CCOMPUTATIONS IN HIGHER TWISTED K -THEORY DAVID BROOK
Abstract.
Higher twisted K -theory is an extension of twisted K -theory introduced byUlrich Pennig which captures all of the homotopy-theoretic twists of topological K -theoryin a geometric way. We give an overview of his formulation and key results, and reformulatethe definition from a topological perspective. We then investigate ways of producing explicitgeometric representatives of the higher twists of K -theory viewed as cohomology classes inspecial cases using the clutching construction and when the class is decomposable. Atiyah–Hirzebruch and Serre spectral sequences are developed and information on their differentialsis obtained, and these along with a Mayer–Vietoris sequence in higher twisted K -theory areapplied in order to perform computations for a variety of spaces. Contents
1. Introduction 2Background 2Higher twists 3Links with physics 3Main results 4Outline of this paper 6Acknowledgements 62. Preliminaries and formulation 62.1. Strongly self-absorbing C ∗ -algebras 62.2. Formulation 92.3. Links to cohomology 122.4. Reformulation using Fredholm operators 133. Geometric constructions of higher twists 163.1. The clutching construction 163.2. Decomposable cohomology classes 184. Spectral sequences 214.1. Atiyah–Hirzebruch spectral sequence 224.2. Segal spectral sequence 235. Computations 255.1. Spheres 255.2. Products of spheres 315.3. Special unitary groups 325.4. Real projective spaces 365.5. Lens spaces 39 Mathematics Subject Classification.
Primary 19L50, Secondary 46L80, 55T25.
Key words and phrases.
Higher twisted K-theory, K-theory, Higher geometric twists, Spectral sequences.The author was partially supported by a University of Adelaide Masters Scholarship. a r X i v : . [ m a t h . K T ] J u l .6. SU (2)-bundles 40References 441. Introduction
Higher twisted K -theory is the full twisted cohomology theory of topological K -theory,for which an operator algebraic model was first developed by Ulrich Pennig [47]. Whiletwisted K -theory has been around for many years, it was understood that the classicaltwists corresponding to third-degree integral cohomology classes, or equivalently principal P U -bundles or bundle gerbes, did not correspond to the full set of abstract twists for K -theory; these twists merely had an elegant geometric interpretation. More general twistswere known to exist by homotopy-theoretic arguments, but until recently these twists wereentirely abstract, rendering them unusable in computations and making it difficult to findapplications of these more general twists. For this reason, the limited set of twists for whichgeometric representatives were known have been studied extensively. In this paper, we focuson developing tools for computations involving this general class of twists, and apply theseto compute the higher twisted K -theory groups of a wide variety of spaces. Background.
What has classically been called twisted K -theory is a generalisation of topo-logical K -theory which gradually emerged over the course of the 1960s after Atiyah andHirzebruch’s initial work in topological K -theory [4]. Interest in the area was sparked whenAtiyah, Bott and Shapiro investigated the relationship between topological K -theory andClifford algebras [3], providing a new perspective from which K -theory could be viewed usingClifford bundles associated to vector bundles which was studied in Karoubi’s doctoral thesis[37]. Karoubi and Donovan extended this by replacing Clifford bundles with algebra bun-dles [28], which resulted in what was then called “ K -theory with local coefficients” in whichthe local coefficient systems over X were classified by H ( X, Z ) and the torsion elementsof H ( X, Z ), corresponding to finite-dimensional complex matrix algebra bundles over X .This was the first notion of twisted K -theory, and it was defined geometrically using algebrabundles to represent twists. Note that twists given by elements of H ( X, Z ) were studiedcomprehensively in [8] and are often neglected in recent work.Over the coming years, much work was done on developing twisted K -theory. Rosenbergextended the definition of twisted K -theory to any class in H ( X, Z ) rather than specificallyconsidering torsion classes [49], corresponding to using infinite-dimensional algebra bundlesover X . Work by Bouwknegt, Carey, Mathai, Murray and Stevenson provided twisted K -theory computations and developed a twisted Chern character in the even case [11], andMathai and Stevenson extended this to the odd case [43] and studied the Connes–Cherncharacter for twisted K -theory [44]. Further developments were later made by Atiyah andSegal in formulating twisted K -theory using bundles of Fredholm operators and exploringthe differentials in an Atiyah–Hirzebruch spectral sequence for twisted K -theory [6, 7].Up until this point, twists were geometric objects which could naturally be used to modify K -theory. From a homotopy-theoretic point of view, however, these twists did not capturethe entire picture – there existed more general abstract twists for which no geometric inter-pretation was known. In fact, Atiyah and Segal acknowledged the existence of more generaltwists of K -theory which had not found a geometric realisation [6]. his changed with the introduction of a Dixmier–Douady theory for strongly self-absorbing C ∗ -algebras by Pennig and Dadarlat [24, 26], analogous to the Dixmier–Douady theory forthe compact operators on a Hilbert space which is integral to the construction of the classicaltwists. Pennig was able to use these algebras to propose a geometric model which capturesall of the twists of K -theory, in which abstract twists can be realised as algebra bundles withfibres isomorphic to the stabilised infinite Cuntz algebra O ∞ ⊗ K [47]. Higher twists.
We provide a brief introduction to the homotopy-theoretic arguments show-ing that more general twists of K -theory exist. Recall that there is a notion of spectrum fora cohomology theory h • , which is a sequence of topological spaces { E n } n ∈ N satisfying par-ticular properties such that h n ( X ) = [ X, E n ]. In the case of topological K -theory, this is the2-periodic sequence Z × BU, U, · · · . To each ring spectrum there is an associated unit spec-trum consisting of the units in each space, which we will denote { GL ( E n ) } n ∈ N , and from thisanother cohomology theory denoted gl ( h ) • may be defined via gl ( h ) n ( X ) = [ X, GL ( E n )].The notation gl ( h ) here reflects that this cohomology theory is in some sense constructedout of the unital elements of the cohomology theory h • . Then the twists of h • over somespace X are classified by the first group of this cohomology theory, i.e. gl ( h ) ( X ). As GL ( E ) denotes the units in E , we see that the twists are classified by [ X, BGL ( E )].More technically, a twisting of a cohomology theory over a space X is a bundle of spectraover X with fibre given by the spectrum R of the cohomology theory. Then letting GL ( R )denote the automorphism group of the spectrum R , these bundles of spectra are classifiedby [ X, BGL ( R )]. One may then define the groups of the twisted cohomology theory.Applying these notions to topological K -theory, we see that the invertible elements of thering K ( X ) are represented by virtual line bundles. Unifying this with the spectrum picture,these classes correspond to [ X, Z × BU ] and so in the notation of the previous paragraphwe have GL ( Z × BU ) = Z × BU . Thus the twists of topological K -theory over X areclassified by gl ( KU ) ( X ) = [ X, B ( Z × BU )] . It is known that BU is homotopy equivalent to K ( Z , × BSU [41] and so we see that B ( Z × BU ) (cid:39) K ( Z , × K ( Z , × BBSU . This means that twists of K -theory areclassified by homotopy classes of maps X → K ( Z , × K ( Z , × BBSU, and therefore for a CW complex X the twists of K -theory correspond to elements of H ( X, Z ), H ( X, Z ) and [ X, BBSU ]. The third of these groups is not well-understood,which led to the lack of understanding of this class of twists for K -theory. Furthermore,while this approach may be used to define what a twist of K -theory is, it does not provideany geometric information about twists, rendering any results specific to twisted K -theoryvery difficult to prove. Thus Pennig’s geometric interpretation of the full set of twists of K -theory lays the foundations for significant developments in the field. We remark thatwhile this will not be a focus of this paper, further investigation into the topology of BBSU may prove useful in the study of higher twisted K -theory, as well as being a topic of interestin its own right. Links with physics.
Further to this abstract motivation, higher twisted K -theory is ex-pected to be of great relevance in mathematical physics. Both topological and classicaltwisted K -theory have found application to physics, particularly in string theory, with work f Witten [53] providing a link between charges of D -branes and topological K -theory andlater work by Bouwknegt and Mathai extending this to twisted K -theory in the presence ofa B -field [17], so it is likely that similar ties exist to higher twisted K -theory. In fact, someapplications have already been found. In the classical case, Bouwknegt, Evslin and Mathaishow that the T-duality transformation induces an isomorphism on twisted K -theory, whichwas an unexpected result as the T-duality transformation often leads to significant differ-ences between the topologies of the circle bundles in question [15, 16]. They then generalisethis to the case of spherical T-duality [12, 13, 14] in which the relevant cohomology classis of degree 7, which led to the result that the spherical T-duality transform induces anisomorphism on higher twisted K -theory. In the first of this series of papers, the authorsalso provide some insight into how higher twisted K -theory fits in with the D -brane picture.In the setting of Type IIB string theory, the data in 10-dimensional supergravity includes a10-manifold Y which is commonly diffeomorphic to R × X for an appropriate 9-manifold X .They explain that the K -theory of X twisted by a 7-class corresponds to the set of conservedcharges of a certain subset of branes. While this is an interesting result, it only applies to7-classes in this specialised setting which implies that there may be a richer relationshipbetween D -branes and higher twisted K -theory than what currently exists in the literature.It may be possible to gain greater insight into the behaviour of D -branes by studying highertwisted K -theory.Further work in studying higher twisted K -theory in relation to spherical T-duality isdone by MacDonald, Mathai and Saratchandran [40], who also develop a Chern characterin higher twisted K -theory and give an isomorphism between higher twisted K -theory andhigher twisted cohomology over the reals.Relevance to physics also appears in the study of strongly self-absorbing C ∗ -algebras, assome common algebras in physics such as the canonical anticommutation relations (CAR)algebra are strongly self-absorbing. This algebra is not only relevant in quantum physics,but also in the study of Clifford algebras. By exploring the version of twisted K -theorywhose twists arise from the CAR algebra, a deeper understanding of this algebra and itsapplications may be gained.A final application of twisted K -theory of critical importance in mathematical physicsis the work of Freed, Hopkins and Teleman in proving that the Verlinde ring of positiveenergy representations of a loop group is isomorphic to the equivariant twisted K -theory ofa compact Lie group [32, 33, 34]; a result which is of relevance in conformal field theory.It is likely that there exists an analogous result in the higher twisted setting, where it isas of yet unknown what can replace the Verlinde ring to be isomorphic to higher twisted K -theory, but another object from physics may arise in this case. Pennig and Evans [31]hint at possible approaches. Main results.
The purpose of this paper is to develop computational tools in higher twisted K -theory and use these to perform explicit computations. The main results that we presentare to achieve this goal.One important aspect of computing higher twisted K -theory groups is determining explicitgenerators for the groups. This is difficult using the C ∗ -algebraic formulation, and so weprovide an alternative definition of the higher twisted K -theory groups from a topologicalviewpoint inspired by a result of Rosenberg [49]. In Theorem 2.14 we prove that the higher wisted K -theory groups can be identified with K ( X, δ ) = [ E δ , Fred O ∞ ⊗K ] Aut( O ∞ ⊗K ) ; K ( X, δ ) = [ E δ , Ω Fred O ∞ ⊗K ] Aut( O ∞ ⊗K ) ;where E δ is the principal Aut( O ∞ ⊗ K )-bundle over X representing the abstract twist δ ,Fred O ∞ ⊗K denotes the Fredholm operators on the standard Hilbert ( O ∞ ⊗ K )-module, Ωdenotes the based loop space and [ − , − ] Aut( O ∞ ⊗K ) = π ( C ( − , − ) Aut( O ∞ ⊗K ) ) denotes unbasedhomotopy classes of Aut( O ∞ ⊗ K )-equivariant maps. This result is used to give an explicitdescription of the generator of K ( S n +1 , δ ) in Theorem 5.3.Since the definition of higher twisted K -theory involves sections of algebra bundles withfibre O ∞ ⊗ K , another task critical to computations is to explicitly construct these bun-dles over spaces. When the topological space is an odd-dimensional sphere S n +1 , wenote that the clutching construction can be used to construct principal Aut( O ∞ ⊗ K )-bundles, or equivalently our desired algebra bundles, over S n +1 by specifying a gluing map S n → Aut( O ∞ ⊗ K ). These maps are classified up to homotopy by an integer, and us-ing Pennig and Dadarlat’s link between twists and cohomology classes [26] we see that theprincipal Aut( O ∞ ⊗ K )-bundles over S n +1 are also classified by an integer. This providesan explicit geometric way of describing the twist over S n +1 associated to a particular co-homology class; the details are presented in Section 3.1. Another case that we considerinvolves restricting to decomposable cohomology classes. For a general CW complex X withtorsion-free cohomology, we take a 5-class δ given by the cup product of a 2-class α and a3-class β . By associating a principal U (1)-bundle to α and a principal P U -bundle to β anddefining an effective action of U (1) × P U on O ∞ ⊗ K , we are able to construct a principalAut( O ∞ ⊗ K )-bundle over X corresponding to the cohomology class δ , and obtain a moregeneral result in Theorem 3.5. We do not apply this result in our computations, but it islikely that it can form the basis for further research.More directly relevant to our goal, we investigate spectral sequences for higher twisted K -theory. We show in Theorem 4.1 that when X is a CW complex and δ is any abstract twistover X , there is an analogue of the Atiyah–Hirzebruch spectral sequence which has E -term E p,q = H p ( X, K q ( pt )) and which strongly converges to the higher twisted K -theory K ∗ ( X, δ ).In particular, when the twist δ can be identified with a cohomology class δ ∈ H n +1 ( X, Z ) wesee in Theorem 4.2 that the d n +1 differential is of the form d n +1 ( x ) = d (cid:48) n +1 ( x ) − δ ∪ x where d (cid:48) n +1 is the differential in the ordinary Atiyah–Hirzebruch spectral sequence for topological K -theory, which is an operator whose image is torsion [2]. There is, in fact, a more generalSegal spectral sequence that can be applied in this setting and we generalise a result ofRosenberg [48] to obtain this sequence. Letting F ι −→ E π −→ B be a fibre bundle of CWcomplexes with δ a twist over E , we prove in Theorem 4.3 that there is a spectral sequencewith E -term E p,q = H p ( B, K q ( F, ι ∗ δ )) which strongly converges to the higher twisted K -theory K ∗ ( E, δ ). We also obtain more explicit information about the differentials of thehigher twisted K -homology version of this spectral sequence in Theorem 4.4, which is usefulin proving general results about the higher twisted K -theory of Lie groups.We conclude by using the techniques developed to perform computations. Table 1 containsthe explicit results obtained in Section 5, displaying the higher twisted K -theory groups ofodd-dimensional spheres, the product of an odd-dimensional and an even-dimensional sphere, eal projective spaces and lens spaces. In each case, δ is N (cid:54) = 0 times the generator of thecohomology group specified in the table. X Degree K ( X, δ ) K ( X, δ ) S n +1 n + 1 0 Z N S m × S n +1 n + 1 0 Z N ⊕ Z N S m × S n +1 m + 2 n + 1 Z Z ⊕ Z N R P n +1 n + 1 Z n Z N L ( n, p ) 2 n + 1 Z p n Z N Table 1.
Higher twisted K -theory groups with δ = N ∈ H degree ( X, Z ) = Z We also use the Segal spectral sequence to obtain results on the higher twisted K -theoryof SU ( n ). In Theorem 5.8 we show that for a 5-twist δ given by N times the generatorof H ( SU ( n + 1) , Z ), with N relatively prime to n !, the graded group K ∗ ( SU ( n + 1) , δ ) isisomorphic to Z N tensored with an exterior algebra on n − δ is any twist given by N times a primitive generator of H ∗ ( SU ( n ) , Z ) we show inTheorem 5.9 that K ∗ ( SU ( n ) , δ ) is a finite abelian group with all elements having order adivisor of a power of N .Our final computations are for a class of examples which arise in spherical T-duality;the total spaces of SU (2)-bundles. Bouwknegt, Evslin and Mathai prove that the sphericalT-duality transformation induces a degree-shifting isomorphism on the 7-twisted K -theorygroups of these bundles [12, 13, 14], but they do not consider the 5-twisted K -theory ofthese SU (2)-bundles. We place restrictions on the base space M in order to ensure that the5-twists of the total space of the bundle correspond exactly to its integral 5-classes, and thenuse the Atiyah–Hirzebruch spectral sequence to compute the 5-twisted K -theory in Section5.6. Outline of this paper.
The paper is organised as follows. Section 2 provides the necessarybackground on the Cuntz algebra O ∞ to formulate higher twisted K -theory using Pennig’soriginal formulation, and an alternative topological characterisation is given. In Section 3,we describe methods of producing explicit geometric representatives for twists of K -theoryusing both the clutching construction and by considering decomposable cohomology classes.Section 4 develops spectral sequences for computations, and finally Section 5 contains explicitcomputations of higher twisted K -theory groups for various spaces. Acknowledgements.
The author acknowledges support from the University of Adelaidein the form of an MPhil scholarship. The author would also like to thank his principalMPhil supervisor Elder Professor Mathai Varghese for suggesting the problems in this paper,explaining techniques to construct geometric representatives for twists and his excellentsupport throughout the project in general. The author thanks his co-supervisor Dr PeterHochs for his exceptional guidance and support, as well as his MPhil referees and Dr DavidRoberts for helpful comments.2.
Preliminaries and formulation
Strongly self-absorbing C ∗ -algebras. We begin by presenting the background on theCuntz algebra O ∞ necessary to the formulation of higher twisted K -theory. We will firstly ntroduce Toms and Winter’s class of strongly self-absorbing C ∗ -algebras, which possessa higher Dixmier–Douady theory mirroring the Dixmier–Douady theory of the compactoperators and with which higher twisted K -theory can be defined. Apart from having thisapplication to K -theory, this class of algebras is interesting in its own right as it has proveduseful in the quest of Elliott to classify all simple nuclear C ∗ -algebras [51]. Here we presenta slightly modified but equivalent definition posed by Pennig and Dadarlat [26], which ismore applicable to topological problems. For the original definition, see [51]. Definition 2.1.
A seperable and unital C ∗ -algebra D is called strongly self-absorbing ifthere exists a ∗ -isomorphism ψ : D → D ⊗ D and a path of unitaries u : [0 , → U ( D ⊗ D )such that, for all d ∈ D , lim t → (cid:107) ψ ( d ) − u t ( d ⊗ u ∗ t (cid:107) = 0.We are specifically concerned with explicit examples of these algebras, the most relevantof which will be the Cuntz algebra O ∞ first introduced by Cuntz [23]. Definition 2.2.
The
Cuntz algebra O n with n generators for n = 1 , , · · · is defined to bethe C ∗ -algebra generated by a set of isometries { S i } ni =1 acting on a separable Hilbert spacesatisfying S ∗ i S j = δ ij I for i, j = 1 , · · · , n and n (cid:88) i =1 S i S ∗ i = I. Similarly, the Cuntz algebra O ∞ with infinitely many generators is defined in an analogousway for an infinite sequence { S i } i ∈ N satisfying S ∗ i S j = δ ij I for i, j ∈ N and k (cid:88) i =1 S i S ∗ i ≤ I for all k ∈ N .It is proved in the original reference that this definition is independent of the choice ofHilbert space and of isometries. While we will not discuss twists of K -theory which arisefrom other strongly self-absorbing C ∗ -algebras in detail, these may also be of interest and sowe will list other examples of strongly self-absorbing C ∗ -algebras. These algebras are shownto be strongly self-absorbing in Toms and Winter’s original paper [51]. Example . (1) The Cuntz algebras O and O ∞ are strongly self-absorbing.(2) The Jiang–Su algebra Z introduced by Jiang and Su [36] is strongly-self absorbing.(3) Uniformly hyperfinite (UHF) algebras (defined in III.5.1 of [27]) of infinite type arestrongly-self absorbing.(4) The tensor product of a UHF algebra of infinite type with O ∞ is strongly-self ab-sorbing. Note that this is the only way to form a new class of algebras out of theprevious examples, as O absorbs UHF algebras of infinite type, all of the examplesabsorb Z and O ⊗ O ∞ ∼ = O .Although all of these algebras can be used to formulate a notion of twisted K -theory, it isthe Cuntz algebra O ∞ which is used to realise the most general notion of twist, and so this isthe algebra whose properties we will explore in more detail. We are particularly interested inautomorphisms of O ∞ and its stabilisation, so we will present some of Dadarlat and Pennig’s ain results on the homotopy type of these automorphism groups and explore an action ofthe stable unitary group on O ∞ by outer automorphisms.Firstly, a general result showing that the automorphism group of a strongly self-absorbing C ∗ -algebra does not have an interesting homotopy type. Theorem 2.3 (Theorem 2.3 [26]) . Let D be a strongly self-absorbing C ∗ -algebra. Then thespace Aut( D ) is contractible. Upon stabilisation, the homotopy type of the automorphism group becomes much moreinteresting.
Theorem 2.4 (Theorem 2.18 [26]) . There are isomorphisms of groups π i (Aut( O ∞ ⊗ K )) ∼ = (cid:40) K ( O ∞ ) × + if i = 0; K i ( O ∞ ) if i ≥ Z if i = 0; Z if i > even ;0 if i odd . To gain further insight into the automorphisms of O ∞ , in particular the outer automor-phisms, we investigate an action of the infinite unitary group U ( ∞ ) on O ∞ . Note that thisis one of the significant differences between the Cuntz algebra and the algebra of compactoperators – all automorphisms of K are inner but this is not the case for O ∞ .We follow the work of Enomoto, Fujii, Takehana and Watatani in describing an actionof U ( ∞ ) on O n [30]. Let M n ( C ) be the algebra of all n × n matrices over C . For any u = ( u ij ) ∈ M n ( C ), we define a map α u : O ∞ → O ∞ to act on the generators S , S , · · · of O ∞ by α u ( S j ) = n (cid:80) i =1 u ij S i for j = 1 , · · · , n ; S j for j > n ;and extend this map to O ∞ such that it is a homomorphism. It is clear that α u ◦ α u (cid:48) = α uu (cid:48) ,but for general u this map does not give an automorphism of O ∞ . For example, taking u to be the zero matrix we certainly do not obtain an automorphism, so we seek a class ofmatrices for which α u is an automorphism. Lemma 2.5 (Lemma 4 [30]) . For u = ( u ij ) ∈ M n ( C ) , the map α u defines an automorphismof O ∞ if and only if u ∈ U ( n ) . The proof of this lemma is a straightforward computation. This provides an action of U ( n )on O ∞ for all n = 1 , , · · · which we will see extends to an action of U ( ∞ ) on O ∞ . Here weare taking U ( ∞ ) to be the direct limit of U ( n ) ι (cid:44) −→ U ( n + 1) with ι ( A ) = diag( A, u ∈ U ( ∞ ), there exists a finite representative ˆ u ∈ U ( n ) for some n , and every representativeof u will be of the form diag(ˆ u, , · · · ). Therefore all representatives of u define the sameaction on O ∞ , so we define the action of u on O ∞ to be that of its finite representatives.This provides a map α : U ( ∞ ) → Aut( O ∞ ) which satisfies some desirable properties; inparticular, α lands in the group Out( O ∞ ) of outer automorphisms of O ∞ . Theorem 2.6.
The map α : U ( ∞ ) → Out( O ∞ ) is continuous and injective. his is sketched for O n in Theorem 4 of [30], and more details for continuity in the caseof O ∞ are provided in Theorem 2.2.7 of [19].This action provides further insight into O ∞ – there are non-trivial outer automorphismsof this algebra, and a space as large as U ( ∞ ) can act effectively via these automorphisms.Given these preliminaries, we are now in a position to present the formulation of highertwisted K -theory by Pennig.2.2. Formulation.
The critical step in moving from the topological K -theory of a compactspace X to the twisted K -theory is replacing the algebra of continuous complex-valuedfunctions on X with the algebra of sections of a bundle of compact operators over X . Thenext natural extension of this would be to replace the algebra of compact operators with adifferent C ∗ -algebra, and one might expect that using a strongly self-absorbing C ∗ -algebragives the desired construction. This is not quite the case; in fact, Aut( D ) is contractible forall strongly self-absorbing D as stated in Theorem 2.3 and therefore there are no non-trivialalgebra bundles with fibre D over X . Instead, we take the stabilisation of the strongly self-absorbing C ∗ -algebra, the automorphism group of which has a far more interesting homotopytype as mentioned in Theorem 2.4 for O ∞ specifically. This culminates in one of the maintheorems of Pennig and Dadarlat’s paper. Theorem 2.7 (Theorem 3.8 (a), (b) [26]) . Let X be a compact metrisable space and let D be a strongly self-absorbing C ∗ -algebra. The set Bun X ( D ⊗ K ) of isomorphism classesof algebra bundles over X with fibre D ⊗ K becomes an abelian group under the operationof tensor product. Furthermore, B Aut( D ⊗ K ) is the first space in a spectrum defining acohomology theory E D • . While this result is interesting from a homotopy-theoretic point of view, it does not yet tellus that we can obtain the twists of K -theory using this construction. What we want is forthe cohomology theory E D • to be gl ( KU ) • , so that the twists of K -theory may be identifiedwith algebra bundles with fibre D ⊗ K . The cohomology theory obtained, however, dependson the choice of strongly self-absorbing C ∗ -algebra. In fact, in the introduction of [25] theauthors claim that using the Jiang–Su algebra Z yields a subset of twists of K -theory where Z × BU is replaced by { } × BU , and using a tensor product of a UHF algebra of infinitetype with O ∞ yields twists for localisations of KU . The full set of twists is the subject ofthe main theorem of [25]. Theorem 2.8 (Adapted from Theorem 1.1 [25]) . The twists of K -theory over X are classifiedby algebra bundles over X with fibre O ∞ ⊗ K . To be more precise, E O ∞ • (cid:39) gl ( KU ) • andhence Bun X ( O ∞ ⊗ K ) ∼ = gl ( KU ) ( X ) . The significance of this theorem should not be overlooked. The proof requires heavymachinery from stable homotopy theory, much of which is built up over the series of threepapers by the authors [24, 25, 26]. It is only through this deep understanding of the abstractnotion of twist that the authors were able to determine an appropriate model for the twistsof K -theory using geometry and operator algebras.Given this geometric notion of twist, we are now able to define the higher twisted K -theory groups. Note that our algebra bundles are fibre bundles where the trivialisation mapsrestrict to algebra isomorphisms on the fibres, and since we are working with C ∗ -algebraswe see that the space of continuous sections of an algebra bundle over a compact Hausdorff pace X itself forms a C ∗ -algebra equipped with the induced norm and involution from thefibres. Furthermore, if X is only a locally compact Hausdorff space then there is a sensiblenotion of a continuous section of an algebra bundle A over X vanishing at infinity, defined inmuch the same way as C ( X ) and denoted C ( X, A ). It is by taking the operator algebraic K -theory of this C ∗ -algebra that we wish to define higher twisted K -theory. Definition 2.9.
The higher twisted K -theory of the locally compact Hausdorff space X with twist δ represented by the algebra bundle O ∞ ⊗ K → A δ π −→ X is defined to be K n ( X, δ ) = K n ( C ( X, A δ )).This is actually not the definition originally given by Pennig – he follows the homotopy-theoretic approach of using bundles of spectra and defines the higher twisted K -theory groupsto be colimits of certain homotopy groups. The equivalent characterisation that we presentabove is given in his Theorem 2.7(c) [47]. As discussed earlier, replacing O ∞ with otherstrongly self-absorbing C ∗ -algebras here will result in different versions of twisted K -theorywhich will likely also produce interesting results. For instance, Evans and Pennig use infiniteUHF-algebras corresponding to twists of localisations of K -theory in a recent paper [31].As one might expect, there are also notions of relative higher twisted K -theory and highertwisted K -homology. The relative theory can be used to show that higher twisted K -theoryforms a generalised cohomology theory; we will not need this, but the details are presented inDefinition 2.6 and Theorem 2.7 of [47]. While we will be focusing on the twisted cohomologyversion of K -theory, the twisted homology version will be important in some computationsand so we include this definition here. Definition 2.10.
The higher twisted K -homology of the locally compact Hausdorff space X with twist δ represented by the algebra bundle O ∞ ⊗ K → A δ π −→ X is defined to be K n ( X, δ ) = KK n ( C ( X, A δ ) , O ∞ ).Again, this is not the way that Pennig initially defines higher twisted K -homology – heintroduces the topological definition via ∞ -categories – but the definition using KK -theoryis shown to be equivalent in Corollary 3.5 using a Poincar´e duality homomorphism [47].Of course, these definitions can be used to prove a variety of expected properties abouthigher twisted K -theory, including functoriality, cohomology properties and the existenceof a graded module structure. These are proved in [47], and the graded module structurein particular is explored in detail in Section 4.1 of [19]. We note that the appropriatedomain category for the higher twisted K -theory functor is the category of locally compactHausdorff spaces equipped with algebra bundles with fibre O ∞ ⊗ K , such that a morphism( X, A δ X ) → ( Y, A δ Y ) is a proper map f : X → Y together with an algebra isomorphism θ : f ∗ A δ Y → A δ X , to ensure a relationship between the twist on X and the twist on Y .A corollary to Theorem 2.8 is that taking the K -theory of a space X twisted by thetrivial algebra bundle simply returns the ordinary topological K -theory of X , while takingan algebra bundle which corresponds to a classical twist – the tensor product of a bundleof compact operators with the trivial bundle with fibre O ∞ – returns the ordinary twisted K -theory of X . These results are reproved in Proposition 2.3.10 and Proposition 2.3.11 of[19]. The former is a straightforward application of the K¨unneth theorem in C ∗ -algebraic K -theory, and the latter is proved using results on the sections of tensor product bundles.As expected, this shows that higher twisted K -theory contains all of the information oftopological and classical twisted K -theory, along with a great deal more. nother basic property of higher twisted K -theory which we will make use of is the Mayer–Vietoris sequence, which will be of critical importance for computations. Proposition 2.11 (Theorem 2.7(f) [47]) . Let X = U ∪ U for closed sets U k whose interiorscover X . Let i k : U k → X and j k : U ∩ U → U k denote inclusion, and δ | U k = i ∗ k δ and δ | U ∩ U = ( i ◦ j ) ∗ δ denote restriction of the twist δ over X to the corresponding subspaces.Then there is a six-term Mayer–Vietoris sequence as follows: K ( X, δ ) K ( U , δ | U ) ⊕ K ( U , δ | U ) K ( U ∩ U , δ | U ∩ U ) K ( U ∩ U , δ | U ∩ U ) K ( U , δ | U ) ⊕ K ( U , δ | U ) K ( X, δ ) . ( i ∗ ,i ∗ ) j ∗ − j ∗ ∂ ∂ j ∗ − j ∗ ( i ∗ ,i ∗ ) The following is an original proposition from [19] relating higher twisted K -theory andhigher twisted K -homology, which we will later use in computations. Proposition 2.12.
Assume that the algebra of sections vanishing at infinity of any algebrabundle with fibres isomorphic to O ∞ ⊗ K over a locally compact space X is contained in thebootstrap category of C ∗ -algebras defined in Definition 22.3.4 of [10] . If the higher twisted K -theory of X is a direct sum of finite torsion groups, then the higher twisted K -theory andhigher twisted K -homology of X are isomorphic with a degree shift.Proof. The higher twisted K -theory and K -homology groups can be related by the universalcoefficient theorem in KK -theory as in Theorem 23.1.1 of [10], which states that0 → Ext Z ( K ∗ ( A ) , K ∗ ( B )) → KK ∗ ( A, B ) → Hom Z ( K ∗ ( A ) , K ∗ ( B )) → A and B are seperable and A is in the bootstrap category of C ∗ -algebras defined in Definition 22.3.4 of [10]. In order to obtain higher twisted K -homologyas the KK -group in this sequence, we let A be the space of sections of the algebra bundlerepresenting the twist and let B = O ∞ . Then assuming that A is in the bootstrap category,the short exact sequence becomes0 → Ext Z ( K n +1 ( X, δ ) , K ( O ∞ )) ⊕ Ext Z ( K n ( X, δ ) , K ( O ∞ ) → K n ( X, δ ) → Hom Z ( K n ( X, δ ) , K ( O ∞ ) ⊕ Hom Z ( K n +1 ( X, δ ) , K ( O ∞ ) → n . Using the K -theory of O ∞ , this reduces to0 → Ext Z ( K n +1 ( X, δ ) , Z ) → K n ( X, δ ) → Hom Z ( K n ( X, δ ) , Z ) → . Now, if K n ( X, δ ) ∼ = (cid:76) k Z m k,n for some finite sequence { m k,n } and n = 0 , Z (cid:32)(cid:77) k Z m k,n , Z (cid:33) = (cid:77) k Ext Z ( Z m k,n , Z )= (cid:77) k Z m k,n = K n ( X, δ ) , ince the Ext functor is additive in the first variable and Ext Z ( Z m , G ) ∼ = G/mG by prop-erties in Section 3.1 of [35]. Therefore the short exact sequence provides an isomorphism K n +1 ( X, δ ) ∼ = K n ( X, δ ) as required. (cid:3)
Remark . We highlight the assumption made in the statement of Proposition 2.12; thatthe algebra of sections vanishing at infinity of any algebra bundle with fibres isomorphic to O ∞ ⊗K over a locally compact space X is contained in the bootstrap category of C ∗ -algebrasdefined in Definition 22.3.4 of [10], and thus satisfies the universal coefficient theorem. Thisassumption is valid when O ∞ ⊗ K is replaced by K , and it is true for both C ( X ) for anylocally compact Hausdorff space X as well as for O ∞ ⊗ K itself. In fact, it is conjectured thatevery seperable nuclear C ∗ -algebra satisfies the universal coefficient theorem in KK -theory.We note that this Proposition is only used in one computation; Theorem 5.8 in the finalsection.2.3. Links to cohomology.
In the classical case, twists of K -theory were not only viewed asalgebra bundles; often this viewpoint was complemented using cohomology classes. Classicaltwists were often viewed as elements of H ( X, Z ), and it is precisely the Dixmier–Douadytheory which provided a link between these cohomology classes and algebra bundles withfibre K . This raises the question as to whether there is any link between algebra bundles overa space X with fibre O ∞ ⊗ K and the cohomology of X , and the higher Dixmier–Douadytheory posed by Pennig and Dadarlat in relation to strongly self-absorbing C ∗ -algebras is thekey to understanding this. The following results are discussed in generality for all stronglyself-absorbing C ∗ -algebras in Section 4 of [26], but we will specifically consider the use of O ∞ .What we desire is a way to interpret the twists of K -theory, i.e. the elements of the firstgroup of some generalised cohomology theory E O ∞ ( X ), in terms of the ordinary cohomologyof X . This is precisely what a spectral sequence allows. As with any generalised cohomologytheory, there is an Atiyah–Hirzebruch spectral sequence converging to E O ∞ • , the coefficientsof which are given in Theorem 2.4. The E -term of this spectral sequence is as follows.0 1 2 30 H ( X, Z ) H ( X, Z ) H ( X, Z ) H ( X, Z ) − − H ( X, Z ) H ( X, Z ) H ( X, Z ) H ( X, Z ) − − H ( X, Z ) H ( X, Z ) H ( X, Z ) H ( X, Z )Note that we use different indexing conventions to distinguish between different Atiyah–Hirzebruch spectral sequences. For the spectral sequence for E O ∞ • we adopt the indexing sed by Pennig and Dadarlat where negative indices are used, while for the spectral sequencefor higher twisted K -theory developed in Section 4.1 we use standard indexing.At this stage there are complications. The differentials in this sequence are unknown,and even if they were known there may be non-trivial extension problems in determining E O ∞ ( X ). At this point, Pennig and Dadarlat restrict to the setting in which X has torsion-free cohomology, as this implies that the differentials of the sequence are necessarily zeroas they are torsion operators, meaning that their image is torsion, as shown in Theorem2.7 of [2]. It is then clear that there will be no extension problems, as there are no non-trivial extensions of free abelian groups and the only torsion will be in the final summand H ( X, Z ). Thus we obtain the following, noting that to apply the spectral sequence wemust be working with a finite connected CW complex. Theorem 2.13 (Corollary 4.7(ii) [26]) . Let X be a finite connected CW complex such thatthe cohomology ring of X is torsion-free. Then E O ∞ ( X ) ∼ = Bun X ( O ∞ ⊗ K ) ∼ = H ( X, Z ) ⊕ (cid:77) k ≥ H k +1 ( X, Z ) . This shows that there is a relationship between the twists of K -theory and cohomology, atleast in the restrictive case when X has torsion-free cohomology. Even when the cohomol-ogy of X has torsion, the spectral sequence argument shows that the twists will correspondto some subset of these odd-degree cohomology groups depending on differentials and ex-tension problems. This also confirms that, in this case, the classical twists classified by H ( X, Z ) ⊕ H ( X, Z ) are indeed twists of K -theory from the perspective of homotopy the-ory, and provides insight into Pennig’s chosen name – “higher” twisted K -theory. The twistsintroduced by Pennig are higher in the sense that they can be represented by higher degreecohomology classes, as opposed to the classical degree 1 and 3 twists.Note that henceforth, when we are in a setting in which Theorem 2.13 applies, we willidentify the twists of K -theory over X with the odd-degree integral cohomology classes of X . Given a twist we may view this as a cohomology class, and given a cohomology class thiswill represent a twist. This will be particularly important in the development of spectralsequences and in computations.There are some slightly more general statements that can be made even if the cohomologyof the base space has torsion. This will be the case if the torsion does not have any effect onthe previous argument, i.e. if the relevant differentials are necessarily zero and there are noextension problems. Since we are only interested in the degree 1 group of this cohomologytheory, only the groups whose row and column index sum to 1 are relevant, and so we onlyneed to worry about the differentials entering and leaving these groups. If, for instance, onlythe odd cohomology groups of X were torsion-free, the differentials between these relevantgroups would all necessarily be zero, and we would be able to reach the same conclusion thattwists correspond to odd-degree cohomology classes. Even these slightly relaxed assumptionsallow for a wider class of spaces to be considered, including real projective spaces and lensspaces.2.4. Reformulation using Fredholm operators.
To conclude this section, we will presentan alternative definition of the higher twisted K -theory groups which will prove useful inexplicitly describing elements of these groups. We follow an approach of Rosenberg presentedin [49] about classical twisted K -theory. e will need to slightly shift our perspective from that of algebra bundles to that ofprincipal bundles. Recall that there is a correspondence between algebra bundles over X with fibre O ∞ ⊗ K and principal Aut( O ∞ ⊗ K )-bundles over X displayed in the followingdiagram, since Aut( O ∞ ⊗ K ) acts effectively on O ∞ ⊗ K by automorphisms.Aut( O ∞ ⊗ K ) E δ E δ × Aut( O ∞ ⊗K ) ( O ∞ ⊗ K ) X X
Associated bundleTransition functions
This is a bijective correspondence; moving from one perspective to the other and back againyields an isomorphic bundle, and therefore we may view either of the objects above as atwist of K -theory over X . Viewing a twist δ as a principal Aut( O ∞ ⊗ K )-bundle as opposedto an algebra bundle, we define the K -theory of X twisted by δ to be the operator algebraic K -theory of the continuous sections vanishing at infinity of the associated algebra bundle toagree with our previous definition.We also need the notion of Fredholm operators on Hilbert C ∗ -modules to present thisresult. The theory of Hilbert modules and the operators which act on them is developed inPart III of [52], including their relevance in C ∗ -algebraic K -theory. Key results relevant tohigher twisted K -theory are summarised in Section 1.2.3 of [19]. We will make use of thestandard Hilbert ( O ∞ ⊗ K )-module and the space of Fredholm operators on this module,denoted H O ∞ ⊗K and Fred O ∞ ⊗K respectively. We are now able to state the main result ofthis section. Theorem 2.14.
Let X be a compact Hausdorff space and E δ a principal Aut( O ∞ ⊗K ) -bundleover X representing a twist δ . There are natural identifications K ( X, δ ) ∼ = [ E δ , Fred O ∞ ⊗K ] Aut( O ∞ ⊗K ) and K ( X, δ ) ∼ = [ E δ , Ω Fred O ∞ ⊗K ] Aut( O ∞ ⊗K ) where [ , ] Aut( O ∞ ⊗K ) denotes the unbased homotopy classes of Aut( O ∞ ⊗ K ) -equivariantmaps, i.e. π ( C ( , ) Aut( O ∞ ⊗K ) ) , and Ω Fred O ∞ ⊗K is the based loop space of Fred O ∞ ⊗K , i.e.the space of continuous maps S → Fred O ∞ ⊗K with ∈ S mapped to I ∈ Fred O ∞ ⊗K . Note that Aut( O ∞ ⊗ K ) acts on E δ as the structure group of the principal bundle and actson Fred O ∞ ⊗K via conjugation in the same way that P U acts on Fred, meaning that(1) F · T = ( T − ) H F T H where T ∈ Aut( O ∞ ⊗ K ), F ∈ Fred O ∞ ⊗K and we denote the map induced on the standardHilbert O ∞ ⊗ K -module H O ∞ ⊗K by applying T pointwise by T H . The action on Ω Fred O ∞ ⊗K is defined to be this action at every point in the loop.Before proceeding with the proof, we need a standard lemma about sections of principalbundles. emma 2.15. Let E be a principal G -bundle over a topological space X and suppose that G has a continuous and effective left action on a topological space F . Then the space of sectionsof the associated fibre bundle E × G F over X can be identified with the space of G -equivariantmaps E → F . This is part of the proof of Proposition 1.3 in [49]. We are now equipped to prove Theorem2.14.
Proof of Theorem 2.14.
Recall that K ( C ( X, A δ )) can be identified with the group of pathcomponents of the Fredholm operators on the standard Hilbert C ( X, A δ )-module. Thismeans that K ( X, δ ) = K ( C ( X, E δ × Aut( O ∞ ⊗K ) ( O ∞ ⊗ K )))= π (Fred C ( X, E δ × Aut(
O∞⊗K ) ( O ∞ ⊗K )) ) , and applying Lemma 2.15 allows us to replace the continuous sections of the algebra bundleassociated to E δ with the Aut( O ∞ ⊗ K )-equivariant maps from E δ to O ∞ ⊗ K . This allowsus to conclude thatFred C ( X, E δ × Aut(
O∞⊗K ) ( O ∞ ⊗K )) = C ( E δ , Fred O ∞ ⊗K ) Aut( O ∞ ⊗K ) , and so K ( X, δ ) = π ( C ( E δ , Fred O ∞ ⊗K ) Aut( O ∞ ⊗K ) ) = [ E δ , Fred O ∞ ⊗K ] Aut( O ∞ ⊗K ) as required. In order to obtain the result for K , we recall that K ( A ) = K ( SA ) for a C ∗ -algebra A where SA denotes the suspension. In this case, we are interested in the C ∗ -algebra SC ( X, E δ × Aut( O ∞ ⊗K ) ( O ∞ ⊗ K )), which can be viewed as { f : S → C ( X, E δ × Aut( O ∞ ⊗K ) ( O ∞ ⊗ K )) | continuous , f (1) = 0 } . We will suppress the continuity of the function and the fact that f (1) = 0 for brevity, but thesame conditions are required to hold in the following sets where 0 is taken to be the additiveidentity in each case. In the same way as above, we can view the Fredholm operators on thestandard Hilbert C ∗ -module of this C ∗ -algebra as { f : S → C ( X, E δ × Aut( O ∞ ⊗K ) Fred O ∞ ⊗K ) } = { f : S → C ( E δ , Fred O ∞ ⊗K ) Aut( O ∞ ⊗K ) } = C ( E δ × S , Fred O ∞ ⊗K ) Aut( O ∞ ⊗K ) = C ( E δ , C ( S , Fred O ∞ ⊗K )) Aut( O ∞ ⊗K ) = C ( E δ , Ω Fred O ∞ ⊗K ) Aut( O ∞ ⊗K ) . Thus we may conclude that K ( X, δ ) = π ( C ( E δ , Ω Fred O ∞ ⊗K ) Aut( O ∞ ⊗K ) ) = [ E δ , Ω Fred O ∞ ⊗K ] Aut( O ∞ ⊗K ) as required. (cid:3) In the case that principal Aut( O ∞ ⊗K )-bundles over a space X can be explicitly described,this provides a useful way of expressing elements in the higher twisted K -theory groups of X . his formulation can also be extended to obtain expressions for the higher twisted K -theorygroups of higher degree, where we see that K n ( X, δ ) = [ E δ , Ω n Fred O ∞ ⊗K ] Aut( O ∞ ⊗K ) . Although this may be a useful expression, the statement of the theorem already coversthe important cases since Bott periodicity implies that everything will reduce to these twogroups. 3.
Geometric constructions of higher twists
Whilst knowing that the twists of K -theory over a space X may be identified with algebrabundles over X with fibres isomorphic to O ∞ ⊗ K is useful in its own right, the method ofproof does not provide us with an explicit construction of a bundle representing a homotopy-theoretic twist. In particular, since the definition of higher twisted K -theory involves thealgebra of sections of such an algebra bundle, it is easier to compute higher twisted K -theorygroups when there is an explicit bundle to work with. In the general case, even classifyingthe O ∞ ⊗ K bundles over X is a difficult task. In the case that twists can be identified withcohomology classes, however, associating an explicit bundle to each cohomology class willallow for simpler methods of computation. We will explore this in two special cases; whenthe base space is an odd-dimensional sphere and when the twisting class is decomposable.3.1. The clutching construction.
We begin by considering spheres; the clutching con-struction is the key to constructing these algebra bundles in this simple case. Recall thatthe clutching construction in general takes a fibre bundle over each hemisphere and gluesthem together using a gluing function from the equatorial sphere into the structure groupof the fibre bundle. More general versions of the construction exist for other spaces whichwe will mention briefly, but it is particularly useful for the spheres because the gluing mapcan be viewed as an element of a homotopy group of the structure group, which providesa way of classifying these maps in cases that the homotopy type of the structure group isunderstood. It is also useful because the hemispheres are contractible, meaning that with-out loss of generality all fibre bundles over the hemispheres can be assumed trivialised. Inparticular, a principal Aut( O ∞ ⊗ K )-bundle over S n may be constructed by specifying amap f : S n − → Aut( O ∞ ⊗ K ) which will glue trivial bundles over the upper and lowerhemispheres D n + and D n − respectively. More precisely, we make the following definition. Definition 3.1.
Let f : S n − → Aut( O ∞ ⊗ K ) be a continuous map. The clutching bundle E f over S n associated to f is defined to be the quotient of the disjoint union( D n + × Aut( O ∞ ⊗ K )) (cid:113) ( D n − × Aut( O ∞ ⊗ K ))under ( x, T ) ∼ ( x, f ( x ) ◦ T ) for all x ∈ S n − and T ∈ Aut( O ∞ ⊗ K ).Note that technically this equivalence is between points ( x + , T ) and ( x − , f ( x ) ◦ T ) where x + ∈ D n + and x − ∈ D n − both represent the same point x ∈ S n − , but we will suppress thesesubscripts. We could equivalently have constructed an algebra bundle with fibres isomorphicto O ∞ ⊗ K over S n by replacing T ∈ Aut( O ∞ ⊗ K ) with o ∈ O ∞ ⊗ K , but the principalbundle construction will be more convenient due to Theorem 2.14 being stated in terms ofprincipal bundles. The following lemma shows that this distinction is unimportant. emma 3.2. The algebra bundle associated to a principal bundle constructed via the clutch-ing construction is isomorphic to the algebra bundle constructed directly via the clutchingconstruction with the same gluing map.
The proof follows easily from definitions; see Lemma 3.1.4 of [19]. An added benefit of theclutching construction is that there is a simple way to describe the sections of a clutchingbundle using sections of the trivial bundles over the hemispheres. In particular, a section of aclutching bundle can be identified with sections of the trivial bundles which interact via thegluing map as follows. Note that we abbreviate C ( D n + , Aut( O ∞ ⊗K )) ⊕ C ( D n − , Aut( O ∞ ⊗K ))as C ( D n + (cid:113) D n − , Aut( O ∞ ⊗ K )) for brevity. Lemma 3.3.
The space of sections of the clutching bundle over S n associated to the gluingmap f : S n − → Aut( O ∞ ⊗ K ) is of the form C ( S n , E f ) = { ( g, h ) ∈ C ( D n + (cid:113) D n − , Aut( O ∞ ⊗ K )) : g ( x ) = f ( x ) · h ( x ) for all x ∈ S n − } . The proof follows using the definition of the clutching bundle; see Lemma 3.1.2 of [19] formore details. This result is particularly useful because the higher twisted K -theory groupsare defined using the algebra of sections, and so having an explicit realisation of this algebrawill allow computations to be performed more easily.Now, we have an explicit construction of a bundle from a gluing map, but we want to beable to explicitly construct a bundle from a cohomology class of the sphere. To move towardsthis goal, we recall that any principal Aut( O ∞ ⊗ K )-bundle over S n can be constructed inthis way, and furthermore that the isomorphism class of the bundle depends only on thehomotopy class of the gluing map. Proposition 3.4.
There is a bijective correspondence between the set of isomorphism classesof principal
Aut( O ∞ ⊗ K ) -bundles over S n and π n − (Aut( O ∞ ⊗ K )) . The proof for vector bundles is given in Theorem 2.7 of [22], which is easily adapted tothis case in Proposition 3.1.3 of [19].The map defined by the bijective correspondence in Proposition 3.4 is an explicit realisationof the isomorphism induced by viewing S n as the suspension Σ S n − :[ S n , B Aut( O ∞ ⊗ K )] = [Σ S n − , B Aut( O ∞ ⊗ K )]= [ S n − , Ω B Aut( O ∞ ⊗ K )] ∼ = [ S n − , Aut( O ∞ ⊗ K )] . Now, since the cohomology groups of the spheres are torsion-free, we see via Theorem 2.13that the twists of K -theory over the spheres are classified by their odd-degree cohomologygroups, i.e. H n +1 ( S n +1 , Z ) ∼ = Z for n ≥
1. Finally, these correspondences H n +1 ( S n +1 , Z ) ∼ = [ S n +1 , B Aut( O ∞ ⊗ K )] ∼ = π n (Aut( O ∞ ⊗ K ))allow us to obtain explicit geometric representatives for twists over S n +1 given in terms ofcohomology classes. Letting [ δ ] ∈ H n +1 ( S n +1 , Z ) ∼ = Z denote a generator and taking any N ∈ Z , we see that the bundle representing the twist N [ δ ] is constructed via a degree N gluing map, i.e. N times the generator of π n (Aut( O ∞ ⊗ K )) corresponding to [ δ ] under theabove identification. Using this result, we are able to explicitly compute the higher twisted K -theory of the odd-dimensional spheres directly from the definition rather than by usinghigher-powered machinery such as spectral sequences. e also note that the construction can only produce trivial principal Aut( O ∞ ⊗K )-bundlesover even-dimensional spheres, which is expected from Theorem 2.13. This is because abundle over S n would come from a gluing map S n − → Aut( O ∞ ⊗ K ), but the homotopygroup π n − (Aut( O ∞ ⊗ K )) is trivial as stated in Theorem 2.4 and so every gluing map ishomotopic to a constant. This means that any gluing map will construct a bundle whichis isomorphic to the trivial bundle, which agrees with the fact that the odd-dimensionalcohomology of S n is trivial.We will briefly make some more general remarks regarding the clutching construction.Given any cover of a space X and a principal bundle over the disjoint union of the cover withcertain isomorphism conditions imposed on points in the disjoint union which are identifiedto construct X , a principal bundle over X can be constructed. For simplicity, we restrictour attention to spaces which can be covered by two sets as was the case with the sphere.For example, by viewing complex projective space C P n as the quotient of the disk D n + bythe equivalence relation identifying antipodal points on the boundary, C P n can be coveredby the image of a set containing a neighbourhood of the boundary of the disk under theprojection map and a set which does not contain the boundary. The latter of these sets iscontractible but the former is topologically more complicated, meaning that the principalbundles over this set would need to be better understood in order to construct any generalprincipal bundle over C P n . This method could still be used to construct some principalAut( O ∞ ⊗ K )-bundles over C P n , even if it is not possible to construct all in this way. Moresimply, this construction can be used to construct principal Aut( O ∞ ⊗ K )-bundles overproducts of spheres containing at least one odd-dimensional sphere where said sphere is splitinto two hemispheres as above, as we will observe in Section 6.2.3.2. Decomposable cohomology classes.
Another approach to constructing geometricrepresentatives for twists of K -theory is to consider general spaces X but to simplify thetwisting cohomology class. One way to do this is to consider decomposable classes, becausethere already exist geometric representatives for some low-dimensional cohomology classes.To begin with, let X be a finite connected CW complex with torsion-free cohomologyso that we are in the setting of Theorem 2.13. Suppose that δ ∈ H ( X, Z ) decomposesas the cup product δ = α ∪ β with α ∈ H ( X, Z ) and β ∈ H ( X, Z ). By the standardidentification H n ( X, Z ) ∼ = [ X, K ( Z , n )] where K ( G, n ) denotes an Eilenberg–Mac Lane space,i.e. a space whose only non-trivial homotopy group is G in degree n , and the fact that thereexist simple geometric models for K ( Z , n ) in the case that n = 1 , ,
3, we identify δ with H : X → K ( Z ,
5) such that H = H ∧ H with H : X → BU (1) and H : X → BP U .Then H determines a principal U (1)-bundle U (1) P H X π H with Chern class α , and similarly H determines a principal P U -bundle
P U Q H X π H ith Dixmier–Douady invariant β . We form the fibred product bundle over X , whose totalspace is P H × X Q H = { ( p, q ) ∈ P H × Q H : π H ( p ) = π H ( q ) } , and this gives us a principal U (1) × P U -bundle U (1) × P U P H × X Q H X π of which α ∪ β will be an invariant. This is because the fibred product of principal bundlesis the pullback of the direct product of principal bundles under the diagonal map, and thecup product in cohomology is the pullback of the external product under the diagonal map.The direct product of principal bundles corresponds to the external product of cohomol-ogy classes, so the fibred product of principal bundles corresponds to the cup product incohomology.Now, to this principal bundle we wish to associate a principal Aut( O ∞ ⊗ K )-bundle over X , to obtain a twist of K -theory. This is done by defining an injective group homomorphismfrom the structure group U (1) × P U into the automorphism group Aut( O ∞ ⊗ K ), or equiva-lently an effective action of the structure group U (1) × P U on the algebra O ∞ ⊗ K . Since P U is isomorphic to the automorphism group of K by conjugation, we have the obvious action P U ∼ = −→ Aut( K ). We seek an effective action of U (1) on O ∞ .As noted in Section 3 of [39], there is a one-parameter automorphism group of O ∞ obtainedby scaling the generators as follows. Letting λ k for k = 1 , , · · · be a sequence of realconstants, we obtain a map γ : R → Aut( O ∞ ) defined by γ t ( S k ) = e iλ k t S k for k = 1 , , · · · where the S k are the generators in the definition of the Cuntz algebra O ∞ . Then taking λ k = 2 kπ we see that γ is periodic in t with a period of 1. In fact, this is a specialcase of the action that we described in Theorem 2.6, and thus we can view it as a map γ : U (1) → Out( O ∞ ), yielding the desired action.Out of our maps U (1) → Aut( O ∞ ) and P U → Aut( K ), we obtain the product map U (1) × P U → Aut( O ∞ ) × Aut( K ). Then by Corollary T.5.19 of [52] we see that the tensorproduct of two automorphisms of C ∗ -algebras is an automorphism of the tensor productalgebra, so Aut( O ∞ ) × Aut( K ) ⊂ Aut( O ∞ ⊗ K ). Finally, since the map U (1) → Aut( O ∞ )is given by scaling generators whereas P U = U ( H ) /U (1) acts by conjugation on K , thereis no non-trivial action of the U (1) factor on the K component or of the P U factor on the O ∞ component and hence the map U (1) × P U → Aut( O ∞ ⊗ K ) that we have constructedis injective.Thus we may form the associated bundle ( P H × X Q H ) × U (1) × P U
Aut( O ∞ ⊗ K ), which isa principal Aut( O ∞ ⊗ K )-bundle over X . As the 5-class α ∪ β is an invariant of the principal U (1) × P U -bundle, this is a prime candidate for the principal Aut( O ∞ ⊗ K )-bundle over X which corresponds to the class α ∪ β under the isomorphism of Dadarlat and Pennig. Due tothe inexplicit nature of the isomorphism, however, it is not immediate that this will indeedbe the correct bundle. In order to get around this issue, we note that Pennig and Dadarlatuse an Atiyah–Hirzebruch spectral sequence to determine E O ∞ ( X ) = [ X, B
Aut( O ∞ ⊗ K )]in terms of cohomology. Following their argument, the same spectral sequence yields the tandard isomorphism [ X, K ( Z , ∼ = H ( X, Z ). This allows us to conclude that the diagram(2) [ X, K ( Z , H ( X, Z )[ X, B
Aut( O ∞ ⊗ K )] H ( X, Z ) ⊕ (cid:76) k ≥ H k +1 ( X, Z ) ∼ = ∼ = commutes, where we are viewing the elements of [ X, K ( Z , U (1) × P U -bundlesover X obtained using the fibred product construction so the left vertical map takes such abundle to the associated Aut( O ∞ ⊗ K )-bundle via our injective group homomorphism.This allows us to conclude that the principal bundle ( P H × X Q H ) × U (1) × P U
Aut( O ∞ ⊗ K )truly does correspond to the 5-class α ∪ β under the isomorphism of Dadarlat and Pennig.We now extend this argument to the case in which δ is a general element of the cupproduct of H ( X, Z ) and H ( X, Z ), i.e. δ is given by a sum of N decomposable classesof the form considered above. We take α ∈ H ( X, Z N ) and β ∈ H ( X, Z N ) such that δ = (cid:104) α | β (cid:105) where (cid:104)· |·(cid:105) is the pairing H ( X, Z N ) × H ( X, Z N ) → H ( X, Z ) given by the cupproduct and the standard inner product Z N × Z N → Z . As above, we identify α with amap H : X → BU (1) N and β with a map H : X → BP U N , and form the principal torusbundle U (1) N P H X π H with Chern class α and the principal P U N -bundle P U N Q H X π H with Dixmier–Douady invariant β . Once again we take the fibred product to obtain theprincipal U (1) N × P U N -bundle P H × X Q H over X with invariant δ . We now require aninjective map U (1) N × P U N → Aut( O ∞ ⊗ K ) to adapt the previous argument and constructthe associated bundle.Again applying Corollary T.5.19 of [52] and using the fact that O ∞ is nuclear, we seethat Aut( O ∞ ) N ⊂ Aut( O ⊗ N ∞ ) = Aut( O ∞ ). We may then combine two copies of the action γ, γ (cid:48) : U (1) → Aut( O ∞ ) described previously to form an injective map U (1) → Aut( O ⊗ ∞ )given by ( γ ⊗ γ (cid:48) ) ( t,t (cid:48) ) ( S k ⊗ S k (cid:48) ) = e πi ( kt + k (cid:48) t (cid:48) ) S k ⊗ S k (cid:48) . This argument can be extendedto N copies of the action in the same way, giving an injective map U (1) N → Aut( O ∞ ).More simply, the map P U N → Aut( K ) is injective because the automorphisms do notinvolve scaling by constants. Thus we obtain our desired injective group homomorphism U (1) N × P U N → Aut( O ∞ ⊗ K ).We then construct the associated bundle ( P H × X Q H ) × U (1) N × P U N Aut( O ∞ ⊗ K ) over X ,at which point the commutative diagram (2) again allows us to conclude that this principalbundle does correspond to δ under the isomorphism. We have proved the following. heorem 3.5. Let X be a finite connected CW complex with torsion-free cohomology, andtake α ∈ H ( X, Z N ) and β ∈ H ( X, Z N ) with δ = (cid:104) α | β (cid:105) . Denote by P α the total spaceof the principal U (1) -bundle with Chern class α and by Q β the total space of the principal P U -bundle with Dixmier–Douady invariant β . Then ( P α × X Q β ) × U (1) N × P U N Aut( O ∞ ⊗ K ) is a principal Aut( O ∞ ⊗ K ) -bundle over X which corresponds to δ under the isomorphismof Dadarlat and Pennig. A natural extension of this result would be to consider decomposable classes in higherdegrees. For example, we could take α ∪ α ∪ β ∈ H ( X, Z ) ∪ H ( X, Z ) ∪ H ( X, Z ) andaim to construct a bundle in much the same way. One could follow the same constructionsas above in order to do so, and would obtain an extension of the result.There are, however, limitations to this approach. It is difficult to determine whethertwo bundles constructed in this way are isomorphic, and in particular it is even difficultto tell whether a bundle constructed in this way is trivial. For instance, taking a spacesuch as S × S which has non-trivial second- and third-degree integral cohomology, thisconstruction can be used to form a principal Aut( O ∞ ⊗ K )-bundle over S × S . It is notobvious from the construction, however, whether this bundle would correspond to a twist in H ( S × S , Z ) ∼ = Z or whether it would correspond to an integral 5-class on S × S , all ofwhich are trivial.This motivates a great deal of future research in constructing geometric representativesfor twists. Whilst Theorem 3.5 provides geometric representatives for a specific class ofdecomposable twists, the majority of twists cannot be decomposed into pieces as simple asthese. Thus it would be desirable to obtain a more general theorem which is applicable toa wider class of decomposable twists, but it is difficult to obtain geometric representativesfor higher degree cohomology classes. If models for higher Eilenberg–Mac Lane spaces couldbe obtained, and injective group homomorphisms from these into Aut( O ∞ ⊗ K ) could bedetermined, then the methods used would directly generalise to provide geometric represen-tatives in higher degrees. Alternatively, if results in representing cohomology classes usingmaps into the stable unitary group could be obtained in some special cases, then the actionby outer automorphisms given in Theorem 2.6 could be used to extend the results presentedhere. The sections of the constructed bundle can also be explored to determine whether thereis any relationship between the sections of the bundle and the sections of the two bundlesused in the construction, which would aid in computations.In general, the problem of associating geometric representatives to cohomology classesis very difficult. A large amount of work has been done on this for classical twists, forinstance Brylinski uses the theory of loop groups and transgression of cohomology classes toobtain geometric representatives [20] and Bouwknegt, Carey, Mathai, Murray and Stevensonproduce bundle gerbes representing twists [11], but it is not apparent how this work can becarried over to the higher twisted setting. Further research in this area following these ideasmay yield more general results. 4. Spectral sequences
We develop spectral sequences for higher twisted K -theory which will improve our abilityto perform computations in the final section. The arguments to obtain the existence of boththe Atiyah–Hirzebruch and the Segal spectral sequences for generalised cohomology theoriesare standard in the literature, for instance in [21] and [50] respectively. More interesting are esults regarding the differentials in these sequences which are unique to higher twisted K -theory, and which we will present here. Owing to the difficulty in determining the differentialsin these spectral sequences, these results are fairly specialised, but will still be useful incomputations.4.1. Atiyah–Hirzebruch spectral sequence.
The existence of the Atiyah–Hirzebruchspectral sequence for higher twisted K -theory can be established using a filtration of the K -theory group determined by the skeletal filtration of the space, and following the standardargument presented in Chapter XV of [21]. The full argument is given in Section 4.2.1 of[19] and culminates in the following. Theorem 4.1.
Let X be a CW complex with δ a twist over X . There exists an Atiyah–Hirzebruch spectral sequence converging strongly to K ∗ ( X, δ ) with E p,q = H p ( X, K q ( x )) . Note that this spectral sequence is of the exact same form as the Atiyah–Hirzebruchspectral sequence for topological K -theory first developed in [4] and that for twisted K -theory constructed both by Rosenberg [49] and by Atiyah and Segal [7]. All of these se-quences have the same E -term, but the difference lies in the differentials. In the untwistedcase it was found that the first nontrivial differential was given by the Steenrod operation Sq : H p ( X, Z ) → H p +3 ( X, Z ), and upon extending to the classical twisted setting the firstnontrivial differential became the Steenrod operation twisted by the class δ ∈ H ( X, Z ),i.e. the differential is expressed by Sq − ( − ) ∪ δ : H p ( X, Z ) → H p +3 ( X, Z ). We obtain ananalogous result in this setting, where we are now forced to restrict to the case that the twist δ can be represented by a cohomology class. We also lose some information in passing tothe higher twisted setting, as the higher differentials of even the Atiyah–Hirzebruch spectralsequence for topological K -theory are not well-understood. Theorem 4.2.
In the setting of the Atiyah–Hirzebruch spectral sequence, if a twist δ can berepresented by δ ∈ H n +1 ( X, Z ) then the d n +1 differential is the differential d (cid:48) n +1 in the spec-tral sequence for topological K -theory twisted by δ , i.e. d n +1 : H p ( X, Z ) → H p +2 n +1 ( X, Z ) is given by d n +1 ( x ) = d (cid:48) n +1 ( x ) − x ∪ δ .Proof. We follow the argument given in [6]. By definition, the d n +1 differential must be auniversal cohomology operation raising degree by 2 n + 1, defined for spaces with a givenclass δ ∈ H n +1 ( X, Z ). Standard arguments in homotopy theory show that these operationsare classified by H p +2 n +1 ( K ( Z , p ) × K ( Z , n + 1) , Z ) , where the K ( Z , p ) factor represents cohomology operations raising degree by 2 n + 1 andthe K ( Z , n + 1) factor comes from X being equipped with a class δ ∈ H n +1 ( X, Z ). Thiscohomology group is isomorphic to H p +2 n +1 ( K ( Z , p ) , Z ) ⊕ H p +2 n +1 ( K ( Z , n + 1) , Z ) ⊕ Z where the third summand is generated by the product of the generators of H p ( K ( Z , p ) , Z )and H n +1 ( K ( Z , n + 1) , Z ). The only factor which will actually result in an operation H p ( X, Z ) → H p +2 n +1 ( X, Z ) is the first, and so we conclude that the differential is of theform d n +1 ( x ) = d (cid:48) n +1 ( x ) + kx ∪ δ for some k ∈ Z , since the operation must agree withthe spectral sequence for topological K -theory when δ = 0. We determine k by explicitlycomputing the spectral sequence for X = S n +1 as follows. he filtration for this case is particularly simple, with X = X = · · · = X n eachconsisting of a single point and X = S n +1 . Then the spectral sequence reduces to thelong exact sequence for the pair ( X, X ), and the d n +1 differential is the boundary map K ( X , δ | X ) → K ( X, X ; δ ). Equivalently, using the excision property of higher twisted K -theory applied to the compact pair ( S n +1 , D n +1+ ) and excising the interior of D n +1+ wesee that K ( X, X ; δ ) ∼ = K ( D n +1 − , S n ; δ | D n +1 − ) and so d n +1 can be viewed as the boundarymap K ( D n +1+ , δ | D n +1+ ) → K ( D n +1 − , S n ; δ | D n +1 − ). This map is the passage from top-leftto bottom-right in the commutative diagram K ( D n +1+ , δ | D n +1+ ) K ( S n +1 , D n +1+ ; δ ) K ( S n , δ | S n ) K ( D n +1 − , S n ; δ | D n +1 − ) . By studying the six-term exact sequence in higher twisted K -theory associated to thepair ( D n +1 − , S n ), it is clear that the lower horizontal map takes the generator (1 ,
0) of K ( S n , δ | S n ) ∼ = K ( S n ), corresponding to the trivial line bundle over S n , to 0 and thegenerator (0 , n -fold reduced external product of ( H −
1) with H thetautological line bundle over S , to the generator of K ( D n +1 − , S n ; δ | D n +1 − ). All that remainsis to determine the left-hand vertical map. This is done in the proof of Proposition 5.1, inparticular this is the top horizontal map in (5) because in order to identify K ( S n , δ | S n )with Z ⊕ Z we are using the trivialisation of δ over D n +1 − as opposed to D n +1+ . The map isshown to be n (cid:55)→ ( n, − N n ) where the twist δ ∈ H n +1 ( S n +1 , δ ) is given by N ∈ Z times agenerator. Hence the composition sends 1 ∈ Z to − N ∈ Z . Since we see that d n +1 (1) = − N then we may conclude that k = − (cid:3) Whilst this is not quite as explicit as the differential in the classical twisted case, since the d differential of the Atiyah–Hirzebruch spectral sequence for topological K -theory is explic-itly known, it is still useful as all differentials in the Atiyah–Hirzebruch spectral sequencefor topological K -theory are torsion operators [2]. Since this result is only applicable whenthe twist can be represented by cohomology, it will frequently be the case that the space hastorsion-free cohomology and so these torsion differentials will have no effect.Atiyah and Segal are also able to show in [6] that the higher differentials of the spectralsequence for classical twisted K -theory are given rationally by higher Massey products, andthey do so by generalising the Chern character to the twisted setting. This work can likelybe generalised to the higher twisted setting, and in fact the Chern character has alreadybeen generalised to this setting [40], but since it only gives the differentials rationally it isnot highly applicable to computations.4.2. Segal spectral sequence.
A more powerful version of the Atiyah–Hirzebruch spectralsequence is the Segal spectral sequence, which we will use for computing higher twisted K -theory in more complicated settings. One may work through the details of the constructionvia a skeletal filtration which induces a filtration of the higher twisted K -theory group, butwe will not present these details. At this point, we also bring higher twisted K -homology backinto the picture, because it is in the Segal spectral sequence for higher twisted K -homologythat the strongest information about the differentials can be easily obtained. heorem 4.3. Let F ι −→ E π −→ B be a fibre bundle of CW complexes, and suppose that atwist δ over E can be represented by a class δ ∈ H n +1 ( E, Z ) . Then there is a homologicalSegal spectral sequence H p ( B, K q ( F, ι ∗ δ )) ⇒ K ∗ ( E, δ ) and a corresponding cohomological Segal spectral sequence H p ( B, K q ( F, ι ∗ δ )) ⇒ K ∗ ( E, δ ) . These spectral sequences are strongly convergent if the ordinary (co)homology of B is bounded.Proof. The proof follows from standard methods, for instance Rosenberg’s proof of Theorem3 in [49] can be adapted which employs Segal’s original proof in Proposition 5.2 of [50]. (cid:3)
Remark . The ordinary (co)homology of B will be bounded if B is weakly equivalent toa finite-dimensional CW complex and this will cover all of the cases that we consider, so weobtain strong convergence from this spectral sequence.Note that we refer to this as a Segal spectral sequence because the method of proof employsSegal’s original techniques from [50].As mentioned above, there is more that can be said about the differentials in the homologyspectral sequence. Note that the following theorem uses a Hurewicz map in higher twisted K -homology which we have not developed. We will not need to use this map explicitlyat any time, and so we do not present the details of its construction. The construction ofthe map is standard for extraordinary homology theories; see for instance Section II.6 of [1]for homology in general and Section II.14 for ordinary K -homology. In the higher twistedsetting we do not obtain a simple interpretation of the Hurewicz map as Rosenberg is ableto for classical twisted K -homology in the statement of Theorem 6 [49], as Aut( O ∞ ⊗ K ) ishomotopically more complicated than an Eilenberg–Mac Lane space. Theorem 4.4.
In the setting of the homology Segal spectral sequence of Theorem 4.3, supposethat • ι ∗ : H n +1 ( E, Z ) → H n +1 ( F, Z ) is an isomorphism, so that the twisting class δ on E can be identified with the restricted twisting class ι ∗ δ on F , • the differentials d , · · · , d r − leave E r, = H r ( B, K ( F, ι ∗ δ )) unchanged, or equiva-lently E r, = E r, = · · · = E rr, , and • there is a class x ∈ E r, which comes from a class α ∈ π r ( B ) under the Hurewiczmap π r ( B ) → H r ( B, K ( F, ι ∗ δ )) .Then d r ( x ) ∈ E r ,r − is given by the image of α under the composition of the boundary map ∂ : π r ( B ) → π r − ( F ) in the long exact sequence of the fibration and the Hurewicz map π r − ( F ) → K r − ( F, ι ∗ δ ) .Proof. Since the class x was not changed by the differentials d , · · · , d r − and the twistingclass comes from the fibre, we can take B to be S r and E = ( R r × F ) ∪ F without loss ofgenerality where R r × F is π − of the open r -cell in B . In this special case, as noted byRosenberg in the proof of Theorem 6 [49] the spectral sequence comes from the long exactsequence · · · → K r ( F, ι ∗ δ ) ι ∗ −→ K r ( E, δ ) → K r ( E, F, δ ) ∼ = K ( F, ι ∗ δ ) ∂ −→ K r − ( F, ι ∗ δ ) → · · · here we identify K ( F, ι ∗ δ ) with H r ( B, K ( F, ι ∗ δ )). Hence the differential d r is simply theboundary map in this sequence, and the result follows from the naturality of the Hurewiczhomomorphism which implies the commutativity of the diagram π r ( B ) π r − ( F ) H r ( B, K ( F, ι ∗ δ )) K r − ( F, ι ∗ δ ) . ∂ Hurewicz Hurewicz ∂ (cid:3) With these tools developed, we are well-equipped to compute higher twisted K -theory fora variety of spaces. 5. Computations
This final section is dedicated to computation, allowing for the Mayer–Vietoris sequenceand the spectral sequences to be applied. As higher twisted K -theory forms a generalisationof both topological and classical twisted K -theory, we will see that computations for thesevariants of K -theory will fall out as a result of our computations.As discussed in the Introduction, computations in higher twisted K -theory may be ofphysical interest in the realms of string theory and M-theory. While we will not give explicitphysical descriptions of our computations here, further research into the relationship betweenhigher twisted K -theory and physics may help to provide insight into both fields and allowthese results to lead to a greater understanding of M-theory.5.1. Spheres.
We have an explicit description of the bundles of interest over the spheres viathe clutching construction, and so we will begin by computing the higher twisted K -theoryof the odd-dimensional spheres. This should reduce to known results in the case that trivialtwists are used or in the case of classical twists over S . We will provide a highly detailedcomputation for the spheres to illustrate the method of using the Mayer–Vietoris sequenceand determining the maps in the sequence, and present other computations more concisely.Note that in this section we refer to “the generator” of various cohomology groups iso-morphic to Z . The choice of generator is unimportant here, because if a twist is N timesone generator then it will be − N times the other. The integer N only appears in our resultsin the form Z N , and since the groups Z N and Z − N are isomorphic, the results will be trueregardless of the choice of generator. Proposition 5.1.
Let δ ∈ H n +1 ( S n +1 , Z ) be a twist of K -theory for S n +1 which is N times the generator, where n ≥ . The higher twisted K -theory of S n +1 is then K ( S n +1 , δ ) = 0 and K ( S n +1 , δ ) = Z N if N (cid:54) = 0 , or K ( S n +1 ) = Z and K ( S n +1 ) = Z when N = 0 .Proof. Given a twist δ as above, we take a degree N gluing map f : S n → Aut( O ∞ ⊗ K ) andform the clutching bundle E f to represent δ . We then construct a short exact sequence of C ∗ -algebras including the algebra of sections of the associated algebra bundle A δ , allowing he higher twisted K -theory groups to be computed via a six-term exact sequence. As shownin Lemma 3.3, the space of sections of this algebra bundle is of the form A δ = { ( h + , h − ) ∈ C ( D n +1+ (cid:113) D n +1 − , O ∞ ⊗ K ) : h + ( x ) = f ( x ) · h − ( x ) ∀ x ∈ S n } , where again we denote C ( D n +1+ , O ∞ ⊗K ) ⊕ C ( D n +1 − , O ∞ ⊗K ) by C ( D n +1+ (cid:113) D n +1 − , O ∞ ⊗K )for brevity. Then we may define the short exact sequence0 → A δ ι −→ C ( D n +1+ (cid:113) D n +1 − , O ∞ ⊗ K ) π −→ C ( S n , O ∞ ⊗ K ) → , where ι denotes inclusion and π ( h + , h − )( x ) = h + ( x ) − f ( x ) · h − ( x ) to make the sequenceexact. Applying the six-term exact sequence gives K ( A δ ) K ( C ( D n +1+ (cid:113) D n +1 − , O ∞ ⊗ K )) K ( C ( S n , O ∞ ⊗ K )) K ( C ( S n , O ∞ ⊗ K )) K ( C ( D n +1+ (cid:113) D n +1 − , O ∞ ⊗ K )) K ( A δ ) . ι ∗ π ∗ ∂∂ π ∗ ι ∗ We are able to simplify several terms in this sequence using trivialisations of the algebrabundle. Firstly, since the hemispheres D n +1+ and D n +1 − are contractible, the algebra bundle A δ can be assumed trivial over these hemispheres. To be more precise, using a trivialisation t + over the upper hemisphere we are able to identify K n ( C ( D n +1+ , O ∞ ⊗ K )) with Z for n = 0 and 0 for n = 1, and similarly trivialising via t − over the lower hemisphere identifies K ( C ( D n +1 − , O ∞ ⊗ K )) with Z for n = 0 and 0 for n = 1.We may also simplify the terms involving the equatorial sphere S n , since the restrictionof A δ to S n will be necessarily trivial due to S n having trivial odd-degree cohomology. Atthis point we must make a choice of trivialisation, since we have both t + and t − which cantrivialise A δ over S n . We choose t + , and in doing so we identify K ( S n , O ∞ ⊗ K ) with Z ⊕ Z for n = 0 and 0 for n = 1.This reduces the six-term exact sequence to0 → K ( S n +1 , δ ) → Z ⊕ Z π ∗ −→ Z ⊕ Z → K ( S n +1 , δ ) → . The only map to determine here is π ∗ , and to do so we must study the differing triviali-sations of A δ over S n since π ∗ is a priori a map between higher twisted K -theory groups.Using the Mayer–Vietoris sequence in Proposition 2.11, the map π ∗ is given by the difference j ∗ + − j ∗− , where j ± : S n → D n +1 ± is inclusion and j ∗± denotes the induced map on highertwisted K -theory. Since we have trivialised the bundle over S n using t + , we will need to takethe differing trivialisations into account when determining the map j ∗− . The trivialisationsof A δ over S n fit into the commuting diagram(3) K ( S n , δ | S n ) K ( S n ) K ( S n ) ( t + ) ∗ ∼ =( t − ) ∗ ∼ = where the map ( t + ) ∗ ◦ ( t − ) ∗− must be determined to change coordinates from D n +1 − to D n +1+ . In order to do this, we write the trivialisations explicitly.Firstly, observe that the restriction of A δ to D n +1+ is the quotient of( D n +1+ × ( O ∞ ⊗ K )) (cid:113) ( S n × ( O ∞ ⊗ K )) nder the usual equivalence relation on the equatorial sphere. Then t + will be the mapsending the class of ( x, o ) using the representative x ∈ D n +1+ to the point ( x, o ). Similarly, t − sends the class of ( x, o ) using the representative x ∈ D n +1 − to ( x, o ). So taking theequivalence relation on S n into account, these trivialisations differ by the transition function t + ◦ ( t − ) − : S n × ( O ∞ ⊗ K ) → S n × ( O ∞ ⊗ K )given by ( x, o ) (cid:55)→ ( x, f ( x ) − ( o )).These trivialisations induce maps ( t ± ) ∗ : C ( D n +1 ± , A δ | D n +1 ± ) → C ( S n , A δ | S n ) on thesection algebras in the obvious way, and composition gives( t + ◦ ( t − ) − ) ∗ : C ( S n , O ∞ ⊗ K ) → C ( S n , O ∞ ⊗ K )sending g : S n → O ∞ ⊗ K to the map S n (cid:51) x (cid:55)→ f ( x ) − · g ( x ). These maps then in turninduce maps between K -theory groups as in the commutative diagram (3).We will now determine the maps j ±∗ induced on higher twisted K -theory. Firstly for j + we have the commutative diagram(4) K ∗ ( D n +1+ , δ | D n +1+ ) K ∗ ( S n , δ | S n ) K ∗ ( D n +1+ ) K ∗ ( S n ) j + ∗ ( t + ) ∗ ( t + ) ∗ where the lower map K ∗ ( D n +1+ ) → K ∗ ( S n ) is the map induced by j + on ordinary K -theory.Thus we see that j ∗ + is the same as the map induced by j + on ordinary K -theory, which is j ∗ + ( m ) = ( m, t + . On D n +1 − ,however, we must change coordinates via the transition function t + ◦ ( t − ) − so that we aretrivialising the bundle over S n via t + rather than t − . This gives the diagram(5) K ∗ ( D n +1 − , δ | D n +1 − ) K ∗ ( S n , δ | S n ) K ∗ ( D n +1 − ) K ∗ ( S n ) j −∗ ( t − ) ∗ ( t + ) ∗ and thus j ∗− can be viewed as the map induced by j − on ordinary K -theory followed by( t + ◦ ( t − ) − ) ∗ . Since the map induced by t + ◦ ( t − ) − on section algebras is multiplication by f − , we seek the map on K -theory induced by the composition C ( D n +1 − , A δ | D n +1 − ) res −→ C ( S n , A δ | S n ) × f − −−−→ C ( S n , A δ | S n ) . In the case of topological K -theory when N = 0 this is the map n (cid:55)→ ( n, N (cid:54) = 0then the second component of this map is non-trivial, resulting in n (cid:55)→ ( n, − N n ) with thefactor of − N corresponding to multiplication by f − .Thus π ∗ ( m, n ) = ( m, − ( n, − N n ) = ( m − n, N n ), which has trivial kernel and whosecokernel is ( Z ⊕ Z ) / ( Z ⊕ N Z ) ∼ = Z N when N (cid:54) = 0. So we are able to conclude via the exactsequence that K ( S n +1 , δ ) = 0 while K ( S n +1 , δ ) ∼ = Z N . Note that if N = 0 we instead have π ∗ ( m, n ) = ( m − n,
0) with kernel and cokernel Z corresponding to the standard topological K -theory of S n +1 . (cid:3) hile this computation shows that the higher twisted K -theory of the odd-dimensionalspheres agrees with and extends the classical notion of twisted K -theory for S , it is desirableto have an explicit geometric representative for the generator of K ( S n +1 , δ ). To do so,we shift our viewpoint to the equivalent definition of higher twisted K -theory in termsof generalised Fredholm operators presented in Theorem 2.14. Firstly, we need a lemmaallowing us to view this higher twisted K -theory group in a slightly different way. Lemma 5.2.
The higher twisted K -theory group K ( S n +1 , δ ) can be expressed as π ( { ( h + , h − ) ∈ C ( D n +1+ (cid:113) D n +1 − , Ω Fred O ∞ ⊗K ) : h + ( x ) = f ( x ) · h − ( x ) ∀ x ∈ S n } ) . Proof.
Recall that K ( S n +1 , δ ) = π ( C ( E δ , Ω Fred O ∞ ⊗K ) Aut( O ∞ ⊗K ) ). Since E δ is constructedvia the clutching construction, an element of C ( E δ , Ω Fred O ∞ ⊗K ) Aut( O ∞ ⊗K ) can be viewed asa pair of maps h ± : D n +1 ± → Ω Fred O ∞ ⊗K satisfying h + ( x ) = f ( x ) · h − ( x ) for all x ∈ S n .Conversely, any such pair of maps h ± may be glued together to form the equivariant map h : E δ → Ω Fred O ∞ ⊗K via h ([ x, T ])( t )( v ) = (cid:40) h + ( x )( t )( T · v ) if x ∈ D n +1+ ; h − ( x )( t )( T · v ) if x ∈ D n +1 − ;where x ∈ S n +1 , T ∈ Aut( O ∞ ⊗ K ), t ∈ S and v ∈ H O ∞ ⊗K . Firstly, we note that h iswell-defined. If [( x, T )] is chosen with x ∈ S n , then the two possible representatives of thispoint are ( x, T ) ∈ D n +1+ × Aut( O ∞ ⊗ K ) and ( x, f ( x ) ◦ T ) ∈ D n +1 − × Aut( O ∞ ⊗ K ). But h − ( x )( t )(( f ( x ) ◦ T ) · v ) = ( f ( x ) · h − ( x ))( t )( T · v ) by the definition of the action, and this isequal to h + ( x )( t )( T · v ) by the compatibility of the maps on the equatorial sphere. Similarly,the map h is Aut( O ∞ ⊗ K )-equivariant by the definition of the action as required. (cid:3) Using this viewpoint, we are able to construct a representative for the generator of thehigher twisted K -theory group. In our description of the generator, we use an isomorphismbetween the hemisphere D n +1+ with its boundary identified to a point and the sphere S n +1 .This allows us to view a map on S n +1 as a map on D n +1+ which is constant on the boundary.We illustrate this for clarity in Figure 1 in the case of S which can be visualised. ∼ = ∼ = Figure 1.
Illustration of the isomorphism between the hemisphere D withits boundary identified to a point and the 2-sphere S . Adapted from [9]. Proposition 5.3.
The generator of K ( S n +1 , δ ) can be represented by the pair of maps h ± where h + is obtained by taking the generator k ∈ π n +1 (Ω Fred O ∞ ⊗K ) and viewing this as amap D n +1+ → Ω Fred O ∞ ⊗K which is constant on the equatorial sphere via the isomorphismdisplayed in Figure 1, and h − is a loop which remains constant at the identity operator. roof. In order to obtain a generator, we use a different short exact sequence of C ∗ -algebrasto obtain a six-term exact sequence in K -theory. Here we take the sequence0 → C ( R n +1 , O ∞ ⊗ K ) ι −→ A δ π −→ C ( x , O ∞ ⊗ K ) → x ∈ S n +1 defined so that S n +1 \ { x } ∼ = R n +1 , with the obvious maps for x / ∈ S n .Note that we may take sections of the trivial bundle over R n +1 since there are no non-trivialprincipal Aut( O ∞ ⊗ K )-bundles over R n +1 , and similarly for { x } . This gives rise to thesix-term exact sequence0 = K ( R n +1 ) K ( S n +1 , δ ) K ( { x } ) = Z K ( { x } ) K ( S n +1 , δ ) K ( R n +1 ) = Z , ι ∗ π ∗ ∂∂ π ∗ ι ∗ where twisted K -theory groups equipped with the trivial twisting have been identified withtheir untwisted counterparts, and this reduces to0 → Z ∂ −→ Z ι ∗ −→ K ( S n +1 , δ ) → . By Proposition 5.1 we know that K ( S n +1 , δ ) = Z N and so ι ∗ is a surjective map from Z to Z N . This means that it must be given by reduction modulo N and hence the generator of K ( S n +1 , δ ) is the image of the generator of K ( R n +1 ) ∼ = Z under ι ∗ . The map ι ∗ can beinterpreted by making the following identifications: K ( R n +1 ) = (cid:101) K ( S n +1 ) ∼ = K ( S n +1 ) ∼ = [ S n +1 , Ω Fred O ∞ ⊗K ]= π n +1 (Ω Fred O ∞ ⊗K ) , where we use the fact that the ordinary topological K -theory of S n +1 is the same as thehigher twisted K -theory of S n +1 with the trivial twist.In order to realise the reduction modulo N map from π n +1 (Ω Fred O ∞ ⊗K ) into K ( S n +1 , δ ),we let [ k : S n +1 → Ω Fred O ∞ ⊗K ] ∈ π n +1 (Ω Fred O ∞ ⊗K ) be the generator and by identifying S n +1 with D n +1+ / ∼ as illustrated in Figure 1, we view k as a map h + on D n +1+ whichis constant at the identity on the equatorial sphere. Then defining a map h − on D n +1 − tobe a loop which is constant at the identity gives a pair [ h ± ] ∈ K ( S n +1 , δ ) via Lemma 5.2.Applying this process with M times the generator of π n +1 (Ω Fred O ∞ ⊗K ) yields an elementof K ( S n +1 , δ ) which is M mod N times the generator. Thus the generator of K ( S n +1 , δ )is obtained by applying this process to the generator k itself as required. (cid:3) Note that the choice of generator of π n +1 (Ω Fred O ∞ ⊗K ) is once again unimportant here,and different choices will simply yield different generators of Z N .This formulation of the generator agrees with and extends that given by Mickelsson in theclassical twisted setting [46], and the existence of such an explicit generator in terms of thegenerator of π n +1 (Ω Fred O ∞ ⊗K ) may have a physical interpretation which could be used tofurther investigate relevant areas of physics.It should be noted that obtaining explicit expressions for the generators of higher twisted K -theory groups is difficult in general. In this case we relied on applying the Mayer–Vietoris n Z · · · Z n − · · · n − Z · · · Z ... ... ... . . . ...2 Z · · · Z · · · Z · · · Z · · · n n + 1 d n +1 n · · · Z N n − · · · n − · · · Z N ... ... ... . . . ... ...2 0 0 · · · Z N · · · · · · Z N · · · n n + 1 Figure 2.
Atiyah–Hirzebruch spectral sequence for S n +1 .sequence as well as a useful identification of a topological K -theory group with a homotopygroup. This will not be possible in other cases, leaving potential for future work in findingmore general methods to express the generators of higher twisted K -theory groups.To complete our computations for odd-dimensional spheres, we provide a more straightfor-ward proof of Proposition 5.1 using the twisted Atiyah–Hirzebruch spectral sequence. Recallthat the Mayer–Vietoris proof of Proposition 5.1 was used in the proof of Theorem 4.2; thisexample is simply to illustrate the use of the spectral sequence rather than to provide analternative proof. Example . We use the spectral sequence in Theorem 4.1. Since H p ( S n +1 , Z ) ∼ = Z if p = 0or 2 n +1 and the cohomology is trivial otherwise, we see that E p,q ∼ = E p,qr for 2 ≤ r ≤ n . Theonly non-zero differential is then d n +1 : H ( S n +1 , Z ) → H n +1 ( S n +1 , Z ), as displayed on theleft side of Figure 2. By Theorem 4.2, this differential is given by d n +1 ( x ) = d (cid:48) n +1 ( x ) − x ∪ δ where d (cid:48) n +1 is some torsion operator, i.e. the image of d (cid:48) n +1 is torsion. Thus the differentialis simply cup product with − δ , meaning that the E n +1 ∼ = · · · ∼ = E ∞ term is as shown onthe right of Figure 2. Then by the convergence of the spectral sequence, we may once againconclude that K ( S n +1 , δ ) = 0 while K ( S n +1 , δ ) ∼ = Z N .While this computation is much more manageable, it cannot be used to obtain any infor-mation about the generators of the higher twisted K -theory groups. Due to the simplicityof this method, however, it will prove useful in computing the higher twisted K -theory ofmore complicated spaces. .2. Products of spheres.
We may apply the same techniques to compute the highertwisted K -theory of products of spheres. In theory, it is possible to consider any productof spheres consisting of at least one odd-dimensional sphere, since the product will havetorsion-free cohomology and non-trivial cohomology in at least one odd degree. In practice,however, without developing more general techniques we are limited to a smaller class ofproducts.One such product that we can compute via the same methods as used to prove Proposition5.1 is S m × S n +1 for m, n ≥
1. This space has non-trivial odd-degree cohomology groupsin degrees 2 n + 1 and 2 m + 2 n + 1, both of which are isomorphic to Z . The clutchingconstruction is not applicable to twists coming from (2 m + 2 n + 1)-classes, as these wouldproduce bundles over S m +2 n +1 as opposed to S m × S n +1 . In spite of this, a modified versionof the clutching construction can be used for twists of degree 2 n + 1. We take trivial bundlesover S m × D n +1 ± and modify the gluing map [ f ] ∈ π n (Aut( O ∞ ⊗ K ))) ∼ = H n +1 ( S n +1 , Z ) toa map (cid:101) f : S m × S n → Aut( O ∞ ⊗K ) which is constant over the S m factor, i.e. (cid:101) f ( x, y ) = f ( y ).The resulting bundle constructed using this gluing map will then pull back to the trivialbundle over S m and to the usual clutching bundle associated to f over S n +1 . Using thisexplicit bundle, the Mayer–Vietoris sequence can be used to compute the higher twisted K -theory groups. We will not provide this Mayer–Vietoris proof; in this case the proof is almostidentical to the proof of Proposition 5.1. For the full details, see the proof of Proposition5.1.4 in [19]. Instead, we use a K¨unneth theorem in C ∗ -algebraic K -theory to illustrateanother method of computation. Proposition 5.4.
Let δ ∈ H n +1 ( S m × S n +1 , Z ) be a twist of K -theory for S m × S n +1 which is N times the generator, where m, n ≥ . The higher twisted K -theory of S m × S n +1 is then K ( S m × S n +1 , δ ) = 0 and K ( S m × S n +1 , δ ) = Z N ⊕ Z N if N (cid:54) = 0 , or K ( S m × S n +1 ) = Z ⊕ Z and K ( S m × S n +1 ) = Z ⊕ Z when N = 0 .Proof. We use the K¨unneth theorem given in Theorem 23.1.3 of [10], which states that0 → K ∗ ( A ) ⊗ K ∗ ( B ) → K ∗ ( A ⊗ B ) → Tor Z ( K ∗ ( A ) , K ∗ ( B )) → A belongs in the bootstrap category of C ∗ -algebras defined inDefinition 22.3.4 of [10]. We let A denote the continuous complex-valued functions on S m – which belongs in the bootstrap category as do all commutative C ∗ -algebras – so that K ∗ ( A ) = K ∗ ( S m ), and we take B to be the algebra of sections of the algebra bundle over S n +1 representing the twist δ so that K ∗ ( B ) = K ∗ ( S n +1 , δ ). Since K ∗ ( A ) is torsion-free, theTor term will be trivial and thus we obtain an isomorphism K ∗ ( A ) ⊗ K ∗ ( B ) ∼ = K ∗ ( A ⊗ B ).Now, since the algebra bundle A (cid:101) f is trivial over the factor of S m , the sections of the bundlecan be split into C ( S m × S n +1 , A (cid:101) f ) = C ( S m , O ∞ ⊗ K ) ⊗ C ( S n +1 , A (cid:101) f | S n +1 ) ∼ = C ( S m ) ⊗ C ( S n +1 , A f ) . Therefore A ⊗ B is isomorphic to the space of sections of A (cid:101) f , and so we conclude that K ∗ ( A ⊗ B ) = K ∗ ( S m × S n +1 , δ ). Hence the isomorphism given by the K¨unneth theorem erifies that K ( S m × S n +1 , δ ) ∼ = (( Z ⊕ Z ) ⊗ ⊕ (0 ⊗ Z N ) = 0and K ( S m × S n +1 , δ ) ∼ = (( Z ⊕ Z ) ⊗ Z N ) ⊕ (0 ⊗
0) = Z N ⊕ Z N when N (cid:54) = 0 as required. For the case N = 0 the K¨unneth theorem can be applied directlyto the topological K -theory groups of S m and S n +1 . (cid:3) Note that in this case we have two different generators of order N for the K -group, andso it would be of interest to explicitly write down these generators. The Mayer–Vietoristechnique used for S n +1 , however, does not generalise to this case and so this would requirethe development of further machinery.Although twists of degree 2 m + 2 n + 1 do not have such a simple geometric interpretation,the computation can easily be carried out using the Atiyah–Hirzebruch spectral sequence toobtain the following. Proposition 5.5.
Let δ ∈ H m +2 n +1 ( S m × S n +1 , Z ) be a higher twist of K -theory for S m × S n +1 which is N times the generator, where m, n ≥ . The higher twisted K -theoryof S m × S n +1 is then K ( S m × S n +1 , δ ) = Z and K ( S m × S n +1 , δ ) = Z ⊕ Z N if N (cid:54) = 0 , or K ( S m × S n +1 ) = Z ⊕ Z and K ( S m × S n +1 ) = Z ⊕ Z when N = 0 . The proof is a straightforward application of Theorem 4.2 with no extension problems. Wenote that this computation cannot be verified using the K¨unneth theorem because we cannotexpress the space of sections of the algebra bundle representing δ ∈ H m +2 n +1 ( S m × S n +1 , Z )as a tensor product as we were able to in the previous case. Suppose we were to decompose δ into the cup product of δ m ∈ H m ( S m , Z ) and δ n +1 ∈ H n +1 ( S n +1 , Z ) via a K¨unneththeorem in cohomology, and then we let A be the space of sections of the bundle over S m represented by δ m and B be the space of sections of the bundle over S n +1 represented by δ n +1 . Then A and B would be exactly as in the proof of Proposition 5.4, since there are nonon-trivial algebra bundles over S m with fibres isomorphic to O ∞ ⊗ K , and so the tensorproduct algebra A ⊗ B would remain unchanged.Of course there are many other possible products of spheres that can be investigated, andthe spectral sequence can be used in a straightforward way to draw conclusions about thehigher twisted K -theory groups. In spite of this, most cases involve non-trivial extensionproblems and so it is difficult to obtain results for products of spheres in full generality usingcurrent techniques.5.3. Special unitary groups.
Another particularly useful class of spaces with torsion-freecohomology is formed by certain types of Lie groups. A great deal of work has been done bymany mathematicians and physicists in computing the twisted K -theory of Lie groups in theclassical setting, including Hopkins, Braun [18], Douglas [29], Rosenberg [48] and Moore [45].In the case of SU ( n ), the twisted K -groups were explicitly computed and as a consequence itwas shown that the higher differentials in the twisted Atiyah–Hirzebruch spectral sequenceare non-zero in general, suggesting that this technique will not yield general results for the igher twisted K -groups of SU ( n ). Nevertheless, it is possible to compute these groups viathe Atiyah–Hirzebruch spectral sequence in a special case, and to use this in more generalcomputations.We compute the higher twisted K -theory of SU ( n ) up to extension problems for δ a(2 n − K -theory groups,it gives important information regarding torsion and the maximum order of elements in thegroups. To illustrate the general technique, we will start by computing the higher twisted K -theory of SU (3) for a 5-twist. Note that to simplify the statements of results we willhenceforth only consider non-trivial twists, as the trivial case reduces to topological K -theory. Lemma 5.6.
Let δ ∈ H ( SU (3) , Z ) be a twist of K -theory for SU (3) which is N (cid:54) = 0 timesthe generator. The 5-twisted K -theory of SU (3) is then K ( SU (3) , δ ) = Z N and K ( SU (3) , δ ) = Z N . Proof.
The E -page of the twisted Atiyah–Hirzebruch spectral sequence is as follows.2 Z Z Z Z Z Z Z Z d differential is given by the Steenrod operation Sq which necessarily annihilates H ( SU (3) , Z ) by Theorem 4L.12 of [35] and also annihilates H ( SU (3) , Z ) since the imageof Sq is a 2-torsion element by definition. Hence the only non-trivial differential in thisspectral sequence is d ( x ) = d (cid:48) ( x ) − x ∪ δ . The torsion operator d (cid:48) will have no effect on thecohomology, and cup product with − δ will be multiplication by − N on both H ( SU (3) , Z )and H ( SU (3) , Z ). Hence the E ∞ -term is as shown below.2 0 0 0 0 0 Z N Z N Z N Z N K ( SU (3) , δ ) ∼ = Z N and K ( SU (3) , δ ) ∼ = Z N as required. (cid:3) ote that this computation for SU (3) works specifically for 5-twists δ , as unlike when tak-ing a classical 3-twist there are no non-trivial higher differentials to consider. Furthermore,there are no extension problems to solve and so this is a complete computation. The methodof computation directly generalises to the case of (2 n − SU ( n ), although herewe only obtain the result up to extension problems and so we can only comment on torsionin the group. Lemma 5.7.
Let δ ∈ H n − ( SU ( n ) , Z ) be a twist of K -theory for SU ( n ) which is N (cid:54) = 0 times the generator. The (2 n − -twisted K -theory of SU ( n ) is then a finite abelian groupwith all elements having order a divisor of a power of N .Proof. We use the same Atiyah–Hirzebruch spectral sequence approach as in the proof ofLemma 5.6. The differentials d j for j < n − d n − is cup product with − δ , which is multiplication by − N for each of the 2 n − maps Z ( (cid:86) c i ) → Z ( (cid:86) c i ) ∧ c n − where the c i − ∈ H i − ( SU ( n ) , Z ) for i = 2 , · · · , n denote the primitive generators, and the higher differentials are zero. At thisstage, there are 2 n − extension problems to solve for n >
3, but no extension problems for n = 3 which is how the previous result was obtained. In spite of this, since every group in the E ∞ -term of the spectral sequence is Z N , we can conclude that the higher twisted K -theorygroups will be torsion with all elements having order a divisor of a power of N , even if theextension problems cannot be solved to determine the explicit torsion. (cid:3) To be more explicit about the extension problems involved, we consider the case of a7-twist on SU (4). Example . Following the proof of Lemma 5.7, we need to solve a single extension problemboth for the odd-degree and even-degree groups of the form0 → Z N → K i ( SU (4) , δ ) → Z N → . Although this extension problem has Ext Z ( Z N , Z N ) ∼ = Z N inequivalent solutions, we canconclude that K i ( SU (4) , δ ) is a torsion group whose elements have order a divisor of N .While an explicit computation of the higher twisted K -theory of SU ( n ) is difficult ingeneral, we employ techniques of Rosenberg [48] to obtain structural information aboutthese groups. It is at this point that we must turn to the more powerful Segal spectralsequence given in Theorem 4.3 so that Theorem 4.4 may be employed. We begin with aresult specifically for 5-twists, before presenting a more general result. Remark . We reiterate the concerns raised in Remark 2.1 that the proof of Proposition2.12 relies on an assumption which is based on a conjecture in C ∗ -algebra theory. Theorem5.8 should be viewed in light of this assumption. Theorem 5.8.
For any non-zero δ ∈ H ( SU ( n + 1) , Z ) given by N times the generator with N relatively prime to n ! ( n > ), the graded group K ∗ ( SU ( n + 1) , δ ) is isomorphic to Z N tensored with an exterior algebra on n − odd generators.Proof. We proceed by induction on n . First, note that the case n = 2 has already beenproved in Lemma 5.6, as we have shown that K ( SU (3) , δ ) ∼ = Z N ∼ = K ( SU (3) , δ ) so that K ∗ ( SU (3) , δ ) is of the form Z N tensored with Z x for some odd generator x . Then byProposition 2.12, the same is true for the higher twisted K -homology groups. So assume > K -homology for smaller values of n .Take the Segal spectral sequence for higher twisted K -homology associated to the classicalfibration SU ( n ) ι (cid:44) −→ SU ( n + 1) → S n +1 , which gives E p,q = H p ( S n +1 , K q ( SU ( n ) , ι ∗ δ )) ⇒ K ∗ ( SU ( n + 1) , δ ) . Note that since the map ι ∗ induced on ordinary cohomology by inclusion is an isomorphismin degree 5, we may identify ι ∗ δ ∈ H ( SU ( n ) , Z ) with δ ∈ H ( SU ( n + 1) , Z ). Since N isrelatively prime to ( n − K ∗ ( SU ( n ) , δ ) ∼ = Z N ⊗ ∧ ( x , · · · , x n − )for some odd generators x i . We aim to show that the spectral sequence collapses. The onlypotentially non-zero differential is d n +1 , which is related to the homotopical non-trivialityof the fibration as explained in Theorem 4.4. To determine the explicit differential, we needto understand the Hurewicz maps mentioned in the theorem and the long exact sequence inhomotopy for the fibration SU ( n ) ι (cid:44) −→ SU ( n + 1) → S n +1 . This long exact sequence contains π n +1 ( SU ( n + 1)) → π n +1 ( S n +1 ) ∂ −→ π n ( SU ( n )) → π n ( SU ( n + 1)) , and so we see that the boundary map ∂ : Z → Z n ! has kernel of index n !. Now, the Hurewiczmap of interest is π n ( SU ( n )) → K n ( SU ( n ) , δ ) ∼ = K ( SU ( n ) , δ ) . Although this map is difficult to describe explicitly, since it is a map from Z n ! to a groupgenerated by elements of order N then if gcd( N, n !) = 1 this map must be trivial andhence the differential is trivial. Thus if gcd(
N, n !) = 1 then the spectral sequence collapsesand K ∗ ( SU ( n + 1) , δ ) is isomorphic to Z N tensored with an exterior algebra on n − E ∞ -term of the spectral sequence will consist of Z N ⊗ ∧ ( x , · · · , x n − ) inthe zeroth and (2 n + 1)th columns which will become K ( SU ( n + 1) , δ ) and K ( SU ( n + 1) , δ )respectively.In order to conclude that the same is true for higher twisted K -theory, we see that the E -term of the Segal spectral sequence for higher twisted K -theory consists only of finitetorsion groups, and even though we do not have information about the differentials in thissequence we may conclude that the E ∞ -term will also consist only of finite torsion groupsand thus the limit of the sequence is a direct sum of torsion groups. Therefore we are inthe setting of Proposition 2.12, and we may use this to obtain the result for higher twisted K -theory from the computation for higher twisted K -homology. (cid:3) As we have not developed a simple interpretation of the Hurewicz map, we are forcedto consider only the case in which this map is necessarily zero for other reasons. If theHurewicz map could be better understood then this would likely allow for this result to bestrengthened, leading to descriptions of the higher twisted K -groups when gcd( N, n !) (cid:54) = 1.In the classical case, the torsion in these groups becomes very complicated and so it wouldbe a great achievement to obtain explicit expressions for the torsion in the higher twisted K -theory groups of SU ( n ) in general.We also have a structural theorem which is applicable in a more general setting, but whichprovides slightly less information about the higher twisted K -theory groups. heorem 5.9. If δ k ∈ H k ( SU ( n ) , Z ) is given by N (cid:54) = 0 times any primitive generator of H ∗ ( SU ( n ) , Z ) (all of which have odd degree) then K ∗ ( SU ( n ) , δ k ) is a finite abelian groupand all elements have order a divisor of a power of N .Proof. Again we proceed by induction on n , and observe that the base case has already beenproved in Lemma 5.7. Hence we need only show that under this assumption it is true for δ n − ∈ H n − ( SU ( n + 1) , Z ). To do this, we once again use the classical fibration over S n +1 and apply the Segal spectral sequence, this time simply in higher twisted K -theory, to obtain E p,q = H p ( S n +1 , K q ( SU ( n ) , δ n − )) ⇒ K ∗ ( SU ( n + 1) , δ n − ) . Here we are again identifying ι ∗ δ with δ using the isomorphism induced by inclusion oncohomology. Since K q ( SU ( n ) , δ n − ) is torsion with all elements having order a divisor of apower of N by the inductive assumption, the same is true for E and thus E ∞ . Finally, evenif there are non-trivial extension problems to solve in order to obtain K ∗ ( SU ( n + 1) , δ n − ),the result is still true as argued in the proof of Lemma 5.7. (cid:3) In some cases, this result yields particularly useful information. For instance, if N = 1 then K ∗ ( X, δ ) vanishes identically and if N = p r is a prime power then K ∗ ( X, δ ) is a p -primarytorsion group.Whilst these results are not completely general, they do provide some insight into thecomplicated behaviour of the higher twisted K -theory of SU ( n ) and lay the foundation forfurther research into this area. In the classical setting, Rosenberg is able to draw moregeneral conclusions by using a universal coefficient theorem of Khorami [38]. This universalcoefficient theorem relies on techniques that we have not developed in the higher twisted K -theory setting, but given a greater understanding of the cohomology groups of Aut( O ∞ ⊗ K )it may be possible to obtain a universal coefficient theorem in higher twisted K -theory whichcould allow for the results in this section to be generalised. Note that the approaches usedhere can also be applied to other compact, simply connected, simple Lie groups such as thesymplectic groups Sp ( n ) and G , and similar approaches exist for some non-simply connectedgroups such as the projective special unitary groups as used in [42].5.4. Real projective spaces.
We now consider spaces with torsion in their cohomology,and show how in these cases the twists of K -theory can still be identified with odd-degreeintegral cohomology. We begin with odd-dimensional real projective space R P n +1 , whichhas integral cohomology groups(6) H p ( R P n +1 , Z ) = Z if p = 0 , n + 1; Z if 0 < p < n + 1 is even;0 else;and by the universal coefficient theorem the Z -cohomology is H p ( R P n +1 , Z ) = (cid:40) Z if 0 ≤ p ≤ n + 1;0 else . Recall from the discussion preceding Theorem 2.13 that the full set of twists of K -theory overa space X is given by the first group in a generalised cohomology theory E O ∞ ( X ), whichis computed via a spectral sequence. In order to determine whether we may identify the wists of K -theory over R P n +1 with all odd-degree cohomology classes, we use this spectralsequence to compute E O ∞ ( R P n +1 ). The E -term of this spectral sequence is as follows.0 1 2 3 4 · · · n n + 10 Z Z Z Z Z · · · Z Z − · · · − Z Z Z · · · Z Z − · · · − n + 1 0 0 0 0 0 · · · − n Z Z Z · · · Z Z The only differentials that could potentially be non-zero in this spectral sequence are maps H odd ( R P n +1 , Z ) → H even ( R P n +1 , Z ) and H even ( R P n +1 , Z ) → H n +1 ( R P n +1 , Z ). Thelatter of these will necessarily be zero since the target is torsion-free. Now, since the twistsare determined specifically by the first group in this generalised cohomology theory, the onlygroups of interest in this spectral sequence are those circled and thus the only differentialsthat may have an effect are those from H ( R P n +1 , Z ). It is known, however, that theclassical twists of K -theory are those coming from H ( X, Z ) and H ( X, Z ), and as suchthese groups always form a subgroup of E O ∞ ( X ). This means that the differentials leav-ing H ( R P n +1 , Z ) are all trivial, and thus E O ∞ ( R P n +1 ) ∼ = Z ⊕ Z . As the H twistsare studied in the classical case, the higher twists of interest are those coming from the H n +1 ( R P n +1 , Z ) ∼ = Z factor. The same argument may be applied to even-dimensional realprojective space R P n , but this has no non-trivial odd-dimensional cohomology groups andso the only twists of K -theory are those coming from H ( R P n , Z ) ∼ = Z .We will now compute the higher twisted K -theory of R P n +1 for a (2 n + 1)-twist δ . Weprovide two different proofs of this proposition using the Mayer–Vietoris sequence, the latterof which can be generalised to lens spaces. Proposition 5.10.
Let δ ∈ H n +1 ( R P n +1 , Z ) be a twist of K -theory for R P n +1 which is N (cid:54) = 0 times the generator, with n ≥ . The higher twisted K -theory of R P n +1 is then K ( R P n +1 , δ ) = Z n and K ( R P n +1 , δ ) = Z N . roof 1. Letting A δ be the algebra bundle with fibre O ∞ ⊗ K over R P n +1 corresponding tothe twist δ , we consider the short exact sequence of C ∗ -algebras0 → C ( R n +1 , O ∞ ⊗ K ) ι −→ C ( R P n +1 , A δ ) π −→ C ( R P n , O ∞ ⊗ K ) → , where the maps ι and π come from viewing R n +1 as the quotient of R P n +1 by R P n . Thecorresponding six-term exact sequence is K ( R n +1 ) K ( R P n +1 , δ ) K ( R P n ) K ( R P n ) K ( R P n +1 , δ ) K ( R n +1 ) , ι ∗ π ∗ ∂∂ π ∗ ι ∗ where higher twisted K -theory groups have been identified with topological K -theory groupsvia trivialisations, and this simplifies to0 → K ( R P n +1 , δ ) π ∗ −→ Z n ⊕ Z ∂ −→ Z ι ∗ −→ K ( R P n +1 , δ ) → . In this case, the connecting map is given by ∂ ( m, n ) = nN as in 8.3 of [11], and this haskernel Z n and cokernel Z N when N (cid:54) = 0 as required. (cid:3) Although we have not discussed equivariant higher twisted K -theory, the definitions andproperties from the classical case generalise immediately and we use this in the followingproof. Proof 2.
Viewing R P n +1 as the quotient S n +1 / Z , we take the short exact sequence0 → C ( R × S n , O ∞ ⊗ K ) ι −→ C ( S n +1 , A δ ) π −→ C ( { x , x } , O ∞ ⊗ K ) → S n +1 \ { x , x } ∼ = R × S n and x and x are related by the Z -action. The associatedsix-term exact sequence in Z -equivariant K -theory is K Z ( R × S n ) K Z ( S n +1 , δ ) K Z ( { x , x } ) K Z ( { x , x } ) K Z ( S n +1 , δ ) K Z ( R × S n ) , ι ∗ π ∗ ∂∂ π ∗ ι ∗ where trivialisations have been used to identify higher twisted K -theory groups with topolog-ical K -theory groups. Here, since Z acts freely on { x , x } then K ∗ Z ( { x , x } ) = K ∗ ( { x } ).Furthermore, since Z acts freely on S n +1 we also have K ∗ Z ( S n +1 , δ ) ∼ = K ∗ ( R P n +1 , δ ),where the class δ on R P n +1 is identified with a class on S n +1 using the isomorphism in-duced by the projection map. In order to determine the equivariant K -theory of R × S n ,the untwisted version of this six-term exact sequence may be applied analogously to [11] tofind that K Z ( R × S n ) = Z n while K Z ( R × S n ) = Z . Thus this sequence reduces to0 → Z n ι ∗ −→ K ( R P n +1 , δ ) π ∗ −→ Z ∂ −→ Z ι ∗ −→ K ( R P n +1 , δ ) → , where the connecting map ∂ : Z → Z is once again multiplication by N as in [11]. Thisallows us to conclude that K ( R P n +1 , δ ) ∼ = Z n and K ( R P n +1 , δ ) ∼ = Z N as required. (cid:3) ote that this computation agrees with the classical case of the 3-twisted K -theory of R P . Although the Atiyah–Hirzebruch spectral sequence proved useful in our previous com-putations, it is not so helpful in this case due to the torsion in the cohomology of R P n +1 and so we will not use it.5.5. Lens spaces.
The equivariant computation for R P n +1 generalises nicely to lens spaces.While lens spaces of the form S / Z p for p prime are particularly common, there exists a notionof higher lens space L ( n, p ) = S n +1 / Z p where Z p identified with the p th roots of unity in C acts on S n +1 ⊂ C n +1 by scaling. In this notation, the lens space S / Z p is L (1 , p ) andreal projective space R P n +1 can be viewed as L ( n, S n +1 , they have non-trivial cohomology in degree 2 n + 1 and thus it is naturalto consider their higher twisted K -theory.As we did for R P n +1 , we must determine the group E O ∞ ( L ( n, p )) to see whether the twistsof K -theory for L ( n, p ) may be identified with all odd-degree cohomology classes. To do so,we observe that the cohomology groups of L ( n, p ) are the same as those of R P n +1 in (6),but with Z replaced with Z p . Without needing to determine H ∗ ( L ( n, p ) , Z ) we then followthe same argument to conclude that E O ∞ ( L ( n, p )) ∼ = H ( L ( n, p ) , Z ) ⊕ H n +1 ( L ( n, p ) , Z ).Thus we can view the higher twists of K -theory over L ( n, p ) as integral (2 n + 1)-classes. Proposition 5.11.
Let δ ∈ H n +1 ( L ( n, p ) , Z ) be a twist of K -theory for L ( n, p ) which is N (cid:54) = 0 times the generator, where n ≥ . The higher twisted K -theory of L ( n, p ) is then K ( L ( n, p ) , δ ) = Z p n and K ( L ( n, p ) , δ ) = Z N . Proof.
Letting A ⊂ L ( n, p ) consist of a Z p -orbit and A δ denote the algebra bundle with fibre O ∞ ⊗ K over L ( n, p ) corresponding to the twist δ , we take the short exact sequence0 → C ( S n +1 \ A, O ∞ ⊗ K ) ι −→ C ( S n +1 , A δ ) π −→ C ( A, O ∞ ⊗ K ) → . The associated six-term exact sequence in Z p -equivariant higher twisted K -theory is K Z p ( S n +1 \ A ) K Z p ( S n +1 , δ ) K Z p ( A ) K Z p ( A ) K Z p ( S n +1 , δ ) K Z p ( S n +1 \ A ) , ι ∗ π ∗ ∂∂ π ∗ ι ∗ where higher twisted K -theory groups have been identified with topological K -theory groupsvia trivialisations. Since Z p acts freely on the compact set A we have K ∗ Z p ( A ) = K ∗ ( { x } ),and similarly K ∗ Z p ( S n +1 , δ ) = K ∗ ( L ( n, p ) , δ ) where once again δ ∈ H n +1 ( L ( n, p ) , Z ) is iden-tified with δ ∈ H n +1 ( S n +1 , Z ) via the isomorphism induced by the projection map. Itremains to determine K ∗ Z p ( S n +1 \ A ). To do so, we require some basic properties of equi-variant K -theory which carry over from the standard case. Firstly, identifying K ∗ ( S n +1 , A )with K ∗ ( S n +1 \ A ), we have the short exact sequences0 → K m Z p ( S n +1 \ A ) → K m Z p ( S n +1 ) → K m Z p ( A ) → m = 0 and 1. Since K ∗ Z p ( A ) = K ∗ ( { x } ), this implies that K n Z p ( S n +1 \ A ) ∼ = (cid:101) K n Z p ( S n +1 ),which is simply isomorphic to (cid:101) K n ( L ( n, p )). By a computation analogous to that for R P n +1 sing Corollary 2.7.6 of [5], we observe that K ( L ( n, p )) ∼ = Z p n ⊕ Z while K ( L ( n, p )) ∼ = Z .Thus the sequence reduces to0 → Z p n ι ∗ −→ K ( L ( n, p ) , δ ) π ∗ −→ Z ∂ −→ Z ι ∗ −→ K ( L ( n, p ) , δ ) → , where the connecting map ∂ : Z → Z is multiplication by N as in 8.3 of [11]. Hence K ( L ( n, p ) , δ ) ∼ = Z p n and K ( L ( n, p ) , δ ) ∼ = Z N as required. (cid:3) This provides a generalisation of the result for R P n +1 in Proposition 5.10 as well as forthe 3-dimensional lens spaces in Section 8.4 of [11].5.6. SU (2) -bundles. For the final computation we turn to a class of examples of greaterrelevance in physics. It is of great interest in both string theory and mathematical gaugetheory to investigate SU (2)-bundles over manifolds M . The link with string theory appearsthrough spherical T-duality; a generalisation of T-duality investigated in a series of papersby Bouwknegt, Evslin and Mathai [12, 13, 14]. As shown by the authors, this form ofduality provides a map between certain conserved charges in type IIB supergravity and stringcompactifications. They also find links between spherical T-duality and higher twisted K -theory. Specifically in the case that M is a compact oriented 4-manifold, the authors computethe 7-twisted K -theory of the total space of a principal SU (2)-bundle over M in terms ofits second Chern class up to an extension problem. The ordinary 3-twisted K -theory can becomputed using standard techniques, which leaves the 5-twisted K -theory to be computed.In our computations we will assume that M has torsion-free cohomology, which is true if theadditional assumption that M is simply connected is made but this is not necessary.Note that we are not requiring the SU (2)-bundle P over M to be a principal bundle.The computation will be valid for both principal SU (2)-bundles as used in [12] as well asoriented non-principal SU (2)-bundles used in [13], the former of which correspond to unitsphere bundles of quaternionic line bundles and the latter of which correspond to unit spherebundles of rank 4 oriented real Riemannian vector bundles. The reason that we need notdistinguish between these types of bundles is that there is a Gysin sequence in each caseallowing the cohomology of P to be computed. Given an SU (2)-bundle of either of theseforms π : P → M , there is a Gysin sequence of the form · · · → H k ( M, Z ) π ∗ −→ H k ( P, Z ) π ∗ −→ H k − ( M, Z ) ∪ e ( P ) −−−→ H k +1 ( M, Z ) → · · · where e ( P ) denotes the Euler class of P which may be identified with the second Chernclass of the associated vector bundle in the principal bundle case. In this sequence, thepushforward π ∗ is defined by using Poincar´e duality to change from cohomology to homology,using the pushforward in homology and once again employing Poincar´e duality to switchback, which explains the degree shift. We can use this to compute the integral cohomologyof P in terms of that of M . In what follows, we will assume that all exact sequences split inorder to compute the higher twisted K -theory up to extension problems. Although this willnot be true in all cases, it still yields meaningful results as in [12, 13].Firstly, we assume that e ( P ) = 0 in which case the obstruction to π : P → M having asection vanishes and therefore the Gysin sequence truly does split at π ∗ . This yields H k ( P, Z ) ∼ = H k ( M, Z ) ⊕ H k − ( M, Z ) . hus we obtain H k ( P, Z ) ∼ = Z if k = 0 , H k ( M, Z ) if k = 1 , H ( M, Z ) ⊕ Z if k = 3; H ( M, Z ) ⊕ Z if k = 4; H k − ( M, Z ) if k = 5 , . If e ( P ) = j ∈ Z with j (cid:54) = 0 on the other hand, then the cup product with e ( P ) from H ( M, Z ) to H ( M, Z ) will be multiplication by j . We obtain the same integral cohomologyof P as above in all degrees except 3 and 4, and to compute these we use the Gysin sequenceas follows: → H ( M, Z ) π ∗ −→ H ( P, Z ) π ∗ −→ H ( M, Z ) ∪ e ( P ) −−−−→ H ( M, Z ) π ∗ −→ H ( P, Z ) π ∗ −→ H ( M, Z ) → . Since the cup product here has trivial kernel, we conclude that H ( P, Z ) ∼ = H ( M, Z ).Similarly, the cokernel is Z j and hence H ( P, Z ) ∼ = H ( M, Z ) ⊕ Z j assuming that the sequenceis split at π ∗ : H ( P, Z ) → H ( M, Z ).In the case that e ( P ) = 0, we see that P has torsion-free cohomology and thus the 5-twistsgiven by H ( P, Z ) ∼ = H ( M, Z ) can be considered. If e ( P ) = j (cid:54) = 0, however, then there istorsion in the cohomology of P and so it must be determined whether all of the elements of H ( P, Z ) correspond to twists. The E -term of the Atiyah–Hirzebruch spectral sequence tocompute E O ∞ ( P ) is shown below. Z H ( P, Z ) H ( P, Z ) H ( P, Z ) H ( P, Z ) H ( P, Z ) H ( P, Z ) Z − − Z H ( M, Z ) H ( M, Z ) H ( M, Z ) H ( M, Z ) ⊕ Z j H ( M, Z ) H ( M, Z ) Z − − Z H ( M, Z ) H ( M, Z ) H ( M, Z ) H ( M, Z ) ⊕ Z j H ( M, Z ) H ( M, Z ) Z Although there is torsion in H ( P, Z ) which will affect the computation of E ∗O ∞ ( P ), thiswill not have an effect on E O ∞ ( P ) much like was the case for R P n +1 . Since the differentialsare torsion operators then d : H ( M, Z ) → H ( M, Z ) will be zero, and as H ( P, Z ) makesup a subset of the twists then the differential d : H ( P, Z ) → H ( P, Z ) will also neces-sarily be zero. Thus we may conclude that the twists of K -theory are given by odd-degreecohomology in this case, and so it is sensible to use 5-twists and 7-twists for P .To compute the higher twisted K -theory groups themselves, the twisted Atiyah–Hirzebruchspectral sequence may be used. The E -term for e ( P ) = 0 is as follows. Z H ( M, Z ) H ( M, Z ) H ( M, Z ) ⊕ Z H ( M, Z ) ⊕ Z H ( M, Z ) H ( M, Z ) Z Z H ( M, Z ) H ( M, Z ) H ( M, Z ) ⊕ Z H ( M, Z ) ⊕ Z H ( M, Z ) H ( M, Z ) Z Similarly, the E -term for e ( P ) = j (cid:54) = 0 is below. Z H ( M, Z ) H ( M, Z ) H ( M, Z ) H ( M, Z ) ⊕ Z j H ( M, Z ) H ( M, Z ) Z Z H ( M, Z ) H ( M, Z ) H ( M, Z ) H ( M, Z ) ⊕ Z j H ( M, Z ) H ( M, Z ) Z In both cases, we may argue that the d differential Sq is zero as follows. Firstly, Sq ofcourse annihilates H k ( P, Z ) for 0 ≤ k ≤ ≤ k ≤ k = 3 and k = 4to be considered. But the image of Sq is a Z -torsion element by definition, and there isno torsion in H ( P, Z ) or H ( P, Z ) and hence this differential must be zero. Similarly, thetorsion part of d will annihilate all cohomology classes, leaving only the cup product withthe twisting class − δ ∈ H ( P, Z ) to be considered.To determine how cup product with the twisting class affects the cohomology, the iso-morphisms between the cohomology groups of M and P need to be viewed more explicitly.Observe that π ∗ : H k ( P, Z ) → H k − ( M, Z ) is an isomorphism for k = 5 , , π ∗ : H k ( M, Z ) → H k ( P, Z ) is an isomorphism for k = 0 , ,
2. Fixing a twist δ ∈ H ( P, Z ),there is an η ∈ H ( M, Z ) such that ( π ∗ ) − ( η ) = δ . Then taking α k ∈ H k ( M, Z ) for k = 0 , , π ∗ ( α k ) ∈ H k ( P, Z ), we see that π ∗ ( δ ∪ π ∗ ( α k )) ∈ H k ( M, Z ) will be the image of thecup product in the cohomology of M . But by a property of pushforwards, this is equal to π ∗ ( δ ) ∪ α k = η ∪ α k . Hence the cup product with δ on the cohomology of P is equivalent tocup product with η on the cohomology of M . Since this cup product is injective on H ( P, Z ),all higher differentials will be zero and thus the E ∞ -terms of the spectral sequences can bedetermined. We will not explicitly write the E ∞ -terms due to their size, but they can easilybe determined from above.Finally we may conclude that, up to extension problems, the 5-twisted K -theory of P when e ( P ) = 0 is K ( P, δ ) = ker ∪ η | H ( M, Z ) ⊕ H ( M, Z ) ⊕ Z ⊕ coker ∪ η | H ( M, Z ) ; K ( P, δ ) = ker ∪ η | H ( M, Z ) ⊕ H ( M, Z ) ⊕ Z ⊕ H ( M, Z ) / | η | Z ⊕ coker ∪ η | H ( M, Z ) ; nd when e ( P ) = j (cid:54) = 0 is K ( P, δ ) = ker ∪ η | H ( M, Z ) ⊕ H ( M, Z ) ⊕ Z j ⊕ coker ∪ η | H ( M, Z ) ; K ( P, δ ) = ker ∪ η | H ( M, Z ) ⊕ H ( M, Z ) ⊕ H ( M, Z ) / | η | Z ⊕ coker ∪ η | H ( M, Z ) . To be more explicit about the extension problems involved, we illustrate how these highertwisted K -theory groups may differ if the extension problems are non-trivial.Considering the case of e ( P ) = j (cid:54) = 0 above, determining the even-degree higher twisted K -theory group of P would require solving the following:0 → H ( M, Z ) ⊕ Z j → A → coker ∪ η | H ( M, Z ) → → ker ∪ η | H ( M, Z ) → K ( P, δ ) → A → . These extension problems should be kept in mind when viewing the direct sums in theexpressions above.Of course these higher twisted K -theory groups are heavily dependent on the ring structureof the cohomology of M as is evident by the presence of the kernel and cokernel of the cupproduct on M in the expressions given above, but given a specific 4-manifold M with torsion-free cohomology, e.g. a simply connected 4-manifold satisfying the previous assumptions, thisdetermines the 5-twisted K -theory of P up to extension problems. Example . We apply the formulas given above to the manifold M = S × S . Thisspace has trivial cohomology in degrees 1 and 3, and H ( M, Z ) ∼ = Z ⊕ Z . To specify the ringstructure on cohomology, the cup product of the two generators of H ( M, Z ) is the generatorof H ( M, Z ).Consider a twist δ ∈ H ( M, Z ) given by δ = ( Lα , N β ) with α , β the generators and L, N (cid:54) = 0. Then the cup product map H ( M, Z ) → H ( M, Z ) is ( a, b ) (cid:55)→ Lb + N a , which haskernel { ( Li/k, N i/k ) : i ∈ Z } ∼ = Z and cokernel Z k where k denotes the greatest commondivisor of L and N . From this, we see that when e ( P ) = 0 we have K ( P, δ ) = Z ⊕ Z ; K ( P, δ ) = Z ⊕ Z L ⊕ Z N ⊕ Z k ;and when e ( P ) = j (cid:54) = 0 the groups are K ( P, δ ) = Z ⊕ Z j ; K ( P, δ ) = Z L ⊕ Z N ⊕ Z k . Since there is no specified 5-class in the setting of spherical T-duality, the physical in-terpretation of these 5-twisted K -theory groups is not as clear as in the case of 7-twists.Nevertheless, the work of Bouwknegt, Evslin and Mathai provides a link between sphericalT-duality and supergravity theories in Type IIB string theory, and so further research intothis area may shed light on the physical meaning of the computations that we have per-formed. Tying these computations together with physics may then provide further insightinto certain aspects of string theory.To conclude, we remark that many of the computations presented in this section can beused as the starting point for further investigation, particularly in obtaining more generalresults for the higher twisted K -theory of Lie groups and in exploring the link between highertwisted K -theory and string theory. eferences [1] J. F. Adams. Stable Homotopy and Generalised Homology . University of Chicago Press, 1974. ISBN:978-0226005249.[2] Dominique Arlettaz. The order of the differentials in the Atiyah–Hirzebruch spectral sequence.
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School of Mathematical Sciences, University of Adelaide, Adelaide 5005, Australia
E-mail address : [email protected]@adelaide.edu.au