Twists over étale groupoids and twisted vector bundles
aa r X i v : . [ m a t h . OA ] F e b Twists over ´etale groupoids and twisted vectorbundles
Carla Farsi and Elizabeth GillaspyNovember 5, 2018
Abstract
Inspired by recent papers on twisted K -theory, we consider in thisarticle the question of when a twist R over a locally compact Hausdorffgroupoid G (with unit space a CW-complex) admits a twisted vector bun-dle, and we relate this question to the Brauer group of G . We show thatthe twists which admit twisted vector bundles give rise to a subgroup ofthe Brauer group of G . When G is an ´etale groupoid, we establish condi-tions (involving the classifying space B G of G ) which imply that a torsiontwist R over G admits a twisted vector bundle. C ∗ -algebras associated to dynamical systems have provided motivation and ex-amples for a wide array of topics in C ∗ -algebra theory: representation theory,ideal structure, K -theory, classification, and connections with mathematicalphysics, to name a few. In many of these cases, a complete understanding ofthe theory has required expanding the notion of a dynamical system to allowfor partial actions and twisted actions, as well as actions of group-like objectssuch as semigroups or groupoids.For example, the C ∗ -algebras C ∗ ( G ; R ) associated to a groupoid G and atwist R over G (hereafter referred to as twisted groupoid C ∗ -algebras ) provideimportant insights into mathematical physics as well as the structure of other C ∗ -algebras. First, the collection of twists R over a groupoid G is intimatelyrelated with the cohomology of G , cf. [15, 16, 26]. Another structural resultis due to Kumjian [14] and Renault [24]: groupoid twists classify Cartan pairs.Finally, the papers [27, 4, 2] establish that twisted groupoid C ∗ -algebras classify D -brane charges in many flavors of string theory.We also note, following [22, 27], that groupoid twists constitute an exampleof Fell bundles. Indeed, Fell bundles provide a universal framework for studyingall of the generalized dynamical systems mentioned above.In several recent papers (cf. [3, 27, 10]) on twisted groupoid C ∗ -algebras,the K -theory groups of these C ∗ -algebras have received a good deal of at-tention. Of particular interest is the question of when K ( C ∗ ( G ; R )) can be1ompletely understood in terms of G -equivariant vector bundles. Phillips estab-lished in Chapter 9 of [23] that G -equivariant vector bundles may not suffice todescribe K ( C ∗ ( G ; R )), even when G = M ⋊ G is a transformation group and R is trivial. Vector bundles provide a highly desirable geometric perspective on K ( C ∗ ( G ; R )), however, and so conditions are sought (cf. [1, 2, 5, 9, 10, 18])under which K ( C ∗ ( G ; R )) is generated by G -equivariant vector bundles.In Theorem 5.28 of [27], Tu, Xu, and Laurent-Gengoux study this questionfor proper Lie groupoids G . They establish, in this context, sufficient conditionsfor the K -theory group K ( C ∗ ( G , R )) associated to a twist R over G to be gen-erated by ( R , G )-twisted vector bundles over the unit space of G (see Definition2.4 below). A necessary condition is that R be a torsion element of the Brauergroup of G . Conjecture 5.7 on page 888 of [27] states that, if G is a properLie groupoid acting cocompactly on its unit space, then this condition is alsosufficient.Conjecture 5.7 of [27] has not yet been disproved, but it has only been proventrue in certain special cases: cf. [18, 10, 5] when R = G × T is the trivial twist,[2] for nontrivial twists R over manifolds M , and [1, 9, 19] for nontrivial twistsover representable orbifolds G ⋊ M , where G is a discrete group acting properlyon a compact space M .In hopes of shedding more light on this Conjecture, we present an equivalentformulation in Conjecture 3.5 below, using the Brauer group of G as defined in[16]. Our reformulated conjecture relies on our result (Proposition 3.4) that, forany locally compact Hausdorff groupoid G whose unit space is a CW-complex,the collection of twists R over G which admit twisted vector bundles gives riseto a subgroup T w τ ( G ) of the Brauer group Br( G ).We note that Theorem 3.2 of [13] also establishes a link between twistedvector bundles and the Brauer group, but Karoubi’s approach in [13] differssubstantially from ours, and does not address the group structure of T w τ ( G ).In the second part of the paper, we address the question of when a torsiontwist R over an ´etale groupoid G admits a twisted vector bundle. The existenceof such vector bundles is necessary (but not sufficient) in order for K ( C ∗ ( G ; R ))to be generated by twisted vector bundles.Theorem 4.6 below establishes that if the classifying space B G is a compactCW-complex and if a certain principal P U ( n )-bundle P lifts to a U ( n )-principalbundle ˜ P , then up to Morita equivalence, the torsion twist R admits a twistedvector bundle. To our knowledge, the connection between classifying spaces andtwisted vector bundles has not been explored previously in the literature; weare optimistic that Theorem 4.6 will lead to new insights into the K -theory oftwisted groupoid C ∗ -algebras. We begin in Section 2 by reviewing the basic concepts we will rely on throughoutthis paper: locally compact Hausdorff groupoids, twists over such groupoids,groupoid vector bundles and twisted vector bundles. In Section 3 we showthat, for any locally compact Hausdorff groupoid G whose unit space is a CW-2omplex, the collection of twists over G which admit twisted vector bundlesgives rise to a subgroup of the Brauer group of G , and we use this to presentan alternate formulation of Conjecture 5.7 from [27]. Finally in Section 4 weconsider torsion twists for ´etale groupoids. We establish, in Theorem 4.6, suf-ficient conditions for a torsion twist R over an ´etale groupoid G to admit atwisted vector bundle, and we present examples showing that the hypotheses ofTheorem 4.6 are satisfied in many cases of interest. The authors are indebted to Alex Kumjian for pointing out a flaw in an earlierversion of this paper. We would also like to thank Angelo Vistoli for helpfulcorrespondence.
Recall that a groupoid is a small category with inverses. Throughout this note, G will denote (the space of arrows of) a groupoid with unit space G (0) , withsource, range (or target), and unit maps s, r : G −→ G (0) , u : G (0) −→ G . As usual we denote the set of composable elements of G by G (2) , where G (2) = G × s, G (0) ,t G = { ( g , g ) ∈ G × G | s ( g ) = r ( g ) } . In this paper we will primarily be concerned with locally compact Hausdorffgroupoids . These are groupoids G such that the spaces G (0) , G , G (2) have locallycompact Hausdorff topologies with respect to which the maps s, r : G → G (0) , themultiplication G (2) → G , and the inverse map G → G are continuous. Conjecture3.5 below makes reference to
Lie groupoids , which are locally compact Hausdorffgroupoids such that the spaces G (0) , G , G (2) are smooth manifolds and all of thestructure maps between them are smooth.Theorem 4.6 deals with ´etale groupoids , which are locally compact Hausdorffgroupoids G for which r, s are local homeomorphisms. For example, if a discretegroup Γ acts on a CW-complex M , the associated transformation group Γ ⋉ M is an ´etale groupoid. Definition 2.1.
Let G , G be two locally compact Hausdorff groupoids withunit spaces G (0)1 , G (0)2 respectively. A morphism f : G → G consists of a pairof continuous maps f = ( f , f ) , with f : G (0)1 → G (0)2 , f : G → G , such that, if we denote by s G j and r G j the source and range maps of G j , j = 1 , s G ◦ f = f ◦ s G , and r G ◦ f = f ◦ r G . T -central extension of a groupoid G was originally de-veloped (cf. [14, 21, 27]) to provide a “second cohomology group” for groupoids.Groupoid twists and their associated twisted vector bundles (see Definition 2.4below) are the groupoid analogues of group 2-cocycles and projective represen-tations. Definition 2.2.
Let G be a locally compact Hausdorff groupoid with unit space G (0) . A T -central extension (or “twist”) of G consists of1. A locally compact Hausdorff groupoid R with unit space G (0) , togetherwith a morphism of locally compact Hausdorff groupoids( id, π ) : R → G which restricts to the identity on G (0) .2. A left T –action on R , with respect to which R is a left principal T -bundleover G .3. These two structures are compatible in the sense that( z r )( z r ) = z z ( r r ) , ∀ z , z ∈ T , ∀ ( r , r ) ∈ R (2) = R × s, G (0) ,r R . We write
T w ( G ) for the set of twists over G .These conditions (1)-(3) imply the exactness of the sequence of groupoids G (0) → G (0) × T → R → G ⇒ G (0) , which highlights the parallel between twists over a groupoid G and extensionsof G by T (or elements of the second cohomology group H ( G , T )).If R , R ∈ T w ( G ), we can form their Baer sum R + R := { ( r , r ) ∈ R × R : π ( r ) = π ( r ) } / ∼ , where ( r , r ) ∼ ( zr , zr ) for all z ∈ T . Define an action of T on R + R by z · [( r , r )] = [( zr , r )] = [( r , zr )], and observe that with this action, R + R becomes a twist over G .With this operation T w ( G ) becomes a group; the identity element is thetrivial extension G × T , and the inverse of a twist R is the twist R . As groupoids, R = R ; however, the action of T on R is the conjugate of the action on R . Tobe precise, if r ∈ R , denote by r the corresponding element of R . Then z · r = z · r. In this note, we will consider actions of groupoids G and twists R over G ona variety of spaces. We make this concept precise as follows.4 efinition 2.3. Let G be a locally compact Hausdorff groupoid with unit space G (0) . A G -space is a locally trivial fiber bundle J : P → G (0) such that, setting G ∗ P = { ( g, p ) ∈ G × P : s ( g ) = J ( p ) } and equipping G ∗ P with the subspace topology inherited from G × P , we havea continuous map σ : G ∗ P → P satisfying • σ ( J ( p ) , p ) = p for all p ∈ P ; • J ( σ ( g, p )) = r ( g ) for all ( g, p ) ∈ G ∗ P ; • If ( g, h ) ∈ G (2) and ( h, p ) ∈ G ∗ P , then σ ( g, σ ( h, p )) = σ ( gh, p ).We will often write g · p for σ ( g, p ) ∈ P .Note that, as a consequence of the above definition, the map σ g : P s ( g ) → P r ( g ) given by p σ ( g, p ) must be a homeomorphism, for all g ∈ G . Definition 2.4.
1. Let G be a locally compact Hausdorff groupoid with unitspace G (0) , where G (0) is a CW-complex. A G –vector bundle is a vectorbundle J : E → G (0) which is a G -space in the sense of Definition 2.3.2. Let G (0) → T × G (0) i −→ R j −→ G ⇒ G (0) be a T -central extension of locally compact Hausdorff groupoids. By a( G , R ) –twisted vector bundle , we mean a R -vector bundle J : E → G (0) such that, whenever z ∈ T , r ∈ R , e ∈ E such that s ( r ) = J ( e ), we have( z · r ) · e = z ( r · e ) . (1)Here, the action on the right-hand side of the equation is simply scalarmultiplication (identifying T with the unit circle of C ).3. An equivalent characterization of ( R , G )-twisted vector bundles is the fol-lowing:A R -vector bundle E → G (0) is a ( R , G )-twisted vector bundle if and onlyif the subgroupoid ker j ∼ = G (0) × T of R acts on E by scalar multiplication,where T is identified with the unit circle of C .In Proposition 3.4, we will establish a connection between the twists over G which admit twisted vector bundles and the Brauer group of G , as introduced in[16]. Thus, we review here a few facts about the Brauer group and its connectionto T w ( G ). Definition 2.5.
Let G be a locally compact Hausdorff groupoid. As in Def-inition 8.1 of [16], we will denote by Br ( G ) the group of Morita equivalenceclasses of G -spaces A such that A = G (0) × K ( H ) for some Hilbert space H . Wedenote the class in Br ( G ) of A by [ A , α ], where α is the action of G on A .Also, let E ( G ) be the quotient of T w ( G ) by Morita equivalence or, equiva-lently, the quotient by the subgroup W of elements which are Morita equivalentto the trivial twist. See Definition 3.1 and Corollary 7.3 of [16] for details.5heorem 8.3 of [16] establishes that Br ( G ) ∼ = E ( G ) = T w ( G ) /W. Let G be a locally compact Hausdorff groupoid with unit space a CW-complex.In this section, we will show that the subset T w τ ( G ) of twists over G whichadmit twisted vector bundles gives a subgroup of Br ( G ). Definition 3.1.
For a locally compact Hausdorff groupoid G , let Br τ ( G ) be thesubgroup of Br ( G ) consisting of Morita equivalence classes [ A , α ] of elementary G -bundles A = G (0) × K ( H ) with zero Dixmier-Douady invariant, such that H is finite dimensional.When, in addition, the unit space of G is a CW-complex, we denote by T w τ ( G ) the subset of T w ( G ) consisting of twists R over G that admit a twistedvector bundle. Proposition 3.2.
Let G be a locally compact groupoid whose unit space is aCW-complex. Then T w τ ( G ) is a subgroup of T w ( G ) .Proof.
1. (Closure under operation) Given two twists G (0) → G (0) × T i −→ R j −→ G ⇒ G (0) , G (0) → G (0) × T i −→ R j −→ G ⇒ G (0) that admit twisted vector bundles E and E respectively, it is straight-forward to show that E ∗ E := { ( e , e ) ∈ E ⊕ E | J ( e ) = J ( e ) } / ∼ , is a twisted vector bundle for the Baer sum R + R . The action of R + R on E ∗ E is given by[( r , r )] · [( e , e )] = [( r · e , r · e )] .
2. (Neutral Element) The neutral element of
T w ( G ) is G × T . Note that G × T admits a twisted vector bundle E – namely, E = G (0) × C , with the action( g, z ) · ( s ( g ) , v ) = ( r ( g ) , zv ).3. (Inverses) We must show that, if R ∈
T w τ ( G ), then R ∈
T w τ ( G ).For R ∈
T w τ ( G ), let E → G (0) be a ( R , G )-twisted vector bundle. Write E for the conjugate vector bundle – that is, E = E as sets, and the additiveoperation on E agrees with that on E (in symbols, e + f = e + f ), butthe C action on E is the conjugate of the action on E : z · e = z · e . Definean action of R on E by r · e = r · e . This action makes E into a R -vectorbundle since E is a R -vector bundle. Moreover, for any z ∈ T we have( z · r ) · e = z · r · e = ( zr ) · e = z ( r · e ) = zr · e = z ( r · e ) . T acts by scalars on E , and so E is a ( W , G )-twisted vector bundle. Remark . Recall from Proposition 5.5 of [27] that if a twist R over G admitsa twisted vector bundle, then R must be torsion. Thus, T w τ ( G ) is a subgroupof T w tor ( G ), the torsion subgroup of T w ( G ). T w τ ( G ) in Br ( G ) Recall that if G is a locally compact Hausdorff groupoid with unit space G (0) ,then Br ( G ) consists of Morita equivalence classes of G -spaces of the form A = G (0) × K ( H ).In Section 8 of [16], the authors construct an isomorphism Θ : Br ( G ) → T w ( G ) /W , where W is the subgroup of T w ( G ) consisting of elements which areMorita equivalent to the trivial twist. We will use this isomorphism to studythe subgroup of Br ( G ) corresponding to T w τ ( G ).Proposition 8.7 of [16] describes a homomorphism θ : T w ( G ) → Br ( G )which induces the inverse of Θ. Proposition 3.4.
Suppose G is a locally compact Hausdorff groupoid whoseunit space G (0) is a connected CW-complex, and suppose R ∈
T w τ ( G ) . Thenthere exists a finite-dimensional G -vector bundle V → G (0) such that θ ( R ) =[Aut( V ) , α ] , where α is induced by the action of G on V .Moreover, if [ A , α ′ ] ∈ Br ( G ) and ( A , α ′ ) is Morita equivalent to ( M n , α ) where M n is an M n ( C ) -bundle over G (0) , then [ A , α ′ ] = [ M n , α ] lies in θ ( T w τ ( G )) . In other words, Br τ ( G ) ∼ = T w τ ( G ) .Proof. If R ∈
T w τ ( G ) and V is a ( R , G )-twisted vector bundle, write j : R → G for the projection of R onto G and write σ : R ∗ V → V for the action of R on V . Define α : G ∗
Aut( V ) → Aut( V ) by( α ( g, A )( v ) = σ ( η, A ( σ ( η − , v ))) , where v ∈ V r ( g ) and η ∈ j − ( g ).Note that α ( g, A ) does not depend on our choice of η ∈ j − ( g ): If η, η ′ ∈ j − ( g ), the fact that R is a principal T -bundle over G implies that η = zη ′ forsome z ∈ T . Since V is a ( G , R )-twisted vector bundle, σ ( η, v ) = zσ ( η ′ , v ), andconsequently σ ( η, A ( σ ( η − , v ))) = σ ( η ′ , A ( σ (( η ′ ) − , v ))) . Now, Lemma 8.8 of [16] establishes that [Aut( V ) , α ] = θ ( R ).For the second statement, suppose α is an action of G on a bundle M n of n -dimensional matrix algebras over G (0) . Then Theorem 8.3 of [16] explainshow to construct the twist Θ([ M n , α ]), using a pullback construction. To beprecise, Θ([ M n , α ]) = { ( g, U ) ∈ G × U n ( C ) : α g = Ad U } =: R ( α ) .
7e will construct a ( G , R ( α ))-twisted vector bundle, proving that [ M n , α ] ∈ θ ( T w τ ( G )).The T -action on R ( α ) which makes it into a twist over G is given by z · ( g, U ) = ( g, z · U ) . Consider the sub-bundle GL n of M n obtained by considering only the in-vertible elements of M n ( C ) in each fiber of M n . Notice that GL n ( C ) acts on GL n by right multiplication in each fiber, and that this action is continuous,and free and transitive in each fiber, and hence makes GL n into a principal GL n bundle. We consequently obtain an associated vector bundle over G (0) , V = GL n × GL n ( C ) C n . Moreover, R ( α ) acts on V :( g, U ) · [ A, v ] = [ α g ( A ) , U v ] . To see that this action is well defined, take G ∈ GL n ( C ) and calculate:[ α g ( AG ) , U ( G − v )] = [ U AGU − , U G − v ] = [ U A, v ] = [
U AU − , U v ][ α g ( A ) , v ] = [ U AU − , U v ] . Moreover, ( z · ( g, U )) · [ A, v ] = [ α g ( A ) , zU ( v )]= z · [ α g ( A ) , U v ] = z · (( g, U ) · [ A, v ]) , so V is an ( R ( α ) , G )-twisted vector bundle. Thus, R ( α ) ∈ T w τ ( G ) whenever[ α, M ] ∈ Br ( G ).Proposition 3.4 thus establishes that twists R over G which admit twistedvector bundles correspond to C ∗ -bundles over G (0) with finite-dimensional fibers.Phrased in this way, the parallel between Proposition 3.4 and Theorem 3.2 of[13] becomes evident. However, the two proofs take very different approaches.Moreover, Karoubi does not address the group structure of T w τ ( G ) in Theorem3.2 of [13].Proposition 3.4 also allows us to rephrase Conjecture 5.7 of [27] in terms ofthe Brauer group, as follows. Recall that, in its original form, Conjecture 5.7of [27] asserts that all torsion elements of T w ( G ) should admit twisted vectorbundles, if G is proper and the quotient G (0) / G is compact. Conjecture . Let G be a proper Lie groupoid such thatthe quotient G (0) / G is compact, and let [ A , α ] ∈ Br ( G ) be a torsion element.Then [ A , α ] = [ M , α ′ ] for some finite-dimensional matrix algebra bundle M over G (0) and an action α ′ of G on M . 8 Twisted vector bundles for ´etale groupoids
In this section we consider torsion twists over ´etale groupoids G . We establishin Theorem 4.6 sufficient conditions for a torsion twist R over G to admit (up toMorita equivalence) a twisted vector bundle, and we describe examples meetingthese conditions in Section 4.1. The conditions of Theorem 4.6 are phrased interms of the classifying space B G and in terms of a principal bundle P associatedto R . Using B G to study twisted vector bundles appears to be a new approach;this perspective was inspired by Moerdijk’s result in [20] identifying H ∗ ( G , S )and H ∗ ( B G , ˜ S ) for an abelian G -sheaf S , and the Serre-Grothendieck Theorem(cf. Theorem 1.6 of [11]) relating H ( M, P U ( n )) and H ( M, T ) for M a CW-complex.We begin with some preliminary definitions and results. Definition 4.1.
Let G be a topological groupoid. The simplicial space associ-ated to G is G • = {G ( k ) , ǫ kj , η jk } ≤ j ≤ k ∈ N , where G ( k ) is the space of composable n -tuples in G , ǫ kj : G ( k ) → G ( k − , and η jk : G ( k ) → G ( k +1) are given as follows: ǫ k ( g , . . . , g k ) = ( g , . . . , g k ) ǫ ki ( g , . . . , g k ) = ( g , . . . , g i g i +1 , . . . , g k ) if 1 ≤ i ≤ k − ǫ kk ( g , . . . , g k ) = ( g , . . . , g k − )If k = 1, we have ǫ ( g ) = s ( g ) , ǫ ( g ) = r ( g ).The degeneracy maps η ki are given for k ≥ η ki ( g , . . . , g k ) = ( g , . . . , g i , s ( g i ) , g i +1 , . . . , g k ) if i ≥ η k ( g , . . . , g k ) = ( r ( g ) , g , . . . , g k ) . When k = 0, the map η : G (0) → G (1) is just the standard inclusion of G (0) into G (1) = G .For the definition of a general simplicial space, see e.g. [17] Section 2.1. Definition 4.2 (cf. [20, 29]) . Let G be a topological groupoid. A classifyingspace B G for G is any space which can be realized as a quotient B G = E G / G ofa weakly contractible space E G by a free action of G . When we need an explicitmodel for B G , we will use the geometric realization |G • | of the simplicial spaceassociated to G : B G = |G • | = G k ≥ G ( k ) × ∆ k / ∼ , k denotes the standard k -simplex. The equivalence relation ∼ is de-fined by ( p, δ k − i v ) ∼ ( ǫ ki p, v ) for p ∈ G ( k ) , v ∈ ∆ k − , where δ k − i : ∆ k − → ∆ k is the i th degeneracy map, gluing ∆ k − to the i th face of ∆ k , and ǫ ki : G ( k ) →G ( k − is the i th face map. In other words, we have δ ( ∅ ) = 0 , δ ( ∅ ) = 1, and if k > δ k − i ( t , . . . , t k − ) = (0 , t , . . . , t k − ) if i = 0( t , . . . , t i , t i , t i +1 , . . . , t k ) if 1 ≤ i ≤ k − t , . . . , t k − ,
1) if i = k. The topology on this model of B G is the inductive limit topology inducedby the natural topologies on G ( n ) , ∆ n . Definition 4.3 ([17] Definition 2.2) . Let X • be a simplicial space and let G be a topological group. A principal G -bundle over X • is a simplicial space P • such that, for each k ≥ P k is a principal G -bundle over X k , and the face anddegeneracy maps in P • are morphisms of principal bundles. Remark . Combining [27] Definition 2.1 and Proposition 2.4 of [17], we seethat principal G -bundles over G • are equivalent to generalized morphisms G → G . Proposition 4.5.
Let G be an ´etale groupoid. Suppose that the classifying space B G is (homotopy equivalent to) a compact CW complex. If R → G is a twist oforder n , then R gives rise to a principal P U ( n ) -bundle P → G (0) . Moreover, P admits a left action of G which commutes with the right action of P U ( n ) .Proof. For any ´etale groupoid G , and any twist R → G , Proposition 11.3, Corol-lary 7.3, and Theorem 8.3 of [16] combine to tell us that R determines an elementof H ( G , S ), where S denotes the sheaf of circle-valued functions on G (0) . Themain Theorem of [20] tells us that we then obtain an associated element [ R ] of H ( B G , S ) ∼ = H ( B G , Z ). All of the maps Tw( G ) → H ( G , S ) ∼ = H ( B G , Z )are group homomorphisms, so if R is a torsion twist of order n , then n · [ R ] = 0also in H ( B G , Z ).Now, suppose that B G is a compact CW complex and that R is a torsiontwist of order n . The Serre-Grothendieck theorem (cf. [11] Theorem 1.6, [8]Theorem 8 or [19] Theorem 7.2.11) tells us that R gives rise to a principal P U ( n ) bundle Q over B G .Note that, for each k ∈ N , the map ϕ k : G ( k ) → B G given by ( g , . . . , g k ) [( g , . . . , g k ) , (0 , . . . , B G ensures that the maps ϕ k commute with the face and degeneracymaps ǫ ki , η ki : ∀ i, ϕ k ◦ η k − i = ϕ k − and ϕ k − ◦ ǫ ki = ϕ k . For k >
0, ∆ k can be realized as a subset of R k , namely,∆ k = { ( t , . . . , t k ) : 0 ≤ t ≤ t ≤ · · · ≤ t k ≤ } . If k = 0, ∆ k consists of one point, and we will denote ∆ = ∅ . P U ( n )-bundles over a space X are classified by homotopy classes ofmaps X → BP U ( n ), so the maps ϕ k allow us to pull back our principal P U ( n )-bundle Q over B G to a principal P U ( n )-bundle P k over G ( k ) for each k ≥ ϕ k commute with the face and degeneracy maps for G • , the maps η ki , ǫ ki induce morphisms of principal bundles which make P • into a principal P U ( n )-bundle over G • in the sense of Definition 4.3. Thus, by Proposition 2.4of [17], we have a principal P U ( n )-bundle P over G (0) which admits an actionof G .In what follows, we will combine the bundle P • constructed above with thecanonical T -central extension β : 1 → T → U ( n ) → P U ( n ) π −→ P U ( n ). The Leray spectral sequence for the map BU ( n ) → P U ( n ) impliesthat β is a generator of H ( P U ( n ) , T ) ∼ = Z n . When n is prime, an alternateproof of this fact is given in Theorem 3.6 of [28].These preliminaries completed, we now present the main result of this sec-tion. Theorem 4.6.
Let G be an ´etale groupoid. Suppose that the classifying space B G is (homotopy equivalent to) a compact CW complex. Let R → G be a twistof order n over G such that the associated P U ( n ) -bundle P of Proposition 4.5lifts to a U ( n ) -bundle ˜ P over G (0) . Then there is a twist T such that [ T ] =[ R ] ∈ H ( G , S ) and such that T admits a twisted vector bundle.Proof. Recall from [20] that for all s ∈ N , the inclusion i : G • → B G inducesan isomorphism i ∗ s : H s ( B G , T ) → H s ( G , T ), for all s ∈ N . Moreover, since i iscontinuous, it also induces a pullback homomorphism p : H ( B G , P U ( n )) → H ( G • , P U ( n )), which need not be an isomorphism since P U ( n ) is not abelian.Write v : H ( B G , P U ( n )) → H ( B G , T ) for the Serre map which associatesto a principal P U ( n )-bundle over B G its Dixmier-Douady class in H ( B G , T ) ∼ = H ( B G , Z ). The Serre-Grothendieck Theorem (cf. [8] Theorem 8, [19] Theorem7.2.11, [11] Theorem 1.6) establishes that v : H ( B G , P U ( n )) → H ( B G , Z )is an isomorphism onto the n -torsion subgroup of H ( B G , Z ) which is inducedby the short exact sequence β of Equation (2).If P is the principal P U ( n )-bundle over G which is associated to R by Propo-sition 4.5, examining the constructions employed in the proof of Proposition 4.5reveals that P = p ◦ v − ◦ ( i ∗ ) − ( R ) . Recall from page 860 of [27] that we have a natural mapΦ : H ( G • , P U ( n )) × H ( P U ( n ) , S ) → H ( G , S ) , P U ( n )-bundle over G along a T -centralextension of P U ( n ). We claim thatΦ( P • , β ) = [ R ] . (3)Since Φ is natural, and taking pullbacks preserves cohomology classes, (3) holdsbecause v is induced by β , and β generates H ( P U ( n ) , S ).We will now use the hypothesis that P admits a lift to a principal U ( n )-bundle ˜ P → G (0) to show that Φ( P • , β ) is represented by a twist T whichadmits a twisted vector bundle. As explained in [27] pp. 860-1, this hypothesisallows us to construct an explicit representative T of Φ( P • , β ) as follows.By hypothesis, the quotient map π : U ( n ) → P U ( n ) induces a bundle mor-phism ˜ π : ˜ P → P . Write P × PP U ( n ) for the gauge groupoid of the bundle P , andnotice that, if ρ : P → G (0) is the projection map of the principal bundle P , wecan define a morphism ϕ : G → P × PP U ( n ) as follows. Given g ∈ G , choose p ∈ P with ρ ( p ) = s ( g ), and define ϕ ( g ) = [ g · p, p ] . The fact that P is a principal P U ( n )-bundle implies that ϕ ( g ) is a well definedgroupoid homomorphism.We define the twist T over G by T = { ([ q , q ] , g ) ∈ ˜ P × ˜ PU ( n ) × G : [˜ π ( q ) , ˜ π ( q )] = ϕ ( g ) } . We observe that ([ q , q ] , g ) ∈ T ⇔ g · ˜ π ( q ) = ˜ π ( q ) . The backward implication is evident; for the forward implication, note that([ q , q ] , g ) ∈ T ⇒ ˜ π ( q ) ∈ P s ( g ) ⇒ ϕ ( g ) = [ g · ˜ π ( q ) , ˜ π ( q )] . But also, ([ q , q ] , g ) ∈ T ⇒ ϕ ( g ) = [˜ π ( q ) , ˜ π ( q )] . Note that[˜ π ( q ) , ˜ π ( q )] = [ p , p ] ⇔ ∃ u ∈ P U ( n ) s.t. ˜ π ( q i ) = p i · u ∀ i ;consequently, g · ˜ π ( q ) = ˜ π ( q ) as claimed.The groupoid structure on T is given by s ([ q , q ] , g ) = s ( g ) , r ([ q , q ] , g ) = r ( g );if s ( g ) = r ( h ) then we define the multiplication by([ q , q ] , g )([ p , p ] , h ) = ([ q · u, p ] , gh ) , where u ∈ U ( n ) is the unique element such that q · u = p ∈ ˜ P .Proposition 2.36 of [27] establishes that T is a twist over G such that [ T ] =Φ( P • , β ). The action of T on T is given by z · ([ q , q ] , g ) = ([ q · z, q ] , g ) . (4)12y construction, T admits a generalized homomorphism T → U ( n ) whichis T -equivariant. To be precise, the bundle ˜ P admits a left action of T : if ˜ p ∈ ˜ P lies in the fiber over s ( g ), and ([ q , q ] , g ) ∈ T , there exists a unique u ∈ U ( n )such that q · u = ˜ p . Thus, we define([ q , q ] , g ) · ˜ p = q · u. One checks immediately that this action is continuous, T -equivariant, and com-mutes with the right action of U ( n ) on ˜ P . In other words, the bundle ˜ P equippedwith this action constitutes a T -equivariant generalized morphism T → U ( n ).Thus, Proposition 5.5 of [27] explains how to construct a ( G , T )-twisted vectorbundle. Since [ T ] = Φ( P • , β ) = [ R ], this completes the proof. In this section, we present some examples establishing that the hypotheses ofTheorem 4.6 are satisfied in many cases of interest.
Example . Let M be a compact CW complex, and let α be a homeomorphismof M . If we set G = M ⋊ α Z , the first paragraph of [29] Section 1.4.3 tells usthat B G = M × Z R . Since M is compact, so is B G . Example . (cf. [25] p. 273) Let F be a foliation of a manifold M . Theholonomy groupoid H F of F is an ´etale groupoid; moreover, if the leaves ofthe foliation all have contractible holonomy coverings, B H F = M . Examples ofsuch foliations include the Reeb foliation of S and the Kronecker foliation of T n .In particular, if M is compact, any foliation F of M with contractible leaveshas an associated holonomy groupoid H F with B H F compact. Example . Let M := R P × S . We will identify R P with D / ∼ , where(in polar coordinates) D = { ( ρ, θ ) ∈ R : 0 ≤ θ < π, ≤ ρ ≤ } and(1 , θ ) ∼ (1 , θ + π ).Fix x ∈ R \ Q , and consider the homeomorphism α of R P × S given by α ([ ρ, θ ] , z ) = ([ ρ, θ + (1 − ρ ) x ] , z ) . Let G = M ⋊ α Z . Since M is compact, Example 4.7 tells us that B G is compactas well.By the K¨unneth Theorem, Z / Z ∼ = H ( R P , Z ) ⊗ H ( S , Z ) is a subgroupof H ( M, Z ) ∼ = H ( M, T ). The groupoid G is an example of a Renault-Deaconugroupoid (cf. [6, 7, 12]); thus, by Theorem 2.2 of [7], twists over G = M × α Z are classified by H ( M, T ). It follows that G admits nontrivial torsion twists.The short exact sequence 1 → T → U ( n ) → P U ( n ) → P U ( n )-bundle over M (an element of H ( M, P U ( n )))lifting to a principal U ( n )-bundle over M lies in H ( M, T ) ∼ = H ( M, Z ). How-ever, the K¨unneth Theorem tells us that H ( M, Z ) ∼ = H ( R P , Z ) ⊗ H ( S ) ∼ = H ( R P , Z ) = 0 .
13n other words, every principal
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