On a generalization of the topological Brauer group
aa r X i v : . [ m a t h . K T ] A p r ON A GENERALIZATION OF THE TOPOLOGICAL BRAUER GROUP
ANDREI V. ERSHOV
Abstract.
The present paper is motivated by an attempt to give a geometric description of “higher”twistings of topological K -theory that have finite order. For this purpose we give a cocycle type descriptionof equivalence classes of locally trivial M k ( C )-bundles modulo those that admit central fiberwise embeddinginto a trivial M kl ( C )-bundle (for some coprime k, l ). The local data that arises in this way not necessarilycomes from some locally trivial matrix algebra bundle and we explain how this leads to some generalizationof the topological Brauer group. Introduction
The starting point of the present paper is the observation that locally trivial complex matrix algebrabundles can be described not only as locally trivial bundles with a structural group (we consider the projectiveunitary group which is the retract of PGL as such a group), but also as bundles with more general topologicalgroupoid and this conveys some of their geometric properties.This paper is closely related to paper [2] (especially to its Section 2) and below we shall freely use thenotation from it.This paper is organized as follows.In Section 2 we introduce some notation and recall some constructions related to topological groupoids,in particular, the groupoid counterpart of a group 1-cocycle to glue locally trivial bundles.In Section 3 we give a description of matrix algebra bundles via their generalized trivializations and showits relation to groupoids. We also recall some notation from [2] related to matrix grassmannians.In Section 4 we introduce our basic groupoids and recall some results from [2] (in particular, on homotopytypes of classifying spaces of the groupoids).In Section 5 to any matrix algebra bundle (MAB) we assign so-called pseudobundle and show that thisassignment (more precisely, the forgetful functor) trivializes exactly embeddable MABs.In Section 6 we introduce some slightly different equivalence relation on MABs which will lead to thesame “stable” theory.In Section 7 we show that the (both) introduced equivalence relations on MABs correspond to the mapof classifying spaces BPU( k ∞ ) → B Fr k ∞ ,l ∞ with homotopy fiber Gr k ∞ ,l ∞ . In Section 8 we give the general definition of a pseudobundle which not necessarily comes from a MAB.Note that every pseudobundle locally can be lifted to a MAB. This allows us to define an equivalence relationon general pseudobundles which coincides with the one from Section 6.In Section 9 we give an example of a pseudobundle that does not come from a MAB.In Section 10 we define the generalized Brauer group as the group (with respect to the operation inducedby the tensor product) of equivalence classes of pseudobundles modulo those that come from MABs.In Section 11 we briefly discuss a relation to bundle gerbes and possible application to twisted K -theory. Acknowledgments.
A number of related questions were discussed with Professors V.M Manuilov, A.S.Mishchenko and E.V. Troitsky, and I would like to express my deepest gratitude to them.2.
Some constructions with topological groupoids
Let G be a topological groupoid, G and G its spaces of objects and morphisms respectively. In particular,the groupoid structure specifies source and target maps s, t : G → G , product m : G × G G → G , identityid : G → G and inversion ι : G → G which satisfy some well known relations.An important special case of a groupoid G is an action groupoid G ⋉ X corresponding to a (continuous)action of a topological group G on a (“good”) space X . So G = X, G = G ⋉ X and the source and targetmaps for G are as follows: s ( g, x ) = x and t ( g, x ) = gx respectively.The groupoid G can be regarded (as we shall usually do) as a (topological) category with Ob G = G , Mor G = G . In particular, it has a classifying space B G (defined up to homotopy equivalence). It iswell known that(1) B(G ⋉ X ) ≃ X × G EG . Let U := { U α } α be an open cover of X , ˇC( U ) the corresponding ˇCech groupoid. We shall denote U α ∩ U β by U αβ , etc. Then a (1-)cocycle with values in G is a (continuous) functor ϕ : ˇC( U ) → G . More precisely, ϕ = ( ϕ αβ , ϕ α , U ) , where ϕ α : U α → G , ϕ αβ : U αβ → G subject to some relations: s ◦ ϕ αβ = ϕ α , t ◦ ϕ αβ = ϕ β on U αβ and ϕ αβ ϕ βγ = ϕ αγ on U αβγ . Diagrammatically they can be expressed as the commutativitycondition of the following diagram(2) G s z z ✈✈✈✈✈✈✈✈✈✈ t $ $ ❍❍❍❍❍❍❍❍❍❍ G G U α ∩ U βi α { { ✈✈✈✈✈✈✈✈✈ i β ❍❍❍❍❍❍❍❍❍ ϕ αβ O O U αϕ α O O U βϕ β O O together with the cocycle condition ϕ αβ ϕ βγ = ϕ αγ over triple overlaps U αβγ . Continuing on this line, we say that an equivalence between cocycles is a natural transformation betweenthem. More precisely, a natural transformation χ : ϕ = ( ϕ αβ , ϕ α , U ) ⇒ ψ = ( ψ αβ , ψ α , U ) is a collection ofcontinuous maps χ α : U α → G such that for any x ∈ U αβ the diagram ϕ α ( x ) χ α ( x ) / / ϕ αβ ( x ) (cid:15) (cid:15) ψ α ( x ) ψ αβ ( x ) (cid:15) (cid:15) ϕ β ( x ) χ β ( x ) / / ψ β ( x )commutes.In case of an action groupoid G = G ⋉ X maps ϕ αβ have the form ( g αβ , x αβ ) and one can verify that g αβ ’s form a G-cocycle { g αβ } (with respect to the same open cover U ). In this case we also have theobvious forgetful functor F : G → G (here we regard the group G as a one-object category) which inducesthe natural transformation ϕ F ◦ ϕ (from G -cocycles to G-cocycles; in the above introduced notation( ϕ αβ , ϕ α , U )
7→ { g αβ } ) and hence the corresponding map of classifying spaces B G → BG which is a fibrationwith fiber X (cf. (1)). 3. Matrix algebra bundles
Let A k p → X be a locally trivial bundle over a “good” base space X with fiber a complex matrix algebra M k ( C ). We shall consider A k as a locally trivial bundle with the structure group PU( k ) ⊂ PGL k ( C ) . Forshort we shall abbreviate a matrix algebra bundle by “MAB”. For a trivializing cover U := { U α } α for A k ,it is glued from trivial bundles U α × M k ( C ) by a PU( k )-cocycle g := { g αβ } , g αβ : U αβ → PU( k ) (where U αβ := U α ∩ U β ). More precisely, A k = a α ( U α × M k ( C )) / ∼ , where the equivalence relation is generated by the identification ( x, α, g αβ ( B )) = ( x, β, B ) for x ∈ U αβ , B ∈ M k ( C ). For x ∈ U αβγ (:= U α ∩ U β ∩ U γ )( x, α, g αγ ( C )) = ( x, γ, C ) = ( x, β, g βγ ( C )) = ( x, α, g αβ g βγ ( C )) , so the transitivity of the relation is provided by the cocycle condition.Now fix a positive integer l , ( k, l ) = 1 . In general there are no fiberwise central embeddings µA k µ / / p ❆❆❆❆❆❆❆❆ X × M kl ( C ) p y y ssssssssss X N A GENERALIZATION OF THE TOPOLOGICAL BRAUER GROUP 3 (even for large l , ( k, l ) = 1, see [2]), but locally such embeddings exist. It is quite natural to regard suchlocal embeddings as local “generalized trivializations” (note that the existence µ α A k | U α µ α / / p " " ❋❋❋❋❋❋❋❋ U α × M kl ( C ) p y y rrrrrrrrrr U α does not imply that A k | U α is trivial in the conventional sense if l > U α actually is the same thing as a map ϕ α : U α → Gr k,l , where µ α ( A k,x ) ⊂ M kl ( C )( x ∈ U α ) is identified with ϕ α ( x ) ∈ Gr k,l . In this case A k | U α = ϕ ∗ α ( A k,l ) , where A k,l → Gr k,l is thetautological M k ( C )-bundle over Gr k,l ; recall [2] that there is the canonical embedding A k,l e µ / / p ❋❋❋❋❋❋❋❋❋ Gr k,l × M kl ( C ) p x x ♣♣♣♣♣♣♣♣♣♣♣ Gr k,l . Now over double overlaps U αβ we have isomorphisms ( x ∈ U αβ ) µ αβ ( x ) := µ α,x ◦ µ − β,x : M k,β,x → M k,α,x between k -subalgebras in the fixed algebra M kl ( C ) . Such isomorphisms determine (and are determined by)continuous maps ϕ αβ : U αβ → G k,l to the topological groupoid G k,l with G k,l = Gr k,l (see [2] and the next section) such that the diagram (cf.(2)) G k, ls z z ttttttttt t $ $ ❏❏❏❏❏❏❏❏❏ Gr k, l Gr k, l U α ∩ U βi α z z tttttttttt i β $ $ ❏❏❏❏❏❏❏❏❏❏ ϕ αβ O O U αϕ α O O U βϕ β O O commutes. Over triple overlaps U αβγ ϕ αβ ’s satisfy the cocycle condition ϕ αβ ϕ βγ = ϕ αγ (here we use themultiplication m : G k, l × s G tk, l G k, l → G k, l in the groupoid G k,l ). So we come to the notion of a groupoid-valued cocycle ϕ := ( ϕ αβ , ϕ α , U ) (which isnothing but a continuous functor from the ˇCech groupoid of U to G k, l ). We shall see that every PU( k )-cocycle defines an equivalent (in some exact sense, see below) G k,l -cocycle,and vice versa. In particular two given descriptions of A k (via conventional trivializations and PU( k )-cocyclesand generalized trivializations and G k, l -cocycles) are equivalent.4. Groupoids
First note that all groupoids G k, l m and G k, l n are Morita-equivalent (as topological groupoids). Indeed,the equivalence is implemented by equivalence bimodules M k,l m ; k,l n (see [2], Remark 19). This implies thatB G k, l m ≃ B G k, l n ; in particular, BPU( k ) ≃ B G k, l . Proposition.
Any
PU( k ) -cocycle g := ( g αβ , V ) is equivalent to a unique up to equivalence G k, l -cocycle ϕ g = ( ϕ αβ , ϕ α , U ) , and vice versa. it coincides with the conventional trivialization for l = 1. here and below we freely use the notation from [2]. Recall that Gr k,l denotes a “matrix grassmannian” — the spaceparametrizing all central ∗ -subalgebras in M kl ( C ) isomorphic to M k ( C ) . and an equivalence between two cocycles is exactly a natural transformation between them as functors. ANDREI V. ERSHOV
So a MAB A k p → X can be defined by a G k, l m -cocycle with arbitrary nonnegative m (in particular,for m = 0 by a PU( k )-cocycle), and equivalences between such cocycles (with possibly different m ) areimplemented by equivalence bimodules M k,l m ; k,l n . This picture can naturally be described in the languageof functors from ˇCech groupoids to G k,l m and natural transformations between them given by M k,l m ; k,l n .Finally, let us make some further remarks.Applying the equivalence bimodule M k, k,l = Fr k,l (where Fr k,l denotes the homogeneous space PU( kl ) / ( E k ⊗ PU( l )), see [2]) to the universal principal PU( k )-bundle, we obtain the universal principal G k, l -bundleEPU( k ) × PU( k ) Fr k,l = H k,l ( A univk ) p → BPU( k ) . Its total space H k,l ( A univk ) is homotopy equivalent to Gr k,l (as it should be), and this equivalence identifies A k,l with p ∗ ( A unk ) (where A unk → BPU( k ) is the universal M k ( C )-bundle). Note also that G k, l = H k,l ( A k,l )(see [2]).There are also action groupoids b G k, l := PU( kl ) ⋉ Gr k,l (so B b G k, l ≃ BPU( k ) × BPU( l )). We have thefollowing commutative diagram of classifying spaces:(3) BPU( l ) = / / (cid:15) (cid:15) BPU( l ) (cid:15) (cid:15) Gr k,l = (cid:15) (cid:15) / / B b G k, l / / (cid:15) (cid:15) BPU( kl ) (cid:15) (cid:15) Gr k,l / / B G k, l / / ? . We shall see below that after taking the direct limit of this diagram the label “?” can be replaced by thespace B Fr k ∞ ,l ∞ .4.2. Remark.
It worth to note that if G k, l would be an action groupoid for some topological group H actingon G k, l = Gr k,l , then H ≃ Fr k, l . This result follows from the homotopy equivalence B G k, l ≃ BPU( k ) andthe fact that for an action groupoid G := X ⋊ H corresponding to an action of H on X the classifying spaceB G is homotopy equivalent to X × H EH. Indeed, there is the fibrationFr k,l → Gr k,l → BPU( k ) . Some preliminary considerations
When we replace a PU( k )-cocycle ( g αβ , U ) by an equivalent G k, l -cocycle ϕ A = ( ϕ αβ , ϕ α , U ), we separatethe information about the locally trivial bundle into two parts: one piece (over U α ’s) is encoded by generalizedtrivializations ϕ α ’s, the other (as we shall see the more essential one) over U αβ ’s by ϕ αβ ’s.We would like to describe the right bottom arrow in (3). The middle right arrow can be described asfollows: to a given pair of MABs A k and C l over X (with fibers M k ( C ) and M l ( C ) respectively) it assignstheir tensor product A k ⊗ C l regarded as an M kl ( C )-bundle (so we forget A k and C l themselves). Thissuggests the following construction (we shall freely use the notation from Section 3).So assume that a MAB A k p → X is defined by a G k, l -cocycle ( ϕ αβ , ϕ α , U ). We assign the following datato it: over open sets U α ’s there is a collection of trivial M kl ( C )-bundles U α × M kl ( C ) which are “glued” by ϕ αβ ’s over U αβ ’s.So in fact we forget generalized trivializations ϕ α ’s over U α ’s (equivalently, embeddings µ α ’s over U α ’s).Note that maps ϕ αβ ’s specify embeddings µ ααβ : A k,α | U αβ → U αβ × M kl ( C ) and µ βαβ : A k,β | U αβ → U αβ × M kl ( C ) and isomorphism between images of these embeddings as matrix subbundles in U αβ × M kl ( C ) . Moreover, ϕ αβ ’s must satisfy the cocycle condition on U αβγ ’s.Such a data we shall call a pseudobundle defined by a MAB A k .To lift such a pseudobundle to a MAB one must continuously extend subbundles µ ααβ ( A k,α | U αβ ) ⊂ U αβ × M kl ( C ) from U αβ to µ α : A k,α → U α × M kl ( C ) . Proposition.
Let ( ϕ αβ , ϕ α , U ) be a G k,l -cocycle for a MAB A k → X. If A k is embeddable (i.e. thereexists a central embedding A k → X × M kl ( C ) ) then any MAB corresponding to a G k,l -cocycle of the form ( ϕ αβ , ψ α , U ) (for any choice of ψ α ’s such that ψ α | U αβ = s ◦ ϕ αβ , ψ β | U αβ = t ◦ ϕ αβ ) is also embeddable. Inother words, the embeddability depends only on ϕ αβ ’s. so generalized trivializations are fixed only on double overlaps. Note that more generally we can consider embeddings into U αβ × M kl m ( C ) with different m for µ ααβ and µ βαβ . N A GENERALIZATION OF THE TOPOLOGICAL BRAUER GROUP 5
Proof . A MAB A k is embeddable iff its G k,l -cocycle ( ϕ αβ , ϕ α , U ) can be lifted to a trivial PU( kl )-cocycle(which defines a trivial M kl ( C )-bundle) in the sense of the diagram (cf. (3)) b G k,l (cid:15) (cid:15) / / PU( kl ) G k,l . (In terms of bundles we must find a M l ( C )-MAB C l such that A k ⊗ C l ∼ = X × M kl ( C ) . ) But it is clear thatthis condition depends only on ϕ αβ ’s, more precisely on the existence of the lift b ϕ αβ : U αβ → b G k,l such that g αβ = F ( b ϕ αβ ) (where F is the forgetful functor from the end of Section 2) form a trivial PU( kl )-cocycle.It is natural to consider the following equivalence relation on pseudobundles: two pseudobundles areequivalent iff they can be defined by equivalent G k,l -cocycles. More explicitly, the equivalence relation isgenerated by 1) equivalence between G k,l m -cocycles (with different m and probably different open covers U ),and 2) forgetting (recovering) generalized trivializations { ϕ α } over U α ’s. In particular, it follows from theprevious proposition that pseudobundles coming from embeddable MABs form one (“trivial”) equivalenceclass.5.2. Example.
Let A k → X be an embeddable MAB (i.e. such that there exists a central emdedding µ : A k → X × M kl ( C )). Then the corresponding pseudobundle is trivial (i.e. equivalent to a pseudobundlecoming from a trivial MAB). Indeed, A k can be defined by a cocycle ( ϕ, U ) , where U consists of only oneopen set U = X and ϕ : X → Gr k,l is a classifying map for A k , in the sense that A k = ϕ ∗ ( A k,l ) , where A k,l → Gr k,l is the tautological M k ( C )-bundle over Gr k,l .5.3. Remark.
There is no reason to expect that a “general pseudobundle”, i.e. actually a collection ϕ αβ : U αβ → G k,l that satisfies “cocycle condition” ϕ αβ ϕ βγ = ϕ αγ on U αβγ ’s, comes from a locally trivial bundle in theway described above. But at the moment we restrict our attention to the special case when it does.5.4. Proposition. G k,l has the following homotopy groups in stable dimensions: π n ( G k,l ) ∼ = Z , n ≥ , π n − ( G k,l ) ∼ = Z /k Z , n ≥ . Proof.
There are two obvious fibrations:PU( k ) → G k,l ( s,t ) −→ Gr k,l × Gr k,l and Fr k,l → G k,l s → Gr k,l . For the calculation of homotopy groups the latter is more convenient because s has a section id : Gr k,l → G k,l (the identity map of the groupoid G k,l ). Thus we have0 → π n ( G k,l ) → Z → Z /k Z → π n − ( G k,l ) → π n ( G k,l ) → Z is an isomorphism. Note that the natural inclusion Fr k,l = M k, k,l ⊂ G k, l induces an isomorphism between π n − (Fr k,l ) and π n − ( G k, l ) . Example.
Take X = S n and assume that k and l are large enough comparing to n . Consider the opencover of S n by two open hemispheres U and V , U ∩ V ≃ S n − . The previous proposition shows that thereare exactly Z /k Z homotopy classes of maps ϕ U,V : U ∩ V → G k, l , and standard arguments show that thereis a one-to-one correspondence between them and equivalence classes of pseudobundles over S n .Obstruction theory says that there are Z ∼ = π n (Gr k,l ) ways (up to homotopy) to extend given ϕ U,V to agenuine G k, l -cocycle (i.e. to extend µ UU ∩ V and µ VU ∩ V to µ U : A k | U → U × M kl ( C ) and µ V : A k | V → V × M kl ( C )respectively). (Another way to show this is to observe that the map PU( k ) → Fr k,l induces surjection Z ։ Z /k Z on odd-dimensional homotopy groups.) Equivalently, there are Z ways for a given ϕ U,V to choosegeneralized trivializations over U and V ; equivalently, every pseudobundle over S n can be lifted to a locallytrivial MAB by exactly in countably many ways (in particular, every pseudobundle over X = S n can be liftedto a MAB). For example, if ϕ U,V is homotopic to the map to a point, then lifts of the corresponding trivialpseudobundle are exactly MABs A k → S n that admit fiberwise embeddings µ : A k → S n × M kl ( C ). Recallthat such A k together with specified embedding µ (up to homotopy) is classified by a map S n → Gr k,l . Such bundles can be represented by G k, l -cocycles with trivial ϕ αβ ’s.Actually, the situation we have faced in the previous example is true in general: a MAB can be restoredby the corresponding pseudobundle up to the tensor probuct by an embeddable MAB (i.e. such that thereexists a global µ ). In order to prove this note that there is the tensor product of pseudobundles (defined byusing obvious maps G k r , l m × G k s , l n → G k r + s , l m + n ) which is compatible with the equivalence relation and the ANDREI V. ERSHOV forgetful functor from MABs to pseudobundles. Moreover, equivalence classes of pseudobundles form a groupwith respect to the operation induced by ⊗ (in Example 5.5 over S n it is Z /k Z and the homomorphism fromequivalence classes of MABs to equivalence classes of pseudobundles is the group epimorphism Z ։ Z /k Z ;more generally, for a finite CW-complex X it is a finite abelian group), and the forgetful functor induces agroup homomorphism. (This also allows us to define the stable equivalence relation on pseudobundles as theone generated by the above introduced relation and the tensor product by trivial pseudobundle.) Now tocomplete our reasoning it remains to notice that the kernel of the group homomorphism consists of classes of“stably embeddable” bundles: they are exactly those MABs that can be defined by a G k m , l n -cocycles withtrivial ϕ αβ ’s. Indeed, it follows from our definition of the equivalence relation and Proposition 5.1.The corresponding equivalence relation on MABs is as follows:MABs A k m ∼ B k n iff there are embeddable C k r , D k s such that A k m ⊗ C k r ∼ = B k n ⊗ D k s ( ⇒ m + r = n + s ) . We shall see below that this gives the description of the map π : BPU( k ∞ ) → B Fr k ∞ ,l ∞ with fiber Gr k ∞ .l ∞ from the viewpoint of represented functors.6. Yet another equivalence relation on MABs
Consider a minimal equivalence relation ∼ on MABs generated by the following “elementary equivalences”.Recall that a MAB A k → X can be glued by a G k, l -cocycle ϕ A = ( ϕ αβ , ϕ α , U ) which is defined by A k upto equivalence. Then consider arbitrary lift of ϕ A to a b G k,l m -cocycle b ϕ A = ( b ϕ αβ , ϕ α , U ) (for some m ∈ N ).Since b G k,l m is an action groupoid, b ϕ A defines some PU( kl m )-cocycle F ( b ϕ A ) (see the end of Section 2). Wecan also replace F ( b ϕ A ) by an equivalent PU( kl m )-cocycle and pass the a refinement of the cover.Note that thereby we also have defined some equivalence relation on pseudobundles.It is not difficult to see that A k ∼ A ′ k iff there are M l m ( C )-bundles C l m , C ′ l m such that A k ⊗ C l m ∼ = A ′ k ⊗ C ′ l m (isomorphism of M kl m ( C )-bundles) iff there is an M kl m ( C )-bundle B kl m → X such that A k and A ′ k are unitalalgebra subbundles in it.Let us describe such equivalence classes over a sphere S n , n > . For two classifying maps f : S n → BPU( k ) , g : S n → BPU( l ) let [ h ] be the homotopy class of the composition S n f × g −→ BPU( k ) × BPU( l ) ⊗ → BPU( kl ) , then [ h ] = l [ f ] + k [ g ] . This shows that two M k ( C )-bundles A k and A ′ k over S n are equivalent iff thecorresponding classifying maps satisfy [ f ] − [ f ′ ] ≡ k Z (here we identify π n (BPU( k )) with Z ). In particular, A k is equivalent to a trivial M k ( C )-bundle iff it admits a fiberwise central embedding µ : A k → S n × M kl ( C ) . (Indeed, the classifying map Gr k,l → BPU( k ) for A k,l induces the homomorphism Z → Z , k onhomotopy groups π n , n ≥ l by l m for m ∈ N .6.1. Theorem. A k is equivalent to a trivial M k ( C ) -bundle iff there is a fiberwise central embedding µ : A k → X × M kl n ( C ) for some (large enough) n .Proof. Let A ′ k → X be a trivial M k ( C )-bundle, and assume that A k ⊗ C l m ∼ = A ′ k ⊗ C ′ l m for some M l ( C )-bundles. Choose C ′′ l n − m such that C ′ l m ⊗ C ′′ l n − m ∼ = X × M l n ( C ) (such a bundle exists since X is a finite CW -complex), then A k ⊗ C l m ⊗ C ′′ l n − m ∼ = X × M kl n ( C ) . The converse direction is clear.There is also another operation on the groupoids G k m ,l n × G k r ,l s → G k m + r ,l n + s induced by the tensor product of matrix algebras. One can define the corresponding stabilization on bundles.The corresponding stable equivalence looks as follows: A k m ∼ A ′ k n ⇔ there exist embeddable bundles B k r , B ′ k s such that A k m ⊗ B k r ∼ = A ′ k n ⊗ B ′ k s . Remark.
Note that the equivalence from the previous section (cf. Proposition 5.1) implies the newone. Indeed, the set of equivalence classes of PU( kl m )-cocycles F ( ϕ ) corresponding to lifts ( b ϕ αβ , ϕ α , U ) of( ϕ αβ , ϕ α , U ) depends only on { ϕ αβ } . N A GENERALIZATION OF THE TOPOLOGICAL BRAUER GROUP 7 Relation to classifying spaces
Let us explain the obtained results from the viewpoint of classifying spaces. The key observation is thata MAB A k → X is embeddable iff its classifying map f : X → BPU( k ) has a lift f µ in the fibrationH k,l ( A univk ) ≃ Gr k,l (cid:15) (cid:15) X f µ ♣♣♣♣♣♣♣♣♣♣♣♣♣ f / / BPU( k )(and homotopy classes of such lifts correspond to homotopy classes of embeddings), see [2]. The homotopyfiber of the vertical map is equivalent to Fr k,l . Note that Gr k ∞ ,l ∞ := lim −→ n Gr k n ,l n ≃ BSU ⊗ and BU ⊗ ∼ = K( Z , × BSU ⊗ are infinite loop spaces (in particular, they represent some generalized (co)homologytheories). By BPU( k ∞ ) denote lim −→ n BPU( k n ) (where the direct limit is taken over maps induced by the ten-sor product). Note also that the map Gr k ∞ ,l ∞ → BPU( k ∞ ) induced by forgetting embeddings µ ’s forembeddable bundles is actually the localization in k . We have the fibration(4) Gr k ∞ ,l ∞ → BPU( k ∞ ) π → B Fr k ∞ ,l ∞ which is closely related to the coefficient sequence0 → Z → Z (cid:20) k (cid:21) → Z (cid:20) k (cid:21) / Z → k ∞ ,l ∞ can be deduced from thestandard argument from the theory of generalized cohomologies, the notation emphasizes that Ω B Fr k ∞ ,l ∞ ≃ Fr k ∞ ,l ∞ ). Indeed, since BU ⊗ ∼ = K( Z , × BSU ⊗ [3, 4], it follows that the previous fibration is the product ofBSU ⊗ → BSU ⊗ (cid:20) k (cid:21) → B e Fr k ∞ ,l ∞ , where e Fr k ∞ ,l ∞ is the direct limit of e Fr k m ,l n := SU( k m l n ) / (SU( k m ) ⊗ SU( l n )) (the universal covering ofFr k m ,l n ) and ∗ → K( Z (cid:20) k (cid:21) / Z , → K( Z (cid:20) k (cid:21) / Z , . It follows from Theorem 6.1 that the equivalence classes of MABs form Im π ∗ , where π ∗ : [ X, BPU( k ∞ )] → [ X, B Fr k ∞ ,l ∞ ] , cf (4).Note also that from the point of view of represented functors, fibration (4) corresponds to two forgetfulfunctors: the first forgets embeddings µ in ( A k , µ, X × M kl ( C )) and the second assigns to a MAB thecorresponding (stabilized) equivalence class (or the equivalence class of the corresponding pseudobundle).7.1. Remark.
Let us note that the space Gr k ∞ ,l ∞ ∼ = BSU ⊗ represents the functor that can be considered asa generalized Picard group in the sense that it is the group of equivalence classes of virtual SU-bundles ofvirtual dimension 1 with respect to the operation induced by the tensor product of such bundles. Thereforethe fibration (4) can be considered as an analog of C P ∞ → BU( k ∞ ) → BPU( k ∞ )which leads to the usual tolological Brauer group H k − tors ( X, Z ) . Remark.
Note that there is also the fibrationBPU( l ∞ ) → BPU(( kl ) ∞ ) → B Fr k ∞ ,l ∞ which is closely related to the equivalence relation introduced in the previous section.Our next goal is to describe kind of bundles over X classified by maps X → B Fr k ∞ ,l ∞ (whose homotopyclasses) not belonging to Im π ∗ . ANDREI V. ERSHOV General pseudobundles 1
Here we are going to introduce some new type of bundles whose classifying space is B Fr k ∞ ,l ∞ .Below we shall use the following result.8.1. Theorem.
Let B kl → X be an M kl ( C ) -bundle over a finite CW -complex X and k, l are sufficiently largecomparing to dim X . Then there are M k ( C ) and M l ( C ) -bundles A k and C l over X such that B kl ∼ = A k ⊗ C l . Proof.
Since π k (Gr k,l ) ∼ = Z for k ≥ k,l → BPU( k ) × BPU( l ) ⊗ −→ BPU( kl )vanish.Our general pseudobundles will be glued from elementary blocks by some functions that form a “cocycle”.More precisely, let U := { U α } α ∈ A be a good cover of a topological space X . Over open sets we have trivialbundles U α × M kl ( C ) and two such blocks U α × M kl ( C ) and U β × M kl ( C ) are glued to each other over U αβ by a function ϕ αβ : U αβ → G k,l . We require the cocycle condition ϕ αβ ϕ βγ = ϕ αγ over triple overlaps U αβγ to be satisfied.Note that the collection { ϕ αβ } is not a genuine groupoid G k,l -cocycle as long as a compatible collection { ϕ α } , ϕ α : U α → Gr k,l is not specified, and as we shall see in the next section, it may not exist. Thereforesuch a collection { ϕ αβ } we call a ( G k,l -) pseudococycle .In contrast with G k,l -pseudococycles a (defined in the same way) b G k,l -pseudococycle { b ϕ αβ } can always beextended to a genuine groupoid b G k,l -cocycle. Indeed, since b G k,l is an action groupoid, it follows that such apseudococycle gives rise to a PU( kl )-cocycle { g αβ } which determines an M kl ( C )-bundle B kl and the initialpseudococycle corresponds to its decomposition B kl ∼ = A k ⊗ C l into the tensor product (cf. Theorem 8.1).More precisely, suppose there is a collection { ϕ α } , ϕ α : U α → Gr k,l for a G k,l -pseudococycle { ϕ αβ } suchthat s ◦ ϕ αβ = ϕ α | U αβ , t ◦ ϕ αβ = ϕ β | U αβ . Then this data is exactly a groupoid G k,l -cocycle and as we havealready seen gives rise to a MAB A k → X together with specified embeddings µ α : A k | U α → U α × M kl ( C )(given by ϕ α ’s) and isomorphisms µ α ( A k | U α ) ∼ = µ β ( A k | U β ) over U αβ (given by ϕ αβ ’s). It is clear that a(genuine) G k,l -cocycle can always be lifted to a b G k,l -cocycle and the choice of such a lift is equivalent to thechoice of some M l ( C )-bundle C l → X (which can be arbitrary). In particular, C l is glued from centralizersof µ α ( A k | U α ) ⊂ U α × M kl ( C ) by some lifts b ϕ αβ : U αβ → b G k,l of ϕ αβ : U αβ → G k,l . Finally, we obtain some M kl ( C )-bundle B kl → X, B kl = A k ⊗ C l , or equivalently we can associate to a groupoid b G k,l -cocycle thecorresponding PU( kl )-cocycle g ( b ϕ ) = { g αβ } (recall that b G k,l is an action groupoid).Conversely, according to Theorem 8.1, if k, l are large enough comparing to the dimension of a finite CWcomplex X , every M kl ( C )-bundle over X is the tensor product of some M k ( C )- and M l ( C )-bundles A k and C l and forgetting about C l we come to some data of the type we have started with.To the PU( kl )-cocycle g ( b ϕ ) = { g αβ } we can apply two operations: first, we can replace it by an equiv-alent PU( kl )-cocycle { g ′ αβ } , g ′ αβ = h α g αβ h − β for { h α } , h α : U α → PU( kl ), and second reduce it to somepseudococycle { ϕ ′ αβ } . For the last we choose some M k ( C )-subbundles in U αβ × M kl ( C ) that are identifiedby g αβ and the choice must be compatible with the cocycle condition.Now we are about to define the equivalence relation on general pseudobundles. General pseudobundles atleast locally can be lifted to MABs and the equivalence on the second should induce equivalence on the first.So assume that a pseudococycle { ϕ αβ } globally can not be extended to a genuine G k,l -cocycle ( ϕ αβ , ϕ α , U )(equivalently, can not be lifted to a PU( kl )-cocycle { g αβ } ). At least, as the cover U is assumed to be good,we can lift maps ϕ αβ : U αβ → G k,l to maps b ϕ αβ : U αβ → b G k,l and hence to maps g αβ : U αβ → PU( kl )but this time they can not be chosen in such a way to satisfy the cocycle condition g αβ g βγ = g αγ . Butsince the pseudococycle { ϕ αβ } satisfies the cocycle condition, it follows that there exist continuous functions t αβγ : U αβγ → PU( l ) such that g αβ g βγ = g αγ t αβγ . In other words, at every point left and right sides differby an automorphism of some M l ( C )-subalgebra in M kl ( C ) . Note that if we replace { g αβ } by { g ′ αβ } , where g ′ αβ = h α g αβ h − β , then t ′ αβγ = h γ t αβγ h − γ . This operations (together with the passage to a refinement of thecover) generate the required equivalence relation.9.
General pseudobundles 2
Above we have described the image of π in (4) as (“stable”) equivalence classes of pseudobundles thatcan be lifted to MABs. It is natural to conjecture that the H -space B Fr k ∞ ,l ∞ represents the group ofstable equivalence classes of pseudobundles that are not nesessarily liftable to MABs. Indeed, it is clearthat the corresponding functor satisfies the assumptions of Brown’s representability theorem. In fact, it is a“minimal” extension of the image of the forgetful functor assigning to a MAB the corresponding pseudobundle N A GENERALIZATION OF THE TOPOLOGICAL BRAUER GROUP 9 that satisfies the Mayer-Vietoris axiom. Suppose it is represented by a space Y . Then there is a map Y → B Fr k ∞ ,l ∞ which induces isomorphisms on all homotopy groups, so it is a homotopy equivalence.9.1. Remark.
The restriction of the equivalence relation on general pseudobundles to the subset of “liftable”(to MABs) pseudobundles should coincide with the above defined one. The equivalence on liftable pseu-dobundles generates the equivalence on general pseudobundles because every pseudobundle is locally liftable.Now let us give the promised example of a pseudobundle that can not be lifted to a MAB. The ideaof its construction is based on the obvious observation that the identity map id : Fr k,l → Fr k,l can not befactorized through PU( k ) ⊂ Fr k,l . Example.
We are going to construct a pseudobundle over X := Σ Fr k,l which can not be lifted to aMAB. Consider the cover of X by two contractible open subsets U, V ≃ C Fr k,l (the cone over Fr k,l ) and U ∩ V ≃ Fr k,l (so the cover this time is not good). It is clear that equivalence classes of pseudobundlesover X are classified by homotopy classes of maps ϕ U,V : Fr k,l → G k, l . Consider the pseudobundle whichcorresponds to the map ϕ U,V : Fr k,l id → Fr k,l ⊂ G k, l . It consists of two trivial MABs U × M kl ( C ) and V × M kl ( C ) over U and V respectively which are glued by ϕ U,V . But since t | Fr k,l : Fr k,l → Gr k,l (where t isthe target map for G k, l ) is not equivalent to the map to a point (it is the principal PU( k )-bundle which thetautological bundle A k,l → Gr k,l is associated to), it follows that t | Fr k,l ◦ id can not be extended to a mapC Fr k,l → Gr k,l : Fr k, l ⊂ G k, ls w w ♦♦♦♦♦♦♦♦♦♦♦♦ t ' ' PPPPPPPPPPPP pt ⊂ Gr k,l Gr k, l Fr k,li U w w ♦♦♦♦♦♦♦♦♦♦♦♦ i V ' ' PPPPPPPPPPPPP ϕ U,V =id O O U ≃ C Fr k,l O O V ≃ C Fr k,l . ∄ O O (Another way to prove this is to note that the identity map id : Fr k,l → Fr k,l can not be factorized throughPU( k ) ⊂ Fr k,l . ) This implies that ϕ U,V can not be extended to a genuine G k, l -cocycle and therefore thecorresponding pseudobundle does not come from a MAB over Σ Fr k,l . Note that this can not be fixed bystabilization. 10.
Generalized topological Brauer group
The conventional topological Brauer group Br( X ) of X is H tors ( X ; Z ) . Sincelim −→ k BU( k ) ≃ Y q ≥ K( Q , q ) , lim −→ k BPU( k ) ≃ K( Q / Z , × Y q ≥ K( Q , q ) , it follows thatBr( X ) = coker { [ X, K( Q , → [ X, K( Q / Z , } = coker { [ X, lim −→ k BU( k )] → [ X, lim −→ k BPU( k )] } . Therefore Br( X ) is the group of stable equivalence classes of MABs over X modulo MABs of the form End( ξ )for a vector bundle ξ → X .Thus we can define the generalized topological Brauer group GenBr( X ) of X asGenBr( X ) := coker { [ X, lim −→ k BU( k )] → [ X, lim −→ k,l, ( k,l )=1 B Fr k,l ] } . So GenBr( X ) consists of stable equivalence classes of pseudobundles modulo those that can be lifted toMABs of the form End( ξ ) . GenBr( X ) contains Br( X ) as a direct summand and is also a finite abelian groupfor a finite CW-complex X .11. Remarks on gerbes and twisted K -theory Let us remark that using functions t αβγ introduced in Section 8 one can define a gerbe whose equivalenceclass is the obstruction for the existence of a lift of a pseudobundle to a genuine PU( kl )-bundle.Another way to define similar gerbe is as follows. Let B k,l → Gr k,l be the centralizer M l ( C )-subbundlein Gr k,l × M kl ( C ) for the tautological M k ( C )-bundle A k,l → Gr k,l , i.e. A k,l ⊗ B k,l = Gr k,l × M kl ( C ) [2].Then we have a bundle Hom Alg ( s ∗ ( B k,l ) , t ∗ ( B k,l )) → G k,l with fiber PU( l ) as a bitorsor. By ξ αβ → U αβ denote its pullback via ϕ αβ : U αβ → G k,l . Using some kind of bimodule structure we can define a product ξ αβ ∗ ξ βγ → ξ αγ .One possible application of pseudobundles is that they should give rise to twists in K -theory. Let us recallthat one possible approach to twisted K -theory in case of twists of finite order is via bundle gerbe modules[1]. Our pseudobundles also give rise to finite order twists (it is easy to see that for all such twists) and wecan expect that the corresponding twisted K -theory also admits analogous description. References [1]
Bouwknegt, Peter, Carey, Alan L., Mathai, Varghese, Murray, Michael K., Stevenson, Danny , Twisted K -theory and K -theory of bundle gerbes, Comm. Math. Phys., volume 228, (2002), number 1, pages 17–45, issn 0010-3616[2] A.V. Ershov , Topological obstructions to embedding of a matrix algebra bundle into a trivial one, arXiv:0807.3544[math.KT][3]
Madsen, I., Snaith, V., Tornehave, J. , Infinite loop maps in geometric topology, Math. Proc. Cambridge Philos. Soc.,volume 81, (1977), number 3, pages 399–430, issn 0305-0041[4]
Segal, Graeme , Categories and cohomology theories, Topology, volume 13, (1974), pages 293–312, issn 0040-9383
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