aa r X i v : . [ m a t h . K T ] A p r FREED-MOORE K -THEORY KIYONORI GOMI
Abstract.
The twisted equivariant K -theory given by Freed and Moore is a K -theory which unifies twisted equivariant complex K -theory, Atiyah’s ‘Real’ K -theory, and their variants. In a general setting, we formulate this K -theoryby using Fredholm operators, and establish basic properties such as the Bottperiodicity and the Thom isomorphism. We also provide formulations of the K -theory based on Karoubi’s gradations in both infinite and finite dimensions,clarifying their relationship with the Fredholm formulation. Contents
1. Introduction 21.1. Freed-Moore K -theory 21.2. The purposes of this paper 31.3. Outline of the paper 62. Twisted vector bundle on groupoid 72.1. Groupoid 72.2. Cohomology of groupoid 82.3. Twisted extension 92.4. Twist 112.5. Twisted vector bundle 122.6. Locally universal bundle 153. Fredholm formulation of Freed-Moore K -theory 163.1. Fredholm family 163.2. The Bott periodicity 203.3. Twist and degree shift 223.4. Reproduction of familiar K -theories 233.5. Finite rank realizability 253.6. The Thom isomorphism 274. Karoubi formulation of Freed-Moore K -theory 304.1. Gradation 304.2. Relationship with Fredholm formulation 314.3. Finite-dimensional Karoubi formulation 354.4. Relationship of finite-dimensional formulations 40Appendix A. Classification of some twists 41A.1. Classification of some twists 42A.2. Realization by group cocycle 42Appendix B. Mackey decomposition and the periodicity on a point 45B.1. Mackey decomposition 45 Mathematics Subject Classification.
Primary 19L50; Secondary 19L47, 55R70, 47A53.
Key words and phrases.
Twisted equivariant K -theory, twisted vector bundle, gradation. B.2. The space of Fredholm operators 50B.3. Postponed proof 56Appendix C. Quotient of monoid 58References 581.
Introduction
Freed-Moore K -theory. The conventional complex K -theory K ( X ) of atopological space X , introduced by Atiyah and Hirzebruch [1], can be constructedfrom complex vector bundles on X . Since its introduction, it admits various gen-eralizations such as: • Equivariant K -theory [29]. For this theory to be defined, we consider aspace X with an action of a compact Lie group G . Then the equivariant K -theory K G ( X ) can be constructed from G -equivariant complex vectorbundles on X , namely, vector bundles that admit G -actions covering the G -action on the base space X which induce complex linear transformationson fibers. • Atiyah’s ‘Real’ K -theory [2]. For this to be defined, we consider a space X with an action of the cyclic group Z of order 2 (i.e. an involution).The ‘Real’ K -theory KR ( X ) can be constructed from ‘Real’ vector bun-dles on X , namely, complex vector bundles that admit Z -actions coveringthe Z -action on the base which induce complex anti-linear transformationson fibers. If one takes up the trivial Z -action on X , then the KR -theoryrecovers KO -theory KO ( X ), which can be constructed from real vectorbundles in the usual sense. A variant of ‘Real’ K -theory is Dupont’s ‘Sym-plectic’ (or ‘Quaternionic’) K -theory [16], which recovers the K -theory ofquaternionic vector bundles if the Z -action on X is trivial. • Twisted K -theory [15, 27] and its equivariant version [10]. For this tobe defined, we need additional data called ‘twists’ τ and c on X whichrespectively define cohomology classes in the Borel equivariant cohomology H G ( X ; Z ) and H G ( X ; Z ). Though is not the case in general [6], the twisted K -theory K ( τ,c ) G ( X ) can be constructed from ( τ, c )-twisted vector bundles.If G = Z , then the twisted equivariant K -theory recovers a variant of K -theory K ± ( X ) introduced in [4, 36], by taking [ τ ] ∈ H Z ( X ; Z ) to be trivialand [ c ] ∈ H Z ( X ; Z ) to be the class given by the identity homomorphism c : Z → Z .The twisted equivariant K -theory of Freed and Moore [11] unifies these gener-alizations. Though is not the most general setting, let us consider a space X withan action of a compact Lie group G , a homomorphism φ : G → Z and a twistrepresented by a group 2-cocycle τ ∈ Z ( G ; C ( X, U (1)) φ ) with local coefficientsassociated to φ . Then the Freed-Moore K -theory φ K τG ( X ) is defined by using finiterank twisted equivariant complex vector bundles [11]. The key datum is the homo-morphism φ that morally indicates which element of G acts on the fibers of complex The terminology is due to the recognition of [11] in the community of condensed matterphysics. The ideas of the K -theory were “largely developed in collaboration with Hopkins andTeleman”, according to Freed. REED-MOORE K -THEORY 3 vector bundles complex anti-linearly. Thus, if φ is trivial, then φ K τG ( X ) recoversthe G -equivariant twisted K -theory K τG ( X ). Atiyah’s ‘Real’ K -theory can be re-covered by taking the cyclic group G = Z , the identity homomorphism φ : G → Z and the trivial twist τ . If we turn on a non-trivial twist τ , then the Freed-Moore K -theory recovers Dupont’s ‘Symplectic’ K -theory as a twisted KR -theory.The introduction of the Freed-Moore K -theory is motivated by recent applica-tions of K -theories to the classification of certain quantum systems such as topolog-ical insulators . A remarkable discovery of Kitaev [19] is that the Bott periodicitiesof K -theories explain the so-called periodic table of topological insulators. Someclasses of topological insulators involve a symmetry called the time-reversal sym-metry, and this serves as the source of the appearance of KR -theory. From theviewpoint of condensed matter physics, it is natural to incorporate other sym-metries which stem from the symmetries of crystals. This leads one to considergeneralizations of KR -theory replacing the Z -action by an action of a larger group G . Some nature of the action of a symmetry on quantum systems naturally pro-duces twisting. Then, as an application of the K -theoretic classification schemeof topological insulators, a calculation of equivariant twisted K -theory results in a‘new’ Z -phase of topological crystalline insulators (see [33] for example). Thereforeone can anticipate calculations of Freed-Moore K -theory lead to further discoveryof interesting topological insulators, and a calculation results in a novel Z -phase[32].1.2. The purposes of this paper.
This paper has two purposes.1.2.1.
Fredholm formulation.
The conventional K -theory can be constructed fromvector bundles. However, an analogous construction of twisted K -theory based ontwisted vector bundles [8, 18] of finite rank fails generally. Instead, an infinite-dimensional formulation is required [6, 10, 27, 35]. Thanks to the work of Atiyahand Singer [7], the infinite-dimensional Fredholm formulation [6, 10, 27] is usefulto define K -theory with degree K n ( X ) and to prove the Bott periodicity. The K -theory of Freed-Moore in [11] is formulated by using finite rank (twisted) vectorbundles. Its Fredholm formulation is sketched, but seems not fully developed in theliterature. One purpose of this paper is therefore to give the Fredholm formulationto lay the foundation of this K -theory.We carry out this formulation under a general setting: Let X be a local quotientgroupoid [10]. Then there is a category Φ( X ) whose objects are classified by thecohomology H ( X ; Z ). A typical object in Φ( X ) is a map of groupoids X → pt // Z , where pt // Z is the quotient groupoid associated to the trivial action of Z on a point. Under a choice of φ ∈ Φ( X ), we can introduce a notion of φ -twists.This is a generalization of the notion of twists [10] based on twisted extensions[11]. The φ -twists form a category φ Twist ( X ), and its objects are classified by H ( X ; Z φ ) × H ( X ; Z ). Then, in a way parallel to the formulation of twistedequivariant complex K -theory in [10], we use skew-adjoint Fredholm families on atwisted Z -graded Hilbert bundle to formulate the K -theory φ K ( τ,c )+ n ( X ), where( τ, c ) represents the data of a φ -twist and n ∈ Z is the grading. We can then provethat the K -theory enjoys the Bott periodicity φ K ( τ,c )+ n ( X ) ∼ = φ K ( τ,c )+ n +8 ( X ) . K. GOMI
A consequence of the periodicity is that φ K ( τ,c )+ n ( X ) satisfies the axioms of gener-alized cohomology theory, formulated suitably in the context of groupoids (Theorem3.11). Another consequence is the existence of particular twists c φ and τ φ whichhave the effects of the degree shift (Theorem 3.12) φ K ( τ,c )+ c φ + n ( X ) ∼ = φ K ( τ,c )+ n +2 ( X ) , φ K ( τ,c )+ τ φ + n ( X ) ∼ = φ K ( τ,c )+ n +4 ( X ) , φ K ( τ,c )+( τ φ ,c φ )+ n ( X ) ∼ = φ K ( τ,c )+ n +6 ( X ) . The effect of the degree shift by τ φ generalizes the fact [16] that ‘Symplectic’ K -theory is isomorphic to KR -theory with its degree shifted by 4.As is mentioned, the Freed-Moore K -theory recovers various K -theories underspecializations. Because of the generality of our formulation, we can introduce atwisted KR -theory, which would reproduce the twisted KR -theory in [24]. Weanticipate that the Freed-Moore K -theory would also reproduces the twisted equi-variant KR -theory in [9]. The generality of our formulation further yields twisted K -theories beyond [11]: A simple example is a twisted K -theory φ K n ( X ) of aspace X whose twisting datum φ ∈ Φ( X ) is classified by H ( X ; Z ). This is differ-ent from the twisted K -theory K c + n ( X ) whose twisting datum c is also classifiedby H ( X ; Z ).The proof of the Bott periodicity is based on the idea in [10]: By nature of localquotient groupoid, we reduce the problem to the case of the quotient groupoidpt //G , where G is a compact Lie group. Then, based on the so-called “Mackeydecomposition”, we further reduce the problem to the case that G is trivial. At thispoint, the periodicity essentially follows from [7], which is the reason that we useskew-adjoint Fredholm operators to formulate φ K ( τ,c )+ n ( X ). It should be noticedthat we topologize the space of Fredholm operators by using the compact opentopology in the sense of [6], as opposed to the operator norm topology as in [7].Accordingly, some analytical details about the space of Fredholm operators are alsosupplied in this paper.Also, based on the idea in [10], the Thom isomorphism theorem for real vectorbundles can be shown in the context of the Freed-Moore K -theory (see §§ φ -twisted Pin c -structure introduced in Definition3.17.1.2.2. Karoubi formulation.
In general, a K -theory of a space can be formulatedin various ways. The formulation by using finite rank vector bundles (and the so-called Grothendieck construction) is a standard one. But this formulation cannotbe applied to twisted K -theory in general, as is pointed out already. An infinite-dimensional formulation is therefore necessary for twisted K -theory including theFreed-Moore K -theory, which motivates the Fredholm formulation. It should benoticed that this formulation is in some sense an infinite-dimensional generaliza-tion of the formulation by using finite rank vector bundles and the Grothendieckconstruction.In view of the classifications of gapped quantum systems like topological insula-tors, Karoubi’s formulation of K -theory by using the notion of triples [17] is very REED-MOORE K -THEORY 5 useful, as is seen in [19]. Concretely, in this formulation of the standard complex K -theory K ( X ) of a space X , its representative is a triple ( E, η , η ) consisting ofa finite rank Hermitian vector bundle E on X and two gradations (or Z -gradings ),namely, self-adjoint involutions η and η acting on E . These self-adjoint involu-tions define subbundles Ker(1 − η i ) ⊂ E , and the pair of these vector bundles isnothing but a representative of the standard formulation of K ( X ). In the contextof the classification of gapped quantum systems, the Hamiltonians of such sys-tems lead to self-adjoint involutions η (see for instance [31]). Hence the K -theoryin Karoubi’s formulation naturally works as a framework to measure the relativetopological phases of two gapped quantum systems.One can generalize Karoubi’s triples to formulate Freed-Moore K -theory. How-ever, its relationship with the finite rank formulation as in [11] and the Fredholmformulation seems to be not fully studied in the literature. It should be noticed alsothat the relationship between Karoubi’s formulation and the standard formulationof K -theory cannot be generalized in the presence of a certain twist. The otherpurpose of this paper is thus to clarify the relationship among the formulations.For this purpose, the key is an infinite-dimensional version of Karoubi’s for-mulation above: Based on the infinite-dimensional Grassmannian in [25, 26], weintroduce a group φ K ( τ,c )+ n ( X ) under the same setting as in the Fredholm formu-lation of the Freed-Moore K -theory. We then prove (Theorem 4.11) that there is anatural isomorphism of groups ϑ : φ K ( τ,c )+ n ( X ) ∼ = −→ φ K (´ τ,c )+ n ( X ) . Here one should notice the change of the φ -twists τ ´ τ . It will be shown in §§ τ ∼ = ´ τ if φ or c are trivial. Hence the essential effect of the twist change isobserved only when non-trivial φ and c are present. The appearance of the twistchange is due to the use of skew-adjoint operators in the Fredholm formulation.Using self-adjoint operators instead, one can avoid the twist change (Remark 4.12).To see the relationship between the infinite-dimensional and finite-dimensionalKaroubi formulations, we suppose that the groupoid X is the quotient groupoid X//G associated to an action of a finite group G on a compact Hausdorff space X , φ ∈ Φ( X//G ) is associated to a homomorphism φ : G → Z , and the φ -twistis realized as a twisted extension of X//G . In this setting, we define a group φ K ( τ,c )+ nG ( X ) fin by using Karoubi triples of finite rank, and show (Theorem 4.20)that there is an isomorphism : φ K ( τ,c )+ nG ( X ) fin ∼ = −→ φ K ( τ,c )+ nG ( X ) = φ K ( τ,c )+ n ( X//G ) . To summarize, we denote by φ K ( τ,c )+ nG ( X ) fin the Freed-Moore K -theory formu-lated by finite rank bundles as in [11], and put φ K ( τ,c )+ nG ( X ) = φ K ( τ,c )+ n ( X//G ).Then we have a diagram φ K ( τ,c )+ nG ( X ) fin ı −−−−→ φ K ( τ,c )+ nG ( X ) ∼ = y ϑφ K (´ τ,c )+ nG ( X ) fin −−−−→ ∼ = φ K (´ τ,c )+ nG ( X ) , in which ı is a homomorphism, and ϑ and are isomorphisms. It is stated in [11] that ı is bijective if n = 0 (Remark 7.37), but its proof (Appendix E) seems to work only K. GOMI when the twist c is trivial (see §§ ı byconstructing the inverse of − ◦ ϑ ◦ ı : φ K ( τ,c )+ nG ( X ) fin → φ K (´ τ,c )+ nG ( X ) fin . Actually,the inverse is induced from the construction ( E, η , η ) (Ker(1 − η ) , Ker(1 − η ))as mentioned above (see §§ C ∗ -algebraic formulations. Such a formula-tion of the Freed-Moore K -theory can be found for example in [20, 34]. Notice that,in [34], Karoubi’s triple formulation is also presented in a context of a C ∗ -algebra.These formulations should produce the same K -theory as formulated in this paper.1.3. Outline of the paper. In §
2, we introduce notions of twists and twistedvector bundles needed for the Freed-Moore K -theory. We start with a brief reviewof groupoids and their cohomology. We then recall the notion of twisted extensionin [11], and use it to define φ -twists and twisted vector bundles along the idea of[10]. We also introduce the notion of locally universal bundles following [10].In §
3, we formulate the Freed-Moore K -theory φ K ( τ,c )+ n ( X ) by using Fredholmoperators. As an intermediate step, we introduce a K -theory φ K ( τ,c )+( p,q ) ( X ) withbigrading as in [17], by using the Clifford algebra. We then prove the Bott pe-riodicity. As explained, the proof consists of reductions to easier cases following[10] and the periodicity on the point [7]. The reduction argument based on theMackey decomposition and the periodicity on the point are separated to Appen-dix. Then, we derive the relation between twists and and degree shifts from theBott periodicity. After that, we review how the Freed-Moore K -theory reproducesknown K -theories. We also treat the finite rank realizability here, introducing φ K ( τ,c )+0 G ( X ) fin . At the end of this section, a notion of φ -twisted Pin c -structuresand the Thom isomorphism in the Freed-Moore K -theory are given. § φ K ( τ,c )+( p,q ) ( X ) inthe infinite-dimensional Karoubi formulation, and relate it with the Fredholm for-mulation φ K ( τ,c )+( p,q ) ( X ). We then relate the infinite-dimensional Karoubi formu-lation φ K ( τ,c )+( p,q ) G ( X ) with its finite-dimensional counterpart φ K ( τ,c )+( p,q ) G ( X ) fin .Finally, two finite-dimensional formulations φ K ( τ,c )+0 G ( X ) fin and φ K ( τ,c )+0 G ( X ) fin are compared.In Appendix A, we summarize the classification of twists in some simple casesneeded. In Appendix B, we provide the Mackey decomposition needed for ourreduction argument, and supply some technical details of the Bott periodicity ona point. Finally, in Appendix C, the quotient monoid is reviewed, which is used togive φ K ( τ,c )+0 G ( X ) fin and φ K ( τ,c )+( p,q ) G ( X ) fin .As a convention, a space is always assumed to be locally contractible, para-compact and completely regular, as in [10]. Vector bundles are always Z -graded,and infinite-dimensional cases are allowed. In the infinite-dimensional case, thefibers are assumed to be separable Hilbert spaces, and operators are assumed to bebounded (continuous). Acknowledgements.
I would like to thank I. Sasaki for discussion about some ana-lytic aspects in this work. The author’s research is supported by JSPS KAKENHIGrant Number JP15K04871.
REED-MOORE K -THEORY 7 Twisted vector bundle on groupoid
In this section, we prepare for the setting for the formulation of the Freed-Moore K -theory. We start with a brief review of local quotient groupoids [10] andtheir cohomology groups. We then recall the notion of twisted extension [11], andintroduce φ -twists and twisted vector bundles.2.1. Groupoid. A groupoid X in this paper means a small category in which allthe arrows (morphisms) are invertible, and the set of objects X as well as thatof invertible morphisms (isomorphisms) X are topological spaces subject to ourconvention. We also assume the continuity of the maps X → X that associatesthe source objects s and the target objects t to morphisms s f → t , the map X → X of taking the inverse of arrows, and the map X → X that associates the identityarrows to objects.We will write ∂ : X → X and ∂ : X → X for the associations of the sourceand the target of a morphism, respectively, ∂ ( s f → t ) = s, ∂ ( s f → t ) = t. For n >
1, we denote by X n the space of n composable morphisms, and define ∂ i : X n → X n − , ( i = 0 , . . . , n ) by ∂ i ( f , . . . , f n ) = ( f , . . . , f n ) , ( i = 0)( f , . . . , f i f i +1 , . . . , f n − ) , (1 ≤ i ≤ n − f , . . . , f n − ) , ( i = n )which satisfy ∂ i ◦ ∂ j = ∂ j − ◦ ∂ i . ( i < j )The spaces X , X , X , . . . and the maps ∂ i above, called the face maps, are part ofthe data of the simplicial space associated to the groupoid X . The remaining datacalled the degeneracy maps will play no essential role in this paper, so we omit theirdefinitions here.A well-known example of a groupoid is the quotient groupoid X = X//G , whichis associated to an action of a compact Lie group G on a compact Hausdorff space X . In this groupoid, the set of objects is identified with ( X//G ) = X , and thatof arrows with ( X//G ) = G × X .A map of groupoids X → Y is given by a functor. Taking the topological settinginto account, we assume the induced map of objects X → Y and that of arrows X → Y are continuous. For example, let us consider a map of groupoids φ : X → pt // Z . Since the map of objects is trivial, this φ amounts to a continuous map φ : X → Z such that φ ( f ◦ f ) = φ ( f ) φ ( f ) for all the composable morphisms f , f ∈ X . In particular, a continuous homomorphism φ : G → Z gives a mapfrom the quotient groupoids X//G to pt // Z , although not every maps X//G → pt // Z come from continuous homomorphisms G → Z .As equivalences of groupoids, we consider local equivalences [10]. Then a localquotient groupoid is defined as a groupoid which is covered by full subgroupoidswhich are locally equivalent to the groupoids associated to actions of compact Liegroups on Hausdorff spaces (see [10] for details). K. GOMI
Cohomology of groupoid.
For any abelian group A (or more generally anyring), the cohomology H n ( X ; A ) of a groupoid X can be defined as the cohomologyof the simplicial space associated to X . A convenient way to realize H n ( X ; A ) is touse a ˇCech cohomology (cf. [13]).Any abelian group A admits the automorphism ι : A → A of taking the inverse.Then, combining ι with a map of groupoids φ : X → pt // Z , we can define thecohomology H n ( X ; A φ ) of X with local coefficients. A definition of H n ( X ; A φ ) interms of ˇCech cohomology uses the notion of a twisting function [23] of the simplicialspace associated to X . The twisting function in the present case is the sequence ofmaps φ n : X → Z , ( n ≥
1) defined by φ = φ and φ n = φ ◦ ∂ ◦ · · · ◦ ∂ n for n ≥ φ n · ∂ ∗ φ n − = ∂ ∗ φ n − , ∂ ∗ i φ n − = φ n . ( i > H n ( X ; A ), byusing the twisting function (cf. [12]). This construction produces another doublecomplex, and its cohomology gives H n ( X ; A φ ).Notice that if φ ′ : X → pt // Z is another map and there is ψ : X → Z such that ∂ ∗ ψ · φ = φ ′ · ∂ ∗ ψ , then ψ defines an isomorphism H n ( X ; A φ ) → H n ( X ; A φ ′ ). If ψ ′ : X → Z is another map such that ∂ ∗ ψ ′ · φ = φ ′ · ∂ ∗ ψ ′ , weget the same isomorphism in cohomology. In view of this fact, we regard that maps φ : X → pt // Z constitute objects of a category in which the set of morphismsMor( φ, φ ′ ) consists of maps ψ as above modulo those satisfying ∂ ∗ ψ = ∂ ∗ ψ .More generally, let us consider the category Φ( X ) such that its object is a pair( F : ˜ X → X , φ ) consisting of a local equivalence F : ˜ X → X and a map ofgroupoids φ : ˜ X → pt // Z . We define the set of morphisms from ( F : ˜ X → X , φ )to ( F : ˜ X → X , φ ) to be the direct limit (colimit)lim −→ Y Mor( π ∗ φ , π ∗ φ ) , where Y runs over groupoids which fill the diagram of local equivalences Y π −−−−→ ˜ X π y y F ˜ X F −−−−→ X . By definition, we can associate an object in Φ( X ) to each map of groupoids φ : X → pt // Z by considering the identity local equivalence X → X . In general, Φ( X )contains objects which are not associated to maps of groupoids φ : X → pt // Z as above. However, for the quotient groupoid pt //G , any object in Φ(pt //G ) isisomorphic to the object associated to a homomorphism φ : G → Z .For each object ( F : ˜ X →
X, φ ) in Φ( X ), we have the cohomology H n ( ˜ X ; A φ ).If there is a morphism between two objects in Φ( X ), then it is unique and inducesa unique isomorphism in cohomology. Therefore we take the colimit to define thecohomology twisted by an isomorphism class [ F, φ ] of (
F, φ ) ∈ Φ( X ) as H n ( X ; A [ F,φ ] ) = lim −→ ( F,φ ) ∈ Φ( X ) H n ( ˜ X ; A φ ) . By abuse of notation, we may write φ to mean an object in Φ( X ), and H n ( X ; A φ )for the above cohomology associated to the isomorphism class of φ . REED-MOORE K -THEORY 9 It should be noticed that the objects in Φ( X ) admit the classification π (Φ( X )) ∼ = H ( X ; Z ) , where π (Φ( X )) denotes the set of isomorphism classes. The identification aboveis actually an isomorphism of groups, where the group structure on π (Φ( X )) isinduced from the obvious product of morphism of groupoids φ : X → pt // Z .If X is a quotient groupoid X = X//G , then H n ( X ; A ) can be identified withthe Borel equivariant cohomology H nG ( X ; A ), which is the cohomology of the Borelconstruction EG × G X with its coefficients in A . By definition, the Borel con-struction is the quotient space EG × G X = ( EG × X ) /G , where EG is the totalspace of the universal G -bundle EG → BG and the action of g ∈ G on the directproduct is ( ξ, x ) ( ξg − , gx ). For a map of groupoids φ : X//G → pt // Z , onemay identify the cohomology H n ( X ; A φ ) with the Borel equivariant cohomology H nG ( X ; A φ ), where the local system on EG × G X is the map EG × G X → B Z associated to φ : X//G → pt // Z . In the case where X = pt, the cohomology canbe identified with a group cohomology. The cochain complex producing H nG (pt; A φ )is explicitly given in Appendix A.2.3. Twisted extension.
We introduce some notations following [11]: Given acomplex number z ∈ C and a sign φ ∈ Z = {± } , we write φ z = (cid:26) z, ( φ = 1)¯ z. ( φ = − . Similarly, for a complex vector bundle E → X on a space X , we write φ E = (cid:26) E, ( φ = 1) E, ( φ = − . where E is the complex conjugate of E . As a generalization, for a continuous map φ : X → Z , we define a vector bundle φ E → X by φ E = E | φ − (1) ⊔ E | φ − ( − , noting that we can express X as the disjoint union X = φ − (1) ⊔ φ − ( − E over a space X is called a Z -graded Hermitianvector bundle if E admits a direct sum decomposition E = E ⊕ E into Hermitianvector bundles E i . We call E the even part (or degree 0 part), and E the oddpart (or degree 1 part). Then, a Z -graded Hermitian line bundle amounts to aHermitian line bundle L → X with a Z -grading (or parity) specified. Generalizingthis, for a continuous map c : X → Z , we define a c -graded Hermitian line bundle L → X to be a Hermitian line bundle such that the restriction to c − (( − i ) ⊂ X has degree i . If L is c -graded and L ′ is c ′ -graded, then their tensor product L ⊗ L ′ is cc ′ -graded as a convention. To the exchange of factors, we apply the Koszulsign rule as in [11], so that a negative sign appears only in the exchange of oddhomogeneous elements. Definition 2.1 ([11]) . Let X be a groupoid, and φ : X → pt // Z a map ofgroupoids. A φ -twisted Z -graded extension ( L, τ, c ) of X consists of • a map of groupoids c : X → pt // Z , • a c -graded Hermitian line bundle L → X , and • a unitary isomorphism τ : ∂ ∗ L ⊗ φ ∂ ∗ L → ∂ ∗ L on X which preserves the Z -grading and makes the following diagram commutative on X , ∂ ∗ ∂ ∗ L ⊗ φ ∂ ∗ ( ∂ ∗ L ⊗ φ ∂ ∗ L ) id ⊗ φ ∂ ∗ τ −−−−−−→ ∂ ∗ ∂ ∗ L ⊗ φ ∂ ∗ ∂ ∗ L (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ∂ ∗ ∂ ∗ L ⊗ ∂ ∗ φ ∂ ∗ L ⊗ ∂ ∗ φ ∂ ∗ L ∂ ∗ ∂ ∗ L ⊗ ∂ ∗ φ ∂ ∗ L ∂ ∗ τ ⊗ id y y ∂ ∗ τ ∂ ∗ ∂ ∗ L ⊗ ∂ ∗ φ ∂ ∗ L ∂ ∗ ∂ ∗ L (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ∂ ∗ ∂ ∗ L ⊗ ∂ ∗ φ ∂ ∗ L ∂ ∗ τ −−−−→ ∂ ∗ ∂ ∗ L. The trivial φ -twisted Z -graded extension ( L, τ, c ) consists of the trivial map c : X → pt // Z (i.e. X → Z is the constant map at 1 ∈ Z ), the product bundle L = X × C and the trivial isomorphism τ .We remark that we apply a convention different from the one in [11]. We alsoremark that a φ -twisted ungraded extension of a groupoid X is defined by forgettingabout the information on the Z -grading specified by c . Every φ -twisted ungradedextension of X can be thought of as a φ -twisted Z -graded extension by taking c : X → pt // Z to be trivial. Definition 2.2.
Let X be a groupoid, and φ : X → pt // Z a map of groupoids.An isomorphism [ K, β, b ] : ( L ′ , τ ′ , c ′ ) → ( L, τ, c ) of φ -twisted Z -graded extensionsof X is the equivalence class of data ( K, β, b ) consisting of • a map b : X → Z , • a b -graded Hermitian line bundle K → X , and • a unitary isomorphism β : L ′ ⊗ φ ∂ ∗ K → ∂ ∗ K ⊗ L on X which preservesthe Z -grading and makes the following diagram commutative on X , ∂ ∗ L ′ ⊗ φ ∂ ∗ ( L ′ ⊗ φ ∂ ∗ K ) id ⊗ φ ∂ ∗ β −−−−−−→ ∂ ∗ L ′ ⊗ φ ∂ ∗ ( ∂ ∗ K ⊗ L ) τ ⊗ y (cid:13)(cid:13)(cid:13) ∂ ∗ L ′ ⊗ φ ∂ ∗ φ ∂ ∗ K ∂ ∗ L ′ ⊗ ∂ ∗ φ ∂ ∗ K ⊗ φ ∂ ∗ L (cid:13)(cid:13)(cid:13) y ∂ ∗ β ⊗ id ∂ ∗ L ′ ⊗ ∂ ∗ φ ∂ ∗ K ∂ ∗ ∂ ∗ K ⊗ ∂ ∗ L ⊗ φ ∂ ∗ L ∂ ∗ β y y id ⊗ τ ′ ∂ ∗ ∂ ∗ K ⊗ ∂ ∗ L ∂ ∗ ∂ ∗ K ⊗ ∂ ∗ L. The data (
K, β, b ) and ( K ′ , β ′ , b ′ ) are equivalent if we have • b ′ = ba for a map a : X → Z such that ∂ ∗ a = ∂ ∗ a , and REED-MOORE K -THEORY 11 • a unitary isomorphism α : K → K ′ on X which preserves the Z -gradingand makes the following diagram commutative on X , L ′ ⊗ φ ∂ ∗ K β −−−−→ ∂ ∗ K ⊗ L id ⊗ φ ∂ ∗ α y y ∂ ∗ α ⊗ id L ′ ⊗ φ ∂ ∗ K ′ β ′ −−−−→ ∂ ∗ K ′ ⊗ L. We here examine a special type of a φ -twisted Z -graded extension ( L, τ, c ) of X such that L → X is the product bundle. In this case, the unitary isomorphism τ of Hermitian line bundles amounts to a function τ : X → U (1) satisfying ∂ ∗ τ · ∂ ∗ τ = φ ∂ ∗ τ · ∂ ∗ τ. Moreover, if X is the quotient groupoid X = X//G , then τ is a function τ : G × G × X → U (1) satisfying τ ( g, h ; kx ) · τ ( gh, k ; x ) = φ ( g ) τ ( h, k ; x ) · τ ( g, hk ; x ) . Thus, in terms of group cohomology, we have τ ∈ Z ( G ; C ( X, U (1)) φ ), namely, τ is a 2-cocycle of G with values in the group C ( X, U (1)) of U (1)-valued functionsregarded as a two-sided G -module by the homomorphism φ : G → Z and thepull-back action of G (see Appendix A for details). Under the same assumption,the unitary isomorphism β in an isomorphism [ K, β, b ] : (
L, τ, c ) → ( L ′ , τ ′ , c ′ ) of φ -twisted Z -graded central extensions amounts to a 1-cochain β of G such that τ ( g, h ; x ) β ( gh ; x ) = φ ( g ) β ( h ; x ) β ( g ; hx ) τ ′ ( g, h ; x ) , modulo the coboundary of a 0-cochain a of G .2.4. Twist.
Generalizing [10], we define twists involving φ as follows. Definition 2.3.
Let X be a groupoid, and φ = ( F : ˜ X → X , φ ) an object of Φ( X ).(a) A graded φ -twist (or a twist for short) on X consists of: • a local equivalence ˜ F : ˜˜ X → ˜ X , • ˜ F ∗ φ -twisted Z -graded extension ( L, τ, c ) of ˜˜ X .We may write ( τ, c ) for a φ -twist ( ˜ F : ˜˜ X → ˜ X, L, τ, c ).(b) For φ -twists ( ˜ F : ˜˜ X → ˜ X , L , τ , c ) and ( ˜ F : ˜˜ X → ˜ X , L , τ , c ) on X ,the set of isomorphisms is defined aslim −→ Y Mor(˜ π ∗ ( c , L , τ ) , ˜ π ∗ ( c , L , τ )) , where Y runs over groupoids which fill the diagram of local equivalences˜ Y ˜ π −−−−→ ˜˜ X π y y ˜ F ˜˜ X F −−−−→ ˜ X . As in the case of extensions of groupoids, ungraded twists are defined by forget-ting the information on the Z -grading c in the definition above. Any ungradedtwist can be thought of as a graded twist by the trivial Z -grading. Hence thecategory φ Twist ( X ) of graded φ -twists on X contains the category φ Twist + ( X ) ofungraded φ -twists as its full subcategory. By the tensor product of line bundles, these categories give rise to monoidal categories. While φ Twist + ( X ) is symmetric, φ Twist ( X ) is not, by the Koszul sign rule. Considering the isomorphism classes ofthese monoidal categories, we get the groups π ( φ Twist ( X )) and π ( φ Twist + ( X )).By means of ˇCech cohomology groups (cf. [13, 20]), we can show the followingclassification of twists π ( φ Twist ( X )) ∼ = H ( X ; Z φ ) × H ( X ; Z ) , π ( φ Twist + ( X )) ∼ = H ( X ; Z φ ) . These groups fit into the exact sequence1 → H ( X ; Z φ ) → H ( X ; Z φ ) × H ( X ; Z ) → H ( X ; Z ) → . With some calculations, we can identify the extension class of π ( φ Twist ( X )) withthe cup product ∪ : H ( X ; Z ) × H ( X ; Z ) → H ( X ; Z ) followed by the Bocksteinhomomorphism ˜ β : H ( X ; Z ) → H ( X ; Z φ ) associated to the short exact sequenceof coefficients Z φ → Z φ → ( Z ) φ = Z .2.5. Twisted vector bundle.
As is mentioned, a Z -graded vector bundle E on a space X is a vector bundle with a decomposition E = E ⊕ E . Such adecomposition is in one to one correspondence with an involution ǫ : E → E covering the identity of X . A fiber preserving map f : E → E is said to be degree k if f ◦ ǫ = ( − k ǫf .For p, q ≥
0, we write Cl p,q for the Clifford algebra [3, 21] associated to thequadratic form Q ( x ) = x + · · · + x p − x p +1 − · · · − x p + q on R p + q . Concretely, Cl p,q is the algebra over R generated by e , . . . , e p + q subject to the relations e i e j = − e j e i , ( i = j ) , e i = (cid:26) − , ( i = 1 , . . . , p )1 , ( i = p + 1 , . . . , p + q )As is known, Cl p,q has a natural Z -grading. A representation of Cl p,q or a Cl p,q -module on a Z -graded Hermitian vector space E will mean an algebra homomor-phism γ : Cl p,q → End( E ) of degree 0 such that γ ( e ) is unitary for each vector e ∈ R p + q of unit norm.Now, we introduce the notion of twisted bundles [11] in our convention. Definition 2.4 ([11]) . Let X be a groupoid, φ : X → pt // Z a map of groupoids,and ( L, τ, c ) a φ -twisted Z -graded extension of X . For p, q ≥
0, a ( φ, τ, c ) -twistedvector bundle E over X with Cl p,q -action (or a twisted bundle for short) is a vectorbundle E → X such that its fiber is a separable Hilbert space and is equipped withthe following data: • ( Z -grading) a self-adjoint involution ǫ : E → E which specifies a Z -grading E = E ⊕ E by E k = Ker( ǫ − ( − k ). • (( φ, τ, c )-twisted action) an isometric map ρ : L ⊗ φ ∂ ∗ E → ∂ ∗ E REED-MOORE K -THEORY 13 on X which preserves the Z -grading and makes the following diagramcommutative on X , ∂ ∗ L ⊗ φ ∂ ∗ L ⊗ ( φ ∂ ∗ φ ) ∂ ∗ ∂ ∗ E ∂ ∗ L ⊗ φ ∂ ∗ ( L ⊗ φ ∂ ∗ E ) τ ⊗ id y y id ⊗ φ ∂ ∗ ρ ∂ ∗ L ⊗ ∂ ∗ φ ∂ ∗ ∂ ∗ E ∂ ∗ L ⊗ φ ∂ ∗ ∂ ∗ E (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) ∂ ∗ L ⊗ ∂ ∗ φ ∂ ∗ E ∂ ∗ L ⊗ ∂ ∗ φ ∂ ∗ E ∂ ∗ ρ y y ∂ ∗ ρ ∂ ∗ ∂ ∗ E ∂ ∗ ∂ ∗ E. • ( Cl p,q -action) Unitary maps γ ( e ) : E → E for unit norm elements e ∈ Cl p,q which make each fiber of E into a representation of Cl p,q and the followingdiagram into a commutative one on X , L ⊗ φ ∂ ∗ E ρ −−−−→ ∂ ∗ E ⊗ φ ∂ ∗ γ i y y ∂ ∗ γ i L ⊗ φ ∂ ∗ E ρ −−−−→ ∂ ∗ E. In the definition above, the fiber of a twisted vector bundle can be both infinite-dimensional and finite-dimensional. In the infinite-dimensional case, we assumethat the structure group of E is topologized by the compact open topology in thesense of Atiyah and Segal [6, 11], and the maps ǫ , ρ and γ i are continuous withrespect to the topology.In the case that some of the data φ , c and τ are trivial, we often omit it from themodifier “( φ, τ, c )-twisted”. For example, when φ is trivial, we say ( τ, c )-twistedbundles instead of ( φ, τ, c )-twisted bundles. The same omission will be applied tothe action of Cl , . Definition 2.5.
Let X be a groupoid, φ : X → pt // Z a map of groupoids, and( L, τ, c ) a φ -twisted Z -graded extension of X . A degree k map f : ( E, ǫ, ρ, γ ) −→ ( E ′ , ǫ ′ , ρ ′ , γ ′ )of ( φ, τ, c )-twisted vector bundles on X with Cl p,q -action is a vector bundle map f : E → E which covers the identity of X and satisfies f ◦ ǫ = ( − k ǫ ′ ◦ f, ∂ ∗ f ◦ ρ = ρ ′ ◦ (id L ⊗ φ ∂ ∗ f ) , f ◦ γ ( e ) = ( − k γ ( e ) ◦ f, where e ∈ R p + q is of unit norm.As before, the continuity of f is understood in the compact open topology.It would be helpful to describe the data of a twisted bundle explicitly undera simplifying assumption. Let us consider a quotient groupoid X = X//G and φ associated to a homomorphism φ : G → Z . We further assume that a φ -twisted Z -graded extension ( L, τ, c ) is such that c is associated to a homomorphism c : G → Z and L is the product bundle. Under these assumptions, a ( φ, τ, c )-twisted bundleis a Hilbert space bundle E → X equipped with: • a self-adjoint involution ǫ : E → E defining the Z -grading, • real orthogonal maps ρ ( g ) : E → E which cover the actions of g ∈ G andsatisfy √− ρ ( g ) = φ ( g ) ρ ( g ) √− , ǫρ ( g ) = c ( g ) ρ ( g ) ǫ, ρ ( g ) ρ ( h ) = τ ( g, h ) ρ ( gh ) . • unitary maps γ j : E → E which cover the identity of X and satisfy γ i γ j = − γ j γ i , ( i = j ) , γ i = (cid:26) − , ( i = 1 , . . . , p )1 , ( i = p + 1 , . . . , p + q ) γ i ǫ = − ǫγ i , γ i ρ ( g ) = c ( g ) ρ ( g ) γ i . A degree k map f from this twisted bundle to another ( φ, τ, c )-twisted bundle E ′ → X with the data ǫ ′ , ρ ′ and γ ′ i as above is a vector bundle map f : E → E ′ on X satisfying f ◦ ǫ = ( − k ǫ ′ ◦ f, f ◦ ρ ( g ) = c ( g ) k ρ ′ ( g ) ◦ f, f ◦ γ i = ( − k γ ′ i ◦ f. As usual, a map of vector bundles can be regarded as a section of a vector bundle.
Lemma 2.6.
Let X be a groupoid, φ : X → pt // Z a map of groupoids, and ( L, τ, c ) a φ -twisted Z -graded extension of X . For ( φ, τ, c ) -twisted vector bundles ( E, ǫ, ρ, γ ) and ( E ′ , ǫ ′ , ρ ′ , γ ′ ) on X with Cl p,q -action, there is a φ -twisted vectorbundle Hom Cl p,q ( E, E ′ ) on X such that the sections of its degree k part are in oneto one correspondence with degree k maps f : ( E, ǫ, ρ, γ ) → ( E ′ , ǫ ′ , ρ ′ , γ ′ ) .Proof. We first consider the case without the Clifford actions ( p = q = 0). The φ -twisted vector bundle Hom( E, E ′ ) is constructed as follows. Its underlying vectorbundle is Hom( E, E ′ ) on X . This vector bundle has the Z -grading ε by the degreeof maps, and its fiber is a Hilbert space since E and E ′ are. In the compact opentopology, continuous sections of Hom( E, E ′ ) → X are in one to one correspondencewith continuous maps E → E ′ . On X , we define a φ -twisted action ̺ to be thecomposition of the degree 0 maps φ ∂ ∗ Hom(
E, E ′ ) = Hom( L ⊗ φ ∂ ∗ E, L ⊗ φ ∂ ∗ E ′ ) → ∂ ∗ Hom(
E, E ′ ) .f id ⊗ f ρ ′ ◦ (id ⊗ f ) ◦ ρ − This map ̺ satisfies the coherence condition on X , making (Hom( E, E ′ ) , ε, ̺ ) intoa φ -twisted vector bundle on X . By construction, a section of the degree k partHom k ( E, E ′ ) is a section s : X → Hom k ( E, E ′ ) such that ̺ ◦ φ ∂ ∗ s = ∂ ∗ s . Suchsections s are clearly one to one correspondence with degree k maps E → E ′ of( φ, τ, c )-twisted bundles. If the Cl p,q -actions are present, then there clearly existsa subbundle Hom Cl p,q ( E, E ′ ) of Hom( E, E ′ ) respecting the Clifford actions. (cid:3) For a groupoid X , a map of groupoids φ : X → Z , and a φ -twisted Z -gradedextension ( L, τ, c ) of X , we denote the category of ( φ, τ, c )-twisted vector bundleson X with Cl p,q -action by φ Vect ( τ,c )+( p,q ) ( X ) . In the case that X is a quotient groupoid X//G , we may write φ Vect ( τ,c )+( p,q ) ( X//G ) = φ Vect ( τ,c )+( p,q ) G ( X ) . The tensor product of twisted bundles induces a functor ⊗ : φ Vect ( τ,c )+( p,q ) ( X ) × φ Vect ( τ ′ ,c ′ )+( p ′ ,q ′ ) ( X ) → φ Vect (( τ,c )+( τ ′ ,c ′ ))+( p + p ′ ,q + q ′ ) ( X ) . REED-MOORE K -THEORY 15 A map of groupoids f : X ′ → X also induces by pull-back a functor f ∗ : φ Vect ( τ,c )+( p,q ) ( X ) −→ f ∗ φ Vect ( f ∗ τ,f ∗ c )+( p,q ) ( X ′ ) , and a representative ( K, β, b ) of an isomorphism [
K, β, b ] : ( L ′ , τ ′ , c ′ ) → ( L, τ, c ) of φ -twisted Z -graded extensions induces( K, β, b ) ∗ : φ Vect ( τ,c )+( p,q ) ( X ) −→ φ Vect ( τ ′ ,c ′ )+( p,q ) ( X )by the assignment of twisted bundles E K ⊗ E . We remark that, in general, anautomorphism of ( L, τ, c ) acts non-trivially on φ Vect ( τ,c )+( p,q ) ( X ).For maps f i : X ′ → X of groupoids ( i = 0 , f : X ′ × [0 , → X such that ˜ f | X ×{ i } = ˜ f i , where X ′ × [0 ,
1] is the groupoid suchthat ( X ′ × [0 , j = X ′ j × [0 , Lemma 2.7 (homotopy property) . Let X be a local quotient groupoid, and φ : X → pt // Z a map of groupoids. Suppose that there is a homotopy ˜ f : X ′ × [0 , → X of maps ˜ f and ˜ f from another groupoid X ′ to X . (a) Let ( L, τ, c ) be a φ -twisted Z -graded extension of X . Then there is a uniqueisomorphism β : f ∗ ( L, τ, c ) → f ∗ ( L, τ, c ) . (b) Let E be a ( φ, τ, c ) -twisted vector bundle on X with Cl p,q -action. Then f ∗ E and f ∗ E are isomorphic under the identification of twisting data by β .Proof. For (a), we have a homotopy ˜ f : X ′ × [0 , → X between maps f i : X ′ → X on the space of morphisms. This homotopy induces an isomorphism β : f ∗ L → f ∗ L ′ of c -graded line bundles. Together with the product bundle K and the trivial map b , the isomorphism β gives rise to an isomorphism of the φ -twisted extensions.For (b), we can generalize the argument in [1] to prove the fact that a homotopicmaps induce isomorphic complex vector bundles. The point in this generalizationis that, for any closed full subgroupoid Y ⊂ X , any section of the φ -twisted vectorbundle Hom Cl p,q ( E, E ′ ) | Y extends to X . This is possible, since by design X iscovered by countable open subgroupoids X α which are weakly equivalent to thequotient groupoids X α //G α associated to actions of compact Lie groups G α . On X α , we can apply the usual Tietze extension theorem. Taking the average bythe actions of G α and the Clifford group associated to Cl p,q , we have the Tietzeextension theorem for sections of the φ -twisted vector bundles. Gluing extensionson X α by a partition of unity, we get an extension of a section of Hom Cl p,q ( E, E ′ ) | Y to X . With respect to the compact open topology, isomorphisms Iso Cl p,q ( E, E ′ ) ⊂ Hom Cl p,q ( E, E ′ ) form an open subset. Based on this fact, we can now generalize thestandard argument in [1] to get an isomorphism of the twisted vector bundles. (cid:3) Locally universal bundle.
We introduce here an extension of the notion oflocally universal twisted Hilbert bundles given in [10].
Definition 2.8.
Let X be a groupoid, φ : X → pt // Z a map of groupoids, and( L, τ, c ) a φ -twisted Z -graded extension of X . A ( φ, τ, c )-twisted vector bundle E on X with Cl p,q -action is called locally universal if there is an isometric embedding E ′ → E | X ′ for any open full subgroupoid X ′ ⊂ X and any ( φ | X ′ , τ | X ′ , c | X ′ )-twistedvector bundle E ′ on X ′ with Cl p,q -action.Extending argument in [10], one can show that an embedding E ′ → E | X ′ asabove is unique up to homotopy, and hence E is unique up to unitary isomorphisms.Also, if E is locally universal, then so is E ⊕ E . Lemma 2.9.
Let X be a local quotient groupoid, φ : X → pt // Z a map ofgroupoids, and ( L, τ, c ) a φ -twisted Z -graded extension of X . There is a ( φ, τ, c ) -twisted locally universal twisted vector bundle E on X with Cl p,q -action.Proof. The idea of the proof is basically the same as that given in [10].First of all, it is enough to consider the case where (
L, τ ) is trivial. The key to thisreduction is the groupoid L such that its space of objects is L = X and its spaceof isomorphisms L = S ( L ) is the unit sphere bundle of L → X . The pull-backunder the projection π : S ( L ) → X induces a one to one correspondence between( φ, τ, c )-twisted bundles on X with Cl p,q -action and ( π ∗ φ, π ∗ c )-twisted bundles on L with Cl p,q -action which are equivariant under the right U (1)-action on S ( L ).Then, we can further reduce the problem, and it suffices to consider the casewhere the base groupoid X is the quotient groupoid pt //G with G a compact Liegroup. The key to this reduction is that we can glue locally universal twisted bun-dles together to form a locally universal twisted bundles. By design, a local quotientgroupoid is covered by open full subgroupoids X α which are weak equivalent to thequotient groupoids X α //G α , with the cardinality of indices α countable. Here G α are compact Lie groups and X α are Hausdorff spaces which are locally contractible,paracompact and completely regular. Each X α admits locally contractible slices,and each slice is G α -equivariantly homotopy equivalent to the space of the form G α /H with H ⊂ G α a closed subgroup. The inclusion induces a local equivalencept //H → ( G α /H ) //G α . Thus, there is a locally universal bundle on X , if there isa locally universal ( φ, c )-twisted bundle with Cl p,q -action on the quotient groupoidof the form pt //G with G any compact Lie group.Now, the remaining thing to show is the existence of a ( φ, c )-twisted (locally)universal vector bundle on pt //G with Cl p,q -action, where G is any compact Liegroup, and φ : G → Z and c : G → Z are any continuous homomorphisms. Thisexistence is shown in Appendix B (Lemma B.15), by using the so-called Mackeydecomposition, which reduces the consideration of a representation of a group tothat of projective representations of smaller groups. (cid:3) Fredholm formulation of Freed-Moore K -theory In this section, we provide the Fredholm formulation of the Freed-Moore K -theory, and prove its periodicity and the degree shift effects of some twists. Thereproductions of known K -theories, a relationship to the finite rank formulation in[11], and the Thom isomorphism theorem are also provided.3.1. Fredholm family.
Let X be a groupoid, φ : X → pt // Z a map of groupoids,( L, τ, c ) a φ -twisted Z -graded extension of X , and ( E, ǫ, ρ, γ ) a ( φ, τ, c )-twisted vec-tor bundle on X with Cl p,q -action. As in Lemma 2.6, we let End( E ) = Hom( E, E )be the φ -twisted vector bundle on X whose sections are in one to one correspon-dence with continuous maps ( E, ǫ, ρ ) → ( E, ǫ, ρ ), where the Clifford action is ig-nored. The topology on End( E ) is given by the compact open topology [6]. Wealso let K( E ) → X be a fiber bundle defined as follows. • The fiber of the underlying fiber bundle K( E ) → X at x ∈ X consists ofcompact operators K : E x → E x . • The bundle isomorphism ̺ : φ ∂ ∗ K( E ) → ∂ ∗ K( E ) on X is given by ̺ ( K ) = ρ ◦ (id L ⊗ K ) ◦ ρ − , where id L : L → L is the identity map.We topologize K( E ) by using the operator norm topology. REED-MOORE K -THEORY 17 Definition 3.1 (Fredholm family) . Let X be a groupoid, φ : X → pt // Z a mapof groupoids, and ( L, τ, c ) a φ -twisted Z -graded extension of X . For a ( φ, τ, c )-twisted vector bundle ( E, ǫ, ρ, γ ) on X with Cl p,q -action, we define a fiber bundleFred( E ) → X as follows: • The fiber of the underlying fiber bundle Fred( E ) → X at x ∈ X consistsbounded operators A : E x → E x such that(i) A are skew-adjoint: A ∗ = − A .(ii) A + id are compact.(iii) Spec( A ) ⊂ [ − i, i ].(iv) A are degree 1, and anti-commute with the Cl p,q -action, that is, Aǫ = − ǫA, Aγ ( e ) = − γ ( e ) A, for any unit norm element e ∈ R p + q . • The bundle isomorphism ̺ : φ ∂ ∗ Fred( E ) → ∂ ∗ Fred( E ) on X is given by ̺ ( A ) = ρ ◦ (id L ⊗ A ) ◦ ρ − , where id L : L → L is the identity map.The fiber bundle Fred( E ) is topologized by the following mapFred( E ) → End( E ) × K( E ) , A ( A, A + id) , where End( E ) = Hom( E, E ) is topologized by using the compact open topologyand K( E ) by using the operator norm topology. The space of sections is defined byΓ( X , Fred( E )) = { A ∈ Γ( X , Fred( E )) | ̺ ◦ φ ∂ ∗ A = ∂ ∗ A } . We write Fred( E ) ∗ ⊂ Fred( E ) for the subbundle such that the fiber of theunderlying fiber bundle Fred( E ) ∗ → X consists of invertible operators. We alsowrite Fred( E ) † ⊂ Fred( E ) ∗ for the subbundle such that the fiber of the underlyingbundle Fred( E ) † → X consists of operators squaring to − id. Therefore we haveΓ( X , Fred( E ) ∗ ) = { A ∈ Γ( X , Fred( E )) | A x is invertible for each x ∈ X } , Γ( X , Fred( E ) † ) = { A ∈ Γ( X , Fred( E )) | A x = − id for each x ∈ X } . By functional calculus, Γ( X , Fred( E ) † ) ⊂ Γ( X , Fred( E ) ∗ ) is a deformation retract,where the compact open topology are considered in the space of sections. Lemma 3.2.
Let X be a local quotient groupoid, φ : X → pt // Z a map ofgroupoids, and ( L, τ, c ) a φ -twisted Z -graded extension of X . Suppose that E isa ( φ, τ, c ) -twisted locally universal vector bundle on X with Cl p,q -action. Then Γ( X , Fred( E ) † ) is non-empty and weakly contractible.Proof. Let Π E be E with its Z -grading reversed. By the local universality, wehave E ∼ = E ⊕ Π E . It is easy to see that E ⊕ Π E ∼ = E ⊗ C , where C is a Z -gradedvector space such that its even part and odd part are 1-dimensional. On this C ,we can let Cl , act by γ ∗ = (cid:18) −
11 0 (cid:19) . Then 1 ⊗ γ ∗ ∈ Γ( X , Fred( E ⊗ C ) † ). To see that Γ( X , Fred( E ) † ) is weakly con-tractible (i.e. weakly homotopy equivalent to the point), we apply the reductionargument as in the proof of Lemma 2.9 and Proposition A.19 in [10] (which is basedon [28]). Then, it suffices to show that Γ( X , Fred( E ) † ) is weakly contractible when X is the quotient groupoid pt //G with G a compact Lie group, φ : X → pt // Z and c : X → pt // Z are associated to homomorphisms φ : G → Z and c : G → Z , and E is a ( φ, c )-twisted locally universal bundle on pt //G with Cl p,q -action. In thiscase, Γ( X , Fred( E ) † ) is contractible, as shown in Appendix B (Lemma B.16). (cid:3) From a groupoid X , we can construct a groupoid X × [0 ,
1] so as to be (
X × [0 , i = X i × [0 , E is a twisted bundle on X , then the pull-back of E underthe projection X × [0 , → X is identified with E × [0 , homotopy between A , A ∈ Γ( X , Fred( E )) is defined to be a section ˜ A ∈ Γ( X × [0 , , Fred( E × [0 , A | X ×{ i } = A i for i = 0 ,
1. In this case, A and A are said to behomotopic, and we write A ∼ A . Lemma 3.3.
Let X be a local quotient groupoid, φ : X → pt // Z a map ofgroupoids, ( L, τ, c ) a φ -twisted Z -graded extension of X , and E a ( φ, τ, c ) -twistedlocally universal vector bundle E on X with Cl p,q -action. Then the set of homotopyclasses of sections Γ( X , Fred( E )) / ∼ is an abelian group.Proof. We can prove the lemma in a standard manner: The addition is induced fromthe direct sum (
A, A ′ ) A ⊕ A ′ . The zero element is represented by invertiblesections A ∈ Γ( X , Fred( E ) ∗ ). The inverse is realized by reversing the Z -gradingof the underlying twisted vector bundle. To show the axiom about the inversion,let Π E denote the twisted bundle E with its Z -grading reversed. The direct sum E ⊕ Π E is isomorphic to the tensor product E ⊗ ∆ C , of E and an irreducible Z -graded complex Cl , -module ∆ C , . If we denote by γ ∈ Γ( X , Fred( E ⊗ ∆ C , ) † )the action of the generator of Cl , on ∆ C , , then A ⊕ Π A is homotopic to γ by thehomotopy ( A ⊕ Π A ) cos θ + γ sin θ for any A ∈ Γ( X , Fred( E )). (cid:3) Now, suppose that, for a groupoid X and φ = ( F : ˜ X → X , φ ) ∈ Φ( X ), we aregiven a φ -twist on X consisting of a local equivalence ˜ F : ˜˜ X → ˜ X and a ˜ F ∗ φ -twisted Z -graded extension ( L, τ, c ) of ˜˜ X . Suppose also that E is a ( ˜ F ∗ φ, τ, c )-twistedvector bundle on ˜˜ X with Cl p,q -action. By the nature of local equivalences [10],fiber bundles on ˜˜ X are in bijective correspondence with those on ˜ X under the pull-back, and the pull-back also induces a homeomorphism of the spaces of sections.This can be generalized to φ -twisted bundles, so that the ˜ F ∗ φ -twisted vector bundleEnd( E ) → ˜˜ X is isomorphic the pull-back of a φ -twisted vector bundle on ˜ X under˜ F . As a result, the fiber bundle Fred( E ) → ˜˜ X is isomorphic to the pull-back under˜ F of a φ -twisted fiber bundle Fred( τ ) → ˜ X , and Γ( ˜ X , Fred( τ )) ∼ = Γ( ˜˜ X , Fred( E )). Definition 3.4.
Let X be a local quotient groupoid, φ = ( F : ˜ X →
X, φ ) ∈ Φ( X )an object, and ( τ, c ) = ( ˜ F : ˜˜ X → ˜ X , L, τ, c ) a φ -twist on X . We define a group by φ K ( τ,c )+( p,q ) ( X ) = Γ( ˜ X , Fred( τ )) / ∼∼ = Γ( ˜˜ X , Fred( E )) / ∼ where Fred( τ ) → ˜ X is the φ -twisted bundle such that ˜ F ∗ Fred( τ ) ∼ = Fred( E ) for a( ˜ F ∗ φ, τ, c )-twisted locally universal vector bundle E → ˜˜ X with Cl p,q -action. In thecase that X is a quotient groupoid X//G , we may write φ K ( τ,c )+( p,q ) ( X//G ) = φ K ( τ,c )+( p,q ) G ( X ) . REED-MOORE K -THEORY 19 The group φ K ( τ,c )+( p,q ) ( X ) is independent of the choice of E , because of theuniqueness of locally universal bundles up to unitary isomorphisms. As in thecase of twisted complex K -theory [10], a local equivalence of groupoids induces anisomorphism by pull-back, and hence φ K ( τ,c )+( p,q ) ( X ) is an invariant of the weakequivalence class of the groupoid X equipped with the twisting data φ and ( τ, c ). Lemma 3.5 (weak periodicity) . Let X be a local quotient groupoid, φ ∈ Φ( X ) anobject, and ( τ, c ) a φ -twist on X . There are natural isomorphisms φ K ( τ,c )+( p,q ) ( X ) ∼ = φ K ( τ,c )+( p +1 ,q +1) ( X ) ∼ = φ K ( τ,c )+( p +8 ,q ) ( X ) ∼ = φ K ( τ,c )+( p,q +8) ( X ) . In the case that φ is trivial, there are natural isomorphisms K ( τ,c )+( p,q ) ( X ) ∼ = K ( τ,c )+( p +1 ,q +1) ( X ) ∼ = K ( τ,c )+( p +2 ,q ) ( X ) ∼ = K ( τ,c )+( p,q +2) ( X ) . Proof.
Let us consider φ K ( τ,c )+( p,q ) ( X ) ∼ = φ K ( τ,c )+( p +1 ,q +1) ( X ). To realize thisisomorphism, we let ∆ , be an irreducible Z -graded real Cl , -module. Concretely,we can choose ∆ , = R and ǫ = (cid:18) − (cid:19) , γ = (cid:18) −
11 0 (cid:19) , γ = (cid:18) (cid:19) . Its complexification ∆ C , = ∆ , ⊗ C , being irreducible also, has the obvious ‘Real’structure from the complex conjugation on C , so that we can regard it as an id Z -twisted vector bundle on pt // Z with Cl , -action. Furthermore, we pull this bun-dle back to X by the map φ : X → pt // Z to get∆ C , ∈ φ Vect (1 , ( X ) . For a ( φ, τ, c )-twisted locally universal twisted vector bundle E on X with Cl p,q -action, the tensor product E ⊗ ∆ C , is a ( φ, τ, c )-twisted locally universal twistedvector bundle on X with Cl p +1 ,q +1 -action. We then considerΓ( X , Fred( E )) → Γ( X , Fred( E ⊗ ∆ C , )) , a a ⊗ . We can directly see that any odd map A : E ⊗ ∆ C , → E ⊗ ∆ C , such that A (1 ⊗ γ i ) = − (1 ⊗ γ i ) A is uniquely expressed as A = a ⊗ a : E → E . Asa result, Γ( X , Fred( E )) → Γ( X , Fred( E ⊗ ∆ C , )) is a homeomorphism, and henceinduces an isomorphism φ K ( τ,c )+( p,q ) ( X ) → φ K ( τ,c )+( p +1 ,q +1) ( X ) . The other isomorphisms follow from this: Iterating it, we get φ K ( τ,c )+( p,q ) ( X ) → φ K ( τ,c )+( p +4 ,q +4) ( X ) . In general, if γ i , ( i = 1 , . . . ,
4) realize a real Cl , -module, then γ ′ i = γ i γ · · · γ , ( i =1 , . . . ,
4) realize a real Cl , -module. This construction induce natural isomorphisms φ K ( τ,c )+( p +8 ,q ) ( X ) ∼ = φ K ( τ,c )+( p +4 ,q +4) ( X ) ∼ = φ K ( τ,c )+( p,q +8) ( X ) . In the case that φ is trivial, there are natural identifications of complex modulesover Cl , , Cl , and Cl , , so that K ( τ,c )+( p +2 ,q ) ( X ) ∼ = K ( τ,c )+( p +1 ,q +1) ( X ) ∼ = K ( τ,c )+( p,q +2) ( X ) , and the lemma is established. (cid:3) Remark . There are two equivalent variants of the fiber bundle Fred( E ). Avariant is the fiber bundle given by dropping the spectral condition Spec( A ) ⊂ [ − i, i ]in Definition 3.1. The resulting fiber bundle has a deformation retract to Fred( E ),and we can use it to formulate the Freed-Moore K -theory. Another variant is touse self-adjoint operators instead of skew-adjoint operators. Its detail will be givenin Remark 4.12.3.2. The Bott periodicity.
Let X be a groupoid, φ : X → pt // Z a map ofgroupoid, ( L, τ, c ) a φ -twisted Z -graded extension of X , and E a ( φ, τ, c )-twistedvector bundle E of X with Cl p,q -action. For a full subgroupoid Y ⊂ X , we writeΓ( X , Y , Fred( E )) = { A ∈ Γ( X , Fred( E )) | A | Y ∈ Γ( Y , Fred( E ) ∗ ) } for the space of sections A of Fred( E ) such that A x : E x → E x is invertible forall x ∈ Y . A homotopy between such sections is defined by using sections inΓ( X × [0 , , Y × [0 , , Fred( E × [0 , φ ∈ Φ( X ) and a φ -twist( τ, c ), the space of sections Γ( X , Y , Fred( τ )) and their homotopy ∼ are defined inthe obvious way. Definition 3.7.
Let X be a local quotient groupoid, φ = ( F : ˜ X →
X, φ ) ∈ Φ( X )an object, and ( τ, c ) = ( ˜ F : ˜˜ X → ˜ X , L, τ, c ) a φ -twist on X .(a) For a full subgroupoid Y ⊂ X , we define φ K ( τ,c ) ( X , Y ) = Γ( ˜ X ∪ ˜ Y × [0 , , ˜ Y × { } , Fred( τ )) / ∼ where ˜ X ∪ ˜ Y × [0 , ⊂ ˜ X × [0 ,
1] is the mapping cylinder of the full sub-groupoid ˜ Y = ˜ F − ( ˜ Y ) ⊂ ˜ X , Fred( τ ) is the φ -twisted bundle on ˜ X suchthat ˜ F ∗ Fred( τ ) ∼ = Fred( E ) for a ( ˜ F ∗ φ, τ, c )-twisted locally universal vectorbundle E → ˜˜ X with Cl , -action, and τ × [0 ,
1] is the pull-back of τ underthe projection ˜ X × [0 , → ˜ X .(b) For a non-negative integer n ≥
0, we define φ K ( τ,c ) − n ( X , Y ) = φ K ( τ,c ) ( X × [0 , n , Y × [0 , n ∪ X × ∂ [0 , n ) . Theorem 3.8.
Let X be a local quotient groupoid, φ ∈ Φ( X ) an object, and ( τ, c ) a φ -twist on X . For n ≥ , there is a natural isomorphism of groups φ K ( τ,c ) − n ( X ) ∼ = φ K ( τ,c )+( n, ( X ) . Proof.
The proof is essentially the same as in the complex case [10] (PropositionA.41): For a ( φ, τ, c )-twisted locally universal vector bundle E → X with Cl , -action, we have the Atiyah-Singer map [7]AS : Γ( X , Fred( E )) → Γ( X × [0 , , X × { , } , Fred( E × [0 , A γ cos πt + A sin πt , where t ∈ [0 , γ = γ ( e ) is the action ofthe generator e ∈ Cl , . The iteration of this map defines a homomorphism φ K ( τ,c )+( n, ( X ) → φ K ( τ,c ) − n ( X ) . To prove that this homomorphism is bijective, we show that AS induces a weakhomotopy equivalence on the spaces of sections. As before, thanks to the reductionargument as in Lemma 2.9 and [10] (Proposition A.19), it suffices to consider thecase of the quotient groupoid X = pt //G with G a compact Lie group and trivial τ . In this case, the map AS provides a homotopy equivalence, as will be shown inAppendix B (Lemma B.17). (cid:3) REED-MOORE K -THEORY 21 By the periodicities in Lemma 3.5, we get:
Corollary 3.9 (Bott periodicity) . For n ≥ , there is a natural isomorphism φ K ( τ,c ) − n ( X ) ∼ = φ K ( τ,c ) − n − ( X ) . If φ is trivial, then there is a natural isomorphism φ K ( τ,c ) − n ( X ) ∼ = φ K ( τ,c ) − n − ( X ) . Corollary 3.10.
For − p + q ≤ , we have a natural isomorphism φ K ( τ,c ) − p + q ( X ) ∼ = φ K ( τ,c )+( p,q ) ( X ) . Theorem 3.11.
Let X be a local quotient groupoid, φ ∈ Φ( X ) an object, and ( τ, c ) a φ -twist on X . We can extend the K -group φ K ( τ,c )+ n ( X ) to define φ K ( τ,c )+ n ( X , Y ) for a full subgroupoid Y ⊂ X and n ∈ Z so that the Bott periodicity holds true: φ K ( τ,c )+ n ( X , Y ) ∼ = φ K ( τ,c )+ n − ( X , Y ) . These groups have the following properties. (a) (the homotopy axiom) Let X ′ be another local quotient groupoid, and Y ′ ⊂X ′ a full subgroupoid. Let f , f : X ′ → X be maps of groupoids such that f i ( Y ′ ) ⊂ Y . If ˜ f : X ′ × [0 , → X is such that ˜ f ( Y ′ × { t } ) ⊂ Y for all t ∈ [0 , , then there is a unique isomorphism of twists β ˜ f : f ∗ τ → f ∗ τ , andthe following diagram becomes commutative φ K ( τ,c )+ n ( X , Y ) f ∗ −−−−→ f ∗ φ K ( f ∗ τ,f ∗ c )+ n ( X ′ , Y ′ ) (cid:13)(cid:13)(cid:13) x β ∗ ˜ f φ K ( τ,c )+ n ( X , Y ) f ∗ −−−−→ f ∗ φ K ( f ∗ τ,f ∗ c )+ n ( X ′ , Y ′ ) . (b) (the excision axiom) Let A and B be closed full subgroupoids in X . Thenthe inclusion A → A ∪ B induces the isomorphisms φ A ∪ B K ( τ A ∪ B ,c A ∪ B )+ n ( A ∪ B , B ) ∼ = φ A K ( τ A ,c A )+ n ( A , A ∩ B ) , where φ A ∪ B = φ | A∪B , etc. mean the restrictions. (c) (the exactness axiom) For a full subgroupoid
Y ⊂ X , there is a long exactsequence ·· → φ K ( τ,c )+ n − ( Y ) → φ K ( τ,c )+ n ( X , Y ) j ∗ → φ K ( τ,c )+ n ( X ) i ∗ → φ K ( τ,c )+ n ( Y ) → ·· , in which j ∗ is induced from the forgetful functor of Y and i ∗ from theinclusion i : Y → X . The restriction of twisting data ( φ φ | Y ) is omitted. (c) (the additivity axiom) For a family X λ of local quotient groupoids and theirfull subgroupoids Y λ ⊂ X λ , the inclusions X λ → ⊔X λ induce an isomor-phism ⊔ φ λ K ( ⊔ τ λ , ⊔ c λ )+ n ( ⊔X λ , ⊔Y λ ) ∼ = Y λ φ λ K ( τ λ ,c λ )+ n ( X λ , Y λ ) . Proof.
With the Bott periodicity in Corollary 3.9, the proof of the theorem is astandard and rather formal procedure: In view of the so-called the cofibration (orPuppe) sequence, we get the non-positive part of the long exact sequence in (c).Using this part, the map AS in the proof of Theorem 3.8 induces an isomorphism φ K ( τ,c )+ n ( X , Y ) ∼ = φ K ( τ,c )+ n − ( X , Y ) . Based on this periodicity, for a positive integer n >
0, we define φ K ( τ,c )+ n ( X , Y ) = φ K ( τ,c )+ n − k ( X , Y ) , where k is an integer such that n − k <
0. Because of this definition, the non-negative part of the long exact sequence is extended to the complete long exactsequence in (c). The homotopy axiom (a) follows from the homotopy propertyof twisted vector bundles (Lemma 2.7). The excision axiom (b) follows from adeformation and an extension of a section A ∈ Γ( A , A∩B , Fred( τ )) by using Lemma3.2. Finally, (d) just follows from the definition of the K -group. (cid:3) The tensor product functor on the category of twisted vector bundles induces amultiplication on the K -groups ⊗ : φ K ( τ,c )+ n ( X , Y ) × φ K ( τ ′ ,c ′ )+ n ′ ( X , Y ′ ) → φ K (( τ,c )+( τ ′ ,c ′ ))+( n + n ′ ) ( X , Y ∪ Y ′ ) . Thus, for example, φ K ( X ) is a ring, and φ K ( τ,c )+ n ( X , Y ) is a module over φ K ( X ).In the following, for a quotient groupoid X//G , we may apply the notation φ K ( τ,c )+ n ( X//G ) = φ K ( τ,c )+ nG ( X ) . In this case, φ K ( τ,c )+ nG ( X ) is a module over the ring φ G K G (pt), where φ G ∈ Φ(pt //G ) is the restriction of φ ∈ Φ( X//G ), which is always equivalent to theobject associated to a homomorphism φ G : G → Z .3.3. Twist and degree shift.
For the quotient groupoid pt // Z , the identityhomomorphism id : Z → Z defines a non-trivial object id ∈ Φ(pt // Z ). Thereare then two distinguished id-twisted Z -graded extensions of pt // Z . • The id-twisted Z -graded extension τ id = ( Z × C , τ id ,
1) consisting of theproduct line bundle Z × C → Z , the 2-cocycle τ id ∈ Z ( Z ; U (1) id )given by τ id (( − m , ( − m ) = exp πim m , τ id ( g, h ) h = 1 h = − g = 1 1 1 g = − − // Z → pt // Z induced from thetrivial homomorphism 1 : Z → Z . • The id-twisted Z -graded extension c id = ( Z × C , , c id ) consisting of theproduct line bundle Z × C → Z , the trivial 2-cocycle 1 ∈ Z ( Z ; U (1) id )and the non-trivial map of groupoids c id : pt // Z → pt // Z given by theidentity homomorphism c id = id : Z → Z .The id-twisted extension τ id can be seen as an ungraded twist, and generates H (pt // Z ; Z φ ) ∼ = H (pt; Z φ ) ∼ = H (pt; U (1) φ ) ∼ = Z , while c id can be seen as the datum of the Z -grading, and generates H (pt // Z ; Z ) ∼ = H (pt; Z ) = Hom( Z , Z ) ∼ = Z . We have (0 , c id ) + (0 , c id ) = ( τ id ,
0) and ( τ id ,
0) + ( τ id ,
0) = 0 in the group π ( id Twist (pt // Z )) ∼ = H Z (pt; Z id ) × H Z (pt; Z ) ∼ = Z , so that c id is a generator of this group.Let X be a groupoid, and φ = { F : ˜ X → X , φ : ˜
X → pt // Z } an object in Φ( X ).By the pull-back under φ : ˜ X → pt // Z , the id-twisted Z -graded extensions τ idREED-MOORE K -THEORY 23 and c id define φ -twisted Z -graded extensions τ φ = φ ∗ τ id and c φ = φ ∗ c id of ˜ X .Hence we have φ -twists τ φ and c φ on X . Theorem 3.12.
Let X be a local quotient groupoid, φ ∈ Φ( X ) an object, and ( τ, c ) a φ -twist on X . For a full subgroupoid Y ⊂ X and n ∈ Z , there are naturalisomorphisms φ K ( τ,c )+ c φ + n ( X , Y ) ∼ = φ K ( τ,c )+ n +2 ( X , Y ) , φ K ( τ,c )+ τ φ + n ( X , Y ) ∼ = φ K ( τ,c )+ n +4 ( X , Y ) , φ K ( τ,c )+( τ φ ,c φ )+ n ( X , Y ) ∼ = φ K ( τ,c )+ n +6 ( X , Y ) . Proof.
We define ∆ ∈ id Vect c id +(2 , (pt // Z ) as follows: The underling vector spaceis ∆ = C ⊕ C with ∆ k = C . The twisted Z -action C and the Cl , -action γ i are C = (cid:18) (cid:19) K, γ = (cid:18) −
11 0 (cid:19) , γ = (cid:18) ii (cid:19) , where K is the complex conjugation. For φ = { F : ˜ X → X , φ : ˜
X → pt // Z } ∈ Φ( X ), we take the pull-back of the twisted bundle ∆ above under φ to get∆ ∈ φ Vect c φ +(2 , ( ˜ X ) . Thus, for a ( φ, τ, c )-twisted locally universal vector bundle E on ˜ X with Cl p,q -action,the tensor product defines a mapΓ( ˜ X , Fred( E )) → Γ( ˜ X , Fred( E ⊗ ∆)) , A A ⊗ δ : φ K ( τ,c )+ n ( X , Y ) → φ K ( τ,c )+ c φ + n − ( X , Y ) . Recall that c φ + c φ = τ φ and τ φ + τ φ = 0. Therefore the present theorem willbe established when δ is shown to be bijective. To prove the bijectivity of δ , it isenough to consider the case of Y = ∅ , because of the exactness axiom. Now, let usconsider the composition δ : φ K ( τ,c )+ n ( X ) → φ K ( τ,c )+ n − ( X ) . This map is induced from the tensor product with the twisted representation ∆ ⊗ ,which is an id-twisted representation of Z with Cl , -action∆ ⊗ ∈ id Vect (8 , (pt // Z ) . An id-twisted representation of Z is nothing but a complex vector space with ananti-linear involution, or equivalently a real structure. By the operation of takingthe real part, id-twisted representations of Z with Cl , -action are in one to onecorrespondence with Z -graded real representations of Cl , . The Z -graded realrepresentation of Cl , corresponding to ∆ ⊗ has the dimension 2 = 16, and henceis an irreducible representation. Such an irreducible representation realizes theperiodicity in Lemma 3.5, so that δ turns out to be bijective. (cid:3) Reproduction of familiar K -theories. As mentioned in §
1, we can recoverfamiliar K -theories by specifying twists. We review here some examples. Twisted equivariant complex K -theory. For a local quotient groupoid X , if φ ∈ Φ( X ) is trivial, then the twisted K -theory in [10] is recovered. In particular, forthe quotient groupoid X = X//G associated to an action of a compact Lie groupon a compact Hausdorff space X , we recover the twisted G -equivariant complex K -theory φ K ( τ,c )+ n ( X//G ) = K ( τ,c )+ nG ( X ) . In this case, the twists ( τ, c ) are classified by the Borel equivariant cohomology H G ( X ; Z ) × H G ( X ; Z )If G = Z , then there is a distinguished twist (0 , c ) coming from H Z (pt; Z ) ∼ = Hom( Z , Z ) = Z . We can identify the K -theory with this twist with a variant of K -theory K c + n Z ( X ) ∼ = K n ± ( X ) in [4, 36].3.4.2. Twisted equivariant KO -theory. Let G be a compact Lie group acting on acompact Hausdorff space X . By this G -action and the trivial Z -action, we definean action of Z × G on X . We let p Z : Z × G → Z be the projection onto the Z -factor, which defines an object p Z ∈ Φ(pt // ( Z × G )). In this case, we have theidentification with the twisted G -equivariant real K -theory p Z K ( τ,c )+ n ( X// ( Z × G )) ∼ = KO ( τ,c )+ nG ( X ) . The twists ( τ, c ) are classified by the equivariant cohomology H Z × G ( X ; Z p Z ) × H Z × G ( X ; Z ) . We can see that H Z × G ( X ; Z p Z ) ∼ = H G ( X ; Z ) ⊕ H G ( X ; Z ) ,H Z × G ( X ; Z ) ∼ = H G ( X ; Z ) ⊕ H G ( X ; Z ) . The factors H G ( X ; Z ) and H G ( X ; Z ) are consistent with the twists for KO -theory in [15]. Notice that a G -equivariant complex line bundle L → X defines atwisted extension in this case. The twist given by L is classified by the image of theequivariant Chern class c G ( L ) ∈ H G ( X ; Z ) under the mod 2 reduction H G ( X ; Z ) → H G ( X ; Z ). The KO -theory twisted by L is the G -equivariant version of the twisted KO -theory in [5]. The remaining factors H G ( X ; Z ) ⊂ H Z × G ( X ; Z p Z ) , H G ( X ; Z ) ⊂ H Z × G ( X ; Z )are the contributions of the twists c φ and τ φ in §§ φ = p Z . In view of thevalues of the 2-cocycle defining τ φ , we find that the equivariant KO -theory twistedby τ φ is the K -theory of G -equivariant quaternionic vector bundles.3.4.3. KR -theory. Let us consider the quotient groupoid X = X// Z associated toa compact Hausdorff space X with an action of Z . The identity homomorphismid : Z → Z defines an object id ∈ Φ( X// Z ). Then we recover Atiyah’s KR -theory [2] id K n ( X// Z ) ∼ = KR n ( X ) . The twists ( τ, c ) in this case are classified by H Z ( X ; Z id ) × H Z ( X ; Z ) , REED-MOORE K -THEORY 25 and we can define a twisted KR -theory by KR ( τ,c )+ n ( X ) = id K ( τ,c )+ n ( X// Z ) . The twist c id in §§ H Z ( X ; Z id ), and the K -theory with thistwist provides Dupont’s ‘Symplectic’ K -theory [16].We notice that twisted KR -theories are already introduced by Moutuou [24] andalso by Fok [9]. The twisted KR -theory id K ( τ,c )+ n ( X// Z ) above is anticipated toreproduce their twisted KR -theories. In particular, the twisted G -equivariant KR -theory KR τG ( X ) for a ‘Real’ G -space of X in the sense of [9] would be identifiedwith φ K τ ( X// ( Z ⋉ G )), where the semi-direct product Z ⋉ G is defined by usingthe ‘Real’ structure on G and φ : Z ⋉ G → Z is the projection.3.5. Finite rank realizability.
We here compare the Fredholm formulation of φ K ( τ,c )+0 ( X ) with its finite-dimensional formulation in [11]. Definition 3.13.
Let X be the quotient groupoid X//G associated to an actionof a finite group G on a compact Hausdorff space X , φ : X//G → pt // Z the mapof groupoids associated to a homomorphism φ : G → Z , and ( L, τ, c ) a φ -twisted Z -graded extension of X//G .(a) We define φ Vect ( τ,c )+( p,q ) G ( X ) fin ⊂ φ Vect ( τ,c )+( p,q ) ( X//G ) to be the full sub-category whose objects are ( φ, τ, c )-twisted vector bundles E on X//G with Cl p,q -action such that the fibers of E are finite rank.(b) We define φ K ( τ,c )+ nG ( X ) fin as the quotient monoid φ K ( τ,c )+ nG ( X ) fin = π ( φ Vect ( τ,c )+( n, G ( X ) fin ) /π ( φ Vect ( τ,c )+( n +1 , G ( X ) fin ) . Note that φ Vect ( τ,c )+( n +1 , G ( X ) fin is a full subcategory of φ Vect ( τ,c )+( n, G ( X ) fin ,so that π ( φ Vect ( τ,c )+(1 , G ( X ) fin ) ⊂ π ( φ Vect ( τ,c )+(0 , G ( X ) fin ) is a submonoid. Usingthe idea of the proof of Lemma 3.3, we can show that the reversal of the Z -gradingof twisted bundles gives a monoid morphism satisfying the assumptions in LemmaC.1, from which φ K ( τ,c )+ nG ( X ) fin is an abelian group. If c is trivial, then this groupagrees with the Grothendieck construction of the monoid of isomorphism classes ofungraded vector bundles.For any E ∈ φ Vect ( τ,c )+( p,q ) G ( X ) fin , the trivial section 0 ∈ Γ( X//G,
Fred( E ))makes sense. Moreover, any section A ∈ Γ( X//G,
Fred( E )) is homotopic to thetrivial section. If E uni is a locally universal twisted bundle, then we have an em-bedding E → E uni . The orthogonal complement E ⊥ ⊂ E uni can be assumed to belocally universal. Then we have a map Γ( X//G,
Fred( E )) → Γ( X//G,
Fred( E uni ))given by A A ⊕ γ ∗ , where γ ∗ ∈ Γ( X//G,
Fred( E ⊥ ) † ). Thus, taking A = 0, wehave an induced homomorphism ı : φ K ( τ,c )+ nG ( X ) fin −→ φ K ( τ,c )+ nG ( X ) . In the case of X = pt, a standard argument shows that the operation of taking thekernel of skew-adjoint (Fredholm) operators induces a well-defined homomorphism κ : φ K ( τ,c )+ nG (pt) → φ K ( τ,c )+ nG (pt) fin , A Ker( A ) , which is inverse to ı . The theorem of Atiyah-J¨anich [1] is a generalization of thisfact in the case that n = 0 and τ is trivial, and a twisted generalization due to [11]is as follows. Proposition 3.14 ([11]) . Under the assumptions in Definition 3.13, if c : X//G → pt // Z is trivial and n = 0 , then the homomorphism ı is an isomorphism. In [11], the result above is stated in Remark 7.37 and a proof is given in AppendixE, which is essentially the construction of the inverse κ . However, this proof seemsnot to work in the presence of a non-trivial homomorphism c : G → Z , at thepoint that we apply the argument proving the Atiyah-J¨anich theorem in [1]. Forthis reason, c is assumed to be trivial in the above proposition. We remark thatthe proposition will be reproved in a different way in §§ c , we can instead prove the following: Proposition 3.15.
Under the assumptions in Definition 3.13, if c : X//G → pt // Z is associated to a non-trivial homomorphism c : G → Z , then there is anexact sequence of groups φ K τ +0 G ( X ) fin π ∗ −→ φ K τ +0 G ( X × Z ) fin π ∗ −→ φ K ( τ,c )+0 G ( X ) fin −→ , where the original G -action on X and the morphism c : G → Z define the G -actionon X × Z by ( x, r ) ( gx, c ( g ) r ) , and π : X × Z → X is the projection.Proof. First of all, we construct π ∗ : φ K τ +0 G ( X × Z ) fin → φ K ( τ,c )+0 G ( X ) fin . Forthe construction, we notice that, the group φ K τ +0 G ( X × Z ) fin agrees with theGrothendieck construction of the monoid of isomorphism classes of ( φ, τ )-twisted ungraded vector bundles on ( X × Z ) //G (or equivalently ( φ, τ )-twisted vectorbundles with trivial odd parts). For such a twisted vector bundle E = E ⊕ X × Z ) //G , we define a Z -graded Hermitian vector bundle ˆ E = ˆ E ⊕ ˆ E on X by setting ˆ E k = E | X ×{ ( − k } . The ( φ, τ )-twisted G -action on E induces a ( φ, τ, c )-twisted G -action on ˆ E . Then the assignment E ˆ E extends to the homomorphism π ∗ : φ K τ +0 G ( X ) fin → φ K ( τ, cG ( X ) fin .If ˆ E = ˆ E ⊕ ˆ E is a ( φ, τ, c )-twisted vector bundle on X//G , then we can definean ungraded vector bundle E on X × Z by E | X ×{ ( − k } = ˆ E k . The ( φ, τ, c )-twisted G -action on ˆ E induces a ( φ, τ )-twisted G -action on E . We can directlycheck π ∗ E ∼ = ˆ E , so that π ∗ is surjective.If E = π ∗ F for a ( φ, τ )-twisted ungraded vector bundle F on X//G , then wehave ˆ E = ˆ E ⊕ ˆ E with ˆ E = ˆ E = F . This Z -graded vector bundle ˆ E admits a Cl , -action γ = (cid:18) −
11 0 (cid:19) . This Cl , -action is compatible with the ( φ, τ, c )-twisted G -action on ˆ E , so that wehave ˆ E ∈ φ Vect ( τ,c )+(1 , ( X//G ) fin . This proves π ∗ π ∗ F = 0 in φ K ( τ,c )+0 G ( X ) fin . Inthe opposite direction, suppose first that a ( φ, τ, c )-twisted vector bundle ˆ E = π ∗ E on X//G is constructed from a ( φ, τ )-twisted ungraded vector bundle E on ( X × Z ) //G . We express the twisted G -action on E as ρ ( g ) ij : L | { g }× X ⊗ E | X ×{ ( − j } → E | X ×{ ( − i } , where ( − i = c ( g )( − j holds true, and L → G × X is the Hermitianline bundle in the data ( L, τ ) of the φ -twisted (trivially Z -graded, or ungraded)extension τ . With this notation, the ( φ, τ, c )-twisted G -action on ˆ E = ˆ E ⊕ ˆ E is REED-MOORE K -THEORY 27 expressed as ˆ ρ ( g ) = (cid:18) ρ ( g ) ρ ( g ) (cid:19) , ( c ( g ) = 1) (cid:18) ρ ( g ) ρ ( g ) (cid:19) . ( c ( g ) = − E admits a compatible Cl , -action, which is expresses as γ = (cid:18) γ γ (cid:19) , where γ γ = −
1. Then we have a ( φ, τ )-twisted vector bundle F on X//G bysetting F = ˆ E = E | X ×{ } and defining its ( φ, τ )-twisted G -action as ρ F ( g ) = (cid:26) ρ ( g ) , ( c ( g ) = 1) γ ρ ( g ) . ( c ( g ) = − E is isomorphic to π ∗ F . This means that the sequence inquestion is exact at φ K τ +0 G ( X × Z ) fin , and the proof is completed. (cid:3) Let us apply the result above to the case where G = Z acts on X trivially, c : G → Z is the identity, and τ is trivial. Then π ∗ turns out to be surjective and K c +0 Z ( X ) fin = 0, whereas K c +0 Z ( X ) ∼ = K ± ( X ) ∼ = K ( X ) as shown in [4]. Thus, if X = S , then ı : K c +0 Z ( S ) fin → K c +0 Z ( S ) ∼ = Z is not bijective. Remark . Under the assumptions in Definition 3.13 that G is finite and X iscompact, the isomorphism class of the ungraded twist [ τ ] ∈ H G ( X ; Z ) is a torsionclass. Hence Proposition 3.14 is consistent with the conjecture in [35].3.6. The Thom isomorphism.
To state the Thom isomorphism in the Freed-Moore K -theory, let us recall, from [21] for instance, that the Pin c -group Pin c ( r )is a central extension of the orthogonal group O ( r ) by U (1). This group sits inthe complexified Clifford algebra Cl r, ⊗ C , so that there is a natural complexconjugation on Pin c ( r ). Definition 3.17.
Let X be a groupoid, φ ∈ Φ( X ) an object, and π : V → X a realvector bundle of rank r . We write P = ( P, ρ ) for the principal O ( r )-bundle arisingas the frame bundle of V with respect to a Riemannian metric. For φ realized as amap of groupoids φ : X → pt // Z , a φ -twisted Pin c -structure on V consists of • A principal Pin c ( r )-bundle ˜ π : ˜ P → X which is a lift of the structure group O ( r ) of π : P → X to Pin c ( r ) by an equivariant map q : ˜ P → P . • A fiber preserving map ˜ ρ : ∂ ∗ ˜ P → ∂ ∗ ˜ P on X such that – ∂ ∗ q ◦ ˜ ρ = ˜ ρ ◦ ∂ ∗ q , – ˜ ρ (˜ p · ˜ g ) = ˜ ρ (˜ p ) · φ ( f ) ˜ g for f ∈ X , ˜ p ∈ ∂ ∗ ˜ P | f and ˜ g ∈ Pin c ( r ), – ∂ ∗ ˜ ρ ◦ ∂ ∗ ˜ ρ = ∂ ∗ ˜ ρ on X .For general φ consisting of a local equivalence ˜ X → X and a map of groupoids φ : ˜ X → pt // Z , a φ -twisted Pin c -structure on V → X means one on the pull-backof V to ˜ X .To help the understanding of the notion of φ -twisted Pin c -structures, let usassume for a moment that the groupoid X is the quotient groupoid X//G associatedto an action of a compact Lie group G on X and φ ∈ Φ( X//G ) is induced from ahomomorphism φ : G → Z . In this case, a real vector bundle V on X//G means a G -equivariant real vector bundle on X , so that its frame bundle P is a G -equivariantprincipal O ( r )-bundle, provided that the rank of V is r . Then, a φ -twisted Pin c -structure ˜ P of V is a Pin c -structure of the underlying vector bundle V which has,for each f ∈ G , a G -action ˜ ρ f : ˜ P → ˜ P covering the G -action ρ f : P → P such that˜ ρ f (˜ p ˜ g ) = ˜ ρ f (˜ p ) · φ ( g ) ˜ g for all ˜ p ∈ ˜ P and ˜ g ∈ Pin c ( r ). Lemma 3.18.
Let X be a groupoid, φ ∈ Φ( X ) an object, and V a real vector bundleof rank r . There exists a φ -twisted Pin c -structure on V if and only if a cohomologyclass φ W ( V ) ∈ H ( X ; Z φ ) vanishes.Proof. We can assume that φ is realized as a map of groupoids φ : X → pt // Z .Then, from the frame bundle P of V , we can construct a groupoid P //O ( r ) ad-mitting a local equivalence ̟ : P //O ( r ) → X . Further, from the central extensionPin c ( r ) of O ( r ), we can construct a ̟ ∗ φ -twisted (ungraded) extension of P //O ( r )whose trivializations are in bijective correspondence with φ -twisted Pin c -structureson V . In general, the class in H ( X ; Z φ ) that classifies a φ -twisted extension is theobstruction to admitting a trivialization, from which the lemma follows. (cid:3) Theorem 3.19.
Let X be a local quotient groupoid, φ ∈ Φ( X ) an object, and ( τ, c ) a φ -twist of X . For any real vector bundle π : V → X of rank r , we write D ( V ) and S ( V ) for the unit disk bundle and the unit sphere bundle of V with respect toa Riemannian metric. Then there is a natural isomorphism φ K ( τ,c )+ n ( X ) ∼ = π ∗ φ K π ∗ (( τ,c )+( τ V ,c V ))+ n + r ( D ( V ) , S ( V )) , where τ V is classified by the obstruction class φ W ( V ) ∈ H ( X ; Z φ ) for V admittinga φ -twisted Pin c -structure, and c V by the obstruction class w ( V ) ∈ H ( X ; Z ) for V to being orientable.Proof. We sketch the proof following [10]. By replacing X if necessary, we canassume that the object φ is realized as a map of groupoids φ : X → pt // Z and the φ -twist ( τ, c ) as a φ -twisted extension of X . Let E be a locally universal ( φ, τ, c )-twisted vector bundle on X . The disk bundle D ( V ) gives rise to a local quotientgroupoid, and S ( V ) is its subgroupoid. By definition, we have π ∗ φ K π ∗ ( τ,c )+0 ( D ( V ) , S ( V )) = Γ( D ( V ) , S ( V ) , Fred( π ∗ E )) / ∼ . Associated to V is a φ -twisted vector bundle C l ( V ) → X whose fibers are thecomplexified Clifford algebra. Then, we have a homotopy equivalenceΓ( D ( V ) , S ( V ) , Fred( π ∗ E )) ≃ Γ( X , Fred Cl ( V ) ( C l ( V ) ⊗ E )) , where Fred Cl ( V ) ( C l ( V ) ⊗ E ) is defined by replacing the Clifford action in Definition3.1 by the natural left fiberwise Clifford action of Cl ( V ) on C l ( V ) ⊗ E . Except forthe use of the O ( r )-equivariance of the Atiyah-Singer map in Lemma B.12, the proofof the homotopy equivalence is essentially the repetition of that of Theorem 3.8.From the frame bundle P of V , we can construct a groupoid P //O ( r ) and a localequivalence ̟ : P //O ( r ) → X . The pull-back under ̟ ∗ induces a homeomorphismΓ( X , Fred Cl ( V ) ( C l ( V ) ⊗ E )) ∼ = Γ( P //O ( r ) , Fred ̟ ∗ Cl ( V ) ( ̟ ∗ ( C l ( V ) ⊗ E ))) . On P //O ( r ) is a ̟ ∗ φ -twisted ungraded extension L V whose trivializations arebijective correspondence with the φ -twisted Pin c -structures on V . The determi-nant det : O ( r ) → Z induces a grading c V of L V which is classified by w ( V ) ∈ H ( X ; Z ). Then the product bundle on P //O ( r ) with fiber M = C l ( V ) gives rise REED-MOORE K -THEORY 29 to a ( ̟ ∗ φ, τ V , c V )-twisted vector bundle M . Furthermore, this twisted bundle is a ̟ ∗ Cl ( V ) − Cl r, -bimodule. Then, by a Morita equivalence based on M , we have ahomeomorphismΓ( P //O ( r ) , Fred ̟ ∗ Cl ( V ) ( ̟ ∗ ( C l ( V ) ⊗ E ))) ∼ = Γ( P //O ( r ) , Fred( E ′ )) , where E ′ is the locally universal ̟ ∗ (( τ, c ) − ( τ V , c V ))-twisted vector bundle with Cl r, -action given by E ′ = Hom ̟ ∗ Cl ( V ) ( M , ̟ ∗ ( C l ( V ) ⊗ E )) . Summarizing, we get a natural isomorphism π ∗ φ K π ∗ ( τ,c )+0 ( D ( V ) , S ( V )) ∼ = φ K (( τ,c ) − ( τ V ,c V )) − r ( X ) . Since an action of the Clifford algebra accounts for the degree n ∈ Z , we cangeneralize the argument so far to have π ∗ φ K π ∗ ( τ,c ) − n ( D ( V ) , S ( V )) ∼ = φ K (( τ,c ) − ( τ V ,c V )) − n − r ( X ) , which is equivalent to the isomorphism in question. (cid:3) To provide examples of the Thom isomorphism, let us consider the quotientgroupoid X = X//G associated to an action of a finite group G on a space X and φ ∈ Φ( X//G ) associated to a homomorphism φ : G → Z . We let ( τ, c ) be any φ -twist.Let E be a φ -twisted vector bundle on X of rank r . For its underlying real vectorbundle of rank 2 r , we can show by a direct computation that the ungraded twist τ E is given by the group cocycle τ φ if r = 1 , r = 3 , c E is given by the homomorphism c φ = φ if r = 1 , c φ is trivial if r = 2 , τ φ and c φ have theeffect of degree shifts, we eventually get φ K ( τ,c )+ nG ( X ) ∼ = φ K π ∗ ( τ,c )+ nG ( D ( E ) , S ( E )) . This generalizes the Thom isomorphism for a complex vector bundle in complex K -theory and that for a ‘Real’ vector bundle in ‘Real’ K -theory.Let f : G → Z be any homomorphism, and R f the product real line bundle X × R with the G -action ( x, r ) ( x, f ( g ) r ). We have τ R f = τ f and c R f = c f = f .In the case of f = φ , the Thom isomorphism is φ K ( τ,c )+ nG ( X ) ∼ = φ K π ∗ ( τ,c )+ n − G ( D ( R φ ) , S ( R φ )) . Then, from the long exact sequence for ( D ( R φ ) , S ( R φ )), we get a generalization ofan exact sequence for ‘Real’ K -theory in [2] (p.377, (3.4)) · · · → φ K ( τ,c )+ n +1 G ( X ) → φ K ( τ,c )+ nG ( X ) → K ( τ,c )+ n Ker φ ( X ) → φ K ( τ,c )+ n +2 G ( X ) → · · · . In the case of f = c , the Thom isomorphism is φ K ( τ,c )+ nG ( X ) ∼ = φ K π ∗ ( τ, n +1 G ( D ( R c ) , S ( R c )) . The long exact sequence for the pair ( D ( R c ) , S ( R c )) gives us · · · → φ K τ + nG ( X ) → φ K τ + nG ( X × Z ) → φ K ( τ,c )+ nG ( X ) → φ K τ + n +1 G ( X ) → · · · , where G acts on X × Z by ( x, r ) ( gx, c ( g ) r ). We remark φ K τ + nG ( X × Z ) ∼ = φ K τ + n Ker( c ) ( X ) in this case. We also remark that the above exact sequence extendsthe one in Proposition 3.15. Karoubi formulation of Freed-Moore K -theory This section is devoted to Karoubi’s formulations of the Freed-Moore K -theory:We introduce the infinite-dimensional Karoubi formulation, and relate it with theFredholm formulation. We then introduce the finite-dimensional Karoubi formula-tion, and relate it with the other formulations.4.1. Gradation.Definition 4.1 (gradation) . Let X be a groupoid, φ : X → Z a map of groupoids,and ( L, τ, c ) a φ -twisted Z -graded extension of X . For a ( φ, τ, c )-twisted vectorbundle ( E, ǫ, ρ, γ ) on X with Cl p,q -action, we define a fiber bundle Gr( E ) → X asfollows: • The fiber of the underlying fiber bundle Gr( E ) → X at x ∈ X consistsbounded operators η : E x → E x such that:(i) η are a self-adjoint involution: η ∗ = η and η = id.(ii) η − ǫ are compact.(iii) η anti-commute with the Cl p,q -action, that is, ηγ ( e ) = − γ ( e ) η, for any unit norm element e ∈ R p + q . • The bundle isomorphism ̺ : φ ∂ ∗ Gr( E ) → ∂ ∗ Gr( E ) on X is given by ̺ ( η ) = ρ ◦ (id L ⊗ η ) ◦ ρ − , where id L : L → L is the identity map.The fiber bundle Gr( E ) is topologized by using the operator norm topology. Thespace of sections is defined byΓ( X , Gr( E )) = { η ∈ Γ( X , Gr( E )) | ̺ ◦ φ ∂ ∗ η = ∂ ∗ η } . We call a section η ∈ Γ( X , Gr( E )) a gradation of E .As in the case of K( E ), the φ -twisted action ̺ is continuous with respect to thetopology on E given by the compact open topology.We notice that no commutation relation among η and ǫ is imposed. We alsonotice that ( E, η, ρ, γ ) is a ( φ, τ, c )-twisted vector bundle on X with Cl p,q -action.Thus a gradation η on E is regarded as another choice of a Z -grading of E . Itis clear that ǫ ∈ Γ( X , Gr( E )). A homotopy between gradations η and η of E isdefined by a gradation ˜ η of the twisted bundle E × [0 ,
1] on
X × [0 ,
1] such that˜ η | X ×{ i } = η i for i = 0 ,
1. In this case, η and η are said to be homotopic, and wewrite η ∼ η .Suppose that for a groupoid X and φ = ( F : ˜ X → X , φ ) ∈ Φ( X ), we have a φ -twist on X consisting of a local equivalence ˜ F : ˜˜ X → ˜ X , a ˜ F ∗ φ -twisted extension( L, τ, c ) of ˜˜ X , and a ( ˜ F ∗ φ, τ, c )-twisted vector bundle E on ˜˜ X with Cl p,q -action.As in the case of Fred( E ), there uniquely exists a φ -twisted bundle Gr( τ ) on ˜ X up to isomorphisms such that ˜ F ∗ Gr( τ ) is isomorphic to Gr( E ) and ˜ F ∗ inducesa homeomorphism from Γ( ˜ X , Gr( τ )) to Γ( ˜˜ X , Gr( E )) preserving the equivalencerelations ∼ given by fiberwise homotopies of sections. Definition 4.2.
Let X be a local quotient groupoid, φ = ( F : ˜ X →
X, φ ) ∈ Φ( X )an object, and ( τ, c ) = ( ˜ F : ˜˜ X → ˜ X , L, τ, c ) a φ -twist on X . We define φ K ( τ,c )+( p,q ) ( X ) = Γ( ˜ X , Gr( τ )) / ∼∼ = Γ( ˜˜ X , Gr( E )) / ∼ REED-MOORE K -THEORY 31 where Gr( τ ) → ˜ X is the φ -twisted bundle such that ˜ F ∗ Gr( τ ) ∼ = Gr( E ) for a( ˜ F ∗ φ, τ, c )-twisted locally universal vector bundle E → ˜˜ X with Cl p,q -action. Inthe case that X is a quotient groupoid X//G , we may write φ K ( τ,c )+( p,q ) ( X//G ) = φ K ( τ,c )+( p,q ) G ( X ) . The operation of taking a direct sum makes φ K ( τ,c )+( p,q ) ( X ) into an abelianmonoid, in which the zero element is represented by ǫ . This monoidal structureturns out to be a group structure, as will be seen shortly. Though will not bedetailed, the same construction as in Lemma 3.5 provides isomorphisms, such as φ K ( τ,c )+( p,q ) ( X ) ∼ = φ K ( τ,c )+( p +1 ,q +1) ( X ) . Remark . Let us consider the trivial setting that X = pt // φ , τ and c aretrivial, and p = q = 0. In this setting, a locally universal bundle is just a Z -gradedvector space ( E, ǫ ) such that E k = Ker( ǫ − ( − k ) are separable infinite-dimensionalHilbert spaces. Then Gr( E ) isGr( E ) = { η ∈ End( E ) | η − ǫ ∈ K( E ) , η ∗ = η, η = id } , where End( E ) is the space of bounded operators and K( E ) that of compact oper-ators. All these spaces are topologized by the operator norm. The space Gr( E )appears in [26] as a model of the classifying space of the even complex K -theory,and admits the identificationGr( E ) ∼ = { U ∈ End( E ) | U − id ∈ K( E ) , ǫU ǫ = U ∗ = U − } given by η U = ηǫ . The space Gr( E ) also admits the identificationGr( E ) ∼ = { W ⊂ E | closed, pr : W → E Fredholm, pr : W → E compact } given by η W = Im(1 + η ) = Ker(1 − η ), where pr k : W → E k are theorthogonal projections. This space essentially agrees with the infinite-dimensionalGrassmannian in [25], where Hilbert-Schmidt operators are used in place of compactoperators.4.2. Relationship with Fredholm formulation.
Now, we relate the K -theory φ K ( τ,c )+( p,q ) ( X ) in the Fredholm formulation with φ K ( τ,c )+( p,q ) ( X ) in the Karoubiformulation. As is mentioned in §
1, a change of twists enters into the relationshipof these two formulations.
Definition 4.4.
Let p : Z × Z → Z be the projection onto the first factor. Wedefine a group 2-cocycle µ ∈ Z ( Z × Z ; U (1) p ) by µ ((( − m , ( − n ) , (( − m ′ , ( − n ′ )) = exp πinn ′ . The 2-cocycle µ defines a p -twisted Z -graded extension ( Z × Z × C , µ, // ( Z × Z ), in which the line bundle on the space ofmorphisms is the product bundle Z × Z × C → Z × Z with the trivial Z -grading.Thus, if φ : X → pt // Z and c : X → pt // Z are maps of groupoids, then we geta φ -twisted Z -graded extension ( φ, c ) ∗ µ = ( X × C , ( φ, c ) ∗ µ,
1) by the pull-backunder ( φ, c ) :
X → pt // ( Z × Z ). Lemma 4.5.
Let φ : X → pt // Z and c : X → pt // Z be maps of groupoids. If φ or c is trivial, then ( φ, c ) ∗ µ is trivial. Proof. If c is trivial, then ( φ, c ) ∗ µ is clearly trivial by the definition of µ . If φ istrivial, then the map of groupoids ( φ, c ) : X → pt // ( Z × Z ) factors as follows X c −→ pt // Z , id) −→ pt // ( Z × Z ) . The pull-back cocycle (1 , id) ∗ µ ∈ Z ( Z ; U (1)) can be trivialized by the group1-cochain β ∈ C ( Z ; U (1)) given by β (1) = 1 and β ( −
1) = i . Hence ( φ, c ) ∗ µ isalso trivialized. (cid:3) Definition 4.6.
Let X be a groupoid and φ : X → pt // Z a map of groupoids.For a φ -twisted Z -graded extension ( τ, c ) = ( L, τ, c ) of X , we define a φ -twisted Z -graded extension (´ τ , c ) by (´ τ , c ) = ( L, τ, c ) + ( φ, c ) ∗ µ = ( L, τ µ, c ).Since the 2-cocycle µ is clearly trivial, we have ´´ τ = τ .As is seen, for a quotient groupoid X = X//G and a map of groupoids φ : X//G → pt // Z induced from a homomorphism φ : G → Z , a group 2-cocycle τ ∈ Z ( G ; C ( X, U (1)) φ ) defines a φ -twisted Z -graded extension ( τ, c ) = ( G × X × C , τ, c ) of X//G , in which the line bundle on the space of morphisms is theproduct bundle G × X × C → G × X , and c : X → pt // Z is a map of groupoids. Inthis case, (´ τ , c ) = ( G × X × C , ´ τ , c ) is the φ -twisted Z -graded extension of X//G associated to the group 2-cocycle ´ τ ∈ Z ( G ; C ( X, U (1)) φ ) given by´ τ ( g, h ; x ) = (cid:26) τ ( g, h ; x ) , ( c ( g, hx ) = 1 or c ( h, x ) = 1) − τ ( g, h ; x ) . ( c ( g, hx ) = c ( h, x ) = − Lemma 4.7.
Let X be a groupoid X , φ : X → pt // Z a map of groupoids, and ( τ, c ) a φ -twisted Z -graded extension of X . (a) For a ( φ, τ, c ) -twisted vector bundle E on X with Cl p,q -action, there existsa ( φ, ´ τ , c ) -twisted vector bundle ´ E on X with Cl q,p -action such that ´´ E isisomorphic to E . (b) For a degree k map f : E → E of ( φ, τ, c ) -twisted vector bundles on X with Cl p,q -actions, there is a degree k map ´ f : ´ E → ´ E of ( φ, ´ τ , c ) -twistedvector bundles on X with Cl q,p -action such that ´ f ◦ ´ f = ( − k k ´( f ◦ f ) for maps f i of degree k i .Proof. For (a), let E = ( E, ǫ, ρ, γ ) be a ( φ, τ, c )-twisted vector bundles E on X with Cl p,q -action. We then have a ( φ, ´ τ , c )-twisted vector bundle ´ E = ( ´ E, ´ ǫ, ´ ρ, ´ γ )on X with Cl q,p -actions as follows: The underlying Hermitian vector bundle on X is given by ´ E = E , and its Z -grading by ´ ǫ = ǫ . The Cl q,p -action ´ γ on ´ E = E isgiven by ´ γ ( e ) = γ ( e ) ǫ for unit norm vectors e ∈ R p + q . Finally, the twisted action´ ρ : L ⊗ φ ∂ ∗ E → ∂ ∗ E is given by the composition L ⊗ φ ∂ ∗ E id ⊗ ǫ c −→ L ⊗ φ ∂ ∗ E ρ −→ ∂ ∗ E, where ǫ c : ∂ ∗ E → ∂ ∗ E is the identity on the component c − (1) ⊂ X and φ ∂ ∗ ǫ on c − ( − ⊂ X . With some direct calculations, we can verify that ´ E is a ( φ, ´ τ , c )-twisted vector bundle on X with Cl q,p -action. By construction, the identity mapgives an isomorphism ´´ E → E . For (b), we define ´ f : ´ E → ´ E by ´ f = f if k is evenand ´ f = f ǫ if k is odd. We can readily verify that ´ f is a degree k map of ( φ, ´ τ , c )-twisted vector bundles on X with Cl q,p -action. The claim about the compositionis also verified readily. (cid:3) REED-MOORE K -THEORY 33 For a better understanding of the construction of ´ E , let us assume that thegroupoid X is a quotient groupoid X//G , the map of groupoids φ is induced froma homomorphism φ : G → Z , and c : X//G → pt // Z is also induced froma homomorphism c : G → Z . As is reviewed already in §§ φ, τ, c )-twistedvector bundle E is a Z -graded vector bundle ( E, ǫ ) with real orthogonal map ρ ( g ) : E → E , ( g ∈ G ) realizing the twisted action and with unitary maps γ i : E → E realizing the Cl p,q -action. Then the twisted action ´ ρ ( g ) and the Cl q,p -action ´ γ on´ E = E are given by´ ρ ( g ) = (cid:26) ρ ( g ) , ( c ( g ) = 1) ρ ( g ) ǫ, ( c ( g ) = −
1) ´ γ i = γ i ǫ. We now introduce a map relating the Fredholm formulation with the Karoubiformulation, which originates from [7].
Definition 4.8.
Let X be a groupoid, φ : X → pt // Z a map of groupoids, ( τ, c ) a φ -twisted Z -graded extension of X , and ( E, ǫ, ρ, γ ) a ( φ, τ, c )-twisted vector bundleon X with Cl p,q -action. We define a map of fiber bundles on X ϑ : Fred( E ) → Gr( ´ E )by ϑ ( A ) = − e πA ǫ for A : E | x → E | x belonging to the fiber of Fred( E ) at x ∈ X .To see that ϑ is a well-defined continuous map, we start with the simplest case. Lemma 4.9.
For a separable Z -graded Hilbert space ( E, ǫ ) , the map ϑ : Fred( E ) → Gr( E ) is well-defined and continuous.Proof. To see that ϑ is well-defined, we notice that − iA : E → E is a self-adjointoperator. Hence the functional calculus gives a bounded operator e i ( − iπA ) = e πA on E . Because A + id is compact, the spectral set Spec( A ) and hence Spec( A )consist of eigenvalues only. The eigenspaces of A whose eigenvalues differ from ± i are finite-dimensional. The eigenspaces of A with their eigenvalues ± i are theonly possible infinite-dimensional eigenspaces, on which − e πA acts by id. Thisproves that − e πA + id is compact, and so is − e πA ǫ + ǫ . We can directly check that − e πA ǫ is a self-adjoint involution. Hence ϑ is well-defined as a map. To prove that ϑ : Fred( E ) → Gr( E ) is continuous, let A m ∈ Fred( E ) be a sequence convergent to A ∞ ∈ Fred( E ) as m → ∞ . By definition, this means that the sequence of maps A m | C : C → E uniformly converges to A ∞ | C : C → E on any compact subset C ⊂ E and A m converges to A ∞ in the operator norm, i.e. k A m − A ∞ k → m → ∞ . The continuity of ϑ will be established when we prove that − ϑ ( A m ) ǫ converges to − ϑ ( A ∞ ) ǫ in the operator norm. For this aim, we express − ϑ ( A ) ǫ asfollows − ϑ ( A ) ǫ = e i ( − iπA ) = cos( − iπA ) + i sin( − iπA )= X n ≥ ( − n (2 n )! ( − iπA ) n + i ( − iπA ) X n ≥ ( − n (2 n + 1)! ( − iπA ) n = X n ≥ π n (2 n )! ( A ) n + i ( − iA ) π X n ≥ π n (2 n + 1)! ( A ) n . Let C ( z ) and S ( z ) be the following power series in zC ( z ) = X n ≥ π n (2 n )! z n , S ( z ) = π X n ≥ π n (2 n + 1)! z n . These series are convergent on the whole of C , and hence define holomorphic func-tions. As a result, cos( − iπA m ) = C ( A m ) converges to cos( − iπA ∞ ) = C ( A ∞ ) inthe operator norm, since A m converges to A ∞ in the operator norm. Let us definea holomorphic function T ( z ) by T ( z ) = S ( z − T ( z ) in z , theconstant part is absent, because T (0) = S ( −
1) = sin π = 0. Therefore T ( A + 1) isa compact operator for any A ∈ Fred( E ). Now, let us see the estimate k sin( − iπA m ) − sin( − iπA ∞ ) k = k A m T ( A m + 1) − A ∞ T ( A ∞ + 1) k≤ k A m k · k T ( A m + 1) − T ( A ∞ + 1) k + k ( A m − A ∞ ) T ( A ∞ + 1) k . Because A m is skew-adjoint, we have k A m k = k A ∗ m A m k = k A m k . Thus, we have k A m k → k A ∞ k and k T ( A m + 1) − T ( A m + 1) k → m → ∞ , since A m convergesto A ∞ in the operator norm. The compact operator T ( A ∞ + 1) maps the unitsphere { v ∈ E | k v k = 1 } in E to a compact subset in E . On the compact subset, A m converges to A ∞ uniformly, so that k ( A m − A ∞ ) T ( A ∞ + 1) k →
0. In summary, ϑ ( A m ) ǫ converges to ϑ ( A ∞ ) ǫ in the operator norm, and ϑ : Fred( E ) → Gr( E ) is acontinuous map. (cid:3) Lemma 4.10.
The map ϑ in Definition 4.8 is well-defined, continuous, and givesrise to a map of φ -twisted fiber bundles on the groupoid X .Proof. In the same way as in Lemma 4.9, we can show that A
7→ − e πA ǫ is well-defined. It is easy to verify the anti-commutation relation between − e πA ǫ and theClifford action on ´ E . This proves that the map ϑ : Fred( E ) → Gr( ´ E ) is well-definedas a map of the fiber bundles on X . The continuity of ϑ follows from Lemma4.9, because of the local triviality of E . We can also directly verify that ϑ ( A ) iscompatible with the twisted action on ´ E , which means that ϑ : Fred( E ) → Gr( ´ E )is a map of fiber bundles on X . (cid:3) As a result of the lemma above, we have a map of sections ϑ : Γ( X , Fred( E )) → Γ( X , Gr( ´ E )) , which is continuous, provided that the spaces of sections are topologized by thethe compact open topologies. Clearly, this map preserves the operations of takingthe direct sum. If E is locally universal and γ ∗ ∈ Γ( X , Fred( E ) † ), then ϑ ( γ ∗ ) = ǫ ,because Spec( γ ∗ ) = {± i } . Consequently, ϑ induces a well-defined map of monoids ϑ : φ K ( τ,c )+( p,q ) ( X ) → φ K (´ τ,c )+( q,p ) ( X ) . Theorem 4.11.
Let X be a local quotient groupoid, φ ∈ Φ( X ) an object, and ( τ, c ) a φ -twist on X . For any p, q ≥ , the monoid map ϑ : φ K ( τ,c )+( p,q ) ( X ) → φ K (´ τ,c )+( q,p ) ( X ) is bijective. In particular, φ K (´ τ,c )+( q,p ) ( X ) gives rise to an abelian group. Further, ifwe put φ K (´ τ,c )+ n ( X ) = φ K (´ τ,c )+(0 ,n ) ( X ) , then ϑ induces an isomorphism of groups φ K ( τ,c )+ n ( X ) ∼ = φ K (´ τ,c )+ n ( X ) . REED-MOORE K -THEORY 35 Proof.
As before, we apply the reduction argument as in Lemma 2.9 and [10](Proposition A.19). Then, it is enough to see that ϑ : Γ( X , Fred( E )) → Γ( X , Gr( ´ E ))is a weak homotopy equivalence when X is the groupoid pt //G associated to a com-pact Lie group G and τ is trivial. In this case, ϑ is a homotopy equivalence, asshown in Appendix B (Lemma B.18). (cid:3) Remark . One can avoid the twist change τ ´ τ by using self-adjoint operatorsinstead of skew-adjoint operators in Definition 3.1. To be precise, we define ´Fred( E )by replacing (i), (ii) and (iii) in Definition 3.1 by(i)’ A are self-adjoint: A ∗ = A .(ii)’ A − id are compact.(iii)’ Spec( A ) ⊂ [ − , E ), we also define φ ´ K ( τ,c )+( p,q ) ( X ) and φ ´ K ( τ,c ) − n ( X , Y ) as in Definition3.4 and Definition 3.7. By Lemma 4.7, we have an isomorphism of fiber bundlesFred( E ) → ´Fred( ´ E ) , A ´ A = Aǫ, where ǫ is the Z -grading of E . This induces the isomorphisms of groups φ K ( τ,c )+( p,q ) ( X ) ∼ = φ ´ K (´ τ,c )+( q,p ) ( X ) , φ K ( τ,c )+ n ( X ) ∼ = φ ´ K (´ τ,c )+ n ( X ) . Hence Theorem 4.11 provides the isomorphisms without the twist change φ ´ K ( τ,c )+( q,p ) ( X ) ∼ = φ K ( τ,c )+( q,p ) ( X ) , φ ´ K ( τ,c )+ n ( X ) ∼ = φ K ( τ,c )+ n ( X ) . Note that the counterpart of Corollary 3.10 reads φ ´ K ( τ,c )+ p − q ( X ) ∼ = φ ´ K ( τ,c )+( p,q ) ( X ) . Note also that the counterpart of ı in §§ φ ´ K ( τ,c )+ nG ( X ) is defined for φ ´ K ( τ,c )+ n ( X ) fin = π ( φ Vect ( τ,c )+(0 ,n ) G ( X ) fin ) /π ( φ Vect ( τ,c )+(0 ,n +1) G ( X ) fin ) , which is isomorphic to φ K (´ τ,c )+ n ( X ) fin by E ´ E .4.3. Finite-dimensional Karoubi formulation.
Let us consider the same setupas in §§ Definition 4.13 (triple) . Let X be the quotient groupoid X//G associated to anaction of a finite group G on a compact Hausdorff space X , φ : X//G → pt // Z the map of groupoids associated to a homomorphism φ : G → Z , and ( L, τ, c ) a φ -twisted Z -graded extension of X//G .(a) We define a triple ( E, η , η ) on X//G by the requirement that (
E, η , ρ, γ )and ( E, η , ρ, γ ) are objects of φ Vect ( τ,c )+( p,q ) G ( X ) fin .(b) We define an isomorphism of triples f : ( E, η , η ) → ( E ′ , η ′ , η ′ ) to be anisomorphism of vector bundles f : E → E ′ on X which gives isomorphisms f : ( E, η i , ρ, γ ) → ( E ′ , η ′ i , ρ ′ , γ ′ ) in φ Vect ( τ,c )+( p,q ) G ( X ) fin for i = 0 , E, η , η ) can be regarded as a twisted vector bundle( E, η , ρ, γ ) on X//G equipped with a gradation η in the sense of Definition4.1, where the compactness of η − η is automatically satisfied by the finite-dimensionality of E . The direct sum of triples is defined by ( E, η , η ) ⊕ ( E ′ , η ′ , η ′ ) =( E ⊕ E, η ⊕ η ′ , η ⊕ η ′ ). Definition 4.14.
We assume the same setting as in Definition 4.13. (a) We define φ M ( τ,c )+( p,q ) G ( X ) fin to be the monoid of the isomorphism classesof triples on X//G .(b) We define φ Z ( τ,c )+( p,q ) G ( X ) fin ⊂ φ M ( τ,c )+( p,q ) G ( X ) fin to be the submonoidconsisting of triples ( E, η , η ) such that η is homotopic to η as gradations.(c) We define φ K ( τ,c )+( p,q ) G ( X ) fin = φ M ( τ,c )+( p,q ) G ( X ) fin / φ Z ( τ,c )+( p,q ) G ( X ) fin tobe the quotient monoid. Lemma 4.15.
The monoid φ K ( τ,c )+( p,q ) G ( X ) fin is an abelian group, in which theadditive inverse of [ E, η , η ] is given by [ E, η , η ] . It also holds that [ E, η , η ] + [ E, η , η ] = [ E, η , η ] . Proof.
To see that the quotient monoid is an abelian group, we verify the monoidmorphism I ([ E, η , η ]) = [ E, η , η ] satisfies the assumptions in Lemma C.1. Thenthe non-trivial thing is that [ E, η , η ] + [ E, η , η ] = 0, namely, the gradations η ⊕ η and η ⊕ η on E ⊕ E are homotopic. As given in [17] (4.16 Lemma), thefamily of gradations˜ η ( θ ) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) (cid:18) η η (cid:19) (cid:18) cos θ sin θ − sin θ cos θ (cid:19) = (cid:18) η cos θ + η sin θ ( η − η ) cos θ sin θ ( η − η ) cos θ sin θ η sin θ + η cos θ (cid:19) realizes such a homotopy in our setting. The remaining formula can be shown inthe same way as in [17] (4.17 Lemma). (cid:3) Using the idea of the proof of Lemma 3.5, we can also prove φ K ( τ,c )+( p,q ) ( X ) fin ∼ = φ K ( τ,c )+( p +1 ,q +1) ( X ) fin . As in §§ φ K ( τ,c )+( p,q ) G ( X ) fin withthe infinite-dimensional formulation φ K ( τ,c )+( p,q ) G ( X ). Lemma 4.16.
Under the assumptions in Definition 4.13, there is a homomorphismof monoids : φ K ( τ,c )+( p,q ) G ( X ) fin → φ K ( τ,c )+( p,q ) G ( X ) . Proof.
Given a triple (
E, η , η ) on X//G , we put ǫ E = η to regard E as a Z -graded vector bundle. In particular, E ∈ φ Vect ( τ,c )+( p,q ) ( X//G ). Then we canembed E into a locally universal bundle E uni . If ǫ E ⊥ denotes the Z -grading ofthe orthogonal complement E ⊥ of E , then the Z -grading ǫ uni of E uni is expressedas ǫ uni = ǫ E ⊕ ǫ E ⊥ . Now, we have a self-adjoint involution η = η ⊕ ǫ E ⊥ on E uni such that η − ǫ uni = ( η − η ) ⊕ η ∈ Γ( X//G,
Gr( E uni )), and we define by the assignment ( E, η , η ) η . Using theproperty E uni ⊕ E uni ∼ = E uni , we can show that is well-defined. It is then clearthat is a homomorphism. (cid:3) To prove that is bijective, we show that any gradation on a locally universalbundle admits a “finite dimensional approximation”. Lemma 4.17.
Let X be a compact Hausdorff space. REED-MOORE K -THEORY 37 (a) Let E be a separable Hilbert space, and { K x } x ∈ X a family of self-adjointcompact operators on E which are continuous in the operator norm. Then,for any r > , there is a finite rank subspace F ⊂ E such that k K x − P K x k F ⊂ E . (b) We additionally suppose in (a) that E is a Z -graded Cl p,q -module. Thenwe can take the subspace F in (a) to be a submodule of E .Proof. For (a), let B ( v ; R ) ⊂ E denote the open ball centered at v ∈ E and radius R ,and B ( v ; R ) its closure. Then { K x v ∈ E| x ∈ X, v ∈ B (0; 1) } is a compact subset.To see this, we define a continuous map k : X × E → X × E by k ( x, v ) = K x v . Asshown in [30] (Proposition 2.1), the projection X × E → X restricts to a propermap k ( X × B (0; 1)) → X . Since X is compact, so are k ( X × B (0; 1)) and its imageunder the projection X × E → E .As a result, we can find a finite number of vectors v , · · · , v n ∈ E so that the openballs B ( v i ; r/ 2) cover { K x v ∈ E| x ∈ X, v ∈ B (0; 1) } . Let F = Span { v , . . . , v n } bethe subspace spanned by the vectors v , . . . , v n , and P the orthogonal projectiononto F . Then, for any x ∈ X and v ∈ B (0; 1), we can find a vector v i from v , · · · , v n such that k K x v − v i k < r/ 2, so that k K x v − P K x v k ≤ k K x v − v i k + k v i − P K x v k = k K x v − v i k + k P v i − P K x v k≤ k K x v − v i k + k v i − K x v k < r r r. This implies that k K x − P K x k < r for any x ∈ X .For (b), we take the subspace F ⊂ E in (a) to be the Cl p,q -submodule of E generated by v , . . . , v n , F = Span { ξv i ∈ E| ξ ∈ Cl p,q , ≤ i ≤ n } . As a vector space, F still remains finite rank. Since v , . . . , v n ∈ F , the succeedingargument in (a) works without change. (cid:3) Lemma 4.18. Under the assumptions in Definition 4.13, let E be a ( φ, τ, c ) -twisted vector bundle on X//G with Cl p,q -action whose fibers are separable infinite-dimensional Hilbert spaces. For any gradation η ∈ Γ( X//G, Gr( E )) , there exists afinite rank ( φ, τ, c ) -twisted subbundle F ⊂ E with Cl p,q -action such that η is homo-topic to η F ⊕ ǫ E/F within gradations, where η F ∈ Γ( X//G, Gr( F )) is the sectionexpressed as η F = | Rηi | − Rηi by using the inclusion i : F → E and the projec-tion R : E → F , and ǫ E/F is the Z -grading of the orthogonal complement F ⊥ of F ⊂ E = F ⊕ F ⊥ .Proof. We can assume that the vector bundle E → X is trivialized as E = X × E by using a separable infinite-dimensional Hilbert space E which is a Z -graded Cl p,q -module. Under this trivialization, we may express the twisted G -action as ρ ( g )( x, v ) = ( gx, ρ ( g ; x ) v ) by using a real linear map ρ ( g ; x ) : L g,x ⊗ E → E , whichis unitary (resp. anti-unitary) if φ ( g ) = 1 (resp. φ ( g ) = − 1) and is degree 0 (resp.1) if c ( g, x ) = 1 (resp. c ( g, x ) = − η ∈ Γ( X//G, Gr( E )) is expressed as η ( x, v ) = ( x, η x v ) by using a self-adjoint involution η x : E → E which is continuousin x with respect to the operator norm and is subject to γ i η x = − η x γ i , ρ ( g ; x )(id L g,x ⊗ η x ) = c ( g, x ) η gx ρ ( g ; x ) , where L g,x is the fiber of L → G × X at ( g, x ) ∈ G × X . (Here we restore the sign c ( g, x ) from the c -twisted line L g,x .) We take r > such that: for each x ∈ X , any bounded operator η ′ : E → E satisfying k η x − η ′ k < r admits a bounded inverse. Such an r exists because X is compact and the invertiblebounded operators on E form an open subset in the space of bounded operatorsequipped with the operator norm topology.We put K x = η x − ǫ to define a continuous family of self-adjoint compact oper-ators on E . Then, by Lemma 4.17, we have a finite rank Cl p,q -submodule F ⊂ E such that k K x − P K x P k < r/ x ∈ X , where P : E → E is the orthogonalprojection onto F . By construction, P commutes with the Z -grading ǫ and withthe Cl p,q -action γ i on E . Now, taking the “average” over the twisted G -action ρ ( g ; x ), we define P x : E → E as P x = 1 | G | X g ∈ G ρ ( g ; g − x )(id L g,g − x ⊗ P ) ρ ( g ; g − x ) − , where | G | is the order of G . It is easy to see that P x is continuous in the operatornorm, γ i P x = P x γ i , and ρ ( g ; x ) ◦ (id L g,x ⊗ P x ) ◦ ρ ( g ; x ) − = P gx . We can also seethat P x is an orthogonal projection, and a finite rank operator, because G is finite.Using the expression K x = 1 | G | X g ∈ G c ( g, g − x ) ρ ( g ; g − x )(id L g,g − x ⊗ K g − x ) ρ ( g ; g − x ) − , we have k K x − P x K x k≤ | G | X g ∈ X k c ( g, g − x ) ρ ( g ; g − x )(id L g,g − x ⊗ ( K g − x − P ′ K g − x )) ρ ( g ; g − x ) − k = 1 | G | X g ∈ G k K g − x − P ′ K g − x k < | G | X g ∈ G r r , for any x ∈ X . As a result, we get k K x − P x K x P x k ≤ k K x − P x K x k + k P x K x − P x K x P x k≤ k K x − P x K x k + k K x − K x P x k = k K x − P x K x k + k ( K x − K x P x ) ∗ k = k K x − P x K x k + k K x − P x K x k < r r r, since K x as well as P x are self-adjoint.Now, we define F x to be the image of P x : E → E . these vector spaces form afinite rank subbundle F = S x ∈ X F x ⊂ X × E . The twisted G -action and the Cl p,q -action on E = X ×E preserve F , so that F gives rise to a finite rank ( φ, τ, c )-twistedvector bundle on X//G with Cl p,q -action. We put H x,t = ǫ + (1 − t ) K x + tP x K x P x for x ∈ X and t ∈ [0 , k H x,t − η x k = t k K x − P x K x P x k < r, the operator H x,t : E → E is invertible for all x ∈ X and t ∈ [0 , η x,t : E → E by η x,t = | H x,t | − H x,t , which is a self-adjoint involution. Noticethat T = H x,t ǫ is an invertible operator on E which differs from the identity bya compact operator. By the spectral theorem for compact operators, the unitary REED-MOORE K -THEORY 39 operator ( T T ∗ ) − / T differs from the identity by a compact operator. Therefore η x,t = | H x,t | − H x,t = ( T T ∗ ) − / T ǫ differs from ǫ by a compact operator. As a result, we get a gradation η t ∈ Γ( X//G, Gr( E )) by defining the bundle map η t : E → E as η t ( x, v ) = ( x, η x,t v ).This is a homotopy from η = η to η within gradations. Since ǫ + P x K x P x commutes with P x , we can decompose η as η = η F ⊕ η ⊥ F by using gradations η F ∈ Γ( X//G, Gr( F )) and η ⊥ F ∈ γ ( X//G, Gr( F ⊥ )). Because of the expression H x, = P x η x P x , if i : F → E and R : E → F respectively denote the inclusion andthe projection, then η F = Rη i = R ( | P ηP | − P ηP ) i = | Rηi | − ( Rηi ) . Similarly, we find η ⊥ F = ǫ E/F . (cid:3) Lemma 4.19. Let F ′ be another subbundle of E as in Lemma 4.18, and η F ′ ∈ Γ( X//G, Gr( F ′ )) the associated gradation. Then there exists a subbundle F ′′ ⊂ E such that F and F ′ are subbundle of F ′′ and η F ⊕ ǫ F ′′ /F and η F ′ ⊕ ǫ F ′′ /F ′ arehomotopic within the gradations of F ′′ , where ǫ F ′′ /F and ǫ F ′′ /F ′ are the Z -gradingsof the orthogonal complements of F ⊂ F ′′ and F ′ ⊂ F ′′ , respectively.Proof. As in the proof of Lemma 4.18, we take a trivialization E = X × E , express η as η ( x, v ) = ( x, η x v ), and put K x = η x − ǫ . We suppose that the finite ranksubbundle F ′ ⊂ E and the gradation η F ′ are constructed from a certain suitablechoice of a real number r ′ > Cl p,q -submodule F ′ ⊂ E along theproof of Lemma 4.18. Therefore the fibers of F = S x ∈ X F x and F ′ = S x ∈ X F ′ x at x ∈ X are respectively the images of the orthogonal projections P x and P ′ x satisfying k K x − P x K x P x k < r, k K x − P ′ x K x P ′ x k < r ′ . We put H x = ǫ + P x K x P x and H ′ x = ǫ + P ′ x K x P ′ x , which are self-adjoint invertibleoperators. Then the gradations η F and η F ′ at x ∈ X are realized as η F,x = R x | H x | − H x i x , η F ′ ,x = R ′ x | H ′ x | − H ′ x i ′ x , where R x : E → F x and R ′ x : E → F ′ x are the projections, and i x : F x → E and i ′ x : F ′ x → E are the inclusions.To give a subbundle F ′′ ⊂ E , we put r ′′ = max { r, r ′ } . For each x ∈ X , anybounded operator η ′ on E such that k η ′ − η x k < r ′′ admits a bounded inverse. Let F ′′ be a finite rank Cl p,q -module which contains both F and F ′ . In view of the proof ofLemma 4.17, the orthogonal projection P ′′ onto F ′′ satisfies k K x − P ′′ K x k < r ′′ / x ∈ X . Applying the construction in the proof of Lemma 4.18 to P ′′ , we getan orthogonal projection P ′′ x whose image F ′′ x forms a finite rank ( φ, τ, c )-twistedvector bundle F ′′ on X//G with Cl p,q -action. By construction, F ′′ contains both F and F ′ . It also holds that k K x − P ′′ x K x P ′′ x k < r ′′ for all x ∈ X .To show that η F ⊕ ǫ F ′′ /F and η F ′ ⊕ ǫ F ′′ /F ′ are homotopic within gradations of F ′′ , we use an intermediate gradation. We put H ′′ x = ǫ + P ′′ x K x P ′′ x . By the proof ofLemma 4.18, this is a self-adjoint invertible operator for each x ∈ X , and we havea gradation η F ′′ of F ′′ given by η ′′ x = R ′′ x | H ′′ x | − H ′′ x i ′′ x , where R ′′ x : E → F ′′ x is the projection and i ′′ x : F ′′ x → E is the inclusion. We thenput H x,t = (1 − t ) H x + tH ′′ x for x ∈ X and t ∈ [0 , E , andfurther invertible, since k H x,t − η x k = k (1 − t )( P x K x P x − K x ) + t ( P ′′ x K x P ′′ x − K x ) k < (1 − t ) r ′′ + tr ′′ = r ′′ . Then η x,t = R ′′ x | H x,t | − H x,t i ′′ x defines a homotopy of gradations η t on F ′′ whichconnects η = η F ⊕ ǫ F ′′ /F with η = η ′′ . The same construction gives a homotopyof gradations η ′ t from η ′ = η F ′ ⊕ ǫ F ′′ /F ′ to η ′ = η ′′ . Hence η F ⊕ ǫ F ′′ /F and η F ′ ⊕ ǫ F ′′ /F ′ are homotopic within the gradations of F ′′ . (cid:3) Theorem 4.20. Under the assumptions in Definition 4.13, the homomorphism : φ K ( τ,c )+( p,q ) G ( X ) fin → φ K ( τ,c )+( p,q ) G ( X ) is bijective. In particular, if we put φ K ( τ,c )+ nG ( X ) fin ∼ = φ K ( τ,c )+( n, G ( X ) fin , then induces an isomorphism of groups φ K ( τ,c )+ nG ( X ) fin ∼ = φ K ( τ,c )+ nG ( X ) . Proof. We construct a homomorphism of monoids α : φ K ( τ,c )+( p,q ) G ( X ) → φ K ( τ,c )+( p,q ) G ( X ) fin which gives the inverse to . For this construction, let E = E uni be a ( φ, τ, c )-twistedlocally universal bundle on X//G with Cl p,q -action, and η ∈ Γ( X//G, Gr( E )) agradation. Thanks to Lemma 4.18, there is a finite rank subbundle F ⊂ E suchthat η is homotopic to η F ⊕ ǫ E/F , where η F ∈ Γ( X//G, Gr( F )) is a gradation on F and ǫ E/F ∈ Γ( X//G, Gr( F ⊥ )) is the Z -grading of the orthogonal complementof F ⊂ E . Denote by ǫ F the Z -grading of F . Then we have a triple ( F, η F , ǫ F ),and let it represent α ([ η ]) ∈ K ( τ,c )+( p,q ) G ( X ) fin . Once α is shown to be well-defined,it is clear that α is a homomorphism and gives the inverse to . The definition of α is independent of the choice of a subbundle F as in Lemma 4.18. This is a directconsequence of Lemma 4.19. If η and η ′ are homotopic within the gradations of E , then α ([ η ]) = α ([ η ′ ]). This is a consequence of the definition that a homotopybetween η and η ′ is a gradation on E × [0 , (cid:3) Relationship of finite-dimensional formulations. Summarizing the for-mulations of the Freed-Moore K -theory so far, we get the following diagram underthe setting of Definition 3.13 and Definition 4.13: φ K ( τ,c )+ nG ( X ) fin ı −−−−→ φ K ( τ,c )+ nG ( X ) ∼ = y ϑφ K (´ τ,c )+ nG ( X ) fin −−−−→ ∼ = φ K (´ τ,c )+ nG ( X ) . Here has the inverse α given in the proof of Theorem 4.20. Proposition 4.21. Under the assumptions in Definition 3.13 and Definition 4.13,the composition α ◦ ϑ ◦ ı : φ K ( τ,c )+ nG ( X ) fin −→ φ K (´ τ,c )+ nG ( X ) fin is induced from the assignment of ( ´ E, − ´ ǫ, ´ ǫ ) to E ∈ φ Vect ( τ,c )+(0 , G ( X ) fin , where ´ ǫ = ǫ is the Z -grading of E . If c is trivial and n = 0 , then α ◦ ϑ ◦ ı is bijective. REED-MOORE K -THEORY 41 Proof. We can readily see the description of α ◦ ϑ ◦ ı along the definitions of α , ϑ and ı . In the case that c is trivial, the inverse of α ◦ ϑ ◦ ı can be constructed asin [17]: Let ( E, η , η ) be a triple representing an element of φ K τ +(0 , G ( X ) fin . For k = 0 , 1, the subbundle Ker(1 − η k ) ⊂ E gives rise to a ( φ, τ )-twisted ungradedvector bundle on X//G . Therefore we have a ( φ, τ )-twisted (graded) vector bundleKer(1 − η ) ⊕ Ker(1 − η ) ∈ φ Vect τ +(0 , G ( X ). Then the assignment ( E, η , η ) Ker(1 − η ) ⊕ Ker(1 − η ) induces the inverse of α ◦ ϑ ◦ ı . (Because of Lemma 4.5,the difference of τ and ´ τ does not matter in this case.) (cid:3) Since ϑ and are bijective, Proposition 3.14 is reproved: Corollary 4.22. Under the assumptions in Definition 3.13 and Definition 4.13,the homomorphism ı is bijective, if c is trivial and n = 0 . As is clear from the proof above, the construction of the inverse of α ◦ ϑ ◦ ı does not work in the presence of a non-trivial c . An example in which α ◦ ϑ ◦ ı isnot bijective can be constructed from the example in §§ α ◦ ϑ ◦ ı is bijective is as follows: Let G = Z act on the unit circle S ⊂ C trivially.As studied in [12], we have H Z ( S ; Z ) ∼ = Z , and its generator can be representedby a group 2-cocycle τ ∈ Z ( Z ; C ( S , U (1))) which takes the following values: τ ( g, h ; u ) h = 1 h = − g = 1 1 1 g = − u We take φ : Z → Z to be the trivial homomorphism, but c : Z → Z to be theidentity. By using the Mayer-Vietoris exact sequence for example ([31], VIII, E, 2),we have K ( τ,c )+0 Z ( S ) ∼ = Z , K ( τ,c )+1 Z ( S ) = 0 . Let E = S × C be the product bundle on S . This bundle gives rise to a ( τ, c )-twisted vector bundle on S // Z by the following Z -grading ǫ and the ( τ, c )-twisted Z -action ρ ( g ) : E → E , ǫ = (cid:18) − (cid:19) , ρ (1)( u, v ) = ( u, v ) , ρ ( − u, v ) = ( u, (cid:18) u (cid:19) v ) . It is easy to see that any finite rank ( τ, c )-twisted vector bundle on S // Z isisomorphic to the direct sum of some copies of E above. A consequence of thisclassification is that ı : K ( τ,c )+0 Z ( S ) fin → K ( τ,c )+0 Z ( S ) gives rise to an isomorphism K ( τ,c )+0 Z ( S ) fin ∼ = K ( τ,c )+0 Z ( S ) . Therefore α ◦ ϑ ◦ ı : K ( τ,c )+0 Z ( S ) fin → K ( τ,c )+0 Z ( S ) fin is also an isomorphism, wherethe isomorphism τ ∼ = ´ τ due to the triviality of φ is understood. As is seen, theimage of [ E ] ∈ K ( τ,c )+0 Z ( S ) fin is represented by the triple ( ´ E, − ´ ǫ, ´ ǫ ). Since [ E ] is agenerator, so is [ E, − ǫ, ǫ ]. The triple ( E, − ǫ, ǫ ), which turns out to be non-trivial bythe argument here, is essentially the same as the building block of nonsymmorphictopological crystalline insulators in [33]. Appendix A. Classification of some twists This appendix classifies some twists to be used in Appendix B. A.1. Classification of some twists. Let G be a compact Lie group. A typicalspace with G -action is G/H , where H ⊂ G is a closed subgroup and G acts on G/H by the left multiplication. Since the inclusions pt ⊂ G/H and H ⊂ G induce the local equivalence pt //H → ( G/H ) //G , we have H (( G/H ) //G ; Z ) ∼ = H (pt //H ; Z ), and hence any object φ in Φ(( G/H ) //G ) comes from a grouphomomorphism H → Z . Furthermore, we have H (( G/H ) //G ; Z φ ) ∼ = H (pt //H ; Z φ | H ) , where φ | H : H → Z is the homomorphism induced from φ ∈ Φ(( G/H ) //G ) byrestriction. Thus, the classification of (ungraded) twists on ( G/H ) //G amountsthe that of twists on pt //H .Applying the argument above and computations of cohomology groups in [14],we give the classification of (ungraded) twists in the case of G = Z and G = Z × Z , which we will need later on. In these cases, we can assume that an object φ ∈ Φ(( G/H ) //G ) is induced from a homomorphism φ : G → Z . In the below, H ⊂ G will be a subgroup. • The case where φ is trivial. G H H (( G/H ) //G ; Z ) Z any subgroup 0 Z × Z not Z × Z Z × Z Z × Z Z • The case where φ is non-trivial. For G = Z , the identity φ = id is theunique non-trivial homomorphism φ : G → Z . We have: G H φ : G → Z H (( G/H ) //G ; Z φ ) Z Z Z id Z For G = Z × Z , the three non-trivial homomorphisms φ : Z × Z → Z are permuted by the outer automorphisms of Z × Z . Thus, it suffices toconsider a non-trivial homomorphism, for example the first projection φ = p . In Z × Z , there are three non-trivial subgroups of order two: Z × × Z and the image ∆( Z ) of the diagonal embedding ∆ : Z → Z × Z .We then have: G H φ : G → Z H (( G/H ) //G ; Z φ ) Z × Z p Z × Z Z × p Z Z × Z × Z p Z × Z ∆( Z ) p Z Z × Z Z × Z p Z ⊕ Z A.2. Realization by group cocycle. We next realize the non-trivial ungradedtwists classified in §§ A.1. For this aim, we start with a review of group cocycles.Let G be a compact Lie group, and M a two sided G -module. We define thegroup of n -cochains of G with coefficients in M to be C n group ( G ; M ) = { τ : G n → M | continuous } REED-MOORE K -THEORY 43 and the coboundary operator ∂ : C n group ( G ; M ) → C n +1group ( G ; M ) to be( ∂τ )( g , . . . , g n ) = g · τ ( g , . . . , g n ) + n − X i =1 ( − i τ ( g , . . . , g i g i +1 , . . . , g n )+ ( − n τ ( g , . . . , g n − ) · g n , by using the two sided action of G . As usual, the group of n -cocycles Z n group ( G ; M )and that of n -coboundaries B n group ( G ; M ) are defined. Then the group cohomologyof G with coefficients in M is defined as the quotient group H n group ( G ; M ) = Z n group ( G ; M ) /B n group ( G ; M ) . The two sided G -module C ( X, U (1)) φ in the body of this paper is defined whena compact Lie group G acts on a space X from the left and a homomorphism φ : G → Z is given. The underlying group is the group C ( X, U (1)) of U (1)-valued functions on X . The left action of g ∈ G on f ∈ C ( X, U (1)) is f f φ ( g ) ,and the right action is f g ∗ f . In this setting, we identify a group cochain τ ∈ C n grup ( G ; C ( X, U (1)) φ ) with a continuous map τ : G n × X → U (1).An example of a 2-cocycle τ f ∈ Z ( G ; C ( X, U (1)) φ )is constructed from any homomorphism f : G → Z by setting τ f ( g, h ; x ) = (cid:26) , ( f ( g ) = 1 or f ( h ) = 1) − . ( f ( g ) = f ( h ) = − H ∗ group ( G ; C ( X, U (1) φ )) is an invariantof the quotient groupoid X//G under the local equivalences. Thus, if H ⊂ G is aclosed subgroup, then the inclusion i : H → G induces an isomorphism i ∗ : H n group ( G ; C ( G/H, U (1)) φ ) → H n group ( H ; U (1) φ | H ) , where C (pt , U (1)) is identified with U (1).We also remark that the exponential exact sequence of two sided G -modules0 → Z φ → R φ → U (1) φ → ·· → H n group ( G ; Z φ ) → H n group ( G ; R φ ) → H n group ( G ; U (1) φ ) → H n +1group ( G ; Z φ ) → ·· , where, for A = Z , R , the two sided H -module A φ above has A as the underlyinggroup, on which the left action of g ∈ G is defined as a φ ( g ) a by using a homo-morphism φ : G → Z and the right action is trivial. By an averaging argumentbased on the Haar measure on G , we have H n group ( G ; R φ ) = 0 for n > 0, so that H n group ( G ; U (1) φ ) ∼ = H n +1group ( G ; Z φ ) for n > G is a finite group. Then the group cohomology H n group ( G ; A φ )appears as the E -term of a spectral sequence computing H n (pt //G ; Z φ ). Further-more, the spectral sequence collapses at E , so that H n group ( G ; A φ ) ∼ = H n (pt //G ; A φ ) . In view of this isomorphism, we represent below the non-trivial ungraded twistsclassified in §§ A.1 by group 2-cocycles with coefficients in U (1) φ . • In the case that G = Z × Z , H = 1 and φ : Z × Z → Z is trivial, H (pt // ( Z × Z ); Z ) ∼ = H ( Z × Z ; U (1)) ∼ = Z . A group 2-cocycle τ ∈ Z ( Z × Z ; U (1)) representing this non-trivialcohomology class is given by τ ((( − m , ( − n ) , (( − m , ( − n ) = exp πin m . • In the case that G = Z , H = 1 and φ : Z → Z is the identity φ = id, H (pt // Z ; Z id ) ∼ = H ( Z ; U (1) id ) ∼ = Z . A group 2-cocycle τ id ∈ Z ( Z ; U (1) id ) representing this non-trivial co-homology class is τ id (( − m , ( − m ) = exp πim m . The values of this 2-cocycle is as follows: τ id ( g, h ) h = 1 h = − g = 1 1 1 g = − − τ id is the unique cocycle that represents the non-trivialcohomology class and is subject to the normalization condition τ id (1 , g ) = τ id ( g, 1) = 1 for all g ∈ Z . • In the case that G = Z × Z , H = Z × φ : Z × Z → Z is the firstprojection φ = p , we have H (( Z × Z ) // ( Z × Z p ) ∼ = H (pt // Z ; Z id ) ∼ = Z . Thus, this non-trivial cohomology class is essentially represented by τ id . • In the case of G = Z × Z , H = ∆( Z ) and φ : Z × Z → Z is the firstprojection φ = p , we have H (( Z × Z ) // ∆( Z ); Z p ) ∼ = H (pt // Z ; Z id ) ∼ = Z . Thus, the non-trivial class is essentially represented by τ id also. • In the case of G = Z × Z , H = 1 and φ : Z × Z → Z is the firstprojection φ = p , we have H (pt // ( Z × Z ); Z p ) ∼ = H ( Z × Z ; U (1) p ) ∼ = Z ⊕ Z . This group is generated by the cocycles τ p i ∈ Z ( Z × Z ; U (1) p ) asso-ciated to the i th projection p i : Z × Z → Z τ p ((( − m , ( − n ) , (( − m , ( − n ) = exp πim m ,τ p ((( − m , ( − n ) , (( − m , ( − n ) = exp πin n . The cocycle τ p agrees with the pull-back p ∗ τ id , and τ p with the cocycle µ introduced in Definition 4.4.We notice that, for β ∈ C ( Z × Z ; U (1) p ) such that β (1) = 1, its cobound-ary ∂β takes the following values. ∂β ( g, h ) h = (1 , 1) ( − , 1) (1 , − 1) ( − , − g = (1 , 1) 1 1 1 1 g = ( − , 1) 1 1 Y − Y − g = (1 , − 1) 1 X XY Yg = ( − , − 1) 1 X − X − REED-MOORE K -THEORY 45 Here X and Y are given by X = β ( − , β ( − , − − β (1 , − , Y = β ( − , − β ( − , − β (1 , − . Using this fact, we can verify that τ p and τ p generate H ( Z × Z ; U (1) p ). Appendix B. Mackey decomposition and the periodicity on a point This appendix contains the argument needed to complete the proof of Lemma2.9, Lemma 3.2, Theorem 3.8 and Theorem 4.11. The argument is to reduce theproblem of showing certain properties on the quotient groupoid pt //G , with G acompact Lie group, to one in the case with G trivial. The reduction is based on acategorical lift of the so-called Mackey decomposition considered in [11] (Theorem9.8). We then prove the properties on the point, describing some details in theapplication of results in [6, 7].B.1. Mackey decomposition. Let K be a compact Lie group. We denote by b K = { λ } the complete set of (labels of) finite-dimensional irreducible unitary rep-resentations of K . Since K is compact, b K is a discrete set and its cardinality is atmost countable. For each λ ∈ b K , we choose and fix its realization ( V λ , ρ λ ), where V λ is a Hermitian vector space of finite rank and ρ λ : K → U ( V λ ) is a homomorphism.Suppose that K is a closed normal subgroup of a compact Lie group G . Then,for any g ∈ G , we define a representation ( g V λ , g ρ λ ) of K by setting g V λ = V λ and g ρ λ ( k ) = ρ λ ( g − kg ). Since ( g V λ , g ρ λ ) is irreducible, there uniquely existsan element gλ ∈ b K such that ( g V λ , g ρ λ ) is equivalent to ( V gλ , ρ gλ ). Then theassignment λ gλ defines a left action of G on b K which descends to an action of G/K .Given ( V λ , ρ λ ), its complex conjugation ( V λ , ρ λ ) is also an irreducible represen-tation. We write λ ∈ b K for the corresponding label. The assignment λ λ definesa Z -action on b K commuting with the action of G , so that we have an action of G × Z on b K . Lemma B.1. Let G be a compact Lie group, and K ⊂ G a closed normal subgroupsuch that G/K is finite. We write p Z : G × Z → Z for the projection, and π : G → G/K for the quotient. Then the Hermitian vector bundle V → b K given by V = [ λ ∈ b K V λ . can be made into a ( p Z , ( π, id) ∗ τ G/K × Z ) -twisted vector bundle on b K// ( G × Z ) such that: • The subgroup K ⊂ G × Z acts on the fiber of λ ∈ b K by the representation ρ λ : K → U ( V λ ) of K . • τ G/K × Z ∈ Z ( G/K × Z ; C ( b K, U (1)) p Z ) is a group -cocycle, where p Z : G/K × Z → Z is also the projection. We remark that the Z -grading of V is assumed to be trivial, so that the evenpart V = V is V and the odd part V = 0 is trivial. Proof. We choose representatives g i of the coset G/K as well as unitary equivalencesof K -modules α ( g i ; λ ) : g i V λ → V g i λ for all λ ∈ b K . For any g ∈ G , we have the unique decomposition g = g i k for an i . Using this decomposition, we define ρ ( g ; λ ) : V λ → V gλ to be the composition of V λ ρ λ ( k ) → V λ = V kλ = g i V kλ α ( g i ; kλ ) → V g i kλ = V gλ . We also choose unitary equivalences of K -modules β ( λ ) : V λ → V λ for all λ ∈ b K and define ρ ( − λ ) : V λ → V λ to be the composition of V λ → V λ β ( λ ) → V λ , where the first map is v v . The maps ρ ( g ; λ ) and ρ ( − λ ) generate an action of G × Z up to U (1)-phases, since each V λ is irreducible. The U (1)-phase factor yieldsa group 2-cocycle τ G × Z ∈ Z ( G × Z ; C ( b K, U (1)) p Z ), and V is a ( p Z , τ G × Z )-twisted vector bundle on b K// ( G × Z ). By construction, it holds that τ G × Z ( g k , g k ; k λ ) = τ G × Z ( g , g ; λ )for all k , k , k ∈ K , g , g ∈ G and λ ∈ b K . This means τ G × Z = ( π, id) ∗ τ G/K × Z for a cocycle τ G/K × Z , and the lemma is proved. (cid:3) Theorem B.2. Let G be a compact Lie group, K ⊂ G a closed normal subgroupsuch that G/K is finite, and π : G → G/K the projection. For homomorphisms φ : G/K → Z and c : G/K → Z , there is an equivalence of categories Φ : π ∗ φ Vect π ∗ c +( p,q ) (pt //G ) → φ Vect ( c, − τ c K )+( p,q ) ( b K// ( G/K )) , where τ b K = (id , φ ) ∗ τ G/K × Z ∈ Z ( G/K ; C ( b K, U (1)) φ ) .Proof. By means of the embedding (id , π ◦ φ ) : G → G × Z , the vector bundle V → b K in Lemma B.1 gives rise to a ( π ◦ φ, π ◦ c, τ G )-twisted vector bundle on b K//G , where the 2-cocycle τ G is given by the pull-back τ G = (id , π ◦ φ ) ∗ ( π, id) ∗ τ G/K × Z = π ∗ (id , φ ) ∗ τ G/K × Z = π ∗ τ b K . For a ( π ∗ φ, π ∗ c )-twisted vector bundle ( E, ǫ, ρ E , γ ) on pt //G with Cl p,q -action, wedefine a vector bundle ˆ E → b K byˆ E = [ λ ∈ b K Hom K ( V λ , E ) , where Hom K ( V λ , E ) is the space of complex linear maps f : V λ → E commutingwith the actions of K ⊂ G on V λ and E . Note that Hom( V λ , E ) = V ∗ λ ⊗ E is a Hilbert space, and so is its subspace Hom K ( V λ , E ). The topology of ˆ E isgiven by this Hilbert space structure (rather than the compact open topology).By the Z -grading ǫ on E (and the trivial Z -grading on V ), the vector bundle ˆ E has the Z -grading ˆ ǫ ( f ) = ǫ ◦ f . For g ∈ G/K , we choose g ∈ π − ( g ) and putˆ ρ ( g )( f ) = ρ E ( g ) ◦ f ◦ ρ V ( g ) − . This turns out to be independent of the choice of g ,and ˆ E gives rise to a ( φ, c, − τ b K )-twisted vector bundle on b K// ( G/K ), which has a Cl p,q -action defined by the composition with γ . The assignment E ˆ E defines thefunctor Φ, where the Φ : Hom( E, E ′ ) → Hom( ˆ E, ˆ E ′ ) is defined by the compositionof homomorphisms. To complete the proof, we construct a functor in the oppositedirection Ψ : φ Vect ( c, − τ c K )+( p,q ) ( b K// ( G/K )) → π ∗ φ Vect π ∗ c +( p,q ) (pt //G ) . REED-MOORE K -THEORY 47 To construct Ψ, let F be a ( φ, c, − τ b K )-twisted vector bundle on b K// ( G/K ) with Cl p,q -action. We then define a vector space ˇ F byˇ F = dM λ ∈ b K V λ ⊗ F | λ , where b ⊕ means the L -completion of the algebraic direct sum ⊕ . The vector bundle V has a twisted G -action, and F also has a twisted G -action induced from π : G → G/K . With these twisted G -actions, ˇ F gives rise to a ( φ ◦ π, cπ )-twisted vectorbundle on pt //G , which inherits a Cl p,q -action from F . The functor Ψ is inducedfrom the assignment F ˇ F . For objects E ∈ π ∗ φ Vect π ∗ c +( p,q ) (pt //G ) , F ∈ φ Vect ( c, − τ c K )+( p,q ) ( b K// ( G/K )) , we can see the mapsΨΦ( E ) = dM λ ∈ b K V λ ⊗ Hom K ( V λ , E ) → E, b ⊕ λ v λ ⊗ f λ b ⊕ λ f λ ( v λ ) ,F → ΦΨ( F ) = [ λ ∈ b K Hom K ( V λ , V λ ) ⊗ F | λ , f λ id V λ ⊗ f λ , provide the natural equivalences of functors ΨΦ ⇒ id and id ⇒ ΦΨ, which provesthat Φ is an equivalence of categories. (cid:3) We now apply the theorem above to some concrete cases. In the following, weuse ≃ to mean an equivalence of categories. We also use the notation Vect ( p,q ) C =Vect ( p,q ) (pt) for the category of Z -graded complex modules over Cl p,q , and Vect ( p,q ) R for the category of Z -graded real modules over Cl p,q . As in the body of this pa-per, infinite-dimensional modules are allowed, and the vector spaces underlyinginfinite dimensional modules are separable Hilbert spaces. We notice that there isan equivalence of categories id Vect ( p,q ) (pt // Z ) ≃ Vect ( p,q ) R , since an id-twisted vector bundle E on pt // Z is nothing but a complex vectorspace with a real structure (i.e. an anti-unitary involution) T : E → E . Thus, the T -invariant part E T is a real vector space, and E E T defines the equivalence ofcategories. Lemma B.3. Let G be a compact Lie group. (a) There is an equivalence of categories Vect ( p,q ) (pt //G ) ≃ Vect ( p,q ) ( b G ) = Y λ ∈ b G Vect ( p,q ) C . (b) Let c : G → Z be a non-trivial homomorphism, and K = Ker( c ) the kernelof c . Then there is an equivalence of categories Vect c +( p,q ) (pt //G ) ≃ Vect id Z +( p,q ) ( b K// Z ) , and the category Vect id Z +( p,q ) ( b K// Z ) is equivalent to the product of somecopies of the following categories Vect ( p,q ) C , Vect ( p,q +1) C . Proof. For (a), the equivalence of categories just follows from a direct applicationof Theorem B.2. For (b), Theorem B.2 provides the equivalence of categoriesVect c +( p,q ) (pt //G ) ≃ Vect ( − τ c K , id Z )+( p,q ) ( b K// Z ) . Since b K is a discrete set, it is a disjoint union of Z -spaces of the form Z /H ,where H ⊂ Z is a subgroup. On the groupoid ( Z /H ) // Z , all the twists aretrivial, as seen in §§ A.1. Hence a trivialization of the twist − τ b K leads to theequivalence of categories in (b). To show the remaining claim, we focus on the Z -orbits ( Z /H ) // Z of b K// Z . If H = 1, then we have the equivalence of categoriesVect id Z +( p,q ) ( Z // Z ) ≃ Vect ( p,q ) (pt) = Vect ( p,q ) C in view of the local equivalence pt // → Z // Z . If H = Z , then we haveVect id Z +( p,q ) (pt // Z ) ≃ Vect ( p,q +1) C , since an id Z -twisted Z -action can be regarded as an additional Cl , -action. (cid:3) Lemma B.4. Let G be a compact Lie group, φ : G → Z a non-trivial homomor-phism, and b K = Ker( φ ) the kernel of φ . (a) There is an equivalence of categories φ Vect ( p,q ) (pt //G ) ≃ id Z Vect − τ c K +( p,q ) ( b K// Z ) , and the category id Z Vect ( p,q ) ( b K// Z ) is equivalent to the product of somecopies of the following categories Vect ( p,q ) C , Vect ( p,q ) R , Vect ( p +4 ,q ) R . (b) If c = φ : G → Z / , then there is an equivalence of categories φ Vect c +( p,q ) (pt //G ) ≃ id Z Vect ( − τ c K , id Z )+( p,q ) ( b K// Z ) , and the category Vect ( − τ c K , id Z )+( p,q ) ( b K// Z ) is equivalent to the product ofsome copies of the following categories Vect ( p,q ) C , Vect ( p +2 ,q ) R , Vect ( p,q +2) R . Proof. For (a), Theorem B.2 gives the equivalence of categories. The Z -space b K is a disjoint union of Z /H , where H ⊂ Z is a subgroup. By the classificationin §§ A.1, the category id Z Vect ( p,q ) ( b K// Z ) is the product of some copies of thefollowing categories id Z Vect ( p,q ) ( Z // Z ) , id Z Vect ( p,q ) (pt // Z ) , id Z Vect τ id +( p,q ) (pt // Z ) , where τ id represents the non-trivial twist in H (pt // Z ; Z id ). The local equivalencept // → Z // Z induces the equivalence of categories id Z Vect ( p,q ) ( Z // Z ) ≃ Vect ( p,q ) C . As is pointed out already, if E ∈ id Z Vect ( p,q ) (pt // Z ) is a twisted vector bundlewith Cl p,q -action, then the twisted Z -action on E provides a real structure T : E → E commuting with the Cl p,q -action. Thus, E E T provides the equivalence id Z Vect ( p,q ) (pt // Z ) ≃ Vect ( p,q ) R . If E ∈ id Z Vect τ id +( p,q ) (pt // Z ) is a twisted vector bundle with Cl p,q -action, thenthe twisted Z -action on E provides a quaternionic structure (i.e. anti-unitary map REED-MOORE K -THEORY 49 whose square is − T : E → E commuting with the Cl p,q -action. As is known (forexample Proposition B.4, [11]), the category of vector spaces over the skew field H of quaternions are in one to one correspondence with that of Cl , -modules. Withthe Cl p,q -actions, E induces a real Cl p +4 ,q -modules, and we get the equivalence id Z Vect τ id +( p,q ) (pt // Z ) ≃ Vect ( p +4 ,q ) R . For (b), the same argument as in (a) proves that id Z Vect ( − τ c K , id Z )+( p,q ) ( b K// Z )is the product of some copies of the following categories id Vect id+( p,q ) ( Z // Z ) , id Vect id+( p,q ) (pt // Z ) , id Vect ( τ id , id)+( p,q ) (pt // Z ) , where id : Z → Z is the identity homomorphism. The local equivalence pt // → Z // Z induces the equivalence of categories id Vect id+( p,q ) ( Z // Z ) ≃ Vect ( p,q ) (pt) = Vect ( p,q ) C . If E ∈ id Vect id+( p,q ) (pt // Z ) is a twisted vector bundle, then the twisted Z -actionon E induces an odd real structure T : E → E . On real vector space E R underlying E , we have an additional Cl , -action generated by T and iT . Consequently, E R isa real Cl p,q +2 -module, and this construction leads to the equivalence id Vect id+( p,q ) (pt // Z ) ≃ Vect ( p,q +2) R . Similarly, if E ∈ id Vect ( τ id , id)+( p,q ) (pt // Z ) is a twisted vector bundle, then thetwisted Z -action induces an odd quaternionic structure T : E → E . The realvector space E R acquires an additional Cl , -action generated by T and iT . Hencewe get the equivalence of categories id Vect ( τ id , id)+( p,q ) (pt // Z ) ≃ Vect ( p +2 ,q ) R induced by the assignment E E R . (cid:3) Lemma B.5. Let G be a compact Lie group. Suppose that φ : G → Z and c : G → Z are non-trivial homomorphisms such that φ = c . We write b K = Ker( φ, c ) for the kernel of ( φ, c ) : G → Z × Z , and p i : Z × Z → Z the i th projection.Then there is an equivalence of categories φ Vect c +( p,q ) (pt //G ) ≃ p Vect ( − τ c K ,p )+( p,q ) ( b K// ( Z × Z )) , and the category p Vect ( − τ c K ,p )+( p,q ) ( b K// ( Z × Z )) is equivalent to the product ofsome copies of the following categories Vect ( p,q ) C , Vect ( p,q +1) C , Vect ( p,q ) R , Vect ( p +4 ,q ) R , Vect ( p,q +2) R , Vect ( p +2 ,q ) R , Vect ( p,q +1) R , Vect ( p +1 ,q ) R , Vect ( p +4 ,q +1) R , Vect ( p +5 ,q ) R . Proof. To suppress notations, we put G = Z × Z . The equivalence of categoriesis given by Theorem B.2. The groupoid b K//G is the disjoint union of ( G/H ) //G ,where H ⊂ G is a subgroup. Hence p Vect ( − τ c K ,p )+( p,q ) ( b K//G ) is the product ofthe categories of the form p Vect ( τ,p )+( p,q ) (( G/H ) //G ) ≃ p | H Vect ( τ | H ,p | H )+( p,q ) (pt //H ) , where τ is an ungraded twist. There are four subgroups H , as seen in §§ A.1. Inthe case of H = 1, the twist τ is trivial, and p Vect ( τ,p )+( p,q ) ( G//G ) ≃ Vect ( p,q ) (pt) ≃ Vect ( p,q ) C . In the case of H = 1 × Z , the twist τ can be trivialized, and we use the argumentin the proof of Lemma B.3 (b) to get the equivalence of categories p Vect ( τ,p )+( p,q ) (( G/ (1 × Z )) //G ) ≃ Vect id+( p,q ) (pt // Z ) ≃ Vect ( p,q +1) C . In the case of H = Z × 1, the twist τ | H is isomorphic to the trivial twist or τ id .Then, as in the proof of Lemma B.4 (a), we get the equivalence of categories p Vect ( τ,p )+( p,q ) (( G/ ( Z × //G ) ≃ id Vect ( p,q ) (pt // Z ) ≃ Vect ( p,q ) R , p Vect ( τ,p )+( p,q ) (( G/ ( Z × //G ) ≃ id Vect τ id +( p,q ) (pt // Z ) ≃ Vect ( p +4 ,q ) R . In the case of H = ∆( Z ), the twist τ | H is again isomorphic to the trivial twist or τ id . By the proof of Lemma B.4 (b), we get the equivalence of categories p Vect ( τ,p )+( p,q ) (( G/ ∆( Z )) //G ) ≃ id Vect id+( p,q ) (pt // Z ) ≃ Vect ( p,q +2) R , p Vect ( τ,p )+( p,q ) (( G/ ∆( Z )) //G ) ≃ id Vect ( τ id , id)+( p,q ) (pt // Z ) ≃ Vect ( p +2 ,q ) R . Finally, in the case of H = Z × Z , the twist τ | H = τ is isomorphic to 1 (trivialtwist), τ p , τ p or τ p + τ p , where τ p and τ p are the 2-cocycles given in §§ A.2. Let E ∈ p Vect ( τ,p )+( p,q ) (pt //G ) be a twisted vector bundle. We write T = ρ ( − , S = ρ (1 , − 1) for the twisted action of the generators ( − , , (1 , − 1) of G . Byconstruction, T : E → E is even and anti-unitary, whereas S : E → E is odd andunitary. If τ is one of the four twists above, then S and T are commutative. Now,in the case of τ = 1, we have T = 1 and S = 1. Hence T is a real structure on E , and S gives an additional Cl , -action on the real vector space E T . Togetherwith the original Cl p,q -action, E T is a real Cl p,q +1 -module, so that the assignment E E T induces p Vect (1 ,p )+( p,q ) (pt //G ) ≃ Vect ( p,q +1) R . In the case of τ = τ p , we have T = 1 and S = − 1. Hence S defines an additional Cl , -action. Thus, in the same way as above, we have the equivalence p Vect ( τ p ,p )+( p,q ) (pt //G ) ≃ Vect ( p +1 ,q ) R . In the case of τ = τ p , we have T = − S = 1. Hence T defines a quaternionicstructure on E , and S an additional Cl , -action. Then the equivalence of thecategories of quaternionic vector spaces and that of real Cl , -modules induces p Vect ( τ p ,p )+( p,q ) (pt //G ) ≃ Vect ( p +4 ,q +1) R . In the case of τ = τ p + τ p , we have T = − S = − 1. By the sameconsideration as above, we have the equivalence of categories p Vect ( τ p + τ p ,p )+( p,q ) (pt //G ) ≃ Vect ( p +5 ,q ) R , which completes the proof. (cid:3) B.2. The space of Fredholm operators. This subsection summarizes someproperties of the spaces of Fredholm operators as models of the classifying spacesof complex and real K -theories. Lemma B.6. There exists a universal Cl p,q -module in Vect ( p,q ) k . REED-MOORE K -THEORY 51 Proof. Let E ∈ Vect ( p,q ) k be a Z -graded Cl p,q -module over k which contains allthe inequivalent Z -graded Cl p,q -modules over k infinitely many. (Actually, we cantake E = ( Cl p,q ⊗ k ) ⊗ E with E a separable infinite-dimensional Hilbert space over k .) Then any Z -graded Cl p,q -module over k can be embedded into E , and hence E has the universality. (cid:3) From now on, we assume that E ∈ Vect ( p,q ) k is a universal module. As in §§ E ) the space of bounded operators with the compact open topology,and by K( E ) the space of compact operators with the operator norm topology.Changing slightly the notation in Definition 3.1, we denoteFred ( p,q ) k ( E ) = (cid:26) A ∈ End( E ) (cid:12)(cid:12)(cid:12)(cid:12) A ∗ = − A, A + id ∈ K( E ) , Spec( A ) ⊂ [ − i, i ]degree 1 , Aγ i = − γ i A ( i = 1 , . . . , p + q ) (cid:27) , where γ , . . . , γ p + q are the Clifford actions of fixed vectors e , . . . , e p + q ∈ R p + q forming an orthonormal basis. As before, Fred ( p,q ) k ( E ) is topologized byFred ( p,q ) k ( E ) → End( E ) × K( E ) , A ( A, A + id) , where End( E ) has the compact open topology and K( E ) the operator norm topol-ogy. We sometimes omit E to write Fred ( p,q ) k = Fred ( p,q ) k ( E ). We define a subspaceFred ( p,q ) k ( E ) † = { A ∈ Fred ( p,q ) k ( E ) | A = − id } . Lemma B.7. Fred ( p,q ) k ( E ) † is non-empty for any universal E .Proof. The same construction as in the proof of Lemma 3.2 applies: Let Π E bethe Z -graded k -vector space with the Z -grading of E reversed. Since Π E is alsouniversal, we have an isometry E ∼ = E ⊕ Π E . As Z -graded k -vector spaces, wealso have E ⊕ Π E ∼ = E ⊗ k , where k is the Z -graded k -vector space whose degree0 and 1 parts are k . The action of e i ∈ R p + q on E ⊕ Π E is then identified with γ i ⊗ 1. Note that k can be a Z -graded Cl , -module, and hence the Cl p,q -actionon E ⊗ k extends to a Cl p +1 ,q -action. Now the additional Cl , -action provides γ ∈ Fred ( p,q ) k ( E ). (cid:3) Lemma B.8. Fred ( p,q ) k ( E ) † is contractible for any universal E .Proof. First of all, we notice the identificationFred ( p,q ) k ( E ) † = (cid:26) A ∈ End( E ) (cid:12)(cid:12)(cid:12)(cid:12) A ∗ = − A, A + id = 0 , Aǫ = − ǫA,Aγ i = − γ i A ( i = 1 , . . . , p + q ) (cid:27) , where ǫ and γ i are the Z -grading and the Cl p,q -action on E , respectively. Thus,Fred ( p,q ) k ( E ) † is topologized by the inclusion Fred ( p,q ) k ( E ) † → End( E ) and the com-pact open topology on End( E ), which allows us to prove the present lemma as ageneralization of Proposition A2.1 in [6]. Since E is universal, we have an isometry E ∼ = L ([0 , , E ) = L ([0 , ⊗ E , where L ([0 , L ([0 , , k ) is the space of k -valued L -functions on the interval [0 , t ∈ [0 , R t and i t be R t : L ([0 , , E ) → L ([0 , t ] , E ) , restriction ,i t : L ([0 , t ] , E ) → L ([0 , , E ) , inclusion . By construction, the composition P t = i t R t is the orthogonal projection onto L ([0 , t ] , E ) ⊂ L ([0 , , E ), and i t R t is the identity of L ([0 , t ] , E ). For t ∈ (0 , Q t be the isometric isomorphism given by Q t : L ([0 , t ] , E ) → L ([0 , , E ) , ( Q t f )( x ) = t f ( tx ) . As a base point γ ∗ ∈ Fred ( p,q ) k ( L ([0 , , E )) † , we choose γ ∗ = 1 ⊗ γ , where γ ∈ Fred ( p,q ) k ( E ) † . We now define h t : End( L ([0 , , E )) → End( L ([0 , , E )) by h t ( A ) = i t Q − t AQ t R t + (1 − P t ) γ ∗ for t ∈ (0 , h ( A ) = γ ∗ . As in [6], we can see the continuity of h : End( L ([0 , , E )) × [0 , → End( L ([0 , , E )) , ( A, t ) h t ( A ) . We can also see h t ( A ) ∈ Fred ( p,q ) k ( L ([0 , , E )) † for any A ∈ Fred ( p,q ) k ( L ([0 , , E )) † and t ∈ [0 , h contracts Fred ( p,q ) k ( L ([0 , , E )) † to the base point. (cid:3) Lemma B.9 (weak periodicity) . In the real case k = R , there are natural homeo-morphisms Fred ( p,q ) R ∼ = Fred ( p +1 ,q +1) R ∼ = Fred ( p +8 ,q ) R ∼ = Fred ( p,q +8) R . In the complex case k = C , there are natural homeomorphisms Fred ( p,q ) C ∼ = Fred ( p +1 ,q +1) C ∼ = Fred ( p +2 ,q ) C ∼ = Fred ( p,q +2) C . Proof. The proof is essentially the same as Lemma 3.5. In the real case, let ∆ , = R be the Z -graded Cl , -module whose Z -grading ǫ and Cl , -action γ i are ǫ = (cid:18) − (cid:19) , γ = (cid:18) − 11 0 (cid:19) , γ = (cid:18) (cid:19) . Since ∆ , is irreducible, the tensor product induces an equivalence of categories · ⊗ ∆ , : Vect ( p,q ) R → Vect ( p +1 ,q +1) R . Thus, in particular, if E ∈ Vect ( p,q ) R is universal, then so is E ⊗ ∆ , ∈ Vect ( p,q ) R .Now, the functor induces a continuous mapFred ( p,q ) R ( E ) → Fred ( p +1 ,q +1) R ( E ⊗ ∆ , ) , A A ⊗ . By a direct computation for example, we can verify that this map is bijective, andalso a homeomorphism. The iteration of this homeomorphism givesFred ( p,q ) R ( E ) ∼ = Fred ( p +4 ,q +4) R ( E ⊗ ∆ ⊗ , ) . In general, if e , · · · , e generate Cl , , then e ′ , · · · , e ′ generate Cl , , where e ′ i = e i ( e · · · e ). As a result, we have the equivalences of categoriesVect ( p +8 ,q ) R ≃ Vect ( p +4 ,q +4) R ≃ Vect ( p,q +8) R , and also homeomorphismsFred ( p +8 ,q ) R ∼ = Fred ( p +4 ,q ) R ∼ = Fred ( p,q +8) R . In the complex case ( k = C ), we consider ∆ C , = ∆ , ⊗ C , which is an irreducible Z -graded complex module over Cl , . As in the real case, we have a homeomor-phism Fred ( p,q ) C ( E ) → Fred ( p +1 ,q +1) C ( E ⊗ ∆ C , ) , A A ⊗ . REED-MOORE K -THEORY 53 If γ j acts on a (universal) module E ∈ Vect ( p,q ) R by γ i = ± 1, then iγ j acts on E by( iγ j ) = ∓ 1. Hence we have equivalence of categoriesVect ( p +2 ,q ) C ≃ Vect ( p +1 ,q +1) C ≃ Vect ( p,q +2) C , and also homeomorphismsFred ( p +2 ,q ) C ∼ = Fred ( p +1 ,q +1) C ∼ = Fred ( p,q +1) C , which completes the proof. (cid:3) For further analysis of Fred ( p,q ) k ( E ), it is useful to express this space in terms of an ungraded Clifford module. Lemma B.9 allows us to set q = 0. For a universal Cl p, -module E , we can assume that E = E = ˆ E is a separable infinite-dimensionalHilbert space. Then the Z -grading ǫ on E and the actions γ i of the Clifford algebra Cl p, are expressed as ǫ = (cid:18) − (cid:19) , γ = (cid:18) − 11 0 (cid:19) , γ i = (cid:18) γ i ˆ γ i (cid:19) , ( i ≥ γ i : ˆ E → ˆ E , (2 ≤ i ≤ p ) make ˆ E into an ungradedmodule over Cl p − , for p ≥ Lemma B.10. We have the following bijections Fred (0 , k ( E ) ∼ = n ˆ A ∈ End( ˆ E ) | ˆ A ∗ ˆ A − id , ˆ A ˆ A ∗ − id ∈ K( ˆ E ) , k ˆ A k = 1 o , Fred (1 , k ( E ) ∼ = n ˆ A ∈ End( ˆ E ) | ˆ A + id ∈ K( ˆ E ) , k ˆ A k = 1 , ˆ A ∗ = − ˆ A o . If p ≥ , then there is the following bijection Fred ( p, k ( E ) ∼ = (cid:26) ˆ A ∈ End( ˆ E ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ A + id ∈ K( ˆ E ) , k ˆ A k = 1 , ˆ A ∗ = − ˆ A, ˆ γ i ˆ A = − ˆ A ˆ γ i ( i = 2 , . . . , p ) (cid:27) . Proof. Any skew-adjoint bounded operator A : E → E of degree 1 is expressed as A = (cid:18) − ˆ A ∗ ˆ A (cid:19) by using a bounded operator ˆ A : ˆ E → ˆ E . The assignment A ˆ A induces all thebijections stated in the lemma. (cid:3) Let ˆ E be as before. We introduceΩ = n T ∈ End( ˆ E ) | T − id ∈ K( ˆ E ) , T ∗ T = T T ∗ = id o . For p ≥ 2, we follow [7] to introduceΩ p − = (cid:26) T ∈ End( ˆ E ) (cid:12)(cid:12)(cid:12)(cid:12) T − ˆ γ p ∈ K( ˆ E ) , T ∗ T = T T ∗ = id ,T = − id , T ˆ γ i = − ˆ γ i T ( i ≤ p − (cid:27) . For p ≥ 1, we topologize Ω p − by the operator norm topology. Lemma B.11. There is a homotopy equivalence Fred ( p, k ≃ Ω p − for p ≥ . Proof. The lemma can be shown by adapting the argument in [7].In the case of p = 1, we use the bijection in Lemma B.10 to introduce a map ̟ : Fred (1 , k ( ˆ E ) → Ω , ̟ ( ˆ A ) = − e π ˆ A . In the same way as in the proof of Lemma 4.9, we can show that ̟ above is well-defined and continuous. (In the case of k = R , we consider the complexificationˆ E ⊗ C and its obvious real structure.) In view of the spectral decomposition ofunitary operators, ̟ is surjective. We would then like to apply Lemma (3.7) in[7] to ̟ . For this aim, it is enough to check that ̟ : ̟ − ( D ( n )) → D ( n ) is afiber bundle with contractible fiber, where D ( n ) = { T ∈ Ω | rank(id − T ) = n } for n ≥ 0, as defined in [7]. A consideration similar to the proof of Lemma (3.6) in[7] shows that ̟ : ̟ − ( D ( n )) → D ( n ) is the fiber bundle associated to a Hilbertspace subbundle { Ker(id − T ) } T ∈ D ( n ) of D ( n ) × ˆ E whose fiber is identified with { ˆ A ∈ End( ˆ E ′ ) | ˆ A = − id , k ˆ A k = 1 , ˆ A ∗ = − ˆ A } , where ˆ E ′ ⊂ ˆ E is the orthogonal complement of a finite rank subspace of the formKer(id − T ) with T ∈ D ( n ). The space above is identified with Fred (1 , k ( E ′ ) † ⊂ Fred (1 , k ( E ′ ) under Lemma B.10, and is contractible by Lemma B.8, since E ′ =ˆ E ′ ⊕ ˆ E ′ is a universal Cl , -module. As a result, ̟ is a homotopy equivalence.In the case of p ≥ 2, we consider ̟ p : Fred ( p, k ( ˆ E ) → Ω p − , ̟ p ( ˆ A ) = − ˆ γ p e π ˆ A ˆ γ p . We can see that ̟ p is well-defined as in [7]. By the spectral decomposition ofunitary operators, ̟ p is surjective. We can also see ̟ p is continuous as in Lemma4.9. For n ≥ 0, let D ( n ) be D ( n ) = { T ∈ Ω p − | rank(id + ˆ γ p T ) = n } . As in the case of p = 1, the restriction ̟ p : ̟ − p ( D ( n )) → D ( n ) is a fiber bundle.Its fiber is identified with (cid:26) ˆ A ∈ End( ˆ E ) (cid:12)(cid:12)(cid:12)(cid:12) ˆ A = − id , ˆ A ∗ = − ˆ A, ˆ γ i ˆ A = − ˆ A ˆ γ i ( i = 2 , . . . , p ) (cid:27) ∼ = Fred ( p, k ( E ) † , which is contractible by Lemma B.8. As a result, ̟ p is a homotopy equivalence. (cid:3) Lemma B.12. For p ≥ and q ≥ , there is a homotopy equivalence AS : Fred ( p,q ) k → ΩFred ( p − ,q ) k , where ΩFred ( p − ,q ) k is the space of continuous paths in Fred ( p − ,q ) k from γ p to − γ p .Proof. The map AS is as given in [7]:AS( A )( t ) = γ p cos πt + A sin πt, which is continuous in the topology of Fred ( p,q ) k ( E ). This map is also compatiblewith the periodicity in Lemma B.9. Thus, it suffices to consider the case of q = 0.To prove that AS is a homotopy equivalence, we tentatively define a space F pk ( E ) tobe F pk ( E ) = Fred ( p, k ( E ) as a set and topologize it by the operator norm topology. REED-MOORE K -THEORY 55 We then define a subspace F pk ( E ) ⊂ F pk ( E ), which is a model of the classifyingspace of K -theory [7], as follows: For A ∈ F pk ( E ), we put w ( A ) = e · · · e p A, and consider the restriction w ( A ) | E , where E is the degree 0 part of a universal Cl p, -module E = E ⊕ E . • If k = C and p = 1 mod 4, then F pk ( E ) consists of (Fredholm) operators A such that i − w ( A ) | E have essentially positive and negative spectra. • If k = C and p = 3 mod 4, then F pk ( E ) consists of (Fredholm) operators A such that w ( A ) | E have essentially positive and negative spectra. • If k = R and p = 3 mod 4, then F pk ( E ) consists of (Fredholm) operators A such that w ( A ) | E have essentially positive and negative spectra. • Otherwise, F pk ( E ) = F pk ( E ).As shown in [7], if p ≥ 1, then the space F pk = F pk ( E ) is homotopy equivalent to thespace Ω p − considered in Lemma B.11. The inclusion induces a continuous map F pk → Fred ( p, k , and makes the following diagram commutative F pk −−−−→ Fred ( p, k y ≃ ≃ y Ω p − Ω p − , where the left and right vertical maps are the homotopy equivalences in [7] andLemma B.11, respectively. Consequently, the inclusion F pk → Fred ( p, k is a homo-topy equivalence. Now, we also have the Atiyah-Singer map AS : F pk → Ω F p − k ,which is a homotopy equivalence [7]. We clearly have the commutative diagram F pk ≃ −−−−→ Fred ( p, k AS y ≃ y AS Ω F p − k ≃ −−−−→ ΩFred ( p − , k , which implies that AS : Fred ( p, k → ΩFred ( p − , k is a homotopy equivalence. (cid:3) Slightly changing the notation in Definition 4.1, we introduceGr ( p,q ) k ( E ) = (cid:26) η ∈ End( E ) (cid:12)(cid:12)(cid:12)(cid:12) η ∗ = η, η = id , η − ǫ ∈ K( E ) ηγ i = − γ i η ( i = 1 , . . . , p + q ) (cid:27) , where E = ( E, ǫ ) is a Z -graded k -vector space which is a universal Cl p,q -module.The set Gr ( p,q ) k ( E ) is topologized by the operator norm. Then, as shown in Lemma4.9, we have the continuous map ϑ : Fred ( p,q ) k ( E ) → Gr ( q,p ) k ( ´ E ) , ϑ ( A ) = − e πA ǫ, where ´ E = ( E, ǫ ) as a Z -graded k -vector space, whereas the Cl q,p -action ´ γ i isdefined as ´ γ i = γ i ǫ by using the original Cl p,q -action γ i on E . Lemma B.13. For any p, q ≥ , k = R , C and a universal E , the map ϑ : Fred ( p,q ) k ( E ) → Gr ( q,p ) k ( ´ E ) is a homotopy equivalence. Proof. For n ≥ 0, we define C ( n ) = { η ∈ Gr ( p,q ) k ( ´ E ) | rank(id − ηǫ ) ≤ n } ,D ( n ) = { η ∈ Gr ( p,q ) k ( ´ E ) | rank(id − ηǫ ) = n } . In the same way as in Lemma B.11, we can see that ϑ : ϑ − ( D ( n )) → D ( n ) is afiber bundle. The fiber of this bundle is contractible by Lemma B.8. We wouldthen like to apply Lemma (3.7) in [7]. For its application, we need to see that C ( n − ⊂ C ( n ) has a respectable open neighbarhood U . This can be shown inthe same way as in [7], since Gr ( q,p ) k ( ´ E ) is topologized by the operator norm. (cid:3) Remark B.14 . Let F ( p,q ) k denote the set Fred ( p,q ) k endowed with the operator normtopology. It is well-known [7] that F ( p,q ) k admits contractible components when p − q = 1 mod 2 for k = C and p − q = 3 mod 4 for k = R . Though is counter-intuitivefrom the viewpoint of the operator norm topology, Lemma B.8 implies that thespace Fred ( p,q ) k is path connected. Hence we need not care about the “contractiblecomponents” in Fred ( p,q ) k to realize the classifying spaces of K -theories.B.3. Postponed proof. We summarize here the proof postponed from the maintext in the reduction argument. Lemma B.15. Let G be a compact Lie group. For any homomorphisms φ : G → Z and c : G → Z , there exists a (locally) universal ( φ, c ) -twisted vector bundle on pt //G with Cl p,q -action.Proof. By Lemma B.3, Lemma B.4 and Lemma B.5, the equivalence of categoriesin Theorem B.2 can be expressed as φ Vect c +( p,q ) (pt //G ) ≃ Y λ ∈ Λ Vect ( p λ ,q λ ) k λ where k λ = R , C , and Λ is a countable discrete set, since the set of inequivalentirreducible representations of a compact Lie group is at most countable. As inLemma B.6, we can realize a (locally) universal bundle E ( p λ ,q λ ) ∈ Vect ( p λ ,q λ ) k λ .Since Λ is countable, the Hilbert space direct sum of E ( p λ ,q λ ) is separable aswell. The resulting Hilbert space produces a ( φ, c )-twisted universal bundle E ∈ φ Vect c +( p,q ) (pt //G ) through the equivalence of categories in Theorem B.2, becausethis equivalence preserves the local universality. (cid:3) Lemma B.16. Let G be a compact Lie group, φ : G → Z and c : G → Z homomorphisms, and E a ( φ, c ) -twisted locally universal vector bundle on pt //G with Cl p,q -action. Then Γ( X , Fred( E ) † ) is contractible.Proof. As in the proof of Lemma B.15, we have the equivalence of categories φ Vect c +( p,q ) (pt //G ) ≃ Y λ ∈ Λ Vect ( p λ ,q λ ) k λ , which preserves the (local) universality of (twisted) vector bundles. The equivalenceof categories above induces the decompositionΓ(pt //G, Fred( E )) ∼ = Y λ ∈ Λ Fred ( p λ ,q λ ) k λ ( E ( p λ ,q λ ) ) , REED-MOORE K -THEORY 57 where E ( p λ ,q λ ) are universal if E ∈ φ Vect c +( p,q ) (pt //G ) is locally universal. It isclear that the decomposition restricts to giveΓ(pt //G, Fred( E )) † ∼ = Y λ ∈ Λ Fred ( p λ ,q λ ) k λ ( E ( p λ ,q λ ) ) † . Now the proof is completed by Lemma B.8. (cid:3) Lemma B.17. Let G be a compact Lie group, φ : G → Z and c : G → Z homomorphisms, and E a ( φ, c ) -twisted locally universal vector bundle on pt //G with Cl p,q -action. If p > , then the Atiyah-Singer map AS : Γ(pt //G, Fred( E )) → Γ([0 , //G, { , } //G, Fred( E × [0 , , which is given by AS( A )( t ) = γ cos πt + A sin πt , is a homotopy equivalence.Proof. As in the proof of Lemma B.16, we have the decompositionΓ(pt //G, Fred( E )) ∼ = Y λ ∈ Λ Fred ( p λ ,q λ ) k λ ( E ( p λ ,q λ ) ) . Note that p > p λ > 0. We can also decomposeΓ([0 , //G, { , } //G, Fred( E × [0 , ∼ = Y λ ∈ Λ Γ([0 , , { , } , Fred ( p λ − ,q λ ) k λ ) . The Atiyah-Singer map is clearly compatible with these decompositions. The spaceof invertible operators in Fred ( p λ − ,q λ ) k λ is contractible, as a result of Lemma B.8.Hence we get a homotopy equivalenceΓ([0 , , { , } , Fred ( p λ − ,q λ ) k λ ) ≃ ΩFred ( p λ ,q λ ) k λ . Now, the lemma follows from Lemma B.12. (cid:3) Lemma B.18. Let G be a compact Lie group, φ : G → Z and c : G → Z homomorphisms, and E a ( φ, c ) -twisted locally universal vector bundle on pt //G with Cl p,q -action. For any p, q ≥ , the map ϑ : Γ(pt //G, Fred( E )) → Γ(pt //G, Gr( ´ E )) is a homotopy equivalence.Proof. Notice that ´ E is a ( φ, c )-twisted locally universal vector bundle on pt //G with Cl q,p -action. As in the proof of Lemma B.16, we have decompositionsΓ(pt //G, Fred( E )) ∼ = Y λ ∈ Λ Fred ( p λ ,q λ ) k λ ( E ( p λ ,q λ ) ) , Γ(pt //G, Gr( ´ E )) ∼ = Y λ ∈ Λ Gr ( q λ ,p λ ) k λ ( ´ E ( p λ ,q λ ) ) . Under these decompositions, the map ϑ in question is decomposed into the homo-topy equivalences in Lemma B.13, and hence is a homotopy equivalence as well. (cid:3) Appendix C. Quotient of monoid This appendix is about the construction of quotient monoid used in the finite-dimensional formulations of K -theories in Definition 3.13 and Definition 4.14.Let M = ( M, + , 0) be an abelian monoid with zero (the additive unit), thatis, a set M equipped with a distributive and commutative binary operation + : M × M → M such that x + 0 = 0 + x = x for any x ∈ M . Let Z ⊂ M be asubmonoid of M , that is, a subset Z ⊂ M which is closed under the addition andcontains 0 ∈ M . Using Z , we can introduce an equivalence relation ∼ on M bydeclaring x ∼ x ′ if and only if there are z, z ′ ∈ Z such that x + z = x ′ + z ′ . Wewrite the quotient set as M/Z = M/ ∼ . It is easy to see that M/Z inherits anabelian monoid structure from M , in which zero is represented by elements in Z . Lemma C.1. Let M be an abelian monoid, and Z ⊂ M its submonoid. Supposethat there is a monoid homomorphism I : M → M such that (i) I ( Z ) = Z , (ii) I ( x ) + x ∈ Z for any x ∈ M ,Then the quotient monoid M/Z gives rise to an abelian group. We remark that I needs not be an involution on M . Proof. It is enough to verify the existence of inverse elements. We denote by [ x ] ∈ M/Z the element represented by x ∈ M . We define the inverse of [ x ] ∈ M/Z tobe − [ x ] = [ I ( x )]. This is well-defined. Actually, if x ∼ x ′ , then there are z, z ′ ∈ Z such that x + z = x ′ + z ′ , and we have I ( x ) + I ( z ) = I ( x + z ) = I ( x ′ + z ′ ) = I ( x ′ ) + I ( z ′ ) . Since I ( z ) , I ( z ′ ) ∈ Z by (i), it holds that [ I ( x )] = [ I ( x ′ )]. Because of (ii), we seethat − [ x ] + [ x ] = [ I ( x ) + x ] = 0. (cid:3) As an example, we let N be an abelian monoid, and consider the product monoid M = N × N . The diagonal set Z = ∆( N ) ⊂ N × N is a submonoid. If we definea homomorphism I : N × N → N × N by I ( x, y ) = ( y, x ), then I meets theassumptions in the lemma above. The resulting abelian group ( N × N ) / ∆( N ) isexactly the Grothendieck construction of N . References [1] M. F. Atiyah, K -theory . Lecture notes by D. W. Anderson W. A. Benjamin, Inc., NewYork-Amsterdam 1967[2] M. F. Atiyah, K-theory and reality . Quart. J. Math. Oxford Ser. (2) 17 1966 367–386.[3] M. F. Atiyah, R. Bott and A. Shapiro, Clifford modules . Topology 3 1964 suppl. 1, 3–38.[4] M. F. Atiyah and and M. Hopkins, A variant of K-theory: K ± . Topology, geometry andquantum field theory, 5–17, London Math. Soc. Lecture Note Ser., 308, Cambridge Univ.Press, Cambridge, 2004.[5] M. F. Atiyah and E. Rees, Vector bundles on projective 3-space . Invent. Math. 35 (1976),131–153.[6] M. F. Atiyah and G. Segal, Twisted K -theory . Ukr. Mat. Visn. 1 (2004), no. 3, 287–330;translation in Ukr. Math. Bull. 1 (2004), no. 3, 291–334[7] M. F. Atiyah and I. M. Singer, Index theory for skew-adjoint Fredholm operators . Inst.Hautes ´Etudes Sci. Publ. Math. No. 37 1969 5–26.[8] P. Bouwknegt, A. L. Carey, V. Mathai,M. K. Murray, D. Stevenson, Twisted K -theory and K -theory of bundle gerbes . Comm. Math. Phys. 228 (2002), no. 1, 17–45. REED-MOORE K -THEORY 59 [9] C-K. Fok, The real K-theory of compact Lie groups . SIGMA Symmetry Integrability Geom.Methods Appl. 10 (2014), Paper 022, 26 pp.[10] D. S. Freed, M. J. Hopkins and C. Teleman, Loop groups and twisted K-theory I . J. Topol.4 (2011), no. 4, 737–798.[11] D. S. Freed and G. W. Moore, Twisted equivariant matter . Ann. Henri Poincar´e 14 (2013),no. 8, 1927–2023.[12] K. Gomi, A variant of K -theory and topological T-duality for real circle bundles . Comm.Math. Phys. 334 (2015), no. 2, 923–975.[13] K. Gomi, Equivariant smooth Deligne cohomology . Osaka J. Math. 42 (2005), no. 2, 309–337.[14] K. Gomi, Twists on the torus equivariant under the 2-dimensional crystallographic pointgroups . SIGMA Symmetry Integrability Geom. Methods Appl. 13 (2017), Paper No. 014, 38pp.[15] P. Donovan and M. Karoubi, Graded Brauer groups and K -theory with local coefficients .Inst. Hautes ´Etudes Sci. Publ. Math. No. 38 1970 5–25.[16] J. L. Dupont, Symplectic bundles and KR -theory . Math. Scand. 24 1969 27–30.[17] M. Karoubi, K -theory. An introduction . Grundlehren der Mathematischen Wissenschaften,Band 226. Springer-Verlag, Berlin-New York, 1978.[18] M. Karoubi, Twisted bundles and twisted K -theory . Topics in noncommutative geometry,223–257, Clay Math. Proc., 16, Amer. Math. Soc., Providence, RI, 2012.[19] A. Kitaev, Periodic table for topological insulators and superconductors . AIP Conf. Proc.1134, 22–30 (2009).[20] Y. Kubota, Notes on twisted equivariant K -theory for C ∗ -algebras . Internat. J. Math. 27(2016), no. 6, 1650058, 28 pp.[21] H. B. Lawson, Jr. and M.-L. Michelsohn, Spin geometry . Princeton Mathematical Series,38. Princeton University Press, Princeton, NJ, 1989.[22] J. P. May, Equivariant homotopy and cohomology theory . With contributions by M. Cole,G. Comeza˜na, S. Costenoble, A. D. Elmendorf, J. P. C. Greenlees, L. G. Lewis, Jr., R. J.Piacenza, G. Triantafillou, and S. Waner. CBMS Regional Conference Series in Mathematics,91. Published for the Conference Board of the Mathematical Sciences, Washington, DC; bythe American Mathematical Society, Providence, RI, 1996.[23] J. P. May, Simplicial objects in algebraic topology . Reprint of the 1967 original. ChicagoLectures in Mathematics. University of Chicago Press, Chicago, IL, 1992.[24] El-ka¨ıoum M. Moutuou, Twisted groupoid KR -Theory , Ph.D. thesis, Universit´e de Lorraine-Metz, and Universit¨a Paderborn, 2012.[25] A. Pressley and G. Segal, Loop groups . Oxford Mathematical Monographs. Oxford SciencePublications. The Clarendon Press, Oxford University Press, New York, 1986.[26] D. Quillen, Superconnection character forms and the Cayley transform . Topology 27 (1988),no. 2, 211–238.[27] J. Rosenberg, Continuous-trace algebras from the bundle theoretic point of view . J. Austral.Math. Soc. Ser. A 47 (1989), no. 3, 368–381.[28] G. Segal, Classifying spaces and spectral sequences . Inst. Hautes ´Etudes Sci. Publ. Math.No. 34 1968 105–112.[29] G. Segal, Equivariant K -theory . Inst. Hautes ´Etudes Sci. Publ. Math. No. 34 1968 129–151.[30] G. Segal, Fredholm complexes . Quart. J. Math. Oxford Ser. (2) 21 1970 385–402.[31] K. Shiozaki, M. Sato and K. Gomi, Topological Crystalline Materials –General Formulationand Wallpaper Group Classification– . Phys. Rev. B 95, 235425 (2017).[32] K. Shiozaki, M. Sato and K. Gomi, Topology of nonsymmorphic crystalline insulators andsuperconductors . Phys. Rev. B 93, 195413 (2016).[33] K. Shiozaki, M. Sato and K. Gomi, Z -topology in nonsymmorphic crystalline insulators:Mobius twist in surface states . Phys. Rev. B 91, 155120 (2015).[34] G. C. Thiang, On the K -theoretic classification of topological phases of matter . Ann. HenriPoincar´e 17 (2016), no. 4, 757–794.[35] J.-L. Tu, P. Xu and C. Laurent-Gengoux, Twisted K -theory of differentiable stacks . Ann.Sci. ´Ecole Norm. Sup. (4) 37 (2004), no. 6, 841–910.[36] E. Witten, D-branes and K -theory . J. High Energy Phys. 1998, no. 12, Paper 19, 41 pp.(electronic). Department of Mathematical Sciences, Shinshu University, 3–1–1 Asahi, Matsumoto,Nagano 390-8621, Japan. E-mail address ::