The Dixmier-Douady Invariant for Dummies Claude Schochet Wayne State University, [email protected]

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1-1-2009 The Dixmier-Douady Invariant for Dummies Claude Schochet Wayne State University, [email protected]

Recommended Citation C. Shochet, The Dixmier-Douady invariant for dummies, Notices of the American Mathematical Society 56.7 (2009), 809-816. Available at: https://digitalcommons.wayne.edu/mathfrp/16

This Article is brought to you for free and open access by the Mathematics at DigitalCommons@WayneState. It has been accepted for inclusion in Mathematics Faculty Research Publications by an authorized administrator of DigitalCommons@WayneState. The Dixmier-Douady Invariant for Dummies Claude Schochet

he Dixmier-Douady invariant is the pri- a map f to the classifying space BG and there is a mary tool in the classification of contin- pullback diagram ∗ uous trace C -algebras. These algebras G ------→ G have come to the fore in recent years     Tbecause of their relationship to twisted y y K-theory and via twisted K-theory to branes, gerbes, T ------→ EG and string theory.   This note sets forth the basic properties of   y y the Dixmier-Douady invariant using only classical f homotopy and bundle theory. Algebraic topology X ------→ BG enters the scene at once since the algebras in where the right column is the universal principal question are algebras of sections of certain fibre G-bundle. bundles. Suppose further that F is some G-space. Then The results stated are all contained in the orig- following Steenrod [24] we may form the associated inal papers of Dixmier and Douady [5], Donovan fibre bundle and Karoubi [7], and Rosenberg [23]. Our treatment is novel in that it avoids the sheaf-theoretic tech- F -→ T ×G F -→ X niques of the original proofs and substitutes more with fibre F and structural group G. Pullbacks classical algebraic topology. Some of the proofs are commute with taking associated bundles, so there borrowed directly from the recent paper of Atiyah is a pullback diagram and Segal [1]. Those interested in more detail and especially in the connections with analysis should F ------→ F   consult Rosenberg [23], the definitive work of Rae-   burn and Williams [21], as well as the recent paper y y of Karoubi [13] and the book by Cuntz, Meyer, and T ×G F ------→ EG ×G F Rosenberg [4]. We briefly discuss twisted K-theory     itself, mostly in order to direct the interested reader y y to some of the (exponentially-growing) literature f X → BG. on the subject. ------It is a pleasure to acknowledge the assistance of Now suppose that M is some fixed C∗-algebra, Alan Carey, Dan Isaksen, Max Karoubi, N. C. Phillips, soon to be either the matrix ring Mn = Mn(C) and Jonathan Rosenberg in the preparation of this for some n < ∞ or the compact operators K on paper. some separable Hilbert space H . Take G = U(M), the group of unitaries of the C∗-algebra. (If M Fibre Bundles is not unital then we modify by first adjoining Suppose that G is a topological group and G → a unit canonically to form the unital algebra M+ T → X is a principal G-bundle over the compact and then define U(M) to be the kernel of the nat- space X. Then up to equivalence it is classified by ural homomorphism U(M+) → U(M+/M) S1.) Then U(M) acts naturally on M by conjugation; ad Claude Schochet is professor of mathematics at Wayne denote M with this action as M . The center State University, Detroit, MI. His email address is claude@ ZU(M) of U(M) acts trivially, and so the action math.wayne.edu. descends to an action of the projective unitary

August 2009 Notices of the AMS 809 group PU(M) = U(M)/ZU(M) on M, denoted Mad . classifying spaces. The one that concerns us is Note that if M is simple then its center is just C and the tensor product operation. Fix some unitary ZU(M) S1 so that PU(M) is just the quotient isomorphism of vector spaces group (U(M))/S1. r s rs Having fixed M, let C ⊗ C C . ζ : PU(M) -→ T → X (This isomorphism is unique up to homotopy, since be a principal PU(M)-bundle over a compact space the various unitary groups are connected.) Let X. Form the associated fibre bundle Un = U(Mn(C)). This determines a homomorphism ad ad p ⊗ Pζ : M -→ T ×PU(M) M -→ X. Ur × Us -→ Urs This fibre bundle always has non-trivial sections. and the associated map on classifying spaces Define Aζ to be the space of sections: ⊗ ad BUr × BUs -→ BUrs Aζ = Γ (Pζ) = {s : X -→ T ×PU(M) M | ps = 1}. This is a C∗-algebra with pointwise operations that given by the composite are well defined because we are using the adjoint B(⊗) action. It is unital if M is unital. If M = Mn or BUr × BUs B(Ur × Us ) -→ BUrs . M = K then this is a continuous trace C∗-algebra. Let [X, Y ] denote homotopy classes of maps and If X is locally compact but not compact then Aζ is still defined by using sections that vanish at recall that if X is compact and connected then infinity and it is not unital. isomorphism classes of complex n-plane vector Note that if Pζ is a trivial fibre bundle then bundles over X correspond to elements of [X, BUn]. sections correspond to functions X → M and hence Then this construction induces an operation ⊗ Aζ C(X) ⊗ M, [X, BUr ] × [X, BUs ] -→ [X, BUrs ], ∗ where C(X) denotes the C -algebra of continuous which does indeed correspond to the tensor prod- complex-valued functions on X. (If X is only locally uct operation on bundles. Precisely, if E → X and compact then we use C to denote continuous 1 o E → X are represented by f and f respectively, functions vanishing at infinity.) 2 1 2 then the tensor product bundle E ⊗ E → X is Continuous trace C∗-algebras may be defined 1 2 represented by f ⊗ f . (This holds at once for intrinsically, of course. Here is one approach. If A 1 2 compact connected spaces. If X is not connected is a (complex) C∗-algebra, then let Aˆ denote the then one checks this on each component.) set of unitary equivalence classes of irreducible ∗-representations of A with the Fell topology (cf. The inclusion [21]). Ur Ur × {1} → Ur × Us → Urs Definition. Let X be a second countable locally is denoted compact Hausdorff space. A continuous trace C∗- algebra with Aˆ = X is a C∗-algebra A with Aˆ = X αrs : Ur → Urs . such that the set 1 The center of Uk is the group S regarded as ma- {x ∈ A | the map π → tr(π(a)π(a)∗) trices of the form zI, where z is a complex number is finite and continuous on Aˆ} of norm 1. The quotient group PUk is the projec- 1 tive unitary group. The fibration S → Uk → PUk is dense in A. induces the sequence

From the deﬁnition it is easy to see that com- k ∗ 0 → Z -→ Z → Z/k → 0 mutative C -algebras Co(X) as well as stable ∗ K commutative C -algebras Co(X, Mn) and Co(X, ) on fundamental groups, and, in particular, are continuous trace. In fact every continuous trace π (PU ) Z/k. algebra arises as a bundle of sections of the type 1 k There is a natural induced map and commuting we have been discussing. diagram Products S1 × S1 ------→ S1 Vector spaces come equipped with natural direct     sum and tensor product operations, and these pass y y over to vector bundles. Thus if E1 → X and E2 → X are complex vector bundles of dimension r and s Ur × Us ------→ Urs   respectively then we may form bundles E1 ⊕E2 → X   y y of dimension r + s and E1 ⊗ E2 → X of dimension rs. There are two corresponding operations on PUr × PUs ------→ PUrs

810 Notices of the AMS Volume 56, Number 7 and this induces a tensor product operation and a Proposition ([8]). On the category of compact commuting diagram spaces, Cechˇ cohomology is representable. That is,

BUr × BUs ------→ BUrs there is a natural isomorphism   n   Hˇ (X; Z) [X, K(Z, n)]. y y Proof. The natural isomorphism is well known BPU × BPU ------→ BPU . r s rs for X a ﬁnite complex, by the Eilenberg-Steenrod It is easy to see that uniqueness theorem. Suppose that X is a compact π (BPU ) π (PU ) Z/k space. Then write X = lim X for some inverse 2 k 1 k ←------j system of ﬁnite complexes. (See [8], Chapters IX, X, and that the natural map αrs : PUr → PUrs induces a commuting diagram and XI for open covers, nerves, and inverse limits.) Continuity of Cechˇ theory implies that (αrs )∗ π2(BPr ) ------→ π2(BPrs ) n n Hˇ (X; Z) lim H (Xj ; Z).   ------→   y y The maps X → Xj induce natural maps s Z/r ------→ Z/rs. [Xj ,K(Z, n)] → [X, K(Z, n)]

There is a similar structure in infinite dimen- and these coalesce to form sions. Fix some separable Hilbert space H with : lim[X ,K(Z, n)] → [X, K(Z, n)]. Φ ------→ j associated group of unitaries U on which we im- pose the strong operator topology. The group U is Claim: the map Φ is a bijection. The key fact contractible in this topology (cf. [21], Lemma 4.72). needed is the following result of Eilenberg- Steenrod ([8], p. 287, Theorem 11.9): if X = lim X is Fix some unitary isomorphism H ⊗H H . This is ←------j unique up to homotopy since U is path-connected. compact, Y is a simplicial complex, and f : X → Y , Then there is a canonical homomorphism then up to homotopy f factors through one of ⊗ the X . This implies immediately that is onto. U × U -→U j Φ On the other hand, if g : Xj → Y and the com- and associated maps on classifying spaces posite X → Xj → Y is null-homotopic then the ⊗ null-homotopy factors through some X × [0, 1] BU × BU ------→ BU k and hence [g] = 0.     y y ⊗ Bundles with Fibre K BPU × BPU ------→ BPU Recall that K = K(H ) denotes the algebra of where PU denotes the infinite projective unitary compact operators on a separable Hilbert space group. H . Let 1 The action of S on U is free and thus ζ : PU -→ T -→ X 1 PU' BS ' K(Z, 2). be a principal PU-bundle with associated C∗- This implies that algebra Aζ . All automorphisms of K = K(H ) are given by conjugation by unitary operators on the BPU' K(Z, 3). Hilbert space H , so the group of unitaries U acts It is simpler to separate the discussion of finite on K by the adjoint action. The center of the group and infinite dimensional bundles at this point. is just S1, and it acts trivially, of course, and so K U 1 = U A Note on Cohomology for Compact Aut( ) /S P , Spaces the infinite projective unitary group. Thus If X is a finite complex then the Eilenberg-Steenrod uniqueness theorem guarantees for us that singu- [X, BPU] [X, K(Z, 3)] H3(X; Z). ˇ lar, simplicial, representable, and Cech cohomology We may regard maps X → BPU as projec- theories all coincide. Moving up to compact spaces tive vector bundles in analogy with projective one must pause to reconsider the question. The nat- representations. ural choice in the classical Dixmier-Douady context ∗ The resulting C -algebras Aζ are stable in the is Cechˇ cohomology, as this relates best to sheaf sense that Aζ ⊗ K Aζ . theories, and so the Dixmier-Douady invariant Define the Dixmier-Douady invariant δ(Aζ ) of 3 was originally defined to take values in Hˇ (X; Z). ∗ the C -algebra Aζ to be the homotopy class of a However, a classical homotopy approach dictates map defining H3(X; Z) = [X, K(Z, 3)]. Fortunately these f : Aˆ → BPU K(Z, 3) two functors agree on compact spaces; the result is again due to Eilenberg and Steenrod. that classifies the bundle E → X.

August 2009 Notices of the AMS 811 We note that given Aζ then its Dixmier-Douady This result may be viewed in a more bundle- 3 invariant lies naturally in the group H (Aˆζ ; Z). The theoretic manner. The fibration 1 identification of Aˆζ with X is only given mod the S → U → PU group of homeomorphisms of X, and hence the induces an exact sequence Dixmier-Douady invariant is only defined modulo δ [X, K(Z, 2)] → [X, BU] → [X, BPU] → [X, K(Z, 3)] the action of the homeomorphism group of X on H3(X; Z). Of course this action preserves the order which, after identifications, becomes δ of the element δ(Aζ ). 3 V ect1(X) → V ect∞(X) → PV ect∞(X) → Hˇ (X; Z) So we have established the first parts of the following Dixmier-Douady result: where V ectk(X) denotes isomorphism classes of vector bundles over X of dimension k, PV ect∞(X) Theorem ([5], [23]). Let X be a compact space. Then: denotes isomorphism classes of projective vector (1) There is a natural isomorphism bundles over X, and we have identified [X, K( , 2)] Hˇ2(X; ) V ect (X) δ : [X, BPU] -→ Hˇ3(X, Z). Z Z 1 using the first Chern class of the bundle. (2) Suppose we are given a principal PU- The map takes an infinite-dimensional vector bundle ζ, associated fibre bundle Pζ, and bundle V → X and associates to it the matrix ∗ associated C -algebra Aζ . Then δ(Aζ ) = 0 bundle if and only if Pζ is equivalent to a trivial (V → X) = End(V ) → X matrix bundle, and in that case where End(V )x = End(Vx), and so if δ(Aζ ) = 0 then the bundle Pζ is isomorphic to a bundle of Aζ C(X) ⊗ K. endomorphisms: Pζ End(V ) as bundles. Then (3) The Dixmier-Douady invariant is additive, we use the fact that every vector bundle over X in the sense that with infinite-dimensional fibres is trivial as a vector bundle (since U is contractible), and thus there are δ(A ) = δ(A ) + δ(A ). ζ1⊗ζ2 ζ1 ζ2 bundle isomorphisms (4) The invariant respects conjugation: Pζ End(V ) End(X × H ) X × K

δ(Aζ∗ ) = −δ(Aζ ). so that Aζ C(X) ⊗ K ˇ3 (5) Every element of H (X; Z) may be realized as C∗-algebras. In fact, recalling that U is con- as the Dixmier-Douady invariant of some tractible, U ' ∗, we may extend the sequence infinite-dimensional bundle and associated above to read ∗ C -algebra. [X, K(Z, 2)] → [X, ∗] → [X, BPU] δ Proof. Only (3) and (4) remain to be demonstrated. → [X, K(Z, 3)] → [X, ∗] Part (3) comes down to an analysis of the commu- and then deduce that δ is also onto. tative diagram ' BPU × BPU ------→ K(Z, 3) × K(Z, 3) Bundles with Fibre Mn(C) 1   The inclusion of S as the center of Un gives rise   y⊗ y to a fibration sequence ' 1 BPU ------→ K(Z, 3) S → Un → PUn → K(Z, 2) δ → BU → BPU -→ K(Z, 3). which deloops to n n For n ≥ 2 the map Un → PUn induces an isomorphism on homotopy. Passing to classifying spaces, ' PU × PU ------→ K(Z, 2) × K(Z, 2) this yields   π2(BPUn) Z/n ⊗ m y y as previously noted, and ' PU ------→ K(Z, 2). πj (BUn) πj (BPUn), j > 2. The map m is determined up to homotopy by The Dixmier-Douady invariant is defined to be the its representative in induced map ˇ3 Hˇ2(K(Z, 2) × K(Z, 2); Z) Z ⊕ Z δ : [X, BPUn] → [X, K(Z, 3)] H (X; Z). There is a long exact sequence and this class is (1, 1) so that m is indeed the map inducing addition in H2(−; Z). [X, K(Z, 2)] → [X, BUn] δ Part (4) is a similar argument, which we omit. -→ [X, BPUn] -→ [X, K(Z, 3)]

812 Notices of the AMS Volume 56, Number 7 which translates into The Dixmier-Douady map δ : BPUn → K(Z, 3) 2 δ 3 factors as Hˇ (X; Z) → V ectn(X) -→ PV ectn(X) -→ Hˇ (X; Z). γ β The map is defined as follows. Given a complex BPUn -→ K(Z/n, 2) -→ K(Z, 3). vector bundle V → X, then The map β induces the Bockstein homomorphism (V → X) = End(V ) → X β : Hˇ2(X; Z/n) → Hˇ3(X; Z) where End(V ) → X is the endomorphism with bundle of V ; the fibre over a point x is just End(V ). x Ker(β) = ImageHˇ2(X; Z) -→ Hˇ2(X; Z/n) This yields the third Dixmier-Douady result: and Proposition. Suppose that X is compact. Let ζ Image(β) = {x ∈ Hˇ3(X; Z) : nx = 0} be a principal PUn-bundle over X with associat- ∗ ed bundle Pζ and C -algebra Aζ . Suppose that whose image lies in the torsion subgroup of δ(Aζ ) = 0. Then there is a complex vector bun- Hˇ3(X; Z). dle V → X of dimension n over X and a bundle isomorphism Theorem. Let X be a compact space and let n ∈ N. Then: (1) There is a natural exact sequence Pζ ------→ End(V ) 2 σ   0 → Hˇ (X; Z) -→ V ectn(X)   y y δ -→ [X, BPU ] -→ Hˇ3(X; Z). 1 n X ------→ X. (2) Suppose we are given a principal PUn- Note that, in contrast to the infinite-dimensional bundle ζ over a compact space X and ∗ case, endomorphism bundles need not be trivial associated C -algebra Aζ . Then bundles. There is one improvement possible, and (a) If γ(Aζ ) ≠ 0 but δ(Aζ ) = 0 then γ(Aζ ) we are indebted to Peter Gilkey for this explicit lifts to an integral class in Hˇ2(X; Z). construction. (b) If γ(Aζ ) = 0 then Pζ End(V ) with Corollary. The vector bundle V → X in the Propo- c1(V ) = 0. sition may be taken to have trivial first Chern class, (3) The Dixmier-Douady invariant is additive, so that its structural bundle may be reduced to an in the sense that SU -bundle. n δζ1⊗ζ2 = δζ1 + δζ2 . Proof. Suppose that Pζ End(V ) as in the (4) The invariant respects conjugation: Proposition. Let L be a complex line bundle over δζ∗ = −δζ . 0 X with c1(L) = −c1(V ). Let V = V ⊗ L. Then using the fact that L ⊗ L∗ is a trivial line bundle (5) For any Mn-bundle ζ, it is the case that = we have nδζ 0. End(V 0) (V 0)∗ ⊗ V 0 V ⊗ V ∗ ⊗ L ⊗ L∗ Proof. Most of this result has already been established. The map σ takes a class c ∈ Hˇ2(X; Z) and V ⊗ V ∗ End(V ) associates to it a vector bundle of the form L⊕θn−1 so we may replace V by V 0 and obtain the same where L is a line bundle with first Chern class c n−1 endomorphism bundle. and θ is a trivial bundle of dimension n−1. This map is one-to-one since it is split by the first Chern Note that even though V and V 0 have isomor- class map phic endomorphism bundles, in general they will 2 not themselves be isomorphic. In fact, End(V ) c1 : Vectn(X) → Hˇ (X; Z). 0 0 End(V ) if and only if V V ⊗ L for some line Parts (3) and (4) follow as in the infinite-dimensional bundle L. case. We can refine this observation as follows. The diagram above expands to a natural commuting The various maps αrs : PUr → PUrs induce maps diagram (below) on classifying spaces that by abuse of language are also denoted αrs : BPUr → BPUrs . These maps

August 2009 Notices of the AMS 813 ˜ form a directed system {BPUr , αrs }. Write BPU∞ implies that the image of β is exactly the torsion for the colimit. Note that this is not the same as subgroup of Hˇ3(X; Z). This shows (1). BPU = K(Z, 3). For (2), let x be a torsion class. Then it is in the Proposition (Serre, [9], pp. 228–229). image of the Bockstein map (1) The natural map β˜‘: Hˇ2(X; Q/Z) → Hˇ3(X; Z)

lim π (BPU ) → π (BPU∞) ˜ ------→ j n j and thus may be represented as x = β(y) where is an isomorphism. y ∈ [X, K(Q/Z, 2)]. The map (2) π (BPU∞) Q/Z. 2 [X, BPU∞] -→ [X, K(Q/Z, 2)] (3) If j ≥ 2 then π2j (BPU∞) Q. is onto, and so the class y lifts to some class (4) If j ≥ 2 then π2j−1(BPU∞) = 0. (5) There is a natural splitting z ∈ [X, BPU ] lim[X, BPU ]. ∞ ------→ n BPU∞ ' K(Q/Z, 2) × F Choose some ζ ∈ [X, BPUk] representing z. (Note with πj (F) = Q for j ≥ 4 and even and that if x has order n then n divides k but in general π (F) = 0 otherwise. j n ≠ k.) Then δ(Aζ ) = x as required.

Proof. Each map αrs : BPUr → BPUrs is a cofibra- tion and so (1) is immediate. We showed previously Twisted K-theory Twisted K-theory was first introduced by Donovan that π2(BPUr ) Z/r. The map induced by αrs and Karoubi [7] for finite-dimensional bundles and takes the generator of π2(BPUr ) to s times the generator of π2(BPUrs ) and so then by Rosenberg [23] in the general case. In our context the point is to look at the Z/2-graded group π2(BPU∞) = lim(Z/r, α∗) = Q/Z. ------→ K∗(Aζ ). Let Aζ denote a continuous trace algebra For n >> j > 2, π2j (BPUn) π2j (BUn) Z by over X. Recall that K0 is defined for any unital ring Bott periodicity, and it follows easily that as the Grothendieck group of finitely projective modules. For our purposes a topological definition π2j (BPU∞) lim(π2j (BPUn), α∗) = Q. ------→ is cleaner and so we may simply define Similarly, in odd degrees homotopy groups vanish, and calculation yields the result. There results a Kj (Aζ ) πj+1(U(Aζ ⊗ K)), j ∈ Z/2, fibration BPU → K(Q/Z, 2); call the fibre F. Then −j ∞ where in all cases we grade as Kj = K (and then the homotopy of F is as stated, and as the base note that by periodicity there are only two groups space has trivial rational cohomology this implies anyhow). If the bundle is infinite dimensional then that the fibration is trivial. it is not necessary to tensor with K since the The various Dixmier-Douady maps algebra is already stable. These groups are denoted 3 in the literature by (for instance) δ : [X, BPUn] -→ Hˇ (X; Z) K∗(X; ζ) or K∗(X; δ(ζ)) or K∗ (X) or K∗(X). are coherent and hence pass to the limit to produce δ(ζ) ∆ an induced Dixmier-Douady map The point is that once one specifies X and ∆ = 3 ∞ 3 δ(ζ) ∈ Hˇ (X; Z) then Aζ is specified up to equiva- δ : [X, BPU∞] -→ Hˇ (X; Z). lence, and hence K∗(X) makes sense as notation It is obvious that δ∞ takes values in the torsion ∆ for Kj (Aζ ). Here are the basic properties: subgroup of Hˇ3(X; Z). In fact, more is true: Proposition. Proposition. Let X be a compact space. Then: (1) Domain: The groups K∗(X) are defined (1) The image of the map δ∞ is the whole tor- ∆ for locally compact spaces X and principal sion subgroup of Hˇ3(X; Z). PUn or PU-bundles ζ over X with asso- (2) Let x ∈ Hˇ3(X; Z) be a torsion class. Then ciated Dixmier-Douady class ∆ = δ(ζ) ∈ there is some finite n and some principal H3(X; Z). PUn-bundle ζ over X such that δ(Aζ ) = x. (2) Naturality: Given (X, ∆) together with a Proof. The lattice of cyclic subgroups of Q/Z in- continuous function f : Y → X then there is duces an equivalence an induced map ∗ ∗ ∗ lim K(Z/n, 2) -→ K(Q/Z, 2). f : K (X) -→ K ∗ (Y ) ------→ ∆ f ∆ Furthermore, the various Bockstein maps K(Z/n, 2) and twisted K-theory is natural with respect -→ K(Z, 3) all factor as to these maps. ∗ ˜ (3) Periodicity: The groups K (X) are periodic β ∆ K(Z/n, 2) -→ K(Q/Z, 2) -→ K(Z, 3). of period 2. The exactness of the coefficient sequence (4) Product: There is a cup product operation ˜ ∗ ∗ ∗ 2 β 3 3 K (X) × K (X) -→ K (X). Hˇ (X; Q/Z) -→ Hˇ (X; Z) -→ Hˇ (X; Q) ∆1 ∆2 ∆1+∆2

814 Notices of the AMS Volume 56, Number 7 (5) Relation to untwisted K-theory: There is a Rational Homotopy natural isomorphism For stable continuous trace algebras, the groups π∗(UAζ ) are periodic of period 2 and in fact corre- ∗ ∗ K0 (X) K (X), spond to the twisted K-theory groups. However, if ζ is a principal PUn-bundle where n is finite then where if X is locally compact but not com- the natural map pact then K-theory with compact support is π (UA ) → K (A ) intended. j ζ j−1 ζ is neither injective nor surjective in general. Further- more, K-theory obscures the geometric dimension Karoubi notes that the cup product is not canoni- of the space X, since K0(S2n) doesn’t depend on n cally defined at the level of cohomology classes. For and hence cannot detect it. In this situation a more instance, in the finite-dimensional case, one must natural question is to calculate π∗(UAζ ) itself. choose representatives from among the various This is impossible even in very elementary cases algebra bundles; i.e. choose Morita equivalences (e.g., when A = M2(C)). A more reasonable project that are not canonical in general. is to calculate the rational homotopy groups Rosenberg [22] points out the simplest case πj (UAζ ) ⊗ Q, where twisted K-theory actually does something and this has been done in general for X compact by interesting. Take X = S3. Then the Dixmier-Douady [15], [14]. The answer depends upon the individual invariant takes values in H3(S3; Z) Z and hence groups Hj (X; Q) and upon n. It turns out to be is determined by an integer m. Rosenberg shows independent of the principal bundle. This is to be that expected, at least after the fact, since the Dixmier- Douady invariant is finite when the bundle is finite 0 3 −1 3 Km(S ) = 0,Km (S ) = Z/m. dimensional and hence is trivial in the world of rational homotopy. He takes us further by introducing a twisted Atiyah-Hirzebruch spectral sequence (for X a finite Generalizations complex) converging to K∗(X) and with ∆ What if X is assumed to be a CW -complex or, more

∗ ∗ generally, a compactly generated space that is not E = H (X; K∗(C)). 2 necessarily compact? First, the definition of Aζ does not lead to a C∗-algebra since infinite CW - Just as with the classical Atiyah-Hirzebruch spec- complexes are not locally compact. These would tral sequence we have d2 = 0. The differential d3 is be pro-C∗-algebras such as those studied by 3 determined by the integral Steenrod operation SqZ N. C. Phillips [20]. The good news is that some (as is the case classically) and (the first mention of of the proofs in this note generalize. The bundle the twist) the class ∆: classification results require restriction to Dold’s numerable bundles [6], [12]. These are bundles 3 d3(x) = SqZ(x) − ∆x. that are trivial with respect to a locally finite cover, and so one can assume, for instance, that X is This spectral sequence is developed further by paracompact. In the infinite-dimensional case the Atiyah and Segal [1], [2] who show, for instance, isomorphism that the spectral sequence does not collapse after [X, BPU] [X, K(Z, 3)] rationalization. There are other ways to twist K-theory and to is tautological, since BPU' K(Z, 3). make it equivariant, to make it real (rather than In the finite-dimensional situation we obtain complex) or both, and the reader should consult the maps papers of Atiyah and Segal [1], [2], Freed, Hopkins, num δ 3 Vectn (X) → [X, BPUn] → Hsing(X; Z), and Teleman [11], and especially the fine survey num paper of Karoubi [13] before burying oneself in where Vectn (X) denotes isomorphism classes of numerable vector bundles, and the Dixmier- the physics literature. That physics comes into Douady results still hold when singular cohomology the picture goes back to the observation of Witten is understood throughout. that D-brane charges in type IIB string theory 0 What if X is assumed to be locally compact but over a space M are elements of Kδ(M)—see for not necessarily compact? In that case the definition instance [3], where the role of the Dixmier-Douady of Aζ is modified to include only those sections invariant in the classification of bundle gerbes is that vanish at infinity, so that the sup norm is ∗ summarized along with a differential geometric defined and then Aζ is a C -algebra again. The model for twisted K-theory. See [10], [16], [17], [19] Dixmier-Douady results still hold, but it is prob- for further developments. ably better in this setting to shift back to the

August 2009 Notices of the AMS 815 sheaf-theoretic setting of the original proofs, since [10] D. Freed and M. Hopkins, On Ramond-Ramond the classification of vector bundles over locally fields and K-theory, J. High Energy Phys. 2000, no. compact spaces is somewhat awkward. 5, Paper 44, 14 pp. [11] D. Freed, M. Hopkins, and C. Teleman, Twisted References equivariant K-theory with complex coefficients, J. Topology 1 (2008), no. 1, 16–44. [1] M. Atiyah and G. Segal, Twisted K-theory, Ukr. Mat. [12] D. Husemoller, Fibre Bundles, Graduate Texts in Visn. 1 (2004), no. 3, 287–330; translation in Ukr. Math., 20, Second Edition, Springer-Verlag, New York, Math. Bull. 1 (2004), no. 3, 291–334 1975. [2] , Twisted K-theory and cohomology, Inspired [13] M. Karoubi, Twisted K-Theory, old and new, by S. S. Chern, 5–43, Nankai Tracts Math. 11 (2006), http://arxiv.org/abs/math.KT/0701789v3 (to World Sci. Publ., Hackensack, NJ, 2006. appear in Proceedings: K-theory and noncommuta- [3] , , P. Bouwknegt A. L. Carey, V. Mathai M. K. Mur- tive geometry) (G. Cortias, J. Cuntz, M. Karoubi, R. , , Twisted K-theory and K-theory ray D. Stevenson Nest and C.A. Weibel, eds.), European Mathematical of bundle gerbes, Comm. Math. Phys. 228 (2002), no. Society, Zurich, 2008. 1, 17–45. [14] J. Klein, C. L. Schochet, and S. B. Smith, Continuous [4] , , and , Topological J. Cuntz R. Meyer J. Rosenberg trace C∗-algebras, gauge groups and rationalization, and Bivariant K-Theory, Oberwolfach Seminars, 36, arXiv:0811.0771v1 [math.AT]. Birkhäuser Verlag, Basel, 2007. xii+262 pp. [15] G. Lupton, N. C. Phillips, C. L. Schochet, S. B. [5] J. Dixmier and D. Douady, Champs continus Smith, Banach algebras and rational homotopy d’espaces Hilbertiens et de C∗-algébres, Bull. Soc. theory, Trans. Amer. Math. Soc. 361 (2009), 267–295. Math. France 91 (1963), 227–284. [16] V. Mathai, R. B. Melrose, and I. M. Singer, Fraction- [6] , Partitions of unity in the theory of A. Dold al analytic index, J. Differential Geom. 74 (2006), fibrations, Ann. Math. 78 (1963), 223–255. no. 2, 265–292. [7] P. Donovan and M. Karoubi, Graded Brauer groups [17] V. Mathai and J. Rosenberg, T -duality for torus and K-theory with local coefficients, Inst. Hautes bundles with H-fluxes via noncommutative topology, Etudes Sci. Publ. Math. 38 (1970), 5–25. Comm. Math. Phys. 253 (2005), no. 3, 705–721. [8] S. Eilenberg and N. Steenrod, Foundations of [18] J. P. May, A Concise Course in Algebraic Topology, Algebraic Topology, Princeton University Press, Chicago Lectures in Math., University of Chicago Princeton, 1952. Press, Chicago, 1999. [9] Correspondance Grothendieck-Serre. (P. Colmez and [19] V. Nistor and E. Troitzky, An index for J.-P. Serre, eds.), Documents Mathématiques (Paris), gauge-invariant operators and the Dixmier-Douady 2, Soc. Math. de France, Amer. Math. Soc. 2001. invariant, Trans. Amer. Math. Soc. 356 (2004), no. 1, 185–218. ∞ American Mathematical Society [20] N. C. Phillips, C loop algebras and noncommutative Bott periodicity, Trans. Amer. Math. Soc. 325 (1991), no. 2, 631–659. Famous Puzzles [21] I. Raeburn and D. Williams, Morita Equivalence and of Great Continuous Trace C∗-Algebras, Math. Surveys and Monographs, Vol. 60, Amer. Math. Soc., 1998. Mathematicians [22] J. Rosenberg, Homological invariants of extensions ∗ Miodrag S. Petkovic´, of C -algebras, Operator algebras and applications, University of Nis, Serbia Part 1 (Kingston, Ont., 1980), pp. 35–75, Proc. Sym- pos. Pure Math., 38, Amer. Math. Soc., Providence, This is the only collection RI, 1982. in English of puzzles and [23] , Continuous-trace algebras from the bundle- challenging elementary math- theoretic point of view, J. Austral. Math. Soc. (Series ematical problems posed, A) 47 (1989), 368–381. discussed and solved by great [24] N. Steenrod, The Topology of Fibre Bundles, mathematicians. The book is Princeton University Press, Princeton, 1951. intended to amuse and entertain while bringing the reader closer to the distinguished mathematicians through their works and some compelling personal stories. The selected problems simply require pencil and paper and a healthy amount of persistence. 2009; 324 pages; Softcover; ISBN: 978-0-8218-4814-2; List US$36; AMS members US$29; Order code MBK/63

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816 Notices of the AMS Volume 56, Number 7