aa r X i v : . [ m a t h . A T ] M a r A SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT
SIMON KITSON ∗ Abstract.
A g eneralisation of the equivariant Dixmier-Douady in-variant is constructed as a second-degree cohomology class withina new semi-equivariant Čech cohomology theory. This invariantobstructs liftings of semi-equivariant principal bundles that are as-sociated to central exact sequences of structure groups in whicheach structure group is acted on by the equivariance group. Theresults and methods described can be applied to the study of com-plex vector bundles equipped with linear/anti-linear actions, suchas Atiyah’s Real vector bundles. C ontents
1. Introduction A Real vector bundle ( E , τ ) is a complex vector bundle equipped withan anti-linear involution that covers an involution on its base space [1].The U ( n ) -frame bundle Fr ( E ) of a Real vector bundle is equipped withtwo actions: a left action of Z induced by τ , and a right action of U ( n ) .Due to the anti-linearity of τ , these actions do not commute. Rather,they combine into an action of Z ⋉ U ( n ) , where Z acts on U ( n ) byelementwise conjugation.More generally, if G is a Γ -group and P is a principal G -bundleequipped with a left action of Γ that maps fibres to fibres and satisfies γ ( p g ) (cid:3) ( γ p )( γ g ) for all γ ∈ Γ , p ∈ P and g ∈ G , then the actions on P combine into an action of Γ ⋉ G . In this situation, P is described as a ∗ I w ould like to thank the Mathematical Sciences Institute of the Australian Na-tional University for the postdoctoral fellowship which supported this research. a group equipped with an action of Γ by group automorphisms. SIMON KITSON Γ -semi-equivariant principal G -bundle . When the Γ -action on G is trivial, P is an equivariant principal bundle in the usual sense.This paper solves the following lifting problem for semi-equivariantprincipal bundles (see Theorem 43): Given a central short exact sequence A α → B β → C of Γ -groups and a Γ -semi-equivariant principal C -bundle P , classify the liftings of P by β to a Γ -semi-equivariant principal B -bundle. In particular, the obstruction to such liftings is identified as a semi-equivariant Dixmier-Douady invariant . This new invariant lies in a new semi-equivariant Čech cohomology theory , which will be constructed in§4. The semi-equivariant Dixmier-Douady invariant generalises theequivariant Dixmier-Douady invariant, which lies in equivariant co-homology. The relationship of this method to existing work, and itspossible applications will be discussed in §6.
2. Semi-equivariant Principal Bundles
Before examining semi-equivariant principal bundles, the notion of asemi-direct product is breifly reviewed.
Definition 1.
Let Γ be a Lie group. A (smooth) Γ -group ( G , θ ) is a Liegroup equipped with a smooth action θ : Γ → Aut ( G ) . A homomorphism ϕ : G → H of Γ -groups is a homomorphism of Liegroups such that, for γ ∈ Γ and g ∈ G , ϕ ( γ g ) (cid:3) γϕ ( g ) . (1) Definition 2.
Let ( G , θ ) be a Γ -group. The (outer) semi-direct product Γ ⋉ θ G is the Lie group consisting of elements ( γ, g ) ∈ Γ × G withmultiplication defined, for γ i ∈ Γ and g i ∈ G , by ( γ , g )( γ , g ) : (cid:3) ( γ γ , g ( γ g )) . One situation in which semi-direct product groups arise is when G and Γ both act on an object X and satsify the relation γ ( gx ) (cid:3) ( γ g )( γ x ) ,for some action θ of Γ on G . In this case, the two actions combine toform a single action of the group Γ ⋉ θ G by ( γ, g ) x : (cid:3) g ( γ x ) . Example 3.
The standard U ( ) -action on C and the Z -action on C byconjugation, combine into a Z ⋉ κ U ( ) -action on C , where κ is the Z -action on U ( ) by conjugation. SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 3
Semi-equivariant principal bundles generalises equivariant princi-pal bundle by using a Γ -group ( G , θ ) as the structure group. The action θ determines the commutation relation between the left action of Γ andright action of G on the total space of the principal bundle. These ac-tions combine into an action of the semi-direct product Γ ⋉ θ G . Inthe following definitions, let ( G , θ ) be a smooth Γ -group and X be amanifold equipped with a smooth Γ -action. Definition 4. A (smooth) Γ -semi-equivariant principal ( G , θ ) -bundle over X is a smooth principal G -bundle π : P → X equipped with a smoothleft action of Γ such that, for γ ∈ Γ , p ∈ P and g ∈ G , π ( γ p ) (cid:3) γπ ( p ) γ ( p g ) (cid:3) ( γ p )( γ g ) . Definition 5. An isomorphism ϕ : P → Q of Γ -semi-equivariant princi-pal ( G , θ ) -bundles is a diffeomorphism such that, for γ ∈ Γ , p ∈ P and g ∈ G , π P (cid:3) π Q ◦ ϕ ϕ ( p g ) (cid:3) ϕ ( p ) g ϕ ( γ p ) (cid:3) γϕ ( p ) . Next, let λ : ( G , θ ) → ( H , ϑ ) be a homomorphism of Γ -groups, and Q be a Γ -semi-equivariant principal ( H , ϑ ) -bundle. Definition 6. A lifting of Q by λ is a pair ( P , ϕ ) , where P is a Γ -semi-equivariant principal ( G , θ ) -bundle and ϕ : P → Q is a smooth mapsuch that, for γ ∈ Γ , p ∈ P and g ∈ G , π P (cid:3) π Q ◦ ϕ ϕ ( p g ) (cid:3) ϕ ( p ) λ ( g ) ϕ ( γ p ) (cid:3) γϕ ( p ) . Definition 7.
Two liftings ( P , ϕ ) and ( P , ϕ ) of Q by λ are equivalent if there is an isomorphism ψ : P → P such that ϕ ◦ ψ (cid:3) ϕ .The set of smooth Γ -semi-equivariant principal ( G , θ ) -bundles willbe denoted PB Γ ( X , ( G , θ )) , and the isomorphisms classes will be de-noted PB ≃ Γ ( X , ( G , θ )) .
3. Semi-equivariant Transition Cocycles
Transition cocycles are used to extract global topological informationfrom a principal bundle into a form which is more easily analysed.A transition cocycle over an open cover U : (cid:3) { U a } with values in aLie group G is a collection of smooth maps φ a : U a → G . Maps onoverlapping open sets are required to satisfy a cocycle condition . Thiscondition ensures that the cocycle can be used to glue together thepatches U a × G into a principal G -bundle. SIMON KITSON
In the equivariant setting, a transition cocycle consists of maps φ a ( γ, ·) : U a → G for each U a ∈ U and γ ∈ Γ . The equivariant cocyclecondition then ensures that the elements φ a ( , ·) can be used constructthe total space of a principal G -bundle, and that the elements φ a ( γ, ·) can be used to construct a Γ -action. The derivation of the equivariantcocycle condition uses the fact that the left and right actions commute.Semi-equivariant transition cocycles can be defined in a similarfashion to equivariant transition cocycles. However, the left and rightactions on a Γ -semi-equivariant principal ( G , θ ) -bundle form an ac-tion of Γ ⋉ θ G . Thus, the commutation relation between the left andright actions is controlled by θ , and the action θ appears in the semi-equivariant cocycle condition. When this cocycle condition is satisfied,the elements φ a ( , ·) in a cocycle can be used to construct the total spaceof a semi-equivariant principal bundle, and the elements φ a ( γ, ·) canbe used to construct a semi-equivariant Γ -action.Throughout this section, let X be a Γ -space, ( G , θ ) be a Γ -group and U : (cid:3) { U a } be an open cover of X . The cover U is not required to beinvariant. Definition 8. A (smooth) Γ -semi-equivariant ( G , θ ) -valued transition cocy-cle over U is a collection of smooth maps φ : (cid:3) (cid:8) φ ba ( γ, ·) : U a ∩ γ − U b → G | U a ∩ γ − U b , ∅ (cid:9) , satisfying φ aa ( , x ) (cid:3) φ ca ( γ ′ γ, x ) (cid:3) φ cb ( γ ′ , γ x )( γ ′ φ ba ( γ, x )) , (2)for x ∈ U a , γ ′ , γ ∈ Γ and x ∈ U a ∩ γ − U b ∩ ( γ ′ γ ) − U c .Note that the conditions (2) define a non-equivariant cocycle whenrestricted to γ (cid:3)
1, and an equivariant cocycle when θ (cid:3) id. Definition 9. An equivalence of Γ -semi-equivariant ( G , θ ) -valued tran-sition cocycles φ and φ with cover U is a collection of smooth maps µ : (cid:3) (cid:8) µ a : U a → G (cid:9) such that µ b ( γ x ) φ ba ( γ, x ) (cid:3) φ ba ( γ, x )( γµ a ( x )) , for γ ∈ Γ and x ∈ U a ∩ γ − U b .Next, let λ : ( G , θ ) → ( H , ϑ ) be a homomorphism of Γ -groups, and φ be a Γ -semi-equivariant ( H , ϑ ) -valued transition cocycle over U . SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 5
Definition 10. A lifting of φ by λ is a Γ -semi-equivariant ( G , θ ) -valuedtransition cocycle ψ such that λ ◦ ψ ba (cid:3) φ ba . Definition 11.
Two liftings ψ and ψ of φ by λ are equivalent if thereexists an equivalence µ between ψ and ψ .The set of smooth Γ -semi-equivariant ( G , θ ) -valued transition cocy-cles over U will be denoted TC Γ (U , X , ( G , θ )) . The set of equivalenceclasses of smooth Γ -semi-equivariant ( G , θ ) -valued transition cocyclesover U will be denoted by TC ≃ Γ (U , X , ( G , θ )) .The first step toward a correspondence between principal bundleand cocycles, is to show how a semi-equivariant transition cocycle canbe constructed from a semi-equivariant principal bundle. Implicit inthe proof of this result is the derivation of the semi-equivariant cocycleproperty. Proposition 12.
Let P ∈ PB Γ ( X , ( G , θ )) and s : (cid:3) (cid:8) s a : U a → P | U a (cid:9) be achoice of smooth local sections over the cover U . The collection of maps φ s : (cid:3) (cid:8) φ ba ( γ, ·) : U a ∩ γ − U b → G | U a ∩ γ − U b , ∅ (cid:9) defined by γ s a ( x ) (cid:3) s b ( γ x ) φ ba ( γ, x ) . (3) is a smooth Γ -semi-equivariant ( G , θ ) -valued transition cocycle.Proof. The given condition implies the following three identities γ ′ γ s a ( x ) (cid:3) s c ( γ ′ γ x ) φ sca ( γ ′ γ, x ) γ ′ s b ( γ x ) (cid:3) s c ( γ ′ γ x ) φ scb ( γ ′ , γ x ) γ s a ( x ) (cid:3) s b ( γ x ) φ sba ( γ, x ) , which, together, imply s c ( γ ′ γ x ) φ sca ( γ ′ γ, x ) (cid:3) γ ′ γ s a ( x ) (cid:3) γ ′ ( s b ( γ x ) φ sba ( γ, x )) (cid:3) ( γ ′ s b ( γ x ))( γ ′ φ sba ( γ, x )) (cid:3) s c ( γ ′ γ x ) φ scb ( γ ′ , γ x )( γ ′ φ sba ( γ, x )) . Thus, φ s satisfies the cocycle property φ sca ( γ ′ γ, x ) (cid:3) φ scb ( γ ′ , γ x )( γ ′ φ sba ( γ, x )) . SIMON KITSON
Note that (3) is the defining relation for a non-equivariant transitioncocycle when restricted to γ (cid:3)
1. If θ (cid:3) id, then (3) is the definingrelation for an equivariant transition cocycle.The map from semi-equivariant principal bundles to semi-equivarianttransition cocycles, defined by Proposition 12, depends on a choiceof local sections. However, if one passes to isomorphism classes ofprincipal bundles and equivalence classes of transition cocycles thisdependence disappears. The next proposition shows that cocycles as-sociated to isomorphic principal bundles by Proposition 12 are alwaysequivalent, regardless of which sections are chosen. Proposition 13.
Let P i ∈ PB Γ ( X , ( G , θ )) , and φ i ∈ TC Γ (U , X , ( G , θ )) be the cocycles associated to local sections s i : (cid:3) (cid:8) s ia : U a → P i | U a (cid:9) as inProposition 12. If ϕ : P → P is an isomorphism, then the collection of maps µ : (cid:3) (cid:8) µ a : U a → G (cid:9) (4) defined by ϕ ( s a ( x )) : (cid:3) s a ( x ) µ a ( x ) (5) is an equivalence between φ and φ .Proof. The properties of semi-equivariant principal bundle isomor-phisms and the defining property (5) imply that ϕ ( γ s a ( x )) (cid:3) γϕ ( s a ( x )) ϕ ( s b ( γ x ) φ ba ( γ, x )) (cid:3) γ ( s a ( x ) µ a ( x )) ϕ ( s b ( γ x )) φ ba ( γ, x ) (cid:3) ( γ s a ( x ))( γµ a ( x )) s b ( γ x ) µ b ( γ x ) φ ba ( γ, x ) (cid:3) s b ( γ x ) φ ba ( γ, x )( γµ a ( x )) . Thus, µ b ( γ x ) φ ba ( γ, x ) (cid:3) φ ba ( γ, x )( γµ a ( x )) , and µ is an equivalence between φ and φ for any choice of sections s i . Corollary 14.
The map of Proposition 12 induces a well-defined map PB ≃ Γ ( X , ( G , θ )) → TC ≃ Γ (U , X , ( G , θ ))[ P ] 7→ [ φ s ] , where s is any collection of smooth local sections of P . The correspondence between semi-equivariant cocycles and prin-cipal bundles has now been shown in one direction. Next, an inverse
SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 7 map reconstructing a semi-equivariant principal bundle from a semi-equivariant transition cocycle is defined.
Proposition 15.
Let φ ∈ TC Γ (U , X , ( G , θ )) . The bundle P φ defined by π : ( Ä a ∈ A U a × G /∼) → X , where(a) ( a , x , g ) ∼ ( b , x , φ ba ( , x ) g ) defines the equivalence relation ∼ (b) π [ a , x , g ] : (cid:3) x is the projection map(c) [ a , x , g ] g ′ : (cid:3) [ a , x , g g ′ ] defines the right-action of G (d) γ [ a , x , g ] : (cid:3) [ b , γ x , φ ba ( γ, x )( γ g )] defines the left action of Γ ,is a smooth Γ -semi-equivariant principal ( G , θ ) -bundle.Proof. The elements (cid:8) φ ba ( , ·) (cid:9) satisfy φ ca ( , x ) (cid:3) φ cb ( , x ) φ ba ( , x ) and so form a G -valued cocycle in the usual sense. Therefore, theusual proof that P φ is a principal G -bundle applies. The Γ -action iswell-defined on equivalence classes as γ [ b , x , φ ba ( , x ) g ] (cid:3) [ c , γ x , φ cb ( γ, x ) γ ( φ ba ( , x ) g )] (cid:3) [ c , γ x , φ cb ( γ, x )( γφ ba ( , x ))( γ g )] (cid:3) [ c , γ x , φ ca ( γ, x )( γ g )] (cid:3) η γ [ a , x , g ] . The semi-equivariance property γ ( p g ) (cid:3) ( γ p )( γ g ) is satisfied as γ ([ a , x , g ] g ′ ) (cid:3) γ ([ a , x , g g ′ ]) (cid:3) [ b , γ x , φ ba ( γ, x )( γ g g ′ )] (cid:3) [ b , γ x , φ ba ( γ, x )( γ g )( γ g ′ )] (cid:3) ( γ [ a , x , g ])( γ g ′ ) Thus, P φ is a Γ -semi-equivariant principal ( G , θ ) -bundle.This reconstruction map is also well-defined at the level of isomor-phism and equivalence classes. SIMON KITSON
Proposition 16.
Let φ i ∈ TC Γ (U , X , ( G , θ )) and P i ∈ PB Γ ( X , ( G , θ )) be the associated principal bundles, constructed using Proposition 15. If µ : (cid:3) (cid:8) µ a : U a → G (cid:9) is an equivalence between φ and φ then ϕ : P → P [ a , x , g ] 7→ [ a , x , µ a ( x ) g ] . is an isomorphism.Proof. That ϕ is a well-defined isomorphism of principal G -bundlesfollows immediately from the proof in the non-equivarant case. Com-patibility with the Γ -action is satisfied as γϕ ([ a , x , g ]) (cid:3) γ [ a , x , µ a ( x ) g ] (cid:3) [ b , γ x , φ ′ ba ( γ, x ) γ ( µ a ( x ) g )] (cid:3) [ b , γ x , φ ′ ba ( γ, x )( γµ a ( x ))( γ g )] (cid:3) [ b , γ x , µ b ( γ x ) φ ba ( γ, x )( γ g )] (cid:3) ϕ ([ b , γ x , φ ba ( γ, x )( γ g )]) (cid:3) ϕ ( γ [ a , x , g ]) . Thus, ϕ is an isomorphism of Γ -semi-equivariant principal ( G , θ ) -bundles. Corollary 17.
The map of Proposition 15 induces a well-defined map TC ≃ Γ (U , X , ( G , θ )) → PB ≃ Γ ( X , ( G , θ )) (6) [ φ ] 7→ [ P φ ] . (7)Finally, one shows that the two maps defined above are inverse toone another. Proposition 18.
The maps TC ≃ Γ (U , X , ( G , θ )) → PB ≃ Γ ( X , ( G , θ ))[ φ ] 7→ [ P φ ] and PB ≃ Γ ( X , ( G , θ )) → TC ≃ Γ (U , X , ( G , θ ))[ P ] 7→ [ φ s ] are inverse to one another. SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 9
Proof.
Let P ∈ PB Γ ( X , ( G , θ )) , φ : (cid:3) φ s and P ′ : (cid:3) P φ for some collectionof local sections s : (cid:3) (cid:8) s a : U a → P | U a (cid:9) . The sections { s a } define atrivialization { t a } of P by t a : P | U a → U a × Gs a ( x ) 7→ ( a , x , ) and a collection of maps (cid:8) T a : P | U a → G (cid:9) by t a ( p ) (cid:3) : ( a , x , T a ( p )) where x (cid:3) π P ( p ) . Note that T a ( p g ) (cid:3) T a ( p ) g . Define ϕ : P → P ′ p
7→ [ t a ( p )] . That ϕ is a well-defined isomorphism of principal G -bundles followsfrom the proof in the non-equivariant case. To check that ϕ is compat-ible with the Γ -actions first note that t b ◦ η γ ◦ t − a ( a , x , g ) (cid:3) t b ( γ ( s a ( x ) g )) (cid:3) t b (( γ s a ( x ))( γ g )) (cid:3) t b ( s b ( γ x ) φ ba ( γ, x )( γ g )) (cid:3) ( b , γ x , φ ba ( γ, x )( γ g )) where η is the Γ -action on P . Thus, γϕ ( p ) (cid:3) γ [ t a ( p )] (cid:3) γ [ a , x , T a ( p )] (cid:3) [ b , γ x , φ ba ( γ, x ) T a ( p )] (cid:3) [ t b ◦ η γ ◦ t − a ( a , x , T a ( p ))] (cid:3) [ t b ( γ p )] (cid:3) ϕ ( γ p ) . Therefore, ϕ is an isomorphism of Γ -semi-equivariant principal ( G , θ ) -bundles and P φ s P φ s is the identity map at the level of isomor-phism classes.The main theorem of this section has now been proved. Theorem 19.
There is a bijective correspondence PB ≃ Γ ( X , ( G , θ )) ↔ TC ≃ Γ (U , X , ( G , θ )) between semi-equivariant cocycles and principal bundles. clockwiserotation reflectionreflection anti-clockwiserotation (cid:3)(cid:3) Figure 1: This figure corresponds to C equipped with conjugation as a Z -action and U ( ) acting by rotations, as in Example 3. The blue linerepresents the conjugation automorphism on U ( ) . This conjugation isrequired in order to obtain the same final result when the two actionsare applied in reversed order. U a xs a U b γ xs b γ s a γ φ ba ( γ, x ) U c γ ′ γ xs c γ ′ s b γ ′ φ cb ( γ ′ , γ x ) γ ′ γ s c γ ′ γ ′ φ ba ( γ, x ) φ ca ( γ ′ γ, x ) Semi-equivariance Cocycleproperty
Figure 2: This diagram represents the derivation of the semi-equivariant cocycle property, as in Proposition 12. Each node of thediagram represents a local section of a principal bundle. The diagonalarrows represent applications of the Γ -action, while the vertical arrowsrepresent the action of a cocycle φ via the right action of the structuregroup. With the exception of the dashed line, all of the arrows followfrom the definitions. The dashed line follows by the semi-equivarianceproperty of the principal bundle, the blue γ ′ is acting on the element φ ba ( γ, x ) of the structure group. SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 11 U a xs ′ a s a U b γ x U b γ xs ′ b s b γ s a γ φ ba ( γ, x ) ϕ ( s a ) ϕµ a γ s ′ a γ φ ′ ba ( γ, x ) ϕ ( s b ) ϕµ b ϕ ( γ s a ) γµ a ϕγ φ ba ( γ, x ) Γ -equivariance G -equivarianceSemi-equivariance Cocycleequivalence Figure 3: This diagram depicts the derivation of the equivalence prop-erty for semi-equivariant cocycles, see Definition 9. Here ϕ is a semi-equivariant principal bundle isomorphism. Each node of the diagramrepresents a local section of a principal bundle. The arrows runningdownward are applications of a principal bundle isomorphism ϕ . Thearrows running left to right are applications of the Γ -action. The arrowsrunning right to left are right actions by the cocycle φ . Those runningupward are right actions of the cocycle equivalence µ . With the ex-ception of the dashed arrow, all of the arrows follow from definitions.The commutation of the top two squares follows from the propertiesof principal bundle isomorphisms. The dashed arrow is follows fromthe semi-equivariance property of the principal bundle. This twiststhe equivalence µ a by the action of Γ on the structure group, which ismarked in blue. The lower right square is the semi-equivariant cocycleequivalence condition.
4. Semi-equivariant Cohomol ogy
In order to study liftings of semi-equivariant principal bundles, a co-homology theory is needed. The existing notions of equivariant coho-mology are inappropriate for this task, and a new cohomology theorymust be constructed. In this section, a Γ -semi-equivariant Čech co-homology theory is developed with an abelian Γ -group ( G , θ ) as itscoefficient group. The theory makes use of a simplicial space whichencodes the group structure of Γ , and the action of Γ on the mani-fold X . In addition to these actions, the effect of the action θ mustbe incorporated. This is achieved by twisting the coboundary mapusing θ . There are a few details to check, but everything works as onewould wish. This semi-equivariant cohomology theory generalises anequivariant cohomology theory outlined by Brylinski [3, §A] Anotherhelpful reference is [7, §3.3]. One feature of the presentation here is thatit avoids the use of hypercohomology. The second dimension of thebicomplex appearing in [3, §A] is an artifact of the choice to separatethe cocycle into two parts, one encoding the transition functions for thetotal space and one encoding the action. Although this is ultimately anotational matter, the reduced book-keeping is helpful when checkinghigher cocycle conditions.The construction of semi-equivariant Čech cohomology begins withthe definition of a simplicial space. The coboundary map on the un-derlying chain complex of the cohomology theory will be constructedusing the face maps of this space. Definition 20.
Let X be a manifold equipped with a smooth action of Γ . The simplicial space associated to X is defined by X • : (cid:3) { Γ p × X } p ≥ . The simplicial space carries face and degeneracy maps d pi : X p → X p − e pi : X p → X p + defined by d pi ( γ , . . . , γ p , x ) : (cid:3) ( γ , . . . , γ p , x ) for i (cid:3) ( γ , . . . , γ i γ i + , . . . , γ p , x ) for 1 ≤ i ≤ p − ( γ , . . . , γ p − , γ p x ) for i (cid:3) p (8) e pi ( γ , . . . , γ p , x ) : (cid:3) ( γ , . . . , γ i , , γ i + , . . . , γ p , x ) for 0 ≤ i ≤ p + SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 13
Notice that in (8) the face map d p discards the element γ , this ele-ment will be used to define the simplicial twisting maps, in Definition22. Proposition 21.
The face and degeneracy maps satisfy the simplicial identities d i ◦ d j (cid:3) d j − ◦ d i for i < je i ◦ e j (cid:3) e j + ◦ e i for i ≤ j d i ◦ e j (cid:3) e j − ◦ d i for i < j id for i (cid:3) j , j + e j ◦ d i − for i > j + d pi , twisting maps θ i : X p × G → G can be defined. These maps encode the action θ of Γ on G and willbe used to twist the coboundary map. They are the basic ingredientneeded for generalisation to the semi-equivariant setting. Note thatit is only the twisting map θ that has any effect. The rest of thetwisting maps are included for notational convenience when dealingwith simplical identities. Definition 22.
The simplicial twisting maps θ i : X p × G → G are givenby θ ( γ ,...,γ p , x ) i : (cid:3) θ γ for i (cid:3) ≤ i ≤ p − i (cid:3) p The twisting maps also satisfy simplicial identities which help to en-sure that the coboundary map in semi-equivariant cohomology squaresto zero.
Proposition 23.
The simplicial twisting maps satisfy the identities θ x p + j ◦ θ d j ( x p + ) i (cid:3) θ x p + i ◦ θ d i ( x p + ) j − for i < j θ e j ( x p ) i (cid:3) θ x p i for i < j id for i (cid:3) j , j + θ x p i − for i > j + , where x p ∈ X p .Proof. The identities are trivial for most combinations of i and j . Theremaining cases can be checked individually. In particular, the first identity reduces toid ◦ θ γ γ (cid:3) θ γ ◦ θ γ id ◦ θ γ (cid:3) θ γ ◦ idid (cid:3) id for i (cid:3) , j (cid:3) i (cid:3) , j ≥ U • of X • isneeded. Such a cover can be constructed from an appropriate cover U : (cid:3) { U a | a ∈ A } of X . First, the indexing set of the simplicial coveris defined. This indexing set has a simplicial structure defined by faceand degeneracy maps, which will again be denoted by d pi and e pi . Definition 24.
Define the indexing set for U • by A • : (cid:3) { A p } p ≥ where A p : (cid:3) (cid:8) ( a , . . . , a p ) | a i ∈ A (cid:9) . Elements of A p will be denoted by a p . This set carries face and degeneracy maps d pi : A p → A p − e pi : A p → A p + defined by d pi ( a , . . . , a p ) : (cid:3) ( a , . . . , ˆ a i , . . . , a p ) e pi ( a , . . . , a p ) : (cid:3) ( a , . . . , a i , a i , a i + , . . . , a p ) , where ˆ a i denotes the removal of the element a i . Proposition 25.
The face and degeneracy maps of the indexing set A • satisfy d i ◦ d j (cid:3) d j − ◦ d i for i < je i ◦ e j (cid:3) e j + ◦ e i for i ≤ j d i ◦ e j (cid:3) e j − ◦ d i for i < j id for i (cid:3) j , j + e j ◦ d i − for i > j + . Before defining the simplicial cover itself, observe that the elementsof the simplicial space define sequences of points in X . Definition 26.
Let x p (cid:3) ( γ , . . . , γ p , x ) ∈ X p . The associated sequence (cid:8) x pi (cid:9) is defined by x pi : (cid:3) γ p − i · · · γ p x ∈ X . Simplicial covers generalise the nerves of covers. The definition willbe made using the definitions of the sequences x pi and indexing set A • . SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 15
Definition 27.
The simplicial cover U • : (cid:3) {U p } p ≥ associated to U is a sequence of covers U p of X p each indexed by A p .A set U ( a ,..., a p ) ∈ U p consists of all points in X p such that x pi ∈ U a i for 0 ≤ i ≤ p .For example, ( γ , γ , γ , x ) ∈ U ( a , a , a , a ) can be visualised as a path U a x U a γ x γ U a γ γ x γ U a γ γ γ x γ .Note that a refinement of U induces a refinement of U • . Also, theface maps of the simplicial cover are compatible with those of thesimplicial space. This is necessary to ensure that the coboundary mapis well-defined. Proposition 28.
The pullback maps of the simplicial space are compatiblewith those on the indexing set of the cover in the sense that d i ( U a p ) ⊆ U d i ( a p ) . Semi-equivariant Čech cohomology is based on a single cochaincomplex. A p -cochain for this cohomology theory consists of a smoothfunction on each set in the p th level of the simplicial cover. Definition 29.
The group of p -cochains is defined by K p Γ (U , X , ( G , θ )) : (cid:3) Ö a p ∈ A p C ∞ ( U a p , G ) , with the group operation ( φ ′ φ ) a p : (cid:3) φ ′ a p φ a p .These cochains can be pulled back by the face maps. In the semi-equivariant setting, the pullback maps are composed with the twistingmaps. This modifies the pullback by d . Definition 30.
The twisted pullback maps ∂ pi : K p Γ (U , X , ( G , θ )) → K p + Γ (U , X , ( G , θ )) are defined by ( ∂ pi φ ) a p + ( x p + ) : (cid:3) θ x p + i ◦ φ d pi ( a p + ) ◦ d pi ( x p + ) Note that the property d i ( U a p ) ⊆ U d i ( a p ) of the cover ensures that ∂ i ( φ ) is a well-defined element of K p + Γ (U , X , ( G , θ )) . Proposition 31.
The twisted pullback maps are group homomorphisms.Proof.
Using the fact that θ γ is an automorphism for all γ ∈ Γ , ( ∂ i ( φ ′ φ )) a p + ( x p + ) (cid:3) θ x p + i ◦ ( φ ′ φ ) d i ( a p + ) ◦ d i ( x p + ) (cid:3) θ x p + i (cid:16) ( φ ′ d i ( a p + ) ◦ d i ( x p + ))( φ d i ( a p + ) ◦ d i ( x p + )) (cid:17) (cid:3) (cid:16) θ x p + i ◦ φ ′ d i ( a p + ) ◦ d i ( x p + ) (cid:17) (cid:16) θ x p + i ◦ φ d i ( a p + ) ◦ d i ( x p + ) (cid:17) (cid:3) (cid:16) ( ∂ i φ ′ ) a p + ( x p + ) (cid:17) (cid:16) ( ∂ i φ ) a p + ( x p + ) (cid:17) The simplicial identities of the face maps for the simplicial space, thesimplicial cover and the twisting maps combine to produce a simplicialidentity for the twisted pullback maps.
Proposition 32.
For i < j the twisted pullback maps satisfy the identity ∂ j ◦ ∂ i (cid:3) ∂ i ◦ ∂ j − . Proof.
Using the corresponding simplicial identities between face mapson the simplicial complex, those on the simplicial cover, and thosebetween the simplical twisting maps one can directly compute ( ∂ j ( ∂ i φ )) a p + ( x p + ) (cid:3) θ x p + j ◦ ( ∂ i φ ) d j ( a p + ) ◦ d j ( x p + ) (cid:3) θ x p + j ◦ θ d j ( x p + ) i ◦ φ d i ◦ d j ( a p + ) ◦ d i ◦ d j ( x p + ) (cid:3) θ x p + i ◦ θ d i ( x p + ) j − ◦ φ d j − ◦ d i ( a p + ) ◦ d j − ◦ d i ( x p + ) (cid:3) θ x p + i ◦ ( ∂ j − φ ) d i ( a p + ) ◦ d i ( x p + ) (cid:3) ( ∂ i ( ∂ j − φ )) a p + ( x p + ) . Finally, the coboundary maps are defined.
Definition 33.
The coboundary maps ∂ p : K p Γ (U , X , ( G , θ )) → K p + Γ (U , X , ( G , θ )) SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 17 are defined by ∂ p : (cid:3) Õ ≤ i ≤ p (− ) i ∂ pi . Using the simplicial identity for the twisted pullback maps, thesquare of the coboundary map is shown to be zero.
Proposition 34.
The coboundary map satisfies ∂∂ (cid:3) .Proof. First note, using Proposition 32, that Õ i < j , j ≤ p + (− ) i + j ∂ j ∂ i (cid:3) Õ i < j , j ≤ p + (− ) i + j ∂ i ∂ j − (cid:3) Õ i ≤ j , j ≤ p + (− ) i + j ∂ i ∂ j (cid:3) Õ j ≤ i , i ≤ p + (− ) i + j ∂ j ∂ i . Therefore, ∂∂ (cid:3) Õ ≤ j ≤ p + (− ) j ∂ j ( Õ ≤ i ≤ p + (− ) i ∂ i ) (cid:3) Õ ≤ j ≤ p + Õ ≤ i ≤ p + (− ) i + j ∂ j ∂ i (cid:3) Õ j ≤ i , i ≤ p + (− ) i + j ∂ j ∂ i + Õ i < j , j ≤ p + (− ) i + j ∂ j ∂ i (cid:3) . When ( G , θ ) is abelian, Proposition 34 allows the cohomologygroups H p Γ (U , X , ( G , θ )) of the complex ( K • Γ (U , X , ( G , θ )) , ∂ ) to be defined. The restriction toabelian Γ -groups is neccesary to ensure that the coboundary maps ∂ p are group homomorphisms. In order to obtain a cohomology theorywhich is independent of the cover U , the direct limit of these cohomol-ogy groups will be taken with respect to refinements of the cover. Arefinement of U consists of another cover V indexed by some set B , anda refining map r : B → A such that V b ⊂ U r ( b ) for all b ∈ B . Such a re-finement induces a refinement of the associated simplicial covers, andrestriction homomorphisms r ∗ : K p Γ (U , X , ( G , θ )) → K p Γ (V , X , ( G , θ )) defined by ( r ∗ φ ) ( b ,..., b p ) : (cid:3) φ ( r ( b ) ,..., r ( b p )) | V ( b ,..., bp ) . These restriction homomorphisms, in turn, induce maps H p Γ (U , X , ( G , θ )) → H p Γ (V , X , ( G , θ )) on the cohomology of the complexes. In order for the direct limit ofcohomology groups to be well-defined, the maps induced on cohomol-ogy by two different refining maps need to be equal. This is true in theequivariant setting, and in the semi-equivariant setting it just needs tobe checked that the twisting of the coboundary map using θ doesn’tcause any problems. Lemma 35.
Let (V , r ) and (V , s ) be refinements of U with refining maps r , s : B → A . The maps induced on semi-equivariant cohomology by r and s are identical.Proof. By analogy with the proof in the non-equivariant case (see forexample [11, pp. 78-79]), a cochain homotopy K p Γ (U , X , ( G , θ )) h p u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ r ∗ (cid:15) (cid:15) s ∗ (cid:15) (cid:15) ∂ p / / K p + Γ (U , X , ( G , θ )) h p + u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ K p − Γ (V , X , ( G , θ )) ∂ p − / / K p Γ (V , X , ( G , θ )) . is defined by ( h p φ ) ( b ,..., b p − ) (cid:3) p − Õ k (cid:3) (− ) k φ ( r ( b ) ,..., r ( b k ) , s ( b k ) ,..., s ( b p − )) ◦ e k , where e k is the k th degeneracy map. Just as in the non-equivariantcase, expanding the expression ( h p + ∂ p φ ) ( b ,..., b p ) − ( ∂ p − h p φ ) ( b ,..., b p ) ∈ K p Γ (V , X , ( G , θ )) results in a large amount of cancelation. The remaining expression is ( ∂ p φ ) ( r ( b ) , s ( b ) ,..., s ( b p )) ◦ e − ( ∂ pp + φ ) ( r ( b ) ,..., r ( b p ) , s ( b p )) ◦ e p . The twisted coboundary maps ∂ and ∂ pp + involve the Γ -actions θ on G and σ on X , respectively. However, in the above expression, thedegeneracy maps e and e p ensure that θ and σ only ever act via theidentity element of Γ . Thus, the above expression simplifies to φ ( s ( b ) ,..., s ( b p )) − φ ( r ( b ) ,..., r ( b p )) (cid:3) ( s ∗ φ ) ( b ,..., b p ) − ( r ∗ φ ) ( b ,..., b p ) . Therefore, if φ ∈ H p Γ (V , X , ( G , θ )) is a cocycle, then ( s ∗ φ ) − ( r ∗ φ ) (cid:3) h p + ◦ ∂ p ( φ ) − ∂ p − ◦ h p ( φ ) (cid:3) ∂ p − ◦ h p ( φ ) , SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 19 which is a coboundary. Thus, r ∗ and s ∗ induce the same cohomologygroups.It is now possible to define the semi-equivariant cohomology groups. Definition 36.
The (smooth) Γ -semi-equivariant Čech cohomology groups with coefficients in an abelian Γ -group ( G , θ ) are defined by H p Γ ( X , ( G , θ )) : (cid:3) lim → H p Γ (U , X , ( G , θ )) , where H p Γ (U , X , ( G , θ )) are the cohomology groups of the complex ( K • Γ (U , X , ( G , θ )) , ∂ ) , and the direct limit is taken with respect to re-finements of U .Semi-equivariant cohomology is functorial with respect to homo-morphisms of abelian Γ -groups. Proposition 37.
A homomorphism α : A → B of abelian Γ -groups inducesa morphism of complexes α • : ( K • Γ (U , X , A ) , ∂ ) → ( K • Γ (U , X , B ) , ∂ ) defined by ( α p φ ) a p : (cid:3) α ◦ φ a p . Proof.
Let θ be the Γ -action on A and ϑ be the Γ -action on B . As α isa homomorphism of Γ -groups α p ◦ θ x p i (cid:3) ϑ x p i ◦ α p for all x p ∈ X p and0 ≤ i ≤ p . Thus, ( α p + ( ∂ i φ )) a p + ( x p + ) (cid:3) α ◦ ( ∂ i φ ) a p + ( x p + ) (cid:3) α ◦ θ x p + i ◦ φ d i ( a p + ) ◦ d i ( x p + ) (cid:3) ϑ x p + i ◦ α ◦ φ d i ( a p + ) ◦ d i ( x p + ) (cid:3) ϑ x p + i ◦ ( α p φ ) d i ( a p + ) ◦ d i ( x p + ) (cid:3) ( ∂ i ( α p φ )) a p + ( x p + ) . Therefore, α p + ◦ ∂ (cid:3) ∂ ◦ α p and α p defines a morphism of complexes.Given a short exact sequence of abelian Γ -groups, connecting mapsfor a long exact sequence can be constructed. Theorem 38.
A short exact sequence of abelian Γ -groups → A α → B β → C → induces a long exact sequence . . . ∆ p − → H p Γ ( X , A ) α p → H p Γ ( X , B ) β p → H p Γ ( X , C ) ∆ p → H p + Γ ( X , A ) α p + → . . . , where ∆ p ( φ ) : (cid:3) [ ∂ ( ψ )] for any element ψ ∈ K p Γ ( B ) such that β p ( ψ ) (cid:3) φ .Proof. The proposition follows by standard diagram chasing argumentsapplied to the exact sequence of complexes1 → ( K • Γ ( X , A ) , ∂ ) α • → ( K • Γ ( X , B ) , ∂ ) β • → ( K • Γ ( X , C ) , ∂ ) → . For an example, see the proof of [11, Theorem 4.30].
5. The Semi-equivariant Dixmier-Douady Invariant
In order to apply semi-equivariant cohomology to the classification ofsemi-equivariant liftings, its relationship with semi-equivariant prin-cipal bundles must be clarified. By Theorem 19, this reduces to theproblem of relating semi-equivariant transition cocycles and semi-equivariant cohomology classes. In this section, semi-equivariant tran-sition cocycles will be interpreted as degree-1 cocycles which can takevalues in a non-abelian coefficient group. An analogue of Theorem38 will be proved that constructs a connecting map from the transi-tion cocycles into degree-2 cohomology. The theorem can be used toclassify certain liftings of semi-equivariant principal bundles betweennon-abelian structure groups. This method has its origins in the workof Dixmier-Douady on continuous trace C ∗ -algebras [5]. See also [4,§4] and [11, §4.3].To begin, note that the p -cochains of Definition 29 and the twistedpullback maps of Definition 30 are well-defined for non-abelian Γ -groups. Thus, it is possible to make the following definitions. Definition 39. TC Γ (U , X , ( G , θ )) : (cid:3) (cid:8) µ ∈ K Γ (U , X , ( G , θ )) | ( ∂ µ ) − ( ∂ µ ) (cid:3) (cid:9) (10)TC Γ (U , X , ( G , θ )) : (cid:3) (cid:8) φ ∈ K Γ (U , X , ( G , θ )) | ( ∂ φ ) − ( ∂ φ )( ∂ φ ) (cid:3) (cid:9) /∼ (11)where φ ∼ φ if and only if there exists a µ ∈ K Γ (U , X , ( G , θ )) suchthat ( ∂ µ ) φ (cid:3) φ ( ∂ µ ) .The set TC Γ (U , X , ( G , θ )) is just TC ≃ Γ (U , X , ( G , θ )) with the transi-tion cocycle condition and equivalence condition expressed in terms SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 21 of twisted pullback maps. Note that the particular order of the terms ∂ i µ in (10) and ∂ i φ in (11) is important as the elements µ and φ takevalues in G , which is not necessarily abelian. When G is abelian, theseterms may be rearranged to give the corresponding cocycle propertiesin semi-equivariant cohomology. An abelian structure group also en-sures that pointwise multiplication is a well-defined group structureon TC Γ and TC Γ , which, in general, are only pointed sets. Theorem 40.
When G is abelian TC Γ (U , X , ( G , θ )) ≃ H Γ (U , X , ( G , θ )) (12)TC Γ (U , X , ( G , θ )) ≃ H Γ (U , X , ( G , θ )) . (13) Proof.
When G is abelian, the defining condition on TC Γ (U , X , ( G , θ )) and the 0-cocycle condition on cohomology are equivalent as0 (cid:3) −( ∂ µ ) + ( ∂ µ ) (cid:3) ( ∂ µ ) − ( ∂ µ ) (cid:3) ∂µ. This proves (12). Similarly, the defining condition on TC Γ (U , X , ( G , θ )) and the 1-cocycle condition on cohomology are equivalent as0 (cid:3) −( ∂ φ ) + ( ∂ φ ) + ( ∂ φ ) (cid:3) ( ∂ φ ) − ( ∂ φ ) + ( ∂ φ ) (cid:3) ∂φ, and the equivalence relations on TC Γ (U , X , ( G , θ )) and H Γ (U , X , ( G , θ )) are the same as ( ∂ µ ) + φ (cid:3) φ + ( ∂ µ ) φ − φ (cid:3) ( ∂ µ ) − ( ∂ µ ) φ − φ (cid:3) ∂µ. These two facts imply (13).Together, Theorem 38 and Theorem 40 enable liftings of semi-equivariant principal bundles between abelian structure groups to beclassified. However, the construction of a Dirac operator involves theconstruction of liftings between non-abelian groups. The next theoremis a generalisation of Theorem 38 that can be used to classify certainliftings between non-abelian structure groups.
Theorem 41.
A short exact sequence of Γ -groups → A α → B β → C → , where α ( A ) is central in B , induces an exact sequence of pointed sets → H Γ ( X , A ) α → TC Γ ( X , B ) β → TC Γ ( X , C ) ∆ → . . .. . . ∆ → H Γ ( X , A ) α → TC Γ ( X , B ) β → TC Γ ( X , C ) ∆ → H Γ ( X , A ) , where the connecting maps ∆ and ∆ are defined by ∆ ([ µ ]) : (cid:3) [( ∂ η ) − ( ∂ η )] ∆ ([ φ ]) : (cid:3) [( ∂ ψ ) − ( ∂ ψ )( ∂ ψ )] , for any η ∈ K Γ ( X , B ) , ψ ∈ K Γ ( X , B ) satisfying β ( η ) (cid:3) µ , β ( ψ ) (cid:3) φ .Proof. The diagram chasing arguments used in the proof of Theorem38 do not apply directly. However, they can be imitated while carefullyworking around any lack of commutivity in the groups B and C . Notethat Proposition 37 and Proposition 32 continue to hold when thestructure groups involved are non-abelian. Thus, the twisted pullbackmaps ∂ i commute with the maps α i and β i induced by α and β , andalso satisfy the simplicial identity ∂ j ◦ ∂ i (cid:3) ∂ i ◦ ∂ j − for i < j .First, the map ∆ will be considered. Let ν : (cid:3) ( ∂ η ) − ( ∂ η ) ∈ K Γ ( X , B ) . The cochain η is a lifting by β of µ so β ( ν ) (cid:3)
1. Thus, ν takes values in ker ( β ) ≃ A and defines an element of K Γ ( X , A ) . Thesimplicial identity can be used to show that the cochain ν satisfies thecocycle property, ( ∂ ν ) − ( ∂ ν ) (cid:3) h ( ∂ ∂ ν ) − ( ∂ ∂ ν ) i − h ( ∂ ∂ ν ) − ( ∂ ∂ ν ) i (cid:3) ( ∂ ∂ ν ) − ( ∂ ∂ ν )( ∂ ∂ ν ) − ( ∂ ∂ ν ) (cid:3) ( ∂ ∂ ν ) − ( ∂ ∂ ν )( ∂ ∂ ν ) − ( ∂ ∂ ν ) (cid:3) ( ∂ ∂ ν ) − ( ∂ ∂ ν )( ∂ ∂ ν ) − ( ∂ ∂ ν ) (cid:3) . Therefore, ∆ ([ µ ]) : (cid:3) [ ν ] ∈ H Γ ( X , A ) . Next, it needs to be shownthat ∆ ([ µ ]) : (cid:3) [( ∂ η ) − ( ∂ η )] is independent of the choice of η . Let η ′ ∈ K Γ ( X , B ) be another element such that β ( η ′ ) (cid:3) µ . Set ω : (cid:3) η ′ η − and ν ′ : (cid:3) ( ∂ η ′ ) − ( ∂ η ′ ) ∈ K Γ ( X , B ) . Then β ( ω ) (cid:3) β ( η ′ η − ) (cid:3) µµ − (cid:3)
1. Thus, ω defines an element of K Γ ( X , A ) and ∂ω ∈ K Γ ( X , A ) is acoboundary. Using the fact that ν and ∂ω take values in the abelian SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 23 group A , ( ∂ω ) ν (cid:3) ( ∂ω )( ∂ η ) − ( ∂ η ) (cid:3) ( ∂ η ) − ( ∂ω )( ∂ η ) (cid:3) ( ∂ η ) − ( ∂ η )( ∂ η ′ ) − ( ∂ η ′ )( ∂ η ) − ( ∂ η ) (cid:3) ( ∂ η ′ ) − ( ∂ η ′ ) (cid:3) ν ′ . Therefore, [ ν ] (cid:3) [ ν ′ ] ∈ H Γ ( X , A ) .In order to examine the map ∆ , let ν : (cid:3) ( ∂ ψ ) − ( ∂ ψ )( ∂ ψ ) ∈ K Γ ( X , B ) . The cochain ψ ∈ K Γ ( X , B ) is a β -lifting of the cocycle φ ∈ TC Γ ( X , C ) so β ( ν ) (cid:3)
1. Therefore, ν defines an element of K Γ ( X , A ) .Using the simplicial identity, and the fact that ν takes values in thecentre of B , it can be shown that ν satisfies the 2-cocycle propery. First,compute ( ∂ ν )( ∂ ν ) (cid:3) ( ∂ ∂ ψ ) − ( ∂ ∂ ψ )( ∂ ∂ ψ )( ∂ ν ) (cid:3) ( ∂ ∂ ψ ) − ( ∂ ∂ ψ )( ∂ ν )( ∂ ∂ ψ ) (cid:3) ( ∂ ∂ ψ ) − ( ∂ ∂ ψ ) h ( ∂ ∂ ψ ) − ( ∂ ∂ ψ )( ∂ ∂ ψ ) i ( ∂ ∂ ψ ) (cid:3) ( ∂ ∂ ψ ) − ( ∂ ∂ ψ ) h ( ∂ ∂ ψ ) − ( ∂ ∂ ψ )( ∂ ∂ ψ ) i ( ∂ ∂ ψ ) (cid:3) ( ∂ ∂ ψ ) − ( ∂ ∂ ψ )( ∂ ∂ ψ )( ∂ ∂ ψ ) (cid:3) ( ∂ ∂ ψ ) − ( ∂ ∂ ψ ) h ( ∂ ∂ ψ )( ∂ ∂ ψ ) − i ( ∂ ∂ ψ )( ∂ ∂ ψ ) (cid:3) ( ∂ ∂ ψ ) − ( ∂ ∂ ψ ) h ( ∂ ∂ ψ )( ∂ ∂ ψ ) − i ( ∂ ∂ ψ )( ∂ ∂ ψ ) (cid:3) h ( ∂ ∂ ψ ) − ( ∂ ∂ ψ )( ∂ ∂ ψ ) i h ( ∂ ∂ ψ ) − ( ∂ ∂ ψ )( ∂ ∂ ψ ) i (cid:3) ( ∂ ν )( ∂ ν ) . Then ( ∂ν ) (cid:3) ( ∂ ν )( ∂ ν ) − ( ∂ ν )( ∂ ν ) − (cid:3) ( ∂ ν )( ∂ ν )( ∂ ν ) − ( ∂ ν ) − (cid:3) ( ∂ ν )( ∂ ν ) h ( ∂ ν )( ∂ ν ) i − (cid:3) ( ∂ ν )( ∂ ν ) h ( ∂ ν )( ∂ ν ) i − (cid:3) , and so [ ν ] ∈ H Γ ( X , A ) .Next, it needs to be shown that ∆ is well-defined. Specifically, that ∆ ([ φ ]) : (cid:3) [( ∂ ψ ) − ( ∂ ψ )( ∂ ψ )] is independent of the choice of ψ , and depends only on the class of φ inTC Γ ( X , C ) . To prove the first statement, let ψ ′ ∈ K Γ ( X , B ) be another β -lifting of φ and ν ′ : (cid:3) ( ∂ ψ ′ ) − ( ∂ ψ ′ )( ∂ ψ ′ ) be the corresponding elementof H Γ ( X , A ) . If ω : (cid:3) ψ ′ ψ − then β ( ω ) (cid:3) β ( ψ ′ ψ − ) (cid:3) φφ − (cid:3) . Thus, ω ∈ K Γ ( X , A ) and ∂ω ∈ K Γ ( X , A ) is a coboundary . Next, using the factthat ω takes values in the center of B , ( ∂ω ) ν (cid:3) ( ∂ ω )( ∂ ω ) − ( ∂ ω )( ∂ ψ ) − ( ∂ ψ )( ∂ ψ ) (cid:3) ( ∂ ψ ) − ( ∂ ω ) − ( ∂ ω )( ∂ ψ )( ∂ ω )( ∂ ψ ) (cid:3) ( ∂ ψ ) − ( ∂ ψ ′ ψ − ) − ( ∂ ψ ′ ψ − )( ∂ ψ )( ∂ ψ ′ ψ − )( ∂ ψ ) (cid:3) ( ∂ ψ ) − ( ∂ ψ )( ∂ ψ ′ ) − ( ∂ ψ ′ )( ∂ ψ ) − ( ∂ ψ )( ∂ ψ ′ )( ∂ ψ ) − ( ∂ ψ ) (cid:3) ( ∂ ψ ′ ) − ( ∂ ψ ′ )( ∂ ψ ′ ) (cid:3) ν ′ . Therefore, [ ν ] (cid:3) [ ν ′ ] ∈ H Γ ( X , A ) .In order to prove that ∆ ([ φ ]) depends only on the class of φ , sup-pose that φ is a coboundary i.e. that φ (cid:3) ( ∂ ˜ φ ) − ( ∂ ˜ φ ) for some˜ φ ∈ K Γ ( X , C ) . By surjectivity of β , there exists an element ˜ ψ such that β ( ˜ ψ ) (cid:3) ˜ φ . Then ψ : (cid:3) ( ∂ ˜ ψ ) − ( ∂ ˜ ψ ) is a lifting by β of φ as β ( ψ ) (cid:3) β h ( ∂ ˜ ψ ) − ( ∂ ˜ ψ ) i (cid:3) ( β∂ ˜ ψ ) − ( β∂ ˜ ψ ) (cid:3) ( ∂ β ˜ ψ ) − ( ∂ β ˜ ψ ) (cid:3) ( ∂ ˜ φ ) − ( ∂ ˜ φ ) (cid:3) φ. So, again applying the simplicial identity, ∆ ([ φ ]) (cid:3) [( ∂ ψ ) − ( ∂ ψ )( ∂ ψ )] (cid:3) [( ∂ ∂ ˜ ψ ) − ( ∂ ∂ ˜ ψ )( ∂ ∂ ˜ ψ ) − ( ∂ ∂ ˜ ψ )( ∂ ∂ ˜ ψ ) − ( ∂ ∂ ˜ ψ )] (cid:3) [( ∂ ∂ ˜ ψ ) − ( ∂ ∂ ˜ ψ )( ∂ ∂ ˜ ψ ) − ( ∂ ∂ ˜ ψ )( ∂ ∂ ˜ ψ ) − ( ∂ ∂ ˜ ψ )] (cid:3) . Thus, ∆ ([ φ ]) depends only on the class of φ in TC Γ ( X , C ) . SEMI-EQUIVARIANT DIXMIER-DOUADY INVARIANT 25
It is now possible to define the semi-equivariant Dixmier-Douadyinvariant and resolve the main problem of this paper.
Definition 42.
The semi-equivariant Dixmier-Douady invariant of a Γ -semi-equivariant principal C -bundle P associated to a central exactsequence 1 → A α → B β → C → DD ( P ) : (cid:3) ∆ ([ φ ]) ∈ H Γ ( X , A ) , where ∆ is the connecting map provided by Theorem 41 and [ φ ] is thetransition cocycle associated to P by Proposition 12. Theorem 43.
The exact sequence produced by Theorem 41 implies that(a) P can be lifted by β if and only if DD ( P ) (cid:3) ,(b) when DD ( P ) (cid:3) , the liftings of P by β correspond non-canonically tothe classes of H Γ ( X , A ) .
6. Related Work and Applications
Semi-equivariant Čech cohomology H • Γ ( X , ( G , θ )) is closely related toseveral other cohomology theories. For example,(a) When Γ is the trivial group, H • Γ ( X , ( G , θ )) is Čech cohomology.(b) When θ is the trivial action, H • Γ ( X , ( G , θ )) is equivariant Čechcohomology ˇ H • Γ ( X , G ) . When X is a compact manifold actedupon by a finite group, the equivariant Čech cohomology can berelated to Grothendieck’s equivariant sheaf cohomology [8, §5.5]or Borel cohomology [3, §A], [7, §3.3].Note that there is a restriction homomorphism H p Γ ( X , ( G , θ )) → H p Γ G ( X , ( G , θ )) ≃ ˇ H p Γ G ( X , G ) , where Γ G ⊆ Γ is the stabiliser subgroup that acts trivially on G .In this way, the semi-equivariant cohomology can be regarded asa restriction of equivariant cohomology.(c) When X is a point, H • Γ ( X , ( G , θ )) is the group cohomology H • ( Γ , G θ ) of Γ with coefficients in the Γ -module G θ defined by G and θ [2,p. 35]. With this in mind, semi-equivariant cohomology can beviewed as a cross between group cohomology and equivariantcohomology. (d) When X is a Real space and κ is the conjugation action on U ( ) , H • Gal ( C / R ) ( X , ( U ( ) , κ )) is closely related to Real Čech cohomologyof [10], and the Real sheaf cohomology defined in [9]. Note that, inthis case, the semi-equivariant cohomology incorporates aspectsof equivariant Čech cohomology and Galois cohomology for thefield extension C / R .An important application of Theorem 41 arises in the study of Spin c -structures on Real spaces [1] and orientifolds [6]. This is the originalmotivation for the present paper. Such structures correspond to semi-equivariant liftings of equivariant principal SO ( n ) -bundles via the cen-tral exact sequence1 → ( U ( ) , κ ◦ ǫ ) → ( Spin c ( n ) , κ ◦ ǫ ) Ad c → ( SO ( n ) , id ◦ ǫ ) → . Here ǫ : Γ → Z is a homomorphism from a finite group Γ , and κ denotes the conjugation action on Spin c ( n ) and U ( ) . The topic ofSpin c -structures on orientifolds will be treated in a forthcoming paper.
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