aa r X i v : . [ m a t h . A T ] A ug NOTES ON EQUIVARIANT BUNDLES
FOLING ZOU
Abstract.
We compare two notions of G -fiber bundles and G -principal bundles inthe literature, with an aim to clarify early results in equivariant bundle theory thatare needed in current work of equivariant algebraic topology. We also give proofs ofsome equivariant generalizations of well-known non-equivariant results involving theclassifying space. Contents
1. Introduction 1Notations. 3Acknowledgements. 32. Equivariant bundles 32.1. Non-equivariant bundles 32.2. Definitions of equivariant bundles 52.3. Comparisons of definitions 112.4. Examples: split extensions and the V -framing bundle 132.5. Fixed point theorems 143. Classifying spaces 183.1. V -trivial bundles 183.2. Universal equivariant bundles 193.3. The gauge group of an equivariant principal bundle 223.4. Free loop spaces and adjoint bundles 24References 281. Introduction
Non-equivariantly, fiber bundles and principal bundles are closely related. Namely,fix a compact Lie group Π and a space F with an effective Π-action, then one can makesense of a fiber bundle with fiber F to have structure group Π, and there is a structuretheorem providing an equivalence of categories between such fiber bundles and principalΠ-bundles.Let G and Π be compact Lie groups, where G is the ambient action group and Πis the structure group. To obtain such a structure theorem equivariantly, we need toanswer the following two questions:(1) What does it mean for a G -fiber bundle to have structure group Π?(2) What is an equivariant principal Π-bundle?There are different answers, both with interesting examples. The first one is forthright.We assume the Π-action on F is effective throughout. Definition 1.1. (Definition 2.7) Let F be a space with Π-action. A G -fiber bundle withfiber F and structure group Π is a map p : E → B such that the following statementshold:(1) The map p is a non-equivariant fiber bundle with fiber F and structure group Π;(2) Both E and B are G -spaces and p is G -equivariant;(3) The G -action is given by morphisms of bundles with structure group Π.The second one is introduced in [LMSM86, IV1] and is a generalization of an earlierdefinition by Tom Dieck [TD69]. Fix an extension of compact Lie groups 1 → Π → Γ → G → Definition 1.2. (Definition 2.20) Let F be a space with Γ-action. A G -fiber bundlewith fiber F , structure group Π and total group Γ is a map p : E → B such that thefollowing statements hold:(1) The map p is a non-equivariant fiber bundle with fiber F and structure group Π;(2) Both E, B are G -spaces and p is a G -map;(3) For any g ∈ G and admissible maps ψ : F → F b and ζ : F → F gb , the composite F ψ → F b g → F gb ζ − → F is a lift y ∈ Γ of g ∈ G . Example 1.3. (Example 2.25 and Example 2.36) A Real vector bundle as defined byAtiyah [Ati66] is a C -fiber bundle with fiber C n , structure group U ( n ) and total groupΓ = U ( n ) ⋊ α C , where α : C → Aut( U ( n )) sends the non-trivial element of C to theentry-wise complex-conjugation of U ( n ). It is NOT a C -fiber bundle with fiber C n ,structure group U ( n ).Thus, these two concepts of G -fiber bundles are in general different. A refinementof the first concept by specifying an extension Γ and certain Γ-action on the fiber F is a special case of the second concept (See Proposition 2.32). With appropriateversions of principal bundles (the principal G -Π-bundles as in Definition 2.10 or theprincipal (Π; Γ)-bundles as in Definition 2.14), there are structure theorems in bothcases (See Theorem 2.12 and Theorem 2.27). As an example, we study the V -framingbundle Fr V ( E ) of a G - n -vector bundle E , where V is an n -dimensional G -representation.It turns out Fr R n ( E ) ∼ = Fr V ( E ) as ( O ( n ); O ( n ) × G ) ∼ = ( O ( n ); O ( V ) ⋊ G )-principalbundles. Their different G -actions are seen by different splittings in the extension1 → O ( n ) → O ( V ) ⋊ G → G → × G , a principal G -Π-bundle isexactly a principal (Π; Γ)-bundle. Take p : P → B to be such a principal G -Π-bundleand let H ⊂ G be a subgroup. [LM86] studied the fixed points of p (See Theorem 2.44).Each component B of B H has an associated homomorphism ρ : H → Π up to Π-conjugation which is determined by the fixed-point behavior of the total space: LetΛ ρ ⊂ Π × G be the subgroup given by the graph of ρ , then { ρ : H → Π | (cid:0) p − ( B ) (cid:1) Λ ρ = ∅ } form a single conjugacy class of representations. Furthermore, the (non-equivariant)principal Π-bundle p − ( B ) → B has a reduction of the structure group from Π to asubgroup Z Π ( ρ ) ⊂ Π. We apply this theorem to obtain a comparison of principal G -Π-bundles in Theorem 2.46. QUIVARIANT BUNDLES 3
We also prove some results relevant to the equivariant classifying space. Denote theuniversal principal G -Π-bundle by E G Π → B G Π . We study the loop space Ω b B G Π of B G Π based at a G -fixed point b . As ( B G Π) G may not be connected, the G -homotopy type of Ω b B G Π depends on the choice of b .Our greatest interest is the case Π = O ( n ), and it works the same for the generalΠ as discussed in Remark 3.15. Note that B G O ( n ) classifies G - n -vector bundles anda homomorphism ρ : G → O ( n ) is an n -dimensional G -representation V . Suppose b ∈ B G O ( n ) is in the component indexed by [ V ]. In Theorem 3.13, we show that thereis a G -homotopy equivalence Ω b B G O ( n ) ≃ O ( V ), where O ( V ) is the isometric self mapsof V with G acting by conjugation.The morphisms of equivariant principal bundles give another naturally arising exam-ple of an equivariant principal bundle with a non-trivial extension in the total group.Let p : P → B be a principal G - O ( n )-bundle and Π = Aut B ( P ) be the group of auto-morphisms of P over B . Both Π and G act on the space Hom( P, E G O ( n )), but theiractions do not commute, so we do not have a (Π × G )-action. In fact, G acts on Π byconjugation and we have a (Γ = Π ⋊ G )-action on Hom( P, E G O ( n )). In Theorem 3.18,we show that π : Hom( P, E G O ( n )) → Map p ( B, B G O ( n ))is a universal principal (Aut B P ; Aut B P ⋊ G )-bundle, where π sends a bundle map toits map of base spaces and Map p ( B, B G O ( n )) is the image of π in Map( B, B G O ( n )).We show that there is a weak G -equivalence between the free loop space LB G Π andthe adjoint bundle Ad ( E G Π) := E G Π × Π Π ad as G -fibrations over B G Π in Theorem 3.31.
Notations.
For compact Lie groups G , Π and an extension 1 → Π → Γ → G → • E G Π → B G Π is the universal principal G -Π-bundle; • E (Π; Γ) → B (Π; Γ) is the universal principal (Π; Γ)-bundle.For a space X and b ∈ X , • P b X is the path space of X at the base point b ; • Ω b X is the loop space of X at the base point b ; • Λ b X is the Moore loop space of X at the base point b , defined to be Λ b X = { ( l, α ) ∈ R ≥ × X R ≥ | α (0) = b, α ( t ) = b for t ≥ l } . Acknowledgements.
This is part of the author’s PhD thesis. The author is indebtedto her advisor Peter May for explaining his past work in this subject and for his enor-mous help with the writing. 2.
Equivariant bundles
Non-equivariant bundles.
We start with a review of non-equivariant bundles.A fiber bundle with fiber F is a map p : E → B with an open cover { U i } of B andhomeomorphisms φ i : p − ( U i ) ∼ = U i × F . The U i are called coordinate neighborhoods and the φ i are called local trivializations .The structure group of a fiber bundle gives information about how local trivializationschange under changes of coordinate neighborhoods. Let Π be a topological group withan effective action on F . Here, effective means Π → Aut( F ) is an injection. A bundlewith fiber F is said to have structure group Π, if for any two local trivialization U i ∩ U j = ∅ , the composite φ i φ − j : ( U i ∩ U j ) × F → ( U i ∩ U j ) × F is given by ( b, f ) ( b, g ij ( b )( f )) FOLING ZOU for some continuous function g ij : U i ∩ U j → Π, called a coordinate transformation . Wealways topologize Aut( F ) with the compact-open topology of mapping spaces. If F isa compact Hausdorff space, Aut( F ) is a topological group; If F is only locally compact,there are more technical assumptions for the inverse map to be continuous due to Arens(See [Ste51, I.5.4]). Morally, a fiber bundle with fiber F is automatically a fiber bundlewith the implicit structure group Aut( F ). Having an explicit structure group Π is extradata to reduce the structure group to a smaller one.One can associate a principal Π-bundle to a fiber bundle with structure group Π. An admissible map of the bundle is a homeomorphism ψ : F → p − ( b ) for some b ∈ U i ,satisfying φ i ψ ∈ Π. The associated principal Π -bundle of p is the space of admissiblemaps.The following immediate observation about admissible maps hides the local trivial-izations in the background. Lemma 2.1.
A map ψ : F → F b is admissible if and only if for any admissible map ζ : F → F b , the composite ζ − ψ is in Π . (cid:3) Let p : E → B and p : E → B be two fiber bundles with fiber F and structuregroup Π. A morphism between them is a bundle map χ : E → E such that for anylocal trivializations φ U : p − ( U ) ∼ = U × F and φ V : p − ( V ) ∼ = V × F , the composite(2.2) φ V χφ − U : ( U ∩ χ − ( V )) × F → ( χ ( U ) ∩ V ) × F is given by ( b, f ) ( χ ( b ) , g V U ( b )( f )), where g V U : U ∩ χ − ( V ) → Π is some continuousfunction. Such a morphism induces a morphism between the two associated principalΠ-bundles.We pause to clarify a possible confusion regarding how to check that a bundle map χ is a morphism, that is, it respects the structure group. It seems as if one only needto check that χ sends an admissible map to an admissible map. However, this is nottrue, since the set of admissible maps does not see the topology.Steenrod [Ste51, I.5] studied this difference carefully and concluded that the followingAssumption 2.3 will resolve the discrepancy. We include some explanation here forcompleteness: What the set of admissible maps sees is an Ehresmann-Feldbau bundlewith structure group Π, which has now become an obsolete notion. An Ehresmann-Feldbau bundle is a bundle p : E → B with fiber F and a set of homeomorphism ψ : F ∼ = p − ( b ) for all b ∈ B , called admissible maps. It is required that for any b ∈ U i ,the composite F = { b } × F → U i × F φ − i → p − ( U i ) is admissible, and that for any b ∈ B and any admissible map ψ : F → p − ( b ), all the admissible maps F → p − ( b ) areexactly ψ ◦ ν for some ν ∈ Π. While this aligns with Lemma 2.1 when the bundle hasa structure group Π, there is a difference of the two notions, which lies exactly in thatan Ehresmann-Feldbau bundle does not require Π to have a topology. In other words,the coordinate transformations g ij are not asked to be continuous, which is equivalentto putting the trivial topology on Π. If Π does start life with a different topology, thecoordinate transformations g ij obtained from an Ehresmann-Feldbau bundle may notbe continuous. However, [Ste51, I.5.4] shows that if Π has the subspace topology inAut( F ), the g ij ’s are automatically continuous. Assumption 2.3.
We always assume that Π has the subspace topology of Aut( F ). QUIVARIANT BUNDLES 5
With this assumption, a fiber bundle has structure group Π if and only if the theadmissible maps satisfy Lemma 2.1. We have the following criteria:
Proposition 2.4.
A bundle map χ : E → E is a morphism of fiber bundles withstructure group Π if and only if either of the two equivalent conditions is true:(1) If F is a fiber in E and F is a fiber in E such that χ maps F to F , then thecomposite ζ − χψ is in Π for any admissible maps ψ : F → F and ζ : F → F .(2) For any admissible map ψ : F → F to a fiber in E , the composite χψ is anadmissible map to a fiber in E .Proof. We need to check that for any φ U , φ V as in (2.2), the desired g V U exists. WithAssumption 2.3, it suffices to check that for any b ∈ U ∩ χ − ( V ), there exists a desired g V U ( b ) ∈ Π. This is part (1). Part (2) follows from Lemma 2.1. (cid:3)
Example 2.5.
The most familiar case is when F is a vector space ( R n or C n ) and Π = GL n is the corresponding general linear group. By definition of the general linear group, χ being a bundle map is equivalent to it being fiberwise linear and non-degenerate.The following well-known structure theorem turns the problem of classifying fiberbundles into classifying principal bundles. Theorem 2.6.
Let Π be a compact Lie group. Let B, F be spaces. Assume that Π actseffectively on F . Then there is an equivalence of categories between { fiber bundles over B with fiber F and structure group Π } and { principal Π -bundles over B } .Proof. We have already shown how to construct a principal Π bundle from a fiberbundle with fiber F and structural group Π at the beginning of this subsection. In theother direction, given a principal Π-bundle P → B , the map P × Π F → B is a fiberbundle with fiber F and structure group Π. These two constructions are functorialand inverse of each other. Indeed, [Ste51, I] described both types of bundles using localtransformations, called coordinate bundles, where the equivalence becomes transparent. (cid:3) Definitions of equivariant bundles.
When it comes to the equivariant story,there are definitions of different generality, both on the fiber bundle side and on theprincipal bundle side. The reason is that the ambient group G could interact non-trivially with the structure group Π. We start with the simplest definition where “ G and Π commute” [Las82]. Let G, Π be compact Lie groups in this subsection.
Definition 2.7. A G -fiber bundle with fiber F and structure group Π is a map p : E → B such that the following statements hold:(1) The map p is a non-equivariant fiber bundle with fiber F and structure group Π;(2) Both E and B are G -spaces and p is G -equivariant;(3) The G -action is given by morphisms of bundles with structure group Π. Proposition 2.8.
The requirement in (3) above is equivalent to the following: for any g ∈ G and admissible map ψ : F → F b , the composite F ψ → F b g → F gb is also admissible.Proof. By Proposition 2.4. (cid:3)
Remark 2.9.
Let G be a finite group. We take F = R n and Π = GL n ( R ) inDefinition 2.7. Although GL n ( R ) is not compact, the definition still works and weobtain a G - n -vector bundle. FOLING ZOU
Definition 2.10.
A principal G -Π-bundle is a map p : P → B such that the followingstatements hold:(1) The map p is a non-equivariant principal Π-bundle;(2) Both P and B are G -spaces and p is G -equivariant;(3) The actions of G and Π commute on P . Remark 2.11.
This is called a principal ( G, Π)-bundle in [LMSM86, IV1].As in the non-equivariant case, we write the Π-action on the right of a principal G -Π-bundle P ; for convenience of diagonal action, we consider P to have a left Π-action,that is, ν ∈ Π acts on z ∈ P by νz = zν − .The structure theorem formally passes to this equivariant context. Theorem 2.12.
Let G, Π be compact Lie groups and F, B be spaces. Assume that Π acts effectively on F . Then there is an equivalence of categories between { G -fiberbundles over B with fiber F and structure group Π } and { principal G - Π -bundles over B } .Proof. The two types of G -bundles in Definitions 2.7 and 2.10 are indeed objects witha G -action in the corresponding non-equivariant category. So the equivalence in thenon-equivariant structure theorem restricts to give an equivalence on the G -objects. (cid:3) However, Definitions 2.7 and 2.10 are not ideal for studying some interesting cases.In the most general scenario, we want to study a map p : E → B that happens to beboth a fiber bundle with structure group Π and a G -map between G -spaces. It is truethat p is a G -fiber bundle with structure group Aut( F ), but p is usually not a G -fiberbundle with structure group Π. In other words, we can’t reduce the structure groupeven though we know non-equivariantly it reduces to Π. Below, we give two concreteexamples of this sort.The first example is Atiyah’s Real vector bundles [Ati66]. Let G = C . A Real vectorbundle is a map p : E → B such that • The map p is a complex vector bundle of dimension n ; • The non-trivial element of C acts anti-complex-linearly.In this case, p is a C -bundle with structure group O (2 n ), but not U ( n ).The second simple but illuminating example is from [LMSM86]. Example 2.13.
For G -spaces B and F , the projection p : B × F → B is not a G -bundlewith structure group e unless G acts trivially on F . Proof.
The admissible maps for p are only the inclusions of fibers ψ b : { b } × F → B × F .
An element g ∈ G acts by a bundle map if and only if for all b , the composite { b } × F ψ b → p − ( b ) g → p − ( gb ) ψ − gb → { gb } × F is in the structure group. But this map is merely the g action on F . (cid:3) Consequently, we would like a more general version than Definitions 2.7 and 2.10. Towork with Real vector bundles, tom Dieck [TD69] introduced a complex conjugationaction of C on U ( n ). Lashof–May [LM86] had the idea to further introduce a total QUIVARIANT BUNDLES 7 group that is the extension of the structure group Π by G . Tom Dieck’s work becamea special case of a split extension, or equivalently a semidirect product. One good, butrather brief and sketchy, early reference for both is [LMSM86, IV1]; we shall flesh outthat source and come back to the two examples afterwards.We start with the well studied principal bundle story. Definition 2.14. ([LM86]) Let 1 → Π → Γ → G → p : P → B such that the followingstatements hold:(1) The map p is a non-equivariant principal Π-bundle;(2) The space P is a Γ-space; B is a G -space. Viewing B as a Γ-space by pullingback the action, the map p is Γ-equivariant. Remark 2.15.
The total space P does not have a G -action in general. It only doeswhen we specify a splitting G → Γ. An example of this sort is discussed in Section 2.4.
Definition 2.16.
A morphism between two principal (Π; Γ)-bundles p : P → B and p : P → B is a pair of maps ( ¯ f , f ) fitting in the commutative diagram P P B B fp p f such that f is G -equivariant and ¯ f is Γ-equivariant. Example 2.17.
Let y ∈ Γ be with image g ∈ G . The action map ( y, g ) is an automor-phism.Taking Γ = Π × G , we recover the principal G -Π-bundles of Definition 2.10. Inthis case we have two names for the same thing. This could be confusing, but since a“principal G -Π-bundle” looks more natural than a “principal (Π; Π × G )-bundle” forthis thing, we will keep both names.Taking Γ to be a split extension, or equivalently Γ = Π ⋊ α G for some group homo-morphism α : G → Aut(Π), we recover tom Dieck’s principal (
G, α,
Π)-bundles.
Remark 2.18.
L¨uck–Uribe [LU14] worked with those principal G -Π-bundles (whichthey call G -equivariant principal Π-bundles) such that the isotropy subgroups of thetotal space are in R , a prescribed family of subgroups of Γ = Π × G . In our case, R = { Λ ⊂ Γ | Λ ∩ Π = e } , and we will not make use of a general R . Remark 2.19.
To be useful later, we write the elements of Γ = Π ⋊ α G as ( ν, g ) for ν ∈ Π , g ∈ G and write α ( g ) ∈ Aut(Π) as α g . We have the following facts: • The product in Γ is given by ( ν, g )( µ, h ) = ( να g ( µ ) , gh ) (That is, g acts on µ when they interchange); • The identity element is ( e, e ); • The inverse is ( ν, g ) − = ( α g − ( ν − ) , g − ); • The elements ( e, g ) form a subgroup of Γ that is canonically isomorphic to G ; • A space with Γ-action is a space with both Π and G actions such that ν ( g ( − )) = g ( α g ( ν )( − )) , which is indeed ( ν, g )( − ) . FOLING ZOU
The fiber bundle story is not as clear. It turns out that the appropriate fiber ofan equivariant fiber bundle is not just the preimage of any point, but rather with apreassigned action of Γ. This is unnatural at first glance, for example in a G -vectorbundle we won’t expect there to be an ( O ( n ) × G )-action on the fiber R n . We willexplain why this is necessary and how G -vector bundles fit in this context later. Let usstart with the definition: Definition 2.20. ([LMSM86, IV1]) Let 1 → Π → Γ → G → F be a space with Γ-action. A G -fiber bundle with fiber F ,structure group Π and total group Γ is a map p : E → B such that the followingstatements hold:(1) The map p is a non-equivariant fiber bundle with fiber F and structure group Π;(2) Both E, B are G -spaces and p is a G -map;(3) For any g ∈ G and admissible maps ψ : F → F b and ζ : F → F gb , the composite F ψ → F b g → F gb ζ − → F is a lift y ∈ Γ of g ∈ G . In other words, the y in the following diagram is askedto be a lift of g ∈ G in Γ: F FF b F gbψ ∼ = y ζ ∼ = g Proposition 2.21.
The requirement (3) above is equivalent to the following: For each y ∈ Γ with image g ∈ G and admissible map ψ : F → F b , the composite F y − → F ψ → F b g → F gb is also admissible.Proof. For any two lifts y and y ′ of g , y ′ y − is a lift of e ∈ G , so it is in Π. The claimthen follows from Lemma 2.1. (cid:3) Taking g = e in Proposition 2.21, the lifts y are exactly elements of Π, so we just seethe non-equivariant structure group (compare with Lemma 2.1); Taking general g , theassignment ψ gψy − is mimicking the action by an element of Π on the admissiblemap ψ , but it changes the fiber from over b to over gb . In this sense, the extension ofthe structure group Π to the total group Γ is used to regulate admissible maps to fibersover the orbit of b . Definition 2.22.
Let p : E → B and p : E → B be two G -fiber bundles withfiber F , structure group Π and total group Γ. A morphism between them is a pair ofmaps ( ¯ f , f ) fitting in the commutative diagram E E B B fp p f such that the following statements hold: QUIVARIANT BUNDLES 9 (1) The pair ( ¯ f , f ) is a non-equivariant morphism between bundles with fiber F and structure group Π.(2) Both ¯ f and f are G -equivariant. Remark 2.23.
By Proposition 2.4, the condition (1) of Definition 2.22 is explicitly thefollowing: For any admissible map ψ : F → F to a fiber in E , the composite ¯ f ψ is anadmissible map to a fiber in E .We do not have a requirement on a morphism regarding the condition (3) of Definition 2.20because it is automatic: if ψ is admissible, we have that gψy − is admissible and so is¯ f ( gψy − ). But ¯ f g = g ¯ f , so g ( ¯ f ψ ) y − is also admissible.As opposed to Definition 2.7, in Definition 2.20 the Γ-action on the total space E canrestrict to a G -action only when there is a splitting of the extension given by G → Γ.The following example illustrates that varying the splitting map can give different G -fiber bundle descriptions of the same bundle. It will be discussed in Section 2.4. Example 2.24. A G - n -vector bundle is both a G -fiber bundle with fiber R n , structuregroup O ( n ) and total group O ( n ) × G and a G -fiber bundle with fiber V , structuregroup O ( V ) and total group O ( V ) ⋊ G . (Here, we take Γ = O ( n ) × G ∼ = O ( V ) ⋊ G .) Example 2.25.
A Real vector bundle is a C -fiber bundle with fiber C n , structuregroup U ( n ) and total group Γ = U ( n ) ⋊ α C , where α : C → Aut( U ( n )) sends thenon-trivial element of C to the entry-wise complex-conjugation of U ( n ). Proof.
Let the non-trivial element a of C act by complex conjugation on C n . Thisextends the U ( n )-action to a Γ-action by Remark 2.19. We only need to check thatDefinition 2.20 (3) holds for g = a . An automorphism X of C n is anti-complex-linearif and only if A = X ◦ a , the pre-composition of X with conjugation, is complex-linear.So A is an element of U ( n ), and X = ( A, a ) is the lift of a in U ( n ) ⋊ α C . (cid:3) Example 2.26.
For G -spaces B and F , the projection B × F → B is a G -fiber bundlewith fiber F , structure group e and total group Γ = G . Proof.
The proof in Example 2.13 verifies Definition 2.20 (3). (cid:3)
It is unexpected that even when Γ = Π × G , Definitions 2.7 and 2.20 are different.On the one hand, a G -fiber bundle in the first sense needs extra data to be one in thesecond sense, as we will show shortly in Proposition 2.32. On the other hand, as wesaw in Example 2.13, if G acts non-trivially on F , then the projection B × F → F isnot a G -bundle with structure group e in the first sense, but it is a G -fiber bundle withstructure group e and total group G in the second sense.We have the following structure theorem in the context of Definitions 2.14 and 2.20: Theorem 2.27. ( [LMSM86, IV1] ) For any Π -effective Γ -space F and G -space B , thereis an equivalence of categories between { G -fiber bundles with structure group Π , totalgroup Γ and fiber F over B } and { principal (Π; Γ) -bundles over B } .Proof. This is an expansion of the sketchy proof in the reference. For brevity, we referto the two categories as equivariant fiber bundles and equivariant principal bundleswhen there is no confusion.
Given an equivariant fiber bundle E → B , we take the non-equivariant associatedprincipal bundle Fr F ( E ) → B . It suffices to give a Γ-action on Fr F ( E ) such thatFr F ( E ) → B is a G -map. For y ∈ Γ with image g ∈ G and an admissible map ψ : F → F b , let y ( ψ ) = gψy − . By Proposition 2.21, gψy − is an admissible map tothe fiber over gb . This shows that Fr F ( E ) → B is an equivariant principal bundle.Given an equivariant principal bundle P → B , let E = ( P × F ) / Π → B be the fiberbundle with admissible maps ψ p : F → E of the form ψ p ( f ) = [ p, f ] for some p ∈ P .We verify the three conditions for E → B to be an equivariant fiber bundle. Firstly, E → B is a non-equivariant fiber bundle with structure group Π. Secondly, we describethe G -action on E . Take the diagonal Γ-action on P × F . For any space with Γ-action X , we can define a Γ / Π ∼ = G -action on X/ Π by lifting g ∈ G to y ∈ Γ and let g [ x ] = [ yx ]for x ∈ X . Since Π is a normal subgroup of Γ, this is a well defined action independentof choice of y or representative x . For X = P × F , this gives ( P × F ) / Π a G -action.Since P → B is Γ-equivariant, it can be checked that E → B is G -equivariant. Thirdly,we show that the condition in Proposition 2.21 is satisfied. In fact, for y ∈ Γ lifting g ∈ G and p ∈ P , we have gψ p y − = ψ yp . To see this, evaluating on f ∈ F , we have gψ p y − ( f ) = g [ p, y − f ] definition of ψ ;= [ yp, yy − f ] definition of G -action;= [ yp, f ] = ψ yp ( f ) definition of ψ .These two constructions give inverse functors. Given an equivariant fiber bundle E → B , we have a map ξ : (Fr F ( E ) × F ) / Π → E, ξ ([ ψ, f ]) = ψ ( f ) . Non-equivariantly we already know that ( ξ, id B ) is a morphism of fiber bundles withstructure group Π and that ξ is a homeomorphism. To check that ξ is G -equivariant,suppose g ∈ G lifts to y ∈ Γ. Then g ([ ψ, f ]) = [ y ( ψ ) , yf ] = [ gψy − , yf ]and ξ ([ gψy − , yf ]) = ( gψy − )( yf ) = g ( ψ ( f )). So ( ξ, id B ) is a morphism of equivariantfiber bundles by Definition 2.22. It is an isomorphism because the non-equivariantinverse is also an equivariant inverse as it is a homeomorphism. Given an equivariantprincipal bundle P → B , we have a map which we abusively denote by ψ : P → Fr F (( P × F ) / Π) , p ψ p . Here, ψ p is the previously defined admissible map of ( P × F ) / Π, thus an elementof its associated principal bundle. Again, non-equivariantly we know that the map ψ is a homeomorphism (the Π-effectiveness is needed to assure that if p = q in P ,then ψ p = ψ q ). To check that ψ is Γ-equivariant, the definition of the Γ-action onadmissible maps gives yψ p = gψ p y − and we have verified gψ p y − = ψ yp , so we have yψ p = ψ yp . Thus, ( ψ, id B ) is a morphism of equivariant principal bundles. It is also anisomorphism. (cid:3) Remark 2.28.
The isomorphisms ξ and ψ in the proof are natural and provide theunit and counit maps of the adjunctionHom(( P × F ) / Π , E ) ∼ = Hom( P, Fr F ( E )) QUIVARIANT BUNDLES 11 ( − × F ) / Π : (cid:26) principal (Π; Γ)-bundles over B (cid:27) G -fiber bundles over B with structure group Π,total group Γ and fiber F : Fr F ( − ) We can see in the proof of Theorem 2.27 that it is essential for F to have a Γ-action.If P were a principal (Π; Γ)-bundle and the fiber F only had a Π-action, the associatedfiber bundle ( P × F ) / Π would not have a G -action. If we insist on our notion of a G -fiber bundle to be a G -map between G -spaces, this is the price to pay.2.3. Comparisons of definitions.
We have two concepts of G -fiber bundles. One isthe G -fiber bundle with fiber F and structure group Π as in Definition 2.7; the otheris the G -fiber bundle with fiber F , structure group Π and total group Γ for a specificextension of compact Lie groups 1 → Π → Γ → G →
1, as in Definition 2.20. Thedifferences between the concepts are two-fold: in the first one, G acts by bundle maps,but in the second one, the G -action is regulated by Γ; in the first one, F has only aΠ-action, but in the second one, F has a Γ-action. We compare these two concepts andshow that the first concept is a special case of the second where Γ ∼ = Π × G and Γ actson F via the projection Π × G → Π (Proposition 2.32).We start with some simple group theory observations that will come into play.
Definition 2.29.
A retraction Γ → Π is a group homomorphism that restricts to theidentity on the subgroup Π.It turns out that Γ admits a retraction to Π if and only if it is isomorphic to Π × G .We prove this explicitly in the case of a semidirect product first, then for general Γ. Proposition 2.30.
Let
Γ = Π ⋊ α G be a split extension. Then(1) The retractions ˜ β : Γ → Π are in bijection to homomorphisms β : G → Π satisfying α g ( ν ) = β ( g ) νβ ( g ) − for all g ∈ G and ν ∈ Π . (Note that for a given α : G → Aut(Π) , the homomorphism β may not exist.)(2) Each β in (1) specifies an isomorphism Π ⋊ α G ∼ = Π × G .Proof. To see (1), we use the explicit expression for semidirect product as in Remark 2.19.Let β ( g ) be the image ˜ β ( e, g ). Then β is a group homomorphism. We have ˜ β ( ν, e ) = ν and ˜ β ( ν, g ) = ˜ β (( ν, e )( e, g )) = νβ ( g ) . In order for ˜ β to be a homomorphism, it is required that the following two elements areequal for all g, h ∈ G and ν, µ ∈ Π:˜ β ( να g ( µ ) , gh ) = να g ( µ ) β ( gh );˜ β ( ν, g ) ˜ β ( µ, h ) = νβ ( g ) µβ ( h ) . Comparing the two lines gives the conclusion.Given such a β , the group isomorphism in (2) is given byΠ ⋊ α G ∼ = Π × G, ( ν, g ) ( νβ ( g ) , g ) . (cid:3) Proposition 2.31.
There is a bijective correspondence between { retractions ˜ β : Γ → Π } and { isomorphisms of extensions Γ ∼ = Π × G } . Proof.
Consider Π as a subgroup of Γ and denote by q the surjection Γ → G . Given aretraction ˜ β : Γ → Π, the map ( ˜ β, q ) : Γ → Π × G is a group isomorphism, and viceversa. 1 Π Γ G
11 Π Π × G G ( ˜ β,q ) q ˜ β (cid:3) We now compare Definitions 2.7 and 2.20 in the following propositions. Note that wecan think about a retraction Γ → Π as a chosen isomorphism Γ ∼ = Π × G of extensionsby Proposition 2.31. Proposition 2.32.
Let F be a space with an effective Π -action and → Π → Γ → G → be an extension of compact Lie groups. Then one can extend the Π -action on F to a Γ -action such that a G -fiber bundle of Definition 2.7 is always a G -fiber bundleof Definition 2.20 if and only if there is a retraction Γ → Π and the extended Γ -actionon F is via the retraction.Proof. Suppose we have p : E → B as in Definition 2.7 and F has an extended Γ-action.Then the only thing to check for p to be a G -fiber bundle of Definition 2.7 is whetherit satisfies the condition in Proposition 2.21. That is, it suffices to show for each y ∈ Γwith image g ∈ G and admissible homeomorphism ψ : F → F b , the composite gψy − is also admissible. By Proposition 2.8, gψ is admissible. So by Lemma 2.1, for y ∈ Γ, gψy − is admissible if and only if y acts on F as an element in Π. In other words, thegroup homomorphism Γ → Aut( F ) factors through Π → Aut( F ). (cid:3) The converse is also true.
Proposition 2.33.
Let → Π → Γ → G → be an extension of compact Lie groupsand F be a Π -effective Γ -space. Then a G -fiber bundle of Definition 2.20 is always a G -fiber bundle of Definition 2.7 if and only if Γ acts on F via a retraction Γ → Π .Proof. We can reverse the argument in Proposition 2.32. Suppose we have p : E → B as in Definition 2.20; to check whether p is a G -fiber bundle of Definition 2.7, we onlyneed to check whether the condition in Proposition 2.8 holds. Take any admissiblehomeomorphism ψ : F → F b . By Proposition 2.21, for any y ∈ Γ with image g ∈ G , gψy − is admissible. By Lemma 2.1, gψ is admissible if and only if y acts on F as anelement in Π, (cid:3) Using Propositions 2.32 and 2.33, we can identity some special cases when the twonotions of fiber bundles do agree.
Example 2.34.
Let Γ = Π × G and F be a space with an effective Π-action. We give F the trivial G -action. Equivalently, this is viewing F as a space with Γ-action via theprojection Γ → Π. In this perspective, the structure theorem Theorem 2.12 is a specialcase of Theorem 2.27.
Example 2.35.
In particular, let Γ = O ( n ) × G and give R n the usual O ( n )-actionand the trivial G -action. We have an equivalence of the two concepts: • G -vector bundles with fiber R n (the classical G -equivariant vector bundles); • G -fiber bundles with fiber R n , structure group O ( n ) and total group O ( n ) × G . QUIVARIANT BUNDLES 13
Example 2.36 (non-example) . For a Real vector bundle as in Example 2.25, Γ doesnot act on C n via U ( n ) for any n . So a Real vector bundle is not a C -fiber bundlewith fiber C n and structure group U ( n ). Proof.
There is no retraction Γ → U ( n ), because otherwise by Proposition 2.30, wewould need an element β ( a ) of U ( n ) such that β ( a ) A = ¯ Aβ ( a ) for all A ∈ U ( n ), where¯ A is the complex conjugation of A . But this does not exist for any n . (cid:3) Examples: split extensions and the V -framing bundle. In the extension1 → Π → Γ → G →
1, the group G is redundant because it is just Γ / Π. However,due to the special role of the group G in equivariant homotopy theory, we would liketo understand the G -action wherever applicable. Since the total space of a principal(Π; Γ)-bundle has only a Γ-action, we now focus on the case of split extensions, whenwe have a specified group homomorphism G → Γ. This becomes relevant at the end ofthis subsection when we define and study the V -framing bundle of a G -vector bundlefor representations V . It turns out that Fr V ( E ) and Fr R n ( E ) are the same even asprincipal (Π; Γ)-bundles, but they have different G -actions.Using Example 2.34 and Proposition 2.30, one can do some yoga with the fiber F . Fixa group homomorphism β : G → Π. Let α : G → Aut(Π) be the group homomorphismgiven by(2.37) α g ( ν ) = β ( g ) νβ ( g ) − , and the β determines an isomorphism (Proposition 2.30)(2.38) Π ⋊ α G ∼ = Π × G. Let F be a space with an effective Π-action. We can let the groups in (2.38) act on F via the retraction to Π. For clarity, we denote this space by F ′ . Explicitly, (Π × G )acts on F ′ by G acting trivially; (Π ⋊ α G ) acts on F ′ by( ν, g )( x ) = ν (cid:0) β ( g )( x ) (cid:1) for x ∈ F ′ . We point out that inclusion to the second coordinate gives a canonical inclusion of G into both Π × G and Π ⋊ α G , but this is not compatible with the isomorphism (2.38).The second image is in fact the graph subgroup Λ β = { ( β ( g ) , g ) | g ∈ G } . Consequently,the two G -actions on F ′ are different.In summary, we have an isomorphism of extensions in the situation, but it is not anisomorphism of split extensions as ( e, g ) of Π ⋊ α G is sent to ( β ( g ) , g ) in Π × G .1 Π Π ⋊ α G G
11 Π Π × G G ∼ = ( e,g ) ← [ g ( e,g ) ← [ g As a consequence, we get the following trivial corollary of Propositions 2.32 and 2.33:
Corollary 2.39.
In the context above, for a group homomorphism α : G → Aut(Π) given by (2.37) with associated isomorphism (2.38), the following categories are equiv-alent: • A G -fiber bundle with fiber F and structure group Π ; • A G -fiber bundle with fiber F ′ , structure group Π and total group Π × G ; • A G -fiber bundle with fiber F ′ , structure group Π and total group Π ⋊ α G . (cid:3) Similarly, a principal (Π; Π × G )-bundle is literally the same thing as a principal (Π; Π ⋊ α G )-bundle, but they have different specified G -actions. Notation 2.40.
For a principal G -Π-bundle, we call it a principal (Π; Π × G )-bundle ifwe let G act on the total space by G ⊂ Π × G ; we call it a principal (Π; Π ⋊ G )-bundleif we let G act on the total space by Λ β ⊂ Π × G . And similarly for a G -fiber bundlewith fiber F and structure group Π.This trivial observation allows us to define and study the V -framing bundle of anequivariant vector bundle. Let V be an orthogonal G -representation given by ρ : G → O ( n ). In the remainder of this subsection, we write O ( V ) for the group O ( n ) with thedata G → Aut( O ( n )) given by g ( ν ) = ρ ( g ) νρ ( g ) − for g ∈ G and ν ∈ O ( n ), so it isclear what O ( V ) ⋊ G means. This convention coincides with the conjugation G -actionon O ( V ) thought of as a mapping space in Top G . In this case, taking F = R n andpointing aloud the G -action on F ′ , Corollary 2.39 reads: A G - n -vector bundle is a G -fiber bundle with fiber R n , structure group O ( n ) and total group O ( n ) × G , as well asa G -fiber bundle with fiber V , structure group O ( n ) and total group O ( V ) ⋊ G . Definition 2.41.
Let p : E → B be a G - n -vector bundle. Let Fr V ( E ) be the space ofthe admissible maps with the G -action g ( ψ ) = gψρ ( g ) − . In other words, Fr V ( E ) has the same underlying space as Fr R n ( E ), but we think ofadmissible maps as mapping out of V instead of R n . Proposition 2.42. Fr V ( E ) is a principal ( O ( n ); O ( V ) ⋊ G ) -bundle and we have iso-morphisms of G -vector bundles: E ∼ = (Fr V ( E ) × V ) /O ( n ) . Proof.
This is a corollary of the structure theorem Theorem 2.27. Namely, Corollary 2.39and the explanation afterwards have turned the vector bundle p : E → B into a G -fiberbundle with fiber V , structure group O ( n ) and total group O ( V ) ⋊ G . By examination,Fr V ( E ) in Definition 2.41 agrees with the construction Fr V ( E ) in Theorem 2.27. (cid:3) Fixed point theorems.
Non-equivariantly, the long exact sequence of the ho-motopy groups of a fiber sequence is a useful tool to study the homotopy group of oneterm, knowing the other two. To do this equivariantly, we need to know what taking-fixed-points does to equivariant bundles. We focus on Γ = Π × G in this subsection;[LM86] gives the analogue of Theorem 2.44 for general Γ.Let Rep( G, Π) be the set:Rep( G, Π) = { group homomorphism ρ : G → Π } / Π-conjugation . Any subgroup H ⊂ G with a group homomorphism ρ : H → Π gives a subgroup Λ ρ of (Π × G ) via its graph. That is,Λ ρ = { ( ρ ( h ) , h ) | h ∈ H } . For each ρ : H → Π, denote the centralizer of the image of ρ in Π byZ Π ( ρ ) = { ν ∈ Π | νρ ( h ) = ρ ( h ) ν for all h ∈ H } . QUIVARIANT BUNDLES 15
Proposition 2.43.
Let Π be a compact Lie group and H be a subgroup. Then Z Π ( H ) is a closed subgroup of Π , thus also a compact Lie group.Proof. Fix an element h ∈ H . Then the map c h : Π → Π , ν νhν − is continuous.Since the singleton { h } ∈ Π is closed, the set c − h ( { h } ) = { ν ∈ Π | νh = hν } is alsoclosed. So Z Π ( H ) = T h ∈ H c − h ( { h } ) is closed. (cid:3) Theorem 2.44. ( [LM86, Theorem 12] ) Let G and Π be compact Lie groups. Let p : E → B be a principal G - Π -bundle and H ⊂ G be a subgroup. Assume that E iscompletely regular.(1) On the base, B H = a [ ρ ] ∈ Rep( H, Π) p ( E Λ ρ ) . (2) As sets, the preimages over each component of B H are p − ( p ( E Λ ρ )) = a { ρ ′ :Π -conjugate to ρ } E Λ ρ ′ . As spaces, p − ( p ( E Λ ρ )) ∼ = Π × Z Π ( ρ ) E Λ ρ . (3) For a fixed representative ρ of [ ρ ] , we have a principal Z Π ( ρ ) -bundle: Z Π ( ρ ) → E Λ ρ p → p ( E Λ ρ ) . (4) In particular, the following is a principal Π -bundle: Π → E H p → p ( E H ) . Explanation.
In words, part (1) says that the H -fixed points of B are the images of theΛ-fixed points of E for all subgroups Λ ⊂ Π × G that are graphs of a homomorphism H → Π. Furthermore, E Λ and E Λ ′ share the same projection image when Λ andΛ ′ are Π-conjugate, or equivalently the corresponding representations H → Π are Π-conjugate. The assumption that E is completely regular implies that if Λ and Λ ′ arenot Π-conjugate, the images of E Λ and E Λ ′ are disjoint.Parts (2) and (3) imply that E restricted on each component of B H has a reductionof the structure group from Π to Z Π ( ρ ). In the proof of Theorem 3.13(1), we willdescribe in an example how to find the representations ρ when H = G . The idea isthat the fiber over an H -fixed base has an H -action, and ρ tells what this action isin terms of the native Π-action as a principal bundle. Note that the representation ρ is dependent on the choice of a base point z in the fiber; a different choice gives aconjugate representation. From the description of the action, a point in the same fiber,written uniquely as zν for some ν ∈ Π, is Λ ρ -fixed if and only if ρ ( h ) νρ ( h ) − = ν forall h ∈ H . This justifies the first statement of part (2) as well as part (3).For the second statement of part (2), which is not in the reference, we use the map:Π × Z Π ( ρ ) E Λ ρ → E, ( ν, x ) xν − . Here, Z Π ( ρ ) is a subgroup of Π and acts on the right of Π by multiplication; the leftΠ-action on E restricts to a left Z Π ( ρ )-action on E Λ ρ . It is a homeomorphism to itsimage, which is exactly p − ( p ( E Λ ρ )):We have Λ e = H for the trivial representation e : H → Π. Part (4) follows fromtaking ρ = e in part (3). (cid:3) Remark 2.45.
From Theorem 2.44, for a principal G -Π-bundle p : E → B and asubgroup H ⊂ G , each component B of B H has an associated representation class[ ρ ] ∈ Rep( H, Π). It is characterized by the fact that for any representation ρ ′ : H → Π, (cid:0) p − ( B ) (cid:1) Λ ρ ′ = ∅ if and only if [ ρ ′ ] = [ ρ ] . The restricted principal Π-bundle p − ( B ) → B has a reduction of the structure groupfrom Π to Z Π ( ρ ).Non-equivariantly, a map between two principal G -bundles that is an underlyingequivalence on the total spaces will give an equivalence on the base spaces, as can beshown by the long exact sequence of homotopy groups. Equivariantly, we also wantthis tool of knowing when a map of two principal G -Π-bundles gives a G -equivalenceon the base spaces. Theorem 2.46.
Let i : Π → Π ′ be an inclusion of compact Lie groups. Let E, E ′ be principal G - Π - and G - Π ′ - bundles respectively of spaces of G -CW homotopy types.Then E ′ has a (Π × G ) -action by i .Suppose that there is a (Π × G ) -map ¯ f : E → E ′ over a G -map f : B → B ′ , as inthe following commutative diagram: Π Π ′ E E ′ B B ′ i ¯ fp p ′ f such that(1) The map i includes Π as a deformation retract of Π ′ in groups, that is, there existsa group homomorphism j : Π ′ → Π such that j ◦ i = id and i ◦ j ≃ id rel i (Π) intopological groups;(2) On the total spaces, the map ¯ f is a Λ -equivalence for any subgroup Λ ⊂ G × Π suchthat Λ ∩ Π = e .Then, on the base spaces, f : B → B ′ is a G -equivalence.Proof. To simply notation in this proof, we use the same letters to denote the restrictionsof the corresponding maps to a subspace. By the equivariant Whitehead theorem, itsuffices to show that:For any subgroup H ⊂ G, the map f : B H → ( B ′ ) H is an equivalence.We make the following two claims comparing Π and Π ′ :(a) For any group H , the induced map i ∗ : Rep( H, Π) → Rep( H, Π ′ ) is a bijection.(b) For any subgroup K of Π, the inclusion i : Z Π K → Z Π ′ i ( K ) is a homotopy equiva-lence;These two claims follow from the assumption (1). For (a), we take the functor F =Rep( H, − ) from the category of groups to sets. It has equivalent images on Π and Π ′ ,and we skip the details. For (b), we take the functor F = Z ( − ) K from the category of QUIVARIANT BUNDLES 17 groups containing K as a subgroup. It also has equivalent images on Π and Π ′ , and thedetails come later in Lemma 2.50.By Theorem 2.44 (1) and (a), it suffices to show that:For any H and ρ ∈ Rep( H, Π) , the map f : p ( E Λ ρ ) → p ′ (( E ′ ) Λ ρ ) is an equivalence.By Theorem 2.44 (3), taking the Λ ρ -fixed points of E and E ′ yields a map betweenprincipal bundles: Z Π ( ρ ) Z Π ′ ( ρ ) E Λ ρ ( E ′ ) Λ ρ p ( E Λ ρ ) p ′ (( E ′ ) Λ ρ ) i ¯ fp p ′ f By the claim (b) and the assumption (2), both i and ¯ f are equivalences. The long exactsequence of homotopy groups shows that f is an equivalence. (cid:3) Remark 2.47.
In Theorem 2.46, the assumption (1) is true in our applications withΠ ′ = Π or Π ′ = Π I . The assumption (2) is satisfied when ¯ f is a ( G × Π)-equivalence,but is weaker. The weaker version is needed in our applications.From the proof, we also have a version of Theorem 2.46 relaxing the assumption (2).
Corollary 2.48.
Suppose we have ( i, ¯ f , f ) in the context of Theorem 2.46, except thatinstead of the assumption (2), ¯ f : E → E ′ is only a Λ ρ -equivalence for a fixed represen-tation ρ : H → Π . Then on the base spaces, f : p ( E Λ ρ ) → p (( E ′ ) Λ ρ ) is an equivalence. Note that p ( E Λ ρ ) is the space of components of B H that are associated to ρ asdescribed in Remark 2.45. In particular, if ( B ′ ) H is connected for all subgroups H ⊂ G ,then ( B ′ ) H has only one associated representation ρ H . Moreover, ρ H has to be therestriction of ρ G . We have: Corollary 2.49.
Let B ′ be a G -connected space as explained above and ρ G be theassociated representation. Suppose we have ( i, ¯ f , f ) in the context of Corollary 2.48,such that ¯ f is a Λ ρ G -equivalence. Then on the base spaces, f : B → B ′ is a G -equivalence.Proof. Since the map f : B H → ( B ′ ) H preserves the associated representation, we knowthat B H only has one associated representation ρ H as well. The claim then follows byapplying Corollary 2.48 to ρ = ρ H for all H . (cid:3) The following is a lemma for Theorem 2.46:
Lemma 2.50.
Assume i : Π → Π ′ is an inclusion of topological groups with a defor-mation retract j : Π ′ → Π , that is, they satisfy condition (1) in Theorem 2.46. Thenfor any subgroup K of Π , the inclusion i : Z Π K → Z Π ′ i ( K ) is a homotopy equivalence.Proof. We first check that in general, given any group homomorphism f : G → G ′ andsubgroup K ⊂ G , the map f restricts to a map f : Z G K → Z G ′ ( f ( K )) on subspaces. This is because xk = kx for all k ∈ K implies f ( x ) f ( k ) = f ( k ) f ( x ) for all f ( k ) ∈ f ( K ).So, we have i : Z Π K → Z Π ′ ( i ( K )) and j : Z Π ′ ( i ( K )) → Z Π ( ji ( K )) = Z Π K. The map j gives deformation retract data of the inclusion i . It is obvious that j i =id. It remains to show i j ≃ id. The image of i is the subspace Z i (Π) ( i ( K )) ⊂ Z Π ′ ( i ( K )). The homotopy ij ≃ id rel i (Π) restricts to a homotopy i j ≃ id rel Z i (Π) ( i ( K )). (cid:3) Classifying spaces V -trivial bundles. An equivariant bundle E → B is V -trivial for some n -dimensional G -representation V if there is a G -vector bundle isomorphism E ∼ = B × V . Such an iso-morphism is a V -framing of the bundle. This is analogous to the case of non-equivariantvector bundles, except that equivariance adds in the complexity of a representation V that’s part of the data.However, the representation V in the equivariant trivialization of a fixed vector bun-dle may not be unique. We give a lemma to recognize when two trivial bundles areisomorphic, then a counterexample.Let Iso( V, W ) be the space of linear isomorphisms V → W with the conjugation G -action for G -representations V and W . Lemma 3.1.
For a G -space B , there exists a G -vector bundle isomorphism B × V ∼ = B × W if and only if there exists a G -map f : B → Iso(
V, W ) .Proof. Let F : B × V → B × W be a vector bundle map. For b ∈ B , let F b : V → W be such that F b ( v ) = F ( b, v ). Then F is a G -vector bundle isomorphism if and only if(1) F is fiberwise isomorphism: F b ∈ Iso(
V, W );(2) F is a G -map: gF ( b, v ) = F ( gb, gv ), or equivalently, F gb = gF b g − , for all g ∈ G .Taking f ( b ) = F b , it follows that F is an isomorphism if and only if f is a G -map. (cid:3) Corollary 3.2. If B has a G -fixed point, then B × V ∼ = B × W only when V ∼ = W .Proof. The equivariant map f : B → Iso(
V, W ) induces f G : B G → Iso G ( V, W ). Thesource being nonempty implies that the target is nonempty. (cid:3)
Remark 3.3.
More generally, for any two n -dimensional G -vector bundles E, E ′ over B , one can form the non-equivariant bundle H om B ( E, E ′ ) which consists of all bundlemaps E → E ′ over B (not necessarily fiberwise isomorphisms). It has a G -action byconjugation and is indeed an n -dimensional G -vector bundle over B . Let I so B ( E, E ′ )be the subspace consisting of only fiberwise isomorphisms. It is a GL n -bundle over B .Then tautologically E ∼ = E ′ if there is a G -invariant subsection of I so B ( E, E ′ ). Example 3.4 (Counterexample) . Let G = C , σ be the sign representation. The unitsphere, S (2 σ ), is S with the 180 degree rotation action. As C -vector bundles, S (2 σ ) × R ∼ = S (2 σ ) × σ. Proof.
By Lemma 3.1, it suffices to construct a C -map S (2 σ ) → Iso( R , σ ) ∼ = GL ,where the nontrivial element of C acts on GL by multiplying by − Id. We give S (2 σ )a G -CW decomposition of a 0-cell C /e and a 1-cell C /e × D and construct the mapby skeleton. It is obvious that any equivariant map on the 0-skeleton extends to the QUIVARIANT BUNDLES 19 GL , whichis true in this case as − Id and Id lie in the same path component. (cid:3)
Example 3.5. (Counterexample, Gus Longerman) Take G to be any compact Lie groupand V and W to be any two representation of G that are of the same dimension. Then G × V ∼ = G × W , because Map G ( G, Iso(
V, W )) ∼ = Map(pt , Iso(
V, W )) = ∅ . Indeed,the isomorphism can be constructed explicitly by F ( g, x ) = ( g, ρ W ( g ) ρ V ( g ) − x ), where ρ V , ρ W : G → O ( n ) are matrix representations of V, W .3.2.
Universal equivariant bundles.
The universal principal (Π; Γ)-bundle was con-structed and studied by tom Dieck [TD69] and Lashof–May [Las82, LM86]. It can berecognized by the following property:
Theorem 3.6. ( [LM86, Theorem 9] ) A principal (Π; Γ) -bundle p : E → B is universalif and only if E Λ ≃ ∗ , for all subgroups Λ ⊂ Γ such that Λ ∩ Π = e. Notation 3.7.
The universal (Π; Γ)-bundle is denoted E (Π; Γ) → B (Π; Γ). Remark 3.8.
When Γ = Π × G , such a subgroup Λ comes in the form of { ( ρ ( h ) , h ) | h ∈ H } , for H ⊂ G and ρ : H → Π is a group homomorphism . This group is denoted Λ ρ in Theorem 2.44.When Γ = Π ⋊ α G , such a subgroup Λ comes in the form of { ( ρ ( h ) , h ) | h ∈ H } , for H ⊂ G and ρ : H → Π such that ρ ( h h ) = ρ ( h ) · α h ( ρ ( h )) . We mostly specialize to the case Γ = G × O ( n ), when a principal (Π; Γ) is also aprincipal G - O ( n )-bundle. Notation 3.9.
We denote the universal principal G - O ( n )-bundle by E G O ( n ) → B G O ( n ).It is universal in the sense that the equivalence classes of principal G - O ( n )-bundles overa G -space B are classified by G -homotopy classes of G -maps B → B G O ( n ). We denotethe universal G - n -vector bundle by ζ n → B G O ( n ) where ζ n = E G O ( n ) × O ( n ) R n . As an immediate corollary of Theorems 3.6 and 2.44, one gets the G -homotopy typeof the universal base. Recall thatRep( G, O ( n )) = { ρ : G → O ( n ) group homomorphism } /O ( n )-conjugation; ∼ = { V : n -dimensional orthogonal representation of G } / isomorphismand Z O ( n ) ( ρ ) = { a ∈ O ( n ) | aρ ( g ) = ρ ( g ) a, for all g ∈ G } is the centralizer of the imageof ρ in O ( n ). Theorem 3.10. ( [Las82, Theorem 2.17] ) ( B G O ( n )) G ≃ a [ ρ ] ∈ Rep(
G,O ( n )) B Z O ( n ) ( ρ ); ≃ a [ V ] ∈ Rep(
G,O ( n )) B ( O ( V ) G ) . Example 3.11.
Take H = G = C and Π = O (2). ThenRep( C , O (2)) = { id , rotation , reflection } . For ρ = id or ρ = rotation, Z Π ( ρ ) = O (2). For ρ = reflection, Z Π ( ρ ) ∼ = Z / × Z /
2. So( B C O ( n )) C ≃ BO (2) ⊔ BO (2) ⊔ B ( Z / × Z / . From Theorem 3.10, one can make explicit the classifying maps of V -trivial bundlesas follows. Proposition 3.12. A G -map θ : pt → B G O ( n ) lands in one of the G -fixed componentsof B G O ( n ) , indexed by [ V ] . Then the pullback of the universal bundle is θ ∗ ζ n ∼ = V .Proof. It follows from part (1) of the following Theorem 3.13 that θ ∗ ζ n ∼ = O ( R n , V ) × O ( n ) R n ∼ = V. In fact, the n -planes in a complete G -universe with the tautology n -plane bundle is amodel for B G O ( n ) and ζ n ; θ (pt) is just a G -representation isomorphic to V . (cid:3) Theorem 3.13.
Take a G -fixed base point b ∈ B G O ( n ) in the component indexed by [ V ] . Let p : E G O ( n ) → B G O ( n ) be the universal principal G - O ( n ) -bundle. Then(1) The fiber over b , p − ( b ) , is homeomorphic to O ( R n , V ) as an ( O ( n ) × G ) -space.Here, ( G × O ( n )) acts on O ( R n , V ) by G acting on V and O ( n ) acting on R n .(2) The loop space of B G O ( n ) at the base point b , Ω b B G O ( n ) , is G -homotopy equiv-alent to O ( V ) , the isometric self maps of V with G acting by conjugation.Proof. (1) This is due to Lashof and we explain how to find the representation V here. Choose and fix a base point z ∈ p − ( b ). We construct a group homomorphism ρ z : G → O ( n ) as follows. For any g ∈ G , there exists a unique element, ρ z ( g ) ∈ O ( n )such that gz = zρ z ( g ). It is easy to check that g ρ z ( g ) gives a group homomorphism.Suppose z ′ is another base point in p − ( b ), and z ′ = zν for some unique ν ∈ O ( n ).Then gz ′ = gzν = zρ z ( g ) ν = z ′ ( ν − ρ z ( g ) ν ) . So ρ z ′ = ν − ρ z ν is O ( n )-conjugate to ρ z . The different ρ z ’s are the matrix representa-tions of some vector space representation V . From the proof of Theorem 2.17 of [Las82],this is exactly the index V . Without loss of generality, we take V to be given by ρ z asmatrix representation.The following map gives a non-equivariant homeomorphism: O ( R n , V ) ∼ = O ( n ) ∼ = → p − ( b ) ,ν zν. It suffices to check it is an equivariant homeomorphism with the described action. Let( µ, g ) ∈ O ( n ) × G . Then z (( µ, g ) ◦ ν ) = z ( ρ z ( g ) νµ − ) = ( zρ z ( g ))( νµ − ) = ( gz )( νµ − ) = ( µ, g ) ◦ zν. (2) The idea is to compare the path space fibration with the universal bundle. Equiv-ariantly, the base point should be G -fixed. Since the space involved is not G -connected,base points from different components might give inequivalent loop spaces. We usesubscripts in path spaces and loop spaces to indicate the base point. For example, P b B G O ( n ) = { α ∈ Map([0 , , B G O ( n )) | α (0) = b } . QUIVARIANT BUNDLES 21
Fix z ∈ p − ( b ) and ρ = ρ z : G → O ( n ) as above. Take z to be the base point of E G O ( n ). It is a Λ-fixed point, whereΛ = { ( ρ ( g ) , g ) | g ∈ G } ⊂ O ( n ) × G. We prove that E G O ( n ) is Λ-contractible. In fact, let Λ ′ be any subgroup of Λ. ThenΛ ′ ∩ O ( n ) = e , so by Theorem 3.6, ( E G O ( n )) Λ ′ is contractible.So, the contraction map gives a based Λ-equivariant homotopy: E G O ( n ) ∧ I → E G O ( n ) . Here, I = [0 ,
1] is based at 0 and has the trivial Λ-action. (The map sends x ∧ z ∧ t to z for all x ∈ E G O ( n ) and t ∈ I .) We take the adjoint of this homotopy to get E G O ( n ) → P z E G O ( n ), and then compose with P z E G O ( n ) → P b B G O ( n ) induced by p : E G O ( n ) → B G O ( n ). The composite is f : E G O ( n ) → P z E G O ( n ) → P b B G O ( n ) . It sends a point x ∈ E G O ( n ) to a path in B G O ( n ) that starts at b and ends at p ( x ).This yields a commutative diagram:(3.14) E G O ( n ) P b B G O ( n ) B G O ( n ) B G O ( n ) fp p Moreover, this diagram is G -equivariant, where the G -action on P b B G O ( n ) is by point-wise action on the path. It is worth noting that the G -action we take on E G O ( n ) is notthe original one, but via the identification q : Λ ∼ = G . In other words, g ∈ G acts bywhat ( ρ ( g ) , g ) acts. The two vertical maps are non-equivariant fibrations and f mapsthe fiber of p over b ∈ B G O ( n ), denoted F , to the fiber of p over b , denoted F .We first identify the fibers F and F . From part (1), F ∼ = O ( R n , V ) as ( O ( n ) × G )-spaces. So F ∼ = O ( V ) as G -spaces. It is clear that F ∼ = Ω b B G O ( n ) as G -spaces.We claim that f restricts to a G -equivalence F → F . The strategy is to show thatit induces an isomorphism on homotopy groups of H -fixed points for all H ⊂ G , usingthe long exact sequences of homotopy groups of fiber sequences. Without dealing withgeneral G -fibrations, it suffices to work out the following: • Denote by Λ ′ = q − ( H ), the subgroup of Λ that is isomorphic to H . Thecommutative diagram (3.14) restricts to the following commutative diagram:( F ) H ( F ) H ( E G O ( n )) Λ ′ ( P b B G O ( n )) H p (( E G O ( n )) Λ ′ ) p (( P b B G O ( n )) H ) p f H p • On the total space level, f H induces isomorphism on homotopy groups. This istrue because E G O ( n ) is Λ-contractible and P b B G O ( n ) is G -contractible. • The base spaces are equal. In fact, it is easy to see that they are both thecomponent of ( B G O ( n )) H indexed by [ V ] from Theorems 2.44 and 3.10. • The two vertical lines are fiber sequences. For the first, we use Theorem 2.44 (3)with ( F ) H = ( O ( V )) H = Z Π ( ρ | H ); for the second, it is merely the path spacefibration Ω b X → P b X → X , where X denotes the component of ( B G O ( n )) H containing b . (cid:3) Remark 3.15.
The proof of Theorem 3.13 works for general Π placing O ( n ). Take a G -fixed base point b ∈ B G Π in the component indexed by [ ρ : G → Π]. Let Π ad be thespace Π with the adjoint Π-action and consider it as a G -space via ρ . Then there is a G -homotopy equivalence Ω b B G Π ≃ Π ad .3.3. The gauge group of an equivariant principal bundle.
Let EO ( n ) → BO ( n )be the universal principal O ( n )-bundle and p : P → B be any principal O ( n )-bundle.The gauge group of P , Aut B ( P ), is the space of bundle automorphisms of P that areidentity on the base space B ([Hus94, Chap 7, Definition 1.1]). It turns out that thespace of principal bundle maps, Hom( P, EO ( n )), is also universal: The map(3.16) Hom( P, EO ( n )) → Map p ( B, BO ( n ))that restricts a bundle map to its base spaces is known to be the universal princi-pal Aut B ( P )-bundle. Here, Map p ( B, BO ( n )) denotes the component of the classifyingmap of p in Map( B, BO ( n )). A proof of this result can be found in [Hus94, Chap 7,Corollary 3.5]. In this subsection, we show the equivariant generalization of this result(Theorem 3.18).Let E G O ( n ) → B G O ( n ) be the universal principal G - O ( n )-bundle and p : P → B beany principal G - O ( n )-bundle. The restricting-to-the-base map(3.17) π : Hom( P, E G O ( n )) → Map p ( B, B G O ( n ))is a G -map lifting (3.16). Here, Map p ( B, B G O ( n )) is the (non-equivariant) componentof the classifying map of p in Map( B, B G O ( n )); G acts by conjugation on both sides of(3.17). Let Γ = Aut B P ⋊ G , where G acts on Aut B P by conjugation. Then the map π in (3.17) is a universal principal (Aut B ( P ); Γ)-bundle. Note that this is an equivariantprincipal bundle not in the sense of Definition 2.10, but of Definition 2.14 - the totalgroup is a non-trivial extension of Aut B ( P ) by G . Theorem 3.18.
In the context above, the map π : Hom( P, E G O ( n )) → Map p ( B, B G O ( n )) is a universal principal (Aut B P ; Γ) -bundle.Proof. As stated above, it is known non-equivariantly that π is a universal principalAut B P -bundle. One can use the conjugation G -action to get a principal (Aut B P ; Γ)-bundle structure on π . However, later in this proof we want a Γ-action on the bundle P ,so at the risk of elaborating the obvious, we describe the Γ-action on Hom( P, E G O ( n ))by putting a Γ-action on both P and E G O ( n ). The group Aut B P naturally has a leftaction on P ; take its trivial action on E G O ( n ). The group G acts on P and E G O ( n )because they are G -vector bundles. One can check by Remark 2.19 that this gives aΓ-action on P and E G O ( n ), thus by conjugation on Hom( P, E G O ( n )). Explicitly,(Aut B P ⋊ G ) × Hom(
P, E G O ( n )) → Hom(
P, E G O ( n ))(( s, g ) , f ) gf g − s − . QUIVARIANT BUNDLES 23
Since s ∈ Aut B P restricts to identity on B , we have π ( gf g − s − ) = gπ ( f ) g − . By Definition 2.14, the map π is a principal (Aut B P ; Γ)-bundle.It remains to show that π is universal. Although Aut B ( P ) can be fairly large, its sizedoes not matter that much: By Theorem 3.6, it suffices to show thatHom( P, E G O ( n )) Λ ≃ ∗ for any Λ ⊂ Γ such that Λ ∩ Aut B P = e. This follows from various application of the postponed Lemma 3.19, and it is essentiallya consequence of the universality of E G O ( n ).To see it, we first consider the case Λ = H , that is, the case where ρ ( h ) = e for all h ∈ H in Remark 3.8. By restricting the G -action to an H -action, E G O ( n ) is also theuniversal principal H - O ( n )-bundle. Then Hom( P, E G O ( n )) H ≃ ∗ by taking Π = O ( n ), G = H and Γ = O ( n ) × H in Lemma 3.19.In the general case, Λ is isomorphic to a subgroup H ⊂ G by the projection mapΓ → G , with a possibly non-trivial map ρ in Remark 3.8. Here is the crux: the elementsin Aut B P are O ( n )-equivariant maps, so the (Γ = Aut B P ⋊ G )-action on P defined atthe beginning of this proof commutes with the O ( n )-action; and we have Λ ⊂ Γ. Inother words, P is also a principal Λ- O ( n )-bundle. Since Λ acts by H on E G O ( n ), thespace E G O ( n ) is also the universal principal Λ- O ( n )-bundle. Now we are basically in thefirst case again: Hom( P, E G O ( n )) Λ ≃ ∗ by taking Π = O ( n ), G = Λ and Γ = O ( n ) × Λin Lemma 3.19. (cid:3)
The following lemma is a consequence of the universality:
Lemma 3.19.
Let → Π → Γ → G → be an extension of groups. Let p Π;Γ : E (Π; Γ) → B (Π; Γ) be the universal principal (Π; Γ) -bundle and let p : P → B be any principal (Π; Γ) -bundle. Then (cid:0) Hom(
P, E (Π; Γ)) (cid:1) G is contractible.Proof. To clarify the notations, Hom(
P, E (Π; Γ)) is the space of maps of (nonequivari-ant) principal Π-bundles. By definition,Hom(
P, E (Π; Γ)) ∼ = Map Π ( P, E (Π; Γ)) . The space Hom(
P, E (Π; Γ)) has a Γ-action by conjugation. Since Π ⊂ Γ acts trivially,it descends to a G -action, and (cid:0) Hom(
P, E (Π; Γ)) (cid:1) G ∼ = Map Γ ( P, E (Π; Γ)) . By definition, the space Map Γ ( P, E (Π; Γ)) is in fact the space of morphisms betweenprincipal (Π; Γ)-bundles. It is non-empty because it consists of the classifying map of p . It is further path-connected because any two Γ-maps P → E (Π; Γ) will both restrictto a classifying map B → B (Π; Γ) of p , so they are G -homotopic. The pull back of p Π;Γ along this homotopy gives a homotopy, or path, between the two maps.Using the arbitrariness of P in the above argument, one can further show that thespace Map Γ ( P, E (Π; Γ)) is contractible as follows. Let Y be a random G -space. Wedenote by Y × P the principal (Π; Γ)-bundle Y × P → Y × B . Here, Γ acts on Y bypulling back the G -action and acts Y × P diagonally. Then we have an adjunction:(3.20) Map G ( Y, Hom(
P, E (Π; Γ))) ∼ = Map Γ ( Y × P, E (Π; Γ)) . By what has been shown, the right hand side, thus the left hand side of (3.20) is alwaysnon-empty and path-connected for any Y . Taking Y = Hom( P, E (Π; Γ)), we obtainthat Map G ( Y, Y ) is path-connected. In particular, the identity map and the constantmap to a point in Y G are connected by a path. This implies the contractibility of Y G = (cid:0) Hom(
P, E (Π; Γ)) (cid:1) G . (cid:3) Remark 3.21.
Alternatively, one can show Map Γ ( P, E (Π; Γ)) ≃ ∗ using the fact that E (Π; Γ) is an universal space for a family of subgroups of Γ specified by Theorem 3.6,which contains all the isotropy groups of P .3.4. Free loop spaces and adjoint bundles.
We end this section by showing anequivariant equivalence of the free loop space LB G Π and the adjoint bundle Ad( E G Π) inTheorem 3.31. This gives Corollary 3.32, which upgrades the G -equivalence Ω b B G O ( n ) ≃ O ( V ) to a multiplicative one. Our proof follows the non-equivariant treatment in theappendix of Gruher’s thesis [Gru07] and the key equivariant tool is Theorem 2.46.We start with G -fibrations. Definition 3.22. A G -map p : E → B between G -spaces is a G -fibration if for allsubgroups H ⊂ G , the map p H : E H → B H is a Hurewicz fibration.The first examples of G -fibrations are G -fiber bundles. Example 3.23.
Let p : E → B be a principal G -Π-bundle as in Definition 2.10. Then p is also a G -fibration by Theorem 2.44 (4). However, p : E H → B H is not necessarilysurjective. In contrast to the other parts of Theorem 2.44, we do not have control overthe components of B H that are not hit by p ( E H ), at least not obviously. In this sense,the notion of a G -fibration is not as rich as a principal G -Π-bundle. Example 3.24.
Let F be an effective Π-space and q : E ′ → B ′ be a G -fiber bundlewith fiber F , structure group Π as in Definition 2.7. Then q is also a G -fibration. Lemma 3.25.
We have the following results on the fiber of a G -fibration:(1) Let p : E → B be a G -fibration and b ∈ B H be an H -fixed point, then the maps ( p − ( b )) H → E H → B H form a fiber sequence.(2) Let p : D → B and q : E → B be two G -fibrations and f : D → E be a G -map over B . Take an H -fixed point b ∈ B H . If f is a G -equivalence, then p − ( b ) → q − ( b ) is an H -equivalence.Proof. Non-equivariantly ( G = { e } ), this is the fact that a map over B and homo-topy equivalence is a homotopy equivalence of fibrations over B (See [May99, 7.5-7.6]).Equivariantly, the first claim is immediate from the definition; the second claim reducesto the non-equivariant case for each subgroup H ′ ⊂ H . (cid:3) We adopt the language of fiberwise monoids in [Gru07, Definition 4.2.1].
Definition 3.26. A G -fibration p : E → B is a G -fiberwise monoid if there is a unitsection map η : B → E and a multiplication map m : E × B E → E over B , both G -equivariant, that satisfy the unital and associativity conditions. In other words, E is a monoid in the category of G -fibrations over B .We can relax the strict associativity condition and define G -fiberwise A ∞ -monoidsas well. Let A be a reduced A ∞ -operad in Top ( A (0) = ∗ ). QUIVARIANT BUNDLES 25
Definition 3.27. A G -fibration p : E → B is a G -fiberwise A ∞ -monoid if it is analgebra over A in the category of G -fibrations over B . In concrete words, there are G -equivariant structure maps over B for each k ≥ γ k : A ( k ) × Σ k (cid:0) E × B E × B · · · × B E | {z } k times (cid:1) → E that satisfy the unital, associativity and Σ-equivariance conditions of an algebra overan operad. Here, A ( k ) is thought to have the trivial G -action; for k = 0, γ : B → E is just a section of p . Definition 3.28.
A morphism of G -fiberwise A ∞ -monoids over B is a morphism of A ∞ -monoids in the category of G -fibrations over B . An equivalence is a morphism and G -equivalence on the total space.By a G -monoid, we mean a monoid in G -spaces, and similarly for a G - A ∞ -monoid.Notice that the fiber of a G -fiberwise ( A ∞ )-monoid over a point b ∈ B is not a G -( A ∞ )-monoid. Instead, it is a G b -( A ∞ )-monoid, where G b is the isotropy subgroup of b .A morphism of fiberwise G -( A ∞ )-monoids induces a morphism of G b -( A ∞ )-monoids onthe fibers over b ; An equivalence induces a G b -equivalence on the fibers by Lemma 3.25.To clarify this notion, we make the following remarks:(1) A G -fiberwise monoid is a G -fiberwise A ∞ -monoid. In this case, the unit sectionmap η is γ and the multiplication map m is γ .(2) The relevant examples of G -fiberwise A ∞ -monoids here are mostly G -fibrationsover B whose fibers are some sort of loops. The structure maps come fromfiberwise- A ∞ structure of loop spaces. We will abuse terms to refer to thestructure maps as the unit map and the multiplication map.(3) A G -fiberwise monoid or a G -monoid here is not a “genuinely equivariant alge-bra” as it does not have G -set indexed multiplications. Construction 3.29.
For a G -space X , the free loop space LX = X S is a G -fibrationover X by evaluating at a base point of S . It is also a G -fiberwise A ∞ -monoid withthe unit map given by the constant loop and the multiplication map given by theconcatenation of loops. Construction 3.30.
For a principal G -Π-bundle E → B , the adjoint bundle of E is Ad ( E ) = E × Π Π ad . Here, Π ad is the left Π-space Π with adjoint action: for elements µ ∈ Π and ν ∈ Π ad , µ acts on ν by µ ( ν ) = µνµ − . As defined, Ad ( E ) is a G -fiberbundle over B with fiber Π, but no longer a principal G -Π-bundle. We claim that Ad ( E ) has the structure of a G -fiberwise monoid over B . First, Ad ( E ) is the fiberwiseautomorphism bundle I so B ( E, E ), so naturally a fiberwise monoid over B . This is thebundle version of the observation that for a right Π-space S homeomorphic to Π, thereis a homeomorphism Aut Π ( S ) ∼ = S × Π Π ad f ( s ) = sν ↔ [( s, ν )] . Moreover, Ad ( E ) ∼ = I so B ( E, E ) as G -spaces over B , where G acts on Ad ( E ) by actingon E and on I so B ( E, E ) by conjugation. This breaks down to commuting the actionof G and Π on E . Just to clarify,Aut B ( E ) = Iso B ( E, E ) ∼ = Section( I so B ( E, E )) . Theorem 3.31.
Let G, Π be compact Lie groups. Then there is a G -fiberwise A ∞ -monoid ( e P E G Π) / Π over B G Π and equivalences as G -fiberwise A ∞ -monoids over B G Π : LB G Π ( e P E G Π) / Π Ad ( E G Π) ξ ≃ ψ ≃ Proof.
We first construct the space and the map e p : ( e P E G Π) / Π → B G Π . Recall that p : E G Π → B G Π is the universal principal G -Π bundle. Denote the spaceof paths in E G Π that start and end in the same fiber over B G Π to be e P E G Π = { α : I → E G Π | p ( α (0)) = p ( α (1)) } . Then e P E G Π inherits an (Π × G )-action from E G Π. The quotient ( e P E G Π) / Π is a G -space over B G Π by e p ( α ) = p ( α (0)) . The map e p has the structure of a G -fiberwise A ∞ -monoid. The unit map η is givenby the constant path in the fiber of p . There is only one constant path in each fibersince we have taken quotient of the Π-action. The multiplication map m is given asfollows: for two classes of paths [ α ] , [ β ] ∈ ( e P E G Π) / Π, we may choose representativessuch that α (1) = β (0). Let m ([ α ] , [ β ]) = [ α.β ] be the concatenation of the paths: • β (1) • α (1) = β (0) • α (0) βαα.β The class [ α.β ] does not depend on the choice of α, β . Both η and m are G -equivariant.Next, we compare both LB G Π and Ad ( E G Π) with ( e P E G Π) / Π.On one hand, we have LB G Π = ( e P E G Π) / Π I . Here, Π I is the group Map([0 , , Π)and acts on e P E G Π ⊂ ( E G Π) I pointwise in I . The projection e P E G Π → LB G Π is aprincipal G -Π I -bundle, as the Π I action commutes with the G -action on e P E G Π.The projection ξ : ( e P E G Π) / Π → ( e P E G Π) / Π I commutes with the unit map andmultiplication map, so it is a map of G -fiberwise A ∞ -monoids. Moreover, we have thefollowing commutative diagram:Π Π I e P E G Π e P E G Π( e P E G Π) / Π ( e P E G Π) / Π I = LB G Π ξ By Theorem 2.46, ξ is a G -equivalence. (The idea is that Π and Π I are not so different.) QUIVARIANT BUNDLES 27
On the other hand, we may define a (Π × G )-equivariant map¯ ψ : e P E G Π → E G Π × Π ad α ( α (1) , ν )where ν ∈ Π is the unique element such that α (1) = α (0) ν − . We give E G Π × Π ad the G -action on E G Π and the diagonal Π-action. To check the equivariance of ¯ ψ , take any( µ, g ) ∈ Π × G , then ( µ, g ) ◦ α ( t ) = gα ( t ) µ − for t ∈ [0 , ψ (( µ, g ) ◦ α ) = ( gα (1) µ − , µνµ − ) = ( µ, g ) ◦ ¯ ψ ( α ) . Since Ad ( E G Π) = ( E G Π × Π ad ) / Π, we get a map ψ : ( e P E G Π) / Π → Ad ( E G Π). It iseasy to check that ψ commutes with the unit and multiplication maps, and is thus amap of G -fiberwise A ∞ -monoids.To show that ψ is a G -equivalence, we consider the following morphism of principal G -Π-bundles: Π Π e P E G Π E G Π × Π ad ( e P E G Π) / Π Ad ( E G Π) ¯ ψ ψ By Theorem 2.46, it suffices to show that ¯ ψ is a Λ-equivalence for any Λ ⊂ Π × G withΛ ∩ Π = e .We can construct a Λ-homotopy inverse for ¯ ψ : e P E G Π → E G Π × Π ad , called ¯ φ . Theidea is already in Gruher’s proof [Gru07]. But in the equivariant case, ¯ φ is dependenton the subgroup Λ. (In particular, it is not meant to be a (Π × G )-homotopy inverse.)Recall that ¯ ψ records the two endpoints of a path. So an inverse ¯ φ is going to choosea canonical path between any two points in a continuous way. This choice of canonicalpath exists because of the Λ-contractibility of E G Π; it is not meant to be a canonicalchoice.The construction of ¯ φ is as follows: Since E G Π is Λ-contractible, ( E G Π) Λ is non-empty. We fix a Λ-fixed base point z ∈ E G Π. Let E G Π × I → E G Π be a Λ-equivariantcontraction of E G Π to z ; the adjoint of it gives a Λ-map γ : E G Π → P z E G Π. For z ∈ E G Π, we write γ ( z ) as γ z . It is a path connecting z to z . Now, recall that for anelement ( z, ν ) ∈ E G Π × Π ad , the image ¯ φ ( z, ν ) ∈ e P E G Π wants to be a path from zν to z in E G Π. We define it to be¯ φ ( z, ν ) = concatenation of γ zν and the reverse of γ z , as illustrated in the picture on the left: zz zν γ z γ zν ¯ φ ( z,ν ) αz a b γ a γ b ¯ φ ¯ ψ ( α ) γ It remains to verify that ¯ φ is Λ-homotopy inverse of ¯ ψ . It is clear that ¯ ψ ¯ φ = id.The illustration above on the right shows how a Λ-equivariant homotopy ¯ φ ¯ ψ ≃ id isdefined: For a path α ∈ e P E G Π going from a point a to a point b , the path ¯ φ ¯ ψ ( α )is the concatenation of γ a and the reverse of γ b . A homotopy of paths ¯ φ ¯ ψ ( α ) ≃ α isa map H out of the square, such that the value of H has been given on the borderas indicated. To fill the interior, we connect every point x on the border to the pointlabeled by z with line segments and use the map γ H ( x ) on each segment. This homotopy H is “functorial” for elements α ∈ e P E G Π, so it extends to a homotopy ¯ φ ¯ ψ ≃ id; it isΛ-equivariant because the map γ is. (cid:3) As a corollary, we can upgrade Theorem 3.13 (2) into an equivalence of G - A ∞ -monoids Ω b B G O ( n ) ≃ O ( V ). Strictifying Ω b B G O ( n ) to the Moore loop space Λ b B G O ( n ),there is an equivalence of G -monoids Λ b B G O ( n ) ≃ O ( V ): Corollary 3.32.
Take a G -fixed base point b ∈ B G O ( n ) in the component indexed by V . Then Λ b B G O ( n ) is equivalent to O ( V ) as a G -monoid. Here, G acts on Λ b B G O ( n ) by acting on B G O ( n ) and acts on O ( V ) by conjugation.Proof. We explain how the G - A ∞ -monoid statement is a corollary. Take the fiber over b in Theorem 3.31 for Π = O ( n ). Then there are equivalences of the fibers as G - A ∞ -monoids by Lemma 3.25. The fiber of LB G O ( n ) is Ω b B G O ( n ). By Theorem 3.13 (1),the fiber of Ad ( E G O ( n )) is O ( R n , V ) × O ( n ) O ( n ) ad ∼ = O ( V ) as G -monoid. So thereis a zig-zag of equivalences of G - A ∞ -monoids between Ω b B G O ( n ) and O ( V ). For the G -monoid statement, just replace the free loop space and path space in Theorem 3.31by the Moore version, and the proof stays intact.Explicitly, the zigzag of G -monoids is given by(3.33) Λ b B G O ( n ) ( e Λ b E G O ( n )) / Π O ( V ) . ξ ≃ ψ ≃ We use p to denote the universal principal G - O ( n )-bundle E G O ( n ) → B G O ( n ). Wedefine e Λ b E G O ( n ) = { ( l, α ) | l ∈ R ≥ , α : R ≥ → E G O ( n ) , p ( α (0)) = p ( α ( t )) = b for t ≥ l } , so that ( e Λ b E G O ( n )) / Π = [ l, α ] where the equivalence relation is( l, α ) ∼ ( l, β ) if there is ν ∈ O ( n ) such that α ( t ) = β ( t ) ν for all t ≥ . While e Λ b E G O ( n ) does not have the structure of a G -monoid, ( e Λ b E G O ( n )) / Π does.Fix a base point z ∈ p − ( b ) ⊂ E G O ( n ). The maps are given by ξ ([ l, α ]) = ( l, p ( α )) ∈ Λ b B G O ( n ); ψ ([ l, α ]) ∈ O ( V ) is determined by α (0) ψ ([ l, α ]) = α ( l ) . (cid:3) References [Ati66] M. F. Atiyah. K -theory and reality. Quart. J. Math. Oxford Ser. (2) , 17:367–386, 1966.[Gru07] Kate Gruher.
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