

(principal bundles internal to $G$-spaces are equivalent to tom Dieck-bundles)
The $G$-equivariant $(\Gamma,\alpha)$-principal bundles in the internal sense of Def. are equivalent to the $(G,\alpha,\Gamma)$-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 $G$-equivariant actions of equivariant groups $(G,\alpha)$ are equivalent to plain actions of the semidirect product group $\Gamma \rtimes_\alpha G$.

Restriction to fixed loci

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

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

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

Proposition

For $(\Gamma,\alpha)$ an $G$-equivariant topological group (Def. ), $P \to X$ a $G$-equivariant $(\Gamma,\alpha)$-principal bundle (Def. ) and $H \subset G$ any subgroup, the $H$-fixed locus (3) $P^H \to X^H$ is canonically an $N(H)/H$-equivariant $(\Gamma^H, \alpha^H)$-principal bundle.

Proof

The point is that passage to $H$-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 $H \subset G$ be a closed subgroup, and let ${\widehat H}$ be a lift of $H$ to a closed subgroup of the semidirect product group, as shown on the left here:

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

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

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

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

This is tom Dieck 69, Lemma in Sec. 2.1

Proof

The first statement is evident.

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

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

We claim that then

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

is an isomorphism, of $\Gamma$-principal bundles over $U \subset G/H$.

• It is clear by construction that the map is

  1. continuous.

  2. $\Gamma$-equivariant.

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

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

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

    That $[\gamma, g] \in \big( \Gamma \rtimes_\alpha G \big)/{\widehat H}$ is in the fiber over $u$ means that $g^{-1} \sigma(u) \in H \subset G$. Then for $\hat h \in \widehat H$ the unique element with $pr_2(\hat h) = g^{-1} \sigma(u)$ we see that $[\gamma, g] = \big[ (\gamma,g) \cdot q \big] = [\cdots, \sigma(u)]$, 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 $H \subset G$ be a closed subgroup. Then every Hausdorff space $(\Gamma,\alpha)$-principal bundle (Def. ) over the coset space

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

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

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

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

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

Proof

Regard $P$ as equipped with the action of the semidirect product group $\Gamma \rtimes_\alpha G$, by Prop. .

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

$${\widehat H} \;\coloneqq\; Stab_{\Gamma \rtimes_\alpha G}(e)$$

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

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

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

  $$\begin{aligned} [H] &= p(\hat h \cdot e) \\ & = pr_2(\hat h) \cdot p(e) \\ & = pr_2(\hat h) \cdot [H] \end{aligned}$$

• ${\widehat H} \to H$ is

  • a surjection because the action of $\Gamma$ on the fibers of $P$ is a transitive action,

  • an injection because the action of $\Gamma$ on the fibers of $P$ is a free action

  (by principality).

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

Hence to see that (5) is an isomorphism (a homeomorphism of underlying topological spaces) it is sufficient to see that after forgetting the $G$-action we have a morphism between ordinary $\Gamma$-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 $(\Gamma \rtimes_\alpha G)/{\widehat H} \longrightarrow G/H$ is locally trivial as a $\Gamma$-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. Some of these look quite different, but are indeed all equivalent, under mild conditions (SS 2021, Thm. 2.1.2).

under construction, for details see SS 2021, Sec. 2.2

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. )

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

is locally trivial if there exists

1. an index-set $I$,

2. an $I$-indexed set of sub-G-spaces

   $$U_i \;\subset\; B \;\; \in G Actions \big( TopologicalSpaces \big)$$

3. an $I$-indexed set of closed subgroup $H_i \subset G$;

4. an $I$-indexed set of $E_i \overset{p_i}{\to} G/H_i \;\in\; (\Gamma,\alpha)PrincipalBundles$ over their coset spaces;

such that

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

2. for each $i \in I$ there is a homomorphism of $(\Gamma,\alpha)PrincipalBundles$

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

   from the restriction of $E$ to $U_i$ 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. )

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

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

1. an open neighbourhood

   $$U_x \;\subset\; X \;\; \in \; G_x Actions(TopologicalSpaces)$$

2. a $G_x$-equivariant homomorphism of $\Gamma PrincipalBundles$

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

   from the restriction of $P$ to $U_x$ to the Cartesian product of $U_x$ with the fiber of $P$ over $x$.

Definition

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

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

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

1. an index-set $I$;

2. an $I$-indexed set of closed subgroups $H_i \subset G$;

3. an $I$-indexed set of $H_i$ slices $S_i \subset X$;

4. an $I$-indexed set of $\Gamma_i \;\in\; H_i Actions(TopologicalSpaces)$ lifting $\Gamma \in TopologicalSpaces$;

such that

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

2. for each $i \in I$ there is a $G$-equivariant homomorphism of $\Gamma PrincipalBundles$

   (7)$$\array{ P_{\vert G\cdot S_i} &\longrightarrow& G \times_{H_i} \big( S_i \times \Gamma_i \big) \\ \big\downarrow && \big\downarrow \\ G \cdot S_i &=& G \times_H S_i }$$

   from the restriction of $P$ over the orbit of the $i$th slice to …

• Lashof-Rothenberg 78, p. 216

(…)

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 $G$ is a compact Lie group then for every closed subgroup $H \subset G$ and topological H-spaces $S, F$ the canonical projection

$$\array{ G \times_H \big( S \times F \big) \\ \big\downarrow \\ G \times_H S }$$

is a locally trivial $F$-fiber bundle.

(Lashof 82, Cor. 1.2)

Proof

Since $G$ 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 $\epsilon$-neighbourhood to $H$ at $e$ in $G$

$$D_e \,\coloneqq\, D_\epsilon N_e H \,\subset \, N H \; \underoverset{\simeq}{\exp}{\longrightarrow} \; T H$$

i.e. of points that with respect to some choice of tubular neighbourhood of $H \subset G$ are a normal distance $\lt \epsilon$ from $H$.

Then the multiplication action

$$\array{ D_e \times H & \underoverset{ \simeq }{ (-)\cdot(-) }{\longrightarrow} & D_e \cdot H \\ (d,h) &\mapsto& d \cdot h }$$

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

$$\array{ D_e &\overset{}{\longrightarrow}& D_h \\ d &\mapsto& d \cdot h }$$

is even an isometry, for each $h \in H$.

It follows

1) by dimension reasons that

$$D \cdot H \;\subset\; G$$

is an open neighbourhood of $H$ in $G$, and hence that

$$\big(D \cdot H\big) \times_H S \hookrightarrow G \times_H S$$

is an open neighbourhood of $S \subset G_H \times S$.

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

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

Finally, since $G \times_H S$ is covered by left $G$-translates of the open subset $(D \cdot H) \times_H S$, 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 $\alpha = 1$) In the case that $\alpha = 1$ (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 $H \subset G$ be a closed subgroup, $S \subset X$ be an $H$-slice and

(8)$$\phi \colon H \to \Gamma$$

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

$$\array{ G \times_H (S \times \Gamma) \\ \big\downarrow \\ G \times_H S \,. }$$

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

$${\widehat H} \;\coloneqq\; graph(\phi) \;\subset\; \Gamma \times G \,.$$

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

$$(\Gamma \times G)/ graph(\phi) \;\simeq\; G \times_H \Gamma \,,$$

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:

$$\array{ G \times_H (S \times \Gamma) & \longrightarrow & G \times_H \Gamma & = & (\Gamma \times G) \big/ ( graph(H \overset{\phi}{\to} \Gamma) ) \\ \big\downarrow && \big\downarrow & \swarrow \\ G \times_H S & \underoverset {} {} {\longrightarrow} & G \times_H H }$$

(…)

Remark

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

1. $G$ is a finite group; or

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

3. $\Gamma$ is a locally compact separable metric space such that for every $G$-orbit type $G/H \subset X$ the fixed locus $\Gamma^H$ 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 $\Gamma$ on an open neighbourhood of that orbit, which thus constitutes a $G$-equivariant local trivialization.

Universal equivariant principal bundles

Definition

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

(9)$$\mathcal{B}\Gamma \;\coloneqq\; Groupoids(TopSpaces) \left( G \times G \underoverset{pr_2}{pr_1}{\rightrightarrows} G, \; \Gamma \rightrightarrows \ast \right)$$

but regarded as an internal groupoid in topological G-spaces

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

by equipping it with the $G$-action given on functors $F$ and natural transformations $\eta$ in (9) by:

$$(g \cdot F) (g_1, g_2) \;\coloneqq\; \alpha(g) \big( F(g_1 g, g_2 g) \big) \,, {\phantom{AAA}} (g \cdot \eta) (g_1) \;\coloneqq\; \alpha(g)(\eta(g_1)) \,.$$

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 $H \subset G$ the $H$-fixed point space of $\mathcal{B}\Gamma$ is the $H$-homotopy fixed point-space of $B \Gamma$ (Thomason 83, (3.2)).

Proposition

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

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

is a classifying space for $G$-equivariant $(\Gamma,\alpha)$-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 $G$ the construction in Theorem 3.1 may be simplified.

For $G$ discrete and $\Gamma$ 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 $G$.

Proposition

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

1. for trivial $\alpha$: the functor groupoid between the delooping groupoids of $G$ and $\Gamma$:

   $$\big( \mathcal{B}\Gamma \big)^H \;\; \simeq \;\; Groupoids(TopSpaces) \big( H \rightrightarrows \ast \,, \Gamma \rightrightarrows \ast \big)$$

2. for general $\alpha$: the action groupoid of the conjugation action of $\Gamma \overset{\gamma \mapsto (\gamma,e)}{\hookrightarrow} \Gamma \rtimes_\alpha G$ on group homomorphisms $\phi \colon G \to \Gamma \rtimes_\alpha G$ which are sections ($pr_2 \circ \phi = id_G$) of $pr_2 \colon \Gamma \rtimes_\alpha G \to G$:

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

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 $\alpha$, as the presence of $\alpha$ can tend to obscure the simple logic behind it.

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

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

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

Similarly, $G$-invariance of a naturaltrans formation $\eta \colon G \to \Gamma$ means that its components are fixed by its values on $e \in G$ via

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

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

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

Next to see the claim for general $H \subset G$, notice that restriction along $(H \times H \rightrightarrows H) \hookrightarrow (G \times G \rightrightarrows G)$ followed by the above equivalence (now for $G$ replaced by $H$) gives a canonical comparison functor $p$. We claim that a homotopy-inverse is given by the evident inclusion functor $i$ that fills up unspecified data by neutral elements. It is clear that $p \circ i = id$ and one checks that there is a homotopy $(id \Rightarrow i \circ p )$ (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 $\Gamma$ 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.

(…)

Proposition

If $\Gamma$ is a compact Lie group which is also abelian (hence a direct product group of an n-torus with a abelian finite group) and regarded as carrying trivial $G$-action, then a classifying G-space for $G$-equivariant $\Gamma$-principal bundles is given by $Map(E G ,\, B\Gamma)$.

This is Lashof, May & Segal 1983, Lem. 1 with Thm. 2, re-stated more explicitly in May 1990, Thm. 10.

This implies that for abelian compact Lie structure groups $\Gamma$, the isomorphism classes of $G$-equivariant $\Gamma$-principal bundles over some G-CW complex $X$ have the same classification as plain $\Gamma$-principal bundles over its Borel construction $X_G \,\coloneqq\, (E G \times X)/G$:

$$\begin{aligned} \big( G Equ \Gamma PrnBdl_X \big)_{/\sim} & \;\simeq\; \tau_0 \, Map\big( X ,\, \Map(EG ,\, B \Gamma) \big)^G \\ & \;\simeq\; \tau_0 \, Map\big( E G \times X ,\, B \Gamma \big)^G \\ & \;\simeq\; \tau_0 \, Map\big( (E G \times X)/G ,\, B \Gamma \big) \\ & \;\simeq\; \big( \Gamma PrnBdl_{X_G} \big)_{/\sim} \,. \end{aligned}$$

In the form of the resulting composite bijection, this statement appears as May 1990, Thm. 3.

Examples

• equivariant tautological line bundle

• universal equivariant PU(H)-bundle

• in arithmetic geometry: shtuka

Related concepts

• twisted bundle

• equivariant differential topology

• equivariant cohomology

• equivariant connection

• equivariant K-theory

References

Equivariant fiber bundles

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

Precursor discussion:

• T. E. Stewart, Lifting Group Actions in Fibre Bundles, Annals of Mathematics Second Series, Vol. 74, No. 1 (1961), pp. 192-198 (jstor:1970310)

  (for structure group commuting with the equivariance group, but noticing already that a normal subgroup-extension could be considered, more generally)

The original definition:

• Tammo tom Dieck, Faserbündel mit Gruppenoperation, Arch. Math 20, 136–143 (1969) (doi:10.1007/BF01899003)

  (for structure group split-extending the equivariance group)

• Edward Bierstone, The equivariant covering homotopy property for differentiable $G$-fibre bundles, J. Differential Geom. 8(4): 615-622 (1973) (doi:10.4310/jdg/1214431963)

  (for structure group commuting with the equivariance group, but with differentiable structure)

• Goro Nishida, Section 2 of: The transfer homomorphism in equivariant generalized cohomology theories, J. Math. Kyoto Univ. 18(3): 435-451 (1978) (doi:10.1215/kjm/1250522505)

  (for structure group split-extending the equivariance group)

• 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)

  (for structure group commuting with the equivariance group, but with differentiable structure)

• Richard Lashof, Equivariant bundles, Illinois J. Math. 26(2): 257-271, 1982 (doi:10.1215/ijm/1256046796)

  (for structure group commuting with the equivariance group)

• Richard Lashof, Peter May, Generalized equivariant bundles, Bull. Soc. Math. Belg. Sér. A 38, 265-271, 1986 (pdf, pdf)

  (for structure group extending the equivariance group)

• L. Gaunce Lewis, Peter May, and Mark Steinberger, Section IV.1 of: Equivariant stable homotopy theory, Springer Lecture Notes in Mathematics Vol.1213. 1986 (pdf, doi:10.1007/BFb0075778)

  (for structure group extending the equivariance group)

• Tammo tom Dieck, Section I.8 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

  (for structure group split-extending the equivariance group)

With abelian structure group:

• Richard Lashof, Peter May, Graeme Segal, Equivariant bundles with abelian structural group, Contemporary Mathematics, Volume 19, 1983 (doi:10.1090/conm/019, pdf, pdf)

  (for structure group commuting with the equivariance group)

• Andrzej Kozlowski, Equivariant Bundles and Cohomology, Transactions of the American Mathematical Society, Vol. 296, No. 1 (Jul., 1986), pp. 181-190 (jstor:2000568)

  (for structure group split-extending the equivariance group)

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

• Peter May, Some remarks on equivariant bundles and classifying spaces, Théorie de l’homotopie, Astérisque, no. 191 (1990), 15 p. (numdam:AST_1990__191__239_0)

  (for structure group split-extending the equivariance group)

• Mitutaka Murayama, Kazuhisa Shimakawa, Universal equivariant bundles, Proc. Amer. Math. Soc. 123 (1995), 1289-1295 (doi:10.1090/S0002-9939-1995-1231040-9, jstor:2160733)

  (for structure group split-extending the equivariance group)

• Wolfgang Lück, Survey on Classifying Spaces for Families of Subgroups, In: Infinite Groups: Geometric, Combinatorial and Dynamical Aspects Progress in Mathematics, 248 Birkhäuser (2005) (arXiv:math/0312378, doi:10.1007/3-7643-7447-0_7)

• Bernardo Uribe, Wolfgang Lück, Equivariant principal bundles and their classifying spaces, Algebr. Geom. Topol. 14 (2014) 1925-1995 [arXiv:1304.4862, doi:10.2140/agt.2014.14.1925]

  (for structure group commuting with the equivariance group)

• Charles Rezk, Sec. 2.3 in: Global Homotopy Theory and Cohesion, 2014 (pdf, pdf)

  (for structure group commuting with the equivariance group, and in relation to the global orbit category with an eye towards equivariant homotopy theory)

• Bertrand Guillou, Peter May, Mona Merling, Categorical models for equivariant classifying spaces, Algebr. Geom. Topol. 17 (2017) 2565-2602 (arXiv:1201.5178, doi:10.2140/agt.2017.17.2565)

  (for structure group split-extending the equivariance group)

• Charles Rezk, Classifying spaces for 1-truncated compact Lie groups, Algebr. Geom. Topol. 18 (2018) 525-546 (arXiv:1608.02999)

  (for structure group a 1-truncated compact Lie group commuting with the equivariance group)

Review and examples over the 2-sphere:

• Eyup Yalcinkaya, Equivariant Principal Bundles over the 2-Sphere (arXiv:1902.06293)

See also:

• Foling Zou, Notes on equivariant bundles (arXiv:2008.01268)

Discussion in the context of principal $\infty$-bundles:

• Hisham Sati, Urs Schreiber, Equivariant principal $\infty$-bundles (arXiv:2112.13654)

Equivariant vector bundles

On equivariant vector bundles:

• tom Dieck 87, Section I.9

In the context of equivariant K-theory:

• Graeme Segal, Section 1 of: Equivariant K-theory, Inst. Hautes Etudes Sci. Publ. Math. No. 34 (1968) p. 129-151 (numdam:PMIHES_1968__34__129_0)

• Dale Husemöller, Michael Joachim, Branislav Jurčo, Martin Schottenloher, Section 13 of: Basic Bundle Theory and K-Cohomology Invariants, Springer Lecture Notes in Physics 726, 2008, (pdf, doi:10.1007/978-3-540-74956-1)

In a context of equivariant differential topology:

• Arthur Wasserman, section 2 of: Equivariant differential topology, Topology Vol. 8, pp. 127-150, 1969 (doi:10.1016/0040-9383(69)90005-6pdf)

In the context of orientation in generalized cohomology generalized to equivariant complex oriented cohomology theory:

• Steven Costenoble, Peter May, Stefan Waner, Equivariant orientation theory, Homology Homotopy Appl.

  Volume 3, Number 2 (2001), 265-339 (euclid:hha/1139840256)

As twists for twisted equivariant K-theory

On 3-twisted equivariant K-theory via the universal equivariant $PU(\mathcal{H})$-bundle:

• Noé Bárcenas, Jesús Espinoza, Michael Joachim, Bernardo Uribe, Universal twist in Equivariant K-theory for proper and discrete actions, Proceedings of the London Mathematical Society, Volume 108, Issue 5 (2014) (arXiv:1202.1880, doi:10.1112/plms/pdt061)

• Uribe & Lück 2014, Sec. 15

For equivariant elliptic cohomology

In a context of equivariant elliptic cohomology:

• Matthew Ando, John Greenlees, Part I of: Circle-equivariant classifying spaces and the rational equivariant sigma genus, Math. Z. 269, 1021–1104 (2011) (arXiv:0705.2687, doi:10.1007/s00209-010-0773-7)

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

• Petr Hořava, Section 5.3 of: Chern-Simons Gauge Theory on Orbifolds: Open Strings from Three Dimensions, J. Geom. Phys. 21:1-33, 1996 (arXiv:hep-th/9404101)

in a context of Chern-Simons theory on orbifolds.