Classification of "Quaternionic" Bloch-bundles: Topological Quantum Systems of type AII
aa r X i v : . [ m a t h - ph ] J un CLASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES:TOPOLOGICAL QUANTUM SYSTEMS OF TYPE AII
GIUSEPPE DE NITTIS AND KIYONORI GOMIA bstract . We provide a classification of type
AII topological quantum systems in dimension d = , , ,
4. Our analysis is based on the construction of a topological invariant, the
FKMM-invariant ,which completely classifies “Quaternionic” vector bundles ( a.k.a. “symplectic” vector bundles) in di-mension d
3. This invariant takes value in a proper equivariant cohomology theory and, in the caseof examples of physical interest, it reproduces the familiar Fu-Kane-Mele index. In the case d = bona fide characteristic class for the category of“Quaternionic” vector bundles in the sense that it can be realized as the pullback of a universal topolog-ical invariant. C ontents
1. Introduction 22. “Quaternionic” vector bundles 82.1. “Quaternionic” structure on vector bundles 82.2. “Quaternionic” sections 102.3. Local Q -triviality 112.4. Homotopy classification of “Quaternionic” vector bundles 122.5. Stable range condition 143. The FKMM-invariant 163.1. A short reminder of the equivariant Borel cohomology 163.2. The determinant construction 183.3. Construction of the FKMM-invariant 194. Classification in dimension d Q -bundles 234.3. Classification for TR-spheres 254.4. Connection with the Fu-Kane-Mele invariant 264.5. Classification for TR-tori 295. Classification in dimension d = Z -value FKMM-invariant 40Appendix B. Spatial parity and quaternionic vector bundles 42Appendix C. An overview to KQ -theory 43References 44 MSC2010
Primary: 57R22; Secondary: 55N25, 53C80, 19L64.
Keywords.
Topological insulators, Bloch-bundle, “Quaternionic” vector bundle, FKMM-invariant, Fu-Kane-Mele index.
1. I ntroduction
In this paper we continue the classification of topological quantum systems started in [DG1] and, inparticular, we focus on the so-called class
AII according to the Altland-Zirnbauer-Cartan classification(AZC) of topological insulators [AZ, SRFL]. Both classes AI and AII describe (quantum) systems thatare invariant under a time-reversal (TR) symmetry but it is the behavior of the spin that distinguishesbetween the two classes. Systems in class AI describe spinless (as well as integer spin) particles and,from a topological point of view, this class is poor of interesting e ff ects, as it emerges from the accurateanalysis carried out in [DG1]. On the other side, systems in class AII show interesting physicalphenomena of topological origin like the so-called
Quantum Spin Hall E ff ect (QSHE) [KM2, MB, Ro].Phenomena of this type were first described by L. Fu, C. L. Kane and E. J. Mele in a series of seminalworks [KM1, KM2, FK, FKM] and nowadays they are source of great interest among the physicscommunity (see e.g.the recent review [MHZ] and references therein). The principal result obtained byFu, Kane and Mele (at least from a mathematical point of view) was the identification of the SQHE witha topological invariant today known as Fu-Kane-Mele index . This index characterizes the topology ofthe Bloch energy bands for periodic systems of free fermions in the presence of a TR-symmetry in thesame way as the Chern numbers describe the topology of the Bloch bands when the TR-symmetry isbroken. However, a correct mathematical understanding of the topological nature of the Fu-Kane-Meleindex seems to be still missing in the literature and the most recent mathematical works [ASV, GP]only treat the case of two-dimensional lattice systems. Our main goal is to fill this gap.In the absence of disorder, the Bloch-Floquet analysis relates topological insulators of class
AII witha special category of complex vector bundles called “Quaternionic” . These vector bundles, introducedfor the first time by J. L. Dupont in [Du] with the (ambiguous) name of symplectic vector bundles (seealso [Se, DSLF, LLM]), are complex vector bundles defined over an involutive base space ( X , τ ) andendowed with an anti -involutive automorphism of the total space which covers the involution τ andrestricts to an anti -linear map between conjugate fibers. In Section 2 we provide a precise descriptionof this category. Let us just point out that throughout the paper we will often use the short expression Q -bundle instead “Quaternionic” vector bundle. The principal results achieved in this paper can besummarized as follows: • We provide a classification of topological quantum systems of class
AII by inspecting the ho-motopy classification of the underlying “Quaternionic” category of vector bundles. In this waywe obtain a classification which is, in spirit, finer than the usual K -theoretical classification[Ki] (see also Appendix C) since it covers also the unstable case. Moreover, our classificationis a priori valid also for quite general base spaces and not only for spheres or tori. • We introduce a topological invariant that discriminates between non-isomorphic Q -bundlesand which is su ffi ciently fine to provide a complete description of the category of “Quater-nionic” vector bundles in low dimensions, i.e.when the base space is a (closed) manifold X ofdimension d
3. The construction of this invariant is based on an original idea by M. Furuta,Y. Kametani, H. Matsue, and N. Minami described in an unpublished work [FKMM] dated2000 (five years earlier than the first paper by Kane and Mele!) and for this reason we decideto call it the
FKMM-invariant . We provide a precise description of the FKMM-invariant inSection 3 and we prove that in special cases (including all cases of interest for the descriptionof topological insulators) this invariant reproduces the Fu-Kane-Mele index (cf.in particularRemark 4.5). • We prove that the FKMM-invariant is a genuine characteristic class for the category of “Quater-nionic” vector bundles in the sense that there exists a universal version of it which providesby pullback the FKMM-invariant of (almost) each Q -bundle. This point of view is developedin Section 6 and represents a quite important point of novelty with respect to the originaldefinition proposed in [FKMM]. Moreover, our interpretation of the FKMM-invariant is stillliable to further generalizations that provide a notion of characteristic class which turns out LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 3 to be well defined in great generality for Q -bundles over involutive base spaces of each “rea-sonable” type. This point of view will be discussed in a future paper [DG3] (cf.also Remark3.10);The topological classification of the family of “Quaternionic” vector bundles strongly depends onthe nature of the involutive space ( X , τ ) which in turns reflects some physical features of the systemunder consideration. An important aspect of an involutive space is the structure of its fixed point set X τ : = { x ∈ X | τ ( x ) = x } . The family of FKMM-spaces is rich enough to contain all the systems ofinterest in condensed matter physics: Definition 1.1 (FKMM-space) . An FKMM-space is an involutive space ( X , τ ) such that:0. X is a compact and path connected Hausdor ff space which admits the structure of a Z -CW-complex;1. The fixed point set X τ , ∅ consists of a finite number of points;2. H Z ( X , Z (1)) = . For the sake of completeness, let us recall that an involutive space ( X , τ ) has the structure of a Z -CW-complex if it admits a skeleton decomposition given by gluing cells of di ff erent dimensionswhich carry a Z -action. For a precise definition of the notion of Z -CW-complex, the reader canrefer to [DG1, Section 4.5] or [Ma, AP]. In addition, the cohomology group that appears in 2. isthe equivariant Borel cohomolgy of the space ( X , τ ) computed with respect to the local system ofcoe ffi cients Z (1). A short reminder of this theory is sketched in Section 3.1. Here, let us only recallthat the group H Z ( X , Z (1)) provides the classification for “Real” line bundles over ( X , τ ) where theadjective “Real” refers to the category of vector bundles introduced by M. F. Atiyah in [At1] andextensively studied in [DG1]. Given an involutive space ( X , τ ) let us denote by Vec n Q ( X , τ ) the set ofisomorphic classes of Q -bundles with fiber of dimension n . We point out that the existence of fixedpoints X τ , ∅ and the the connectedness of X imply that the dimension of the fibers is forced to beeven, i.e., n = m (cf.Proposition 2.2). The key result at the basis of our classification can be statedas follows: Theorem 1.2 (Injective group homomorphism: low dimensional cases) . Let ( X , τ ) be an FKMM-spacesuch that the dimension of the free Z -cells in the skeleton decomposition of X does not exceed d = .Then, the FKMM-invariant defines an injective map κ : Vec m Q ( X , τ ) −→ H Z ( X | X τ , Z (1)) m ∈ N . Moreover,
Vec m Q ( X , τ ) can be endowed with a group structure in such a way that κ becomes an injectivegroup homomorphism. The proof of this result is postponed to Section 4.1 and a precise description of the map κ is givenin Section 3.3. Let us just comment that the abelian group H Z ( X | X τ , Z (1)) describes the relative equivariant cohomology associated with the pair X τ ⊂ X . The importance of Theorem 1.2 becomesapparent when we compare it with the well-known similar result for the classification of complexvector bundles over a CW-complex X of dimension d
3; In this situation one has the isomorphism c : Vec m C ( X ) −→ H ( X , Z ) m ∈ N induced by the first Chern class. The parallelism between κ and c is quite evocative: The FKMM-invariant is the proper characteristic class that classifies “Quaternionic” vector bundles as elements ofthe cohomology group H Z ( X | X τ , Z (1)). In e ff ect, this turns out to be the correct point of view fordeveloping a more general theory of the FKMM-invariant (cf.Remark 3.10).In order to connect the general result of Theorem 1.2 with the specific problem of the classificationof topological insulators of class AII we need to recall some basic facts. A typical representative ofthis class is a system of quantum particles with half-integer spin and subjected to an odd time-reversalsymmetry (-TR). More in detail, let us consider a (self-adjoint) Hamiltonian ˆ H acting on the Hilbert G. DE NITTIS AND K. GOMI space H : = L ( R d ) ⊗ C L (continuous case) or H : = ℓ ( Z d ) ⊗ C L (tight-binding approximation) wherethe number L ∈ N takes into account the spinorial degrees of freedom. We say that ˆ H is of type AII (cf.[AZ, SRFL]) if there exists an anti- unitary symmetry of type ˆ Θ : = ˆ C ˆ J where ˆ C is the complexconjugation ˆ C ψ = ψ and ˆ J is a unitary operator that verifies ˆ C ˆ J ˆ C = − ˆ J ∗ ˆ J ˆ H ˆ J ∗ = ˆ C ˆ H ˆ C ( AII - symmetry) . (1.1)Equation (1.1) is equivalent to ˆ Θ ˆ H ˆ Θ ∗ = ˆ H and the first condition in (1.1) implies that ˆ Θ is an anti -involution in the sense that ˆ Θ = − (or equivalently ˆ Θ = − ˆ Θ ∗ ). Remark 1.3 (Topological insulators in class AI ) . If in (1.1) one replaces the first condition withˆ C ˆ J ˆ C = ˆ J ∗ , or equivalently ˆ Θ = , one obtains the class AI of topological insulators. Systems of thistype possess an even time-reversal symmetry ( + TR) and have been classified in [DG1]. ◭ For each a ∈ R d let ˆ U a be the unitary operator on H that implements the translation by a in the sensethat ˆ U a ψ ( · ) = ψ ( ·− a ) for ψ ∈ H . The condition [ ˆ H , ˆ U a ] = H under the translation ˆ U a . According to a standard nomenclature, one says that ˆ H describes a free (resp. periodic ) system if it is translational invariant for all a ∈ R d (resp. for all a ∈ Z d ). In both cases onecan represent ˆ H as a fibered operator over the momentum space (one uses the Fourier transform in thefree case or the Bloch-Floquet transform in the periodic case). By repeating verbatim the constructionexplained in [DG1, Section 2] one associates to each gapped spectral region of ˆ H a vector bundlewhich is usually called Bloch-bundle . The presence of the symmetry (1.1) endows the Bloch-bundlewith a “Quaternionic” structure in the sense of [Du] (we refer to Section 2.1 for a precise definition).This construction justifies the following definition.
Definition 1.4 (Topological insulators of type
AII ) . A d-dimensional free system of type
AII is a“Quaternionic” vector bundle over the involutive TR-sphere ˜ S d : = ( S d , τ ) (1.2) where S d : = (cid:8) k ∈ R d + | k k k = (cid:9) and the involution τ is defined by τ ( k , k , . . . , k d ) : = ( k , − k , . . . , − k d ) . (1.3) A d-dimensional periodic system of type
AII is a “Quaternionic” vector bundle over the involutiveTR-torus ˜ T d : = ( T d , τ ) (1.4) where T d : = S × . . . × S (d-times) and the involution τ extends diagonally the involution on ˜ S givenby (1.3) in such a way that ˜ T d = ˜ S × . . . × ˜ S . Let us point out that the involutive spaces ˜ S d and ˜ T d are particular examples of FKMM-spaces.Conditions 0 and 1 come from the explicit description of the Z -CW-complex structure of these spaces[DG1, Examples 4.20 & 4.21] and condition 2 follows from the computation of the equivariant co-homology based on a recursive application of the Gysin sequence [DG1, Sections 5.3 & 5.4]. Inparticular, Theorem 1.2 applies to the description of Vec m Q ( S d , τ ) and Vec m Q ( T d , τ ) when d Theorem 1.5 (Classification of
AII topological insulators: low dimensional case) . Let ( S d , τ ) and ( T d , τ ) be the TR-involutive spaces described in Definition 1.4. Then:(i) “Quaternionic” vector bundles over ( S d , τ ) and ( T d , τ ) can have only even rank. In particularthere are no “Quaternionic” line-bundles for all d ∈ N ;(ii) Vec m Q (cid:0) S , τ (cid:1) = for all m ∈ N ;(iii) For TR-spheres in dimension d = , one has group isomorphisms Vec m Q (cid:0) S , τ (cid:1) = Z , Vec m Q (cid:0) S , τ (cid:1) = Z , given by the FKMM-invariant κ ; LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 5 (iv) For TR-tori in dimension d = , one has group isomorphisms Vec m Q (cid:0) T , τ (cid:1) = Z , Vec m Q (cid:0) T , τ (cid:1) = Z , given by the FKMM-invariant κ . The various items listed in Theorem 1.5 are proved separately in the paper: (i) is a consequenceof Proposition 2.2; The proof of (ii) is contained in Proposition 2.15 and is based on the homotopy classification of “Quaternionic” vector bundles (cf.Theorem 2.13); Item (iii) is proved in Proposition4.6 while the proof of (iv) is contained in Proposition 4.16 and Proposition 4.18. Let us spend fewwords about the strategy of the proofs of (iii) and (iv). Firstly, one computes the relevant cohomologygroups H Z (cid:0) ˜ S d | ( ˜ S d ) τ , Z (1) (cid:1) ≃ Z , H Z (cid:0) ˜ T d | ( ˜ T d ) τ , Z (1) (cid:1) ≃ Z d − ( d + ∀ d > κ inTheorem 1.2 is indeed surjective by constructing suitable explicit realizations of non-trivial Q -bundles.An important remark is that, due to the low dimensionality of the base space, the FKMM-invariant canbe described in terms of a sign map d w : X τ → ± T one recovers thedistinction between week and strong invariants according to the original definition given in [FKM](cf.Remark 4.19). Finally, the non-vanishing of the FKMM-invariant can be also interpreted as theobstruction to the existence of a global frame of (Bloch) sections which supports the “Quaternionic”symmetry (cf.Remarks 4.2 & 4.19) and this fact recovers and generalizes the point of view investigatedin [GP] for the particular case of the involutive space ˜ T .The classification for d > κ is nolonger su ffi cient to establish an injective morphism between Vec m Q ( X , τ ) and some cohomology group.In the case d = second Chern class and, under conditions which are slightly morerestrictive that those in Definition 1.1, one can prove the following result:
Theorem 1.6 (Injective group homomorphism: d = . Let ( X , τ ) be an FKMM-space and assume inaddition that X is a closed and oriented -manifold with an involution τ which is smooth. Then, theFKMM-invariant κ and the second Chern class c define a map ( κ, c ) : Vec m Q ( X , τ ) −→ H Z ( X | X τ , Z (1)) ⊕ H ( X , Z ) m ∈ N that is injective. Moreover, Vec m Q ( X , τ ) can be endowed with a group structure in such a way that thepair ( κ, c ) sets an injective group homomorphism. The proof of a slightly weaker version of this theorem is postponed to Section 5, where we provethe injectivity of the pair ( κ, c ) under some extra hypothesis (cf.Assumption 5.1 & Theorem 5.4)which are still verified by the involutive TR-spaces ˜ S and ˜ T . Let us just comment that the claim ofTheorem 1.6 is true as it is and, at the cost of increasing the technical di ffi culty of the proof (one needsobstruction theory), it can be proved in full generality [DG3].Before discussing the ramifications of Theorem 1.6 for the classification of topological insulators,it is important to comment about the relevance of the dimension d = ff ects.In fact, since d d = piezoelectric polarization (see [DL] and references therein) or the isotropic magneto-electric response [EMV, HPB]. In these cases the relevant topological response ofthe system depends on the full spacetime dimensionality and higher invariants like the second Chernnumber become relevant (see e.g.[LY]). An interesting di ff erence between the case d = d κ and c are not independent. Inparticular, it is possible to show that the strong component of the FKMM-invariant is uniquely fixedby the week components and by the parity of the second Chern number C : = h c , [ X ] i ∈ Z (wedenote by [ X ] ∈ H ( X ) the fundamental class of X ). This fact is made evident in the following result, G. DE NITTIS AND K. GOMI which completes the classification of topological insulators in class
AII for all physically interestingdimensions.
Theorem 1.7 (Classification of
AII topological insulators: d = . Let ( S , τ ) and ( T , τ ) be theTR-involutive spaces described in Definition 1.4.(i) The map ( κ, c ) : Vec m Q ( S , τ ) −→ Z ⊕ Z is injective and the image ( κ, c ) : E ( ǫ, C ) ∈ Z ⊕ Z of each “Quaternionic” vector bundle ( E , Θ ) is contained in the subgroup n ( ǫ, C ) ∈ Z ⊕ Z | ǫ = ( − C o ≃ Z . More precisely, elements in
Vec m Q ( S , τ ) ≃ Z are completely classified by the second Chernclass c ( E ) and the value κ ( E ) ≃ ǫ of the (strong component of the) FKMM-invariant is fixedby the reduction mod. 2 of the second Chern number C = h c ( E ) , [ S ] i .(ii) The map ( κ, c ) : Vec m Q ( T , τ ) −→ Z ⊕ Z is injective and the image ( κ, c ) : E ( ǫ , . . . , ǫ , C ) ∈ Z ⊕ Z of each “Quaternionic”vector bundle ( E , Θ ) is contained in the subgroup ( ǫ , . . . , ǫ , ǫ , C ) ∈ Z ⊕ Z (cid:12)(cid:12)(cid:12)(cid:12) Y j = ǫ j = ( − C ≃ Z ⊕ Z . More precisely, elements in
Vec m Q ( T , τ ) ≃ Z ⊕ Z are completely classified by the sec-ond Chern class c ( E ) and the FKMM-invariant. However, only the first ten week compo-nents of the FKMM-invariant κ ( E ) ≃ ǫ are independent since the strong component ǫ = ( − C Q j = ǫ j is fixed by the values of the week components and the reduction mod. 2 of thesecond Chern number C = h c ( E ) , [ T ] i . The proof of this theorem is explained in Section 5. It should be remarked that item (ii) abovewas originally shown in [FKMM]. The content of Theorem 1.5 and Theorem 1.7 is summarized inTable 1.1 together with the classification of type A topological insulators (systems without symmetries,cf.with [DG1, Table 1.1]).VB AZC d = d = d = d = m C ( S d ) A Z m = Z ( m >
2) FreeVec m Q ( S d , τ ) AII Z Z Z systemsVec m C ( T d ) A Z Z Z ( m = Z ( m >
2) PeriodicVec m Q ( T d , τ ) AII Z Z Z ⊕ Z systemsT able unstable regime ( d > m )which is not covered by the K -theoretical classification. The rank of the “Quater-nionic” vector bundles is forced to be even since the involutive spaces ( S d , τ ) and( T d , τ ) described in Definition 1.4 have fixed points. LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 7
A comparison between Table 1.1 and the calculations in Appendix C shows that our classificationcompletely agrees with the predictions supplied by K -theory. In particular, for “Quaternionic” vectorbundles in dimension d stable and unstable regime and so the K -theory provides a precise description for the “labeling sets” of various isomorphism classes of Q -bundles. Nevertheless, our classification is strictly stronger than the K -theoretical analysis since itprovides an explicit description for the classifying invariants as elements of a proper cohomologytheory. This information is not trivial at all! On the contrary, it plays a prominent role in the descriptionof physical e ff ects like the stability of spin currents.Before ending this introductory part, we would briefly compare our results with the existing physicalliterature on the classification of topological insulators in class AII . Table 1 is certainly not a noveltyin the literature and, as already mentioned, the first classification of the distinct topological phases oftime-reversal invariant fermionic systems has been described in [KM1, KM2, FK] for the case d = d = Z -order is characterized by the existence of stable spin currents(QSHE) or by the net e ff ect of a “spin pump”. Mathematically, the di ff erent phases are distinguishedby the signs, the so-called Fu-Kane-Mele indices, that a suitable (Pfa ffi an) function of the Bloch wavestakes on the fixed points of the torus (see Remark 4.5 for more details). In [MB] the authors proposed adi ff erent description of the Fu-Kane-Mele indices in terms of an implicit version of the Z -equivarianthomotopy classification described in Theorem 2.13. Subsequently, an independent elegant argumenthas been developed in [Ro] where the classification for d = , ff ers from these pervious researchesessentially in two aspects. First of all, our approach based on the definition of the FKMM-invariant isindependent on the particular nature of the base space, while all the previous techniques are designed ad hoc over the peculiar structure of the torus. This clarification is not of secondary importance, sincecondensed matter systems are not the only sources of quantum topological e ff ects; e.g.our results areapplicable to certain topological quantum field theories with symmetries as well as to time-reversalquantum systems perturbed by external fields parametrized by the points of some complicated man-ifold. Second, the FKMM-invariant used for our classification is a cohomological class, a fact thathas two immediate implications: cohomology is algorithmically computable as opposed to homotopyand cohomological classes are liable to be described in terms of di ff erential forms [DG3]. None ofthe invariants previously considered in the literature manifestly shows this important cohomologicalnature, although all of them are described by our FKMM-invariant (cf.Section 4.4).As a final remark let us point out that one of the merits of the present work is that it shows howthe FKMM-invariant can be understood as a bona fide characteristic class for the category of “Quater-nionic” vector bundles (cf.Section 6). This discovery, which in our opinion may have future impli-cations, is an important point of novelty with respect to the original (and certainly inspiring) ideascontained in [FKMM]. Let us also recall that in the literature there already exist works devoted to theconstruction of characteristic classes for “Quaternionic” vector bundles. Among these, we mentionat least the two papers [DSLF, LLM] in which the authors developed the notion of “Quaternionic”Chern classes. We feel that it should be of some interest and utility to understand the link between theFKMM-invariant described in this work and these “Quaternionic” Chern classes. Acknowledgements.
The authors are immensely grateful to M. Furuta for allowing them to use ideascontained in the preprint [FKMM]. GD wants to thank F. Rivera for the excellent hospitality at SJIH,San Juan, Puerto Rico where this investigation has begun. GD’s research is supported by the Alexandervon Humboldt Foundation. KG’s research is supported by the Grant-in-Aid for Young Scientists (B23740051), JSPS.
G. DE NITTIS AND K. GOMI
2. “Q uaternionic ” vector bundles This section is devoted to the description of the category of “Quaternionic” vector bundles intro-duced for the first time in [Du]. Through the paper we often use the shorter expression Q -bundle instead of “Quaternionic” vector bundle.2.1. “Quaternionic” structure on vector bundles. The first ingredient to define a “Quaternionic”structure on a complex vector bundle is an involution on the base space. We recall that an involution τ on a topological space X is a homeomorphism of period 2, i.e. τ = Id X . The pair ( X , τ ) will be calledan involutive space . The spaces ˜ S d and ˜ T d described in Definition 1.4 are examples of involutivespaces. Other examples have been discussed in [DG1, Section 4.1]. We tacitly assume through thepaper that all the involutive spaces ( X , τ ) verify at least condition 0 in Definition 1.1.A “Quaternionic” vector bundle, or Q -bundle, over ( X , τ ) is a complex vector bundle π : E → X endowed with a (topological) homeomorphism Θ : E → E such that:( Q ) the projection π is equivariant in the sense that π ◦ Θ = τ ◦ π ;( Q ) Θ is anti-linear on each fiber, i.e. Θ ( λ p ) = λ Θ ( p ) for all λ ∈ C and p ∈ E where λ is thecomplex conjugate of λ ;( Q ) Θ acts fiberwise as the multiplication by −
1, namely Θ | E x = − E x .It is always possible to endow E with a (essentially unique) Hermitian metric with respect to which Θ is an anti-unitary map between conjugate fibers (cf.Proposition 2.10).A vector bundle morphism f between two vector bundles π : E → X and π ′ : E ′ → X over the samebase space is a continuous map f : E → E ′ which is fiber preserving in the sense that π = π ′ ◦ f andthat restricts to a linear map on each fiber f | x : E x → E ′ x . Complex (resp. real) vector bundles over X together with vector bundle morphisms define a category and we use Vec m C ( X ) (resp. Vec m R ( X )) todenote the set of equivalence classes of isomorphic vector bundles of rank m . Also Q -bundles define acategory with respect to Q -morphisms . A Q -morphism f between two Q -bundles ( E , Θ ) and ( E ′ , Θ ′ )over the same involutive space ( X , τ ) is a vector bundle morphism commuting with the involutions,i.e. f ◦ Θ = Θ ′ ◦ f . The set of equivalence classes of isomorphic Q -bundles of rank m over ( X , τ ) isdenoted with Vec m Q ( X , τ ).The set Vec m C ( X ) is non-empty since it contains at least the product vector bundle X × C m → X with canonical projection ( x , v) x . Similarly, in the real case one has that X × R m → X providesan element of Vec m R ( X ). A complex (resp. real) vector bundle is called trivial if it is isomorphic tothe complex (resp. real) product vector bundle. In order to extend these definitions to “Quaternionic”vector bundles we need to investigate the structure of the fibers of a Q -bundles ( E , Θ ) over fixed pointsof the base space ( X , τ ). Let x ∈ X τ and E x ≃ C m be the related fiber. In this case the restriction Θ | E x ≡ J defines an anti -linear map J : E x → E x such that J = − E x . This means that the fibers E x over fixed points x ∈ X τ are endowed with a quaternionc structure in the following sense: Remark 2.1 (Quaternionic structure over complex vector spaces) . We shall denote with H the skew-field of quaternions and by (1 , i , j , k) its usual basis over R , H = R ⊕ R i ⊕ R j ⊕ R k (cid:0) i = j = k = ijk = − (cid:1) . Similarly, the pair (1 , j) provides a basis of H over C H = C ⊕ C j = ( R ⊕ R i) ⊕ ( R ⊕ R i) j . where the relation ij = k has been used. Let V be a complex vector space of complex dimension n . One says that V has a quaternionic structure if there is an anti -linear map J : V → V such that J = − (cf.[Va, Section 1] and references therein). A complex vector space V admits a quaternionicstructure if and only if it has even complex dimension n = m and in this case the pair ( V , J ) turns outto be isomorphic to the space H m = ( C ⊕ C j) m understood as left -module over C and endowed withthe left multiplication by j. Since ji = − ij this multiplication is automatically anti -linear with respect LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 9 to the complex structure. Said di ff erently, we can identify ( V , J ) with C m endowed with the standard quaternionic structure v Q v where v is the complex conjugate of v and Q denotes the real matrix Q : = −
11 0 . . . . . . −
11 0 . (2.1)Let us recall that the quaternionic unitary group U H ( m ) coincides with the symplectic group S p ( m ).Moreover, U H ( m ) has the following characterization: let µ : U (2 m ) → U (2 m ) be the involutiondefined by µ ( U ) : = − QUQ , then U H ( m ) ≃ U (2 m ) µ where U (2 m ) µ is the set of fixed points under theaction of µ . Finally U (2 m ) µ ⊂ S U (2 m ). Indeed if µ ( U ) = U and U v = λ v for some λ ∈ C and v ∈ C m then also U ( Q v) = λ ( Q v) and the vectors v and Q v are linearly independent (even in the case λ = ± U is given by m pairs { λ i , λ i } i = ,..., m such that | λ i | = U ) = ◭ The first consequence of Remark 2.1 is:
Proposition 2.2.
Let ( X , τ ) be an involutive and path-connected space. If X τ , ∅ then every Q -bundleover ( X , τ ) necessarily has even rank.Proof. Fibers over fixed points needs an even complex dimension in order to support a quaternionicstructure. Moreover, if the base space X is path-connected the dimension of the fibers is constant. (cid:4) Remark 2.3 ( Q -bundles of odd rank) . In Proposition 2.2 the condition X τ , ∅ can not be removed.In fact, if the base space X is endowed with a free involution then it is possible to realize Q -bundleswith fibers of odd rank. For instance, an example of Q -line bundle has been worked out in [Du]: Let Z = {± } endowed with the free involution τ : ǫ
7→ − ǫ , ǫ ∈ {± } . Then the complex line bundle L = Z × C gives rise to a “Quaternionic” vector bundle by ( ǫ, z ) ( − ǫ, ǫ z ). This example, togetherwith Proposition 2.2, shows that a base space with a free involution is a necessary condition for theconstruction of a Q -bundle with odd fibers. ◭ The set Vec m Q ( X , τ ) is non-empty since it contains at least the “Quaternionic” product bundle X × C m → X endowed with the product Q -structure Θ ( x , v) = ( τ ( x ) , Q v) where the matrix Q is thesame as in (2.1). Moreover, as a consequence of Proposition 2.2, this is the only type of product Q -bundle which is possible to build if X τ , ∅ . We say that a Q -bundle is Q -trivial if it is isomorphicto the product Q -bundle in the category of Q -bundles. Since the Whitney sum of a rank 2 product Q -bundle defines a map Vec m Q ( X , τ ) → Vec m + Q ( X , τ ) one introduces the inductive limit Vec Q ( X , τ ) : = S m ∈ N Vec m Q ( X , τ ) which describes isomorphism classes of Q -bundles over involutive spaces with fixedpoints independently of the (even) rank of the fibers.The name of “Quaternionic” vector bundles (in the category of involutive spaces) for elements inVec Q ( X , τ ) is justified by the following result: Proposition 2.4.
Let
Vec m H ( X ) be the set of equivalence classes of vector bundles over X with typicalfiber H m . Then, Vec m H ( X ) ≃ Vec m Q ( X , Id X ) ∀ m ∈ N . sketch of. Let E be an element of Vec m H ( X ). Each fiber of E x is a left H -module. Now, if one considers E x simply as a left C -module endowed with an extra left multiplication by j one obtains, by virtue ofRemark 2.1, a map c : Vec m H ( X ) → Vec m Q ( X , Id X ). On the other side, if E is an element of Vec m Q ( X , Id X )then each fiber E x turns out to be a complex vector space of dimension 2 m endowed with a quaternionicstructure so that E x ≃ H m . This leads to a map q : Vec m Q ( X , Id X ) → Vec m H ( X ). By construction oneverifies that c and q are inverses of each other. (cid:4) Given a “Quaternionic” bundle ( E , Θ ) over the involutive space ( X , τ ) we can “forget” the Q -structure and consider only the complex vector bundle E → X . This forgetting procedure goes throughisomorphism classes. In fact, a Q -isomorphism between two Q -bundles is, in particular, an isomor-phism of complex vector bundles plus an extra condition of equivariance which is lost under the processof forgetting the “Quaternionic” structure. Proposition 2.5.
The process of forgetting the “Quaternionic” structure defines a map : Vec m Q ( X , τ ) −→ Vec m C ( X ) such that : [0] → [0] where [0] denotes the trivial class in the appropriate category. “Quaternionic” sections. Let Γ ( E ) be the set of sections of a Q -bundle ( E , Θ ) over the involutespace ( X , τ ). We recall that a section s is a continuous maps s : X → E such that π ◦ s = Id X where π : E → X is the bundle projection. The set Γ ( E ) has the structure of a module over thealgebra C ( X ) and inherits from the “Quaternionic” structure of ( E , Θ ) an anti -linear anti -involution τ Θ : Γ ( E ) → Γ ( E ) defined by τ Θ ( s ) : = Θ ◦ s ◦ τ . This means that the C ( X )-module Γ ( E ) is endowed with a quaternionic structure (in the jargon ofRemark 2.1) given by τ Θ and the left multiplication by i.The product Q -bundle ( X × C m , Θ ) over ( X , τ ) has a special family of sections { r , . . . , r m } givenby r j : x ( x , e j ) with e j : = (0 , . . . , , , , . . . ,
0) the j -th vector of the canonical basis of C m . Thesesections verify τ Θ ( r j )( x ) = Θ (cid:0) τ ( x ) , e j (cid:1) = ( x , Q e j ) = : ( Q r j )( x ) (2.2)where we exploited the reality of the canonical basis e j = e j . The matrix Q is the one defined in (2.1)and the constant endomorphism Q ∈ Γ (cid:0) End( X × C m ) (cid:1) is specified pointwise by the last equality in(2.2). Let us point out that Q is invertible, real in the sense τ Θ ◦ Q = Q ◦ τ Θ and anti-involutive Q = − Id. Moreover, for all x ∈ X the vectors { r ( x ) , . . . , r m ( x ) } provide a complete basis for the fiber { x } × C m over x .For a Q -bundle ( E , Θ ) this kind of behavior is locally general. Let s ∈ Γ ( E ) and U ⊂ X a τ -invariantopen set such that s ( x ) , x ∈ U . Set s : = τ Θ ( s ) which implies τ Θ ( s ) = − s . Since Θ isa homeomorphism one has also s ( x ) , x ∈ U . Moreover, it is easy to check that s and s are linearly independent. In fact, if one assumes that s = λ s then the application of τ Θ to both sidesprovides s = − ¯ λ s which is possible if and only if λ =
0. We say that ( s , s ) is a “Quaternionic”pair (or a Kramers pair using a physical terminology!) over U . Now, let us add a third section s which is independent from s and s and such that s ( x ) , x ∈ U . The section s : = τ Θ ( s )does not vanish on U and is independent from s , s , s . The last claim can be easily proved alongthe same strategy used for the independence between s and s . If E has rank 2 m we can iterate thisprocedure m − { s , s , s , s . . . , s m − , s m } which areindependent and non-zero over U and that verify the relations s j : = τ Θ ( s j − ) for all j = , . . . , m . Inother words, we have realized a frame for E | U made by “Quaternionic” pairs ( s j − , s j ). This impliesalso τ Θ ( s j ) = Q s j where the endomorphism Q ∈ Γ (cid:0) End( E | U ) (cid:1) acts as the multiplication by thematrix Q with respect to the local basis { s , s , . . . , s m − , s m } . This discussion justifies the followingdefinition: Definition 2.6 (Global Q -frame) . Let ( E , Θ ) be a “Quaternionic” vector bundle of rank m over theinvolutive space ( X , τ ) . We say that ( E , Θ ) admits a global Q -frame if there is a collection of sections { s , s , s , s . . . , s m − , s m } ⊂ Γ ( E ) such that:(a) For each x ∈ X the set of vectors { s ( x ) , s ( x ) , . . . , s m − ( x ) , s m ( x ) } spans the fiber E x over x;(b) s j = τ Θ ( s j − ) for all j = , . . . , m. The existence of a global Q -frame characterizes the Q -triviality of a “Quaternionic” vector bundle. LASSIFICATION OF “QUATERNIONIC”
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Theorem 2.7 ( Q -triviality) . An even rank Q -bundle ( E , Θ ) over ( X , τ ) is Q -trivial if and only if itadmits a global Q -frame.Proof. Let us assume that ( E , Θ ) is Q -trivial. This means that there is a Q -isomorphism h : X × C m → E between ( E , Θ ) and the product Q -bundle ( X × C m , Θ ). Let us define sections s j ∈ Γ ( E ) by s j : = h ◦ r j where { r , r , . . . , r m − , r m } is the global Q -frame of the product bundle. The fact that h isan isomorphism implies that { s , s , . . . , s m − , s m } spans each fiber of E . Moreover, the equivariancecondition Θ ◦ h = h ◦ Θ implies that τ Θ ( s j − ) = Θ ◦ h ◦ r j − ◦ τ = h ◦ τ Θ ( r j − ) = h ◦ r j = s j andthis shows that { s , s , . . . , s m − , s m } is a global Q -frame.Conversely, let us assume that ( E , Θ ) has a global Q -frame { s , s , . . . , s m − , s m } . For each x ∈ X we can set the linear isomorphism h x : { x } × C m → E x defined by h x ( x , e j ) : = s j ( x ). The collectionof h x defines an isomorphism h : X × C m → E between complex vector bundles [MS, Theorem 2.2].Moreover, Θ ◦ h ( x , e j − ) = Θ ◦ s j − ( x ) = s j ( τ ( x )) = h ( τ ( x ) , e j ) = h ◦ Θ ( x , e j − ) for all x ∈ X andall j = , . . . , m and this proves that h is a Q -isomorphism. (cid:4) Local Q -triviality. A Q -bundle is locally trivial in the category of complex vector bundles bydefinition. Less obvious is that a Q -bundle is also locally trivial in the category of vector bundles overan involutive space. In order to discuss this point we start with a classical result. Lemma 2.8 (Extension) . Let ( X , τ ) be an involutive space and assume that X verifies (at least) con-dition 0 of Definition 1.1. Let ( E , Θ ) be a Q -bundle over ( X , τ ) and Y ⊂ X a closed subset such that τ ( Y ) = Y. Then each Q -pair { ˜ s , ˜ s } ∈ Γ ( E | Y ) extends to a Q -pair { s , s } ⊂ Γ ( E ) .Proof. By definition of Q -pair one has ˜ s = τ Θ ( ˜ s ). Using [AB, Lemma 1.1] we know that we canextend ˜ s to a section s ∈ Γ ( E ). By setting s : = τ Θ ( s ) one has that ( s , s ) is a Q -pair and s | Y = ˜ s . (cid:4) Proposition 2.9 (Local Q -triviality) . Let ( E , Θ ) be a Q -bundle of even rank over the involutive space ( X , τ ) such that X verifies (at least) condition 0 of Definition 1.1. Then, π : E → X is locally Q -trivial meaning that for all x ∈ X there exists a τ -invariant neighborhood U of x and a Q -isomorphismh : π − ( U ) → U × C m with respect to the trivial Q -structure on the product bundle U × C m given by Θ ( x , v) = ( τ ( x ) , Q v) (the matrix Q is defined by (2.1) ). Moreover, if x ∈ X τ the neighborhood U canbe chosen connected otherwise, when x , τ ( x ) , U can be taken as the union of two disjoint open sets U : = U ′ ∪ U ′′ with x ∈ U ′ and τ : U ′ → U ′′ an homeomorphism.Proof. This proof is an adaption of the argument in [AB, Lemma 1.2] and of the discussion in [At1, pg.374]. Let us start with the case of a x ∈ X τ . On the fiber E x ≃ C m the procedure described in Section2.2 leads to a basis of vectors { s ( x ) , s ( x ) , . . . , s m − ( x ) , s m ( x ) } such that each ( s j − ( x ) , s j ( x )), j = , . . . , m , is a Q -pair with respect to τ Θ . By the extension Lemma 2.8 we can extend these vectorsto a family of sections { s , s , . . . , s m − , s m } ⊂ Γ ( E ) formed by Q -pairs ( s j − , s j ). Moreover, thereexists an open neighborhood U ′ of x where this family of sections behaves as a global frame for E | U ′ (this is a consequence of the fact that the linear group GL m ( C ) is open). In order to have a τ -invariantneighborhood it is enough to consider U : = U ′ ∩ τ ( U ′ ). Moreover, since X is assumed to be connected, U is at least locally connected and so it can be chosen su ffi ciently small to be connected around thefixed point x . The family of sections { s , s , . . . , s m − , s m } provides a global Q -frame for E | U , henceTheorem 2.7 assures the existence of a Q -isomorphism h between E | U = π − ( U ) and the product Q -bundle U × C m .In the case x , τ ( x ) we can start with the construction of a global Q -frame defined on the closed set Y : = { x , τ ( x ) } . To do this let us consider a nowhere vanishing section ˜ s ∈ Γ ( E | Y ) defined by a pair ofnon-zero vectors ˜ s ( x ) ∈ E x and ˜ s ( τ ( x )) ∈ E τ ( x ) . The section ˜ s : = τ Θ ( ˜ s ) is a well defined nowherevanishing element of Γ ( E | Y ) which is independent of ˜ s for the same argument sketched in Section2.2. After iterating the procedure m − Q -pair, oneends up eventually with a global Q -frame for E | Y . At this point the proof proceeds exactly as before:the extension Lemma 2.8 and a continuity argument assure the existence of a τ -invariant neighborhood U ⊃ Y such that E | U admits a global Q -frame. Clearly we can choose U su ffi ciently localized around x and τ ( x ) in such a way that the requirements in the claim are fulfilled. (cid:4) One of the consequences of Proposition 2.9 is that it allows us to define τ -invariant partitions of unity subordinate to a given τ -invariant covering { U i } associated with a Q -trivialization of the Q -bundle; τ -invariant partitions of unity can be then used to build equivariant Hermitian metrics compatible withthe “Quaternionic” structure (cf.[DG1, Remark 4.11]). The same result can be achieved also with adirect average of the metric. Let ( E , Θ ) be a Q -bundle over the involutive space ( X , τ ) and considerthe set E × X E : = { ( p , p ) ∈ E × E | π ( p ) = π ( p ) } associated with the underlying complex vectorbundle π : E → X . A Hermitian metric is a map m ′ : E × X E → C which is a positive-definiteHermitian form on each fiber. By a standard result [Kar, Chapter I, Theorem 8.7] we know that eachcomplex vector bundle over a paracompact base space admits a Hermitian metric. Moreover if m ′ and m ′′ are two di ff erent Hermitian metrics for π : E → X there exists an isomorphism f : E → E such that m ′ ( p , p ) = m ′′ ( f ( p ) , f ( p )) [Kar, Chapter I, Theorem 8.8]. This mans that the choiceof a Hermitian metric is essentially unique. A Hermitian metric compatible with the “Quaternionic”structure must verify the condition m ( Θ ( p ) , Θ ( p )) = m ( p , p ) for all ( p , p ) ∈ E × X E . Such ametric is called equivariant . With respect to an equivariant metric the involution Θ acts as an “anti-unitary” map. The existence of an equivariant metric m follows directly from the existence of anyHermitian metric m ′ by means of the average procedure m (cid:0) p , p (cid:1) : = h m ′ (cid:0) p , p (cid:1) + m ′ (cid:0) Θ ( p ) , Θ ( p ) (cid:1) i , ( p , p ) ∈ E × X E . Also in this case an equivariant generalization of [Kar, Chapter I, Theorem 8.8] assures that two equi-variant metrics for the Q -bundle ( E , Θ ) are related by a Q -isomorphism f : E → E . Summarizing onehas: Proposition 2.10 (Equivariant metric) . Each Q -bundle ( E , Θ ) over an involutive space ( X , τ ) suchthat X verifies (at least) condition 0 of Definition 1.1 admits an equivariant Hermitian metric which isessentially unique up to Q -isomorphisms. The main implication of Proposition 2.10 is that the problem of the classification of “Quaternionic”vector bundles coincides with the problem of the classification of “Quaternionic” vector bundles en-dowed with an equivariant Hermitian metric. For this reason, we tacitly assume hereafter that:
Assumption 2.11.
Each Q -bundle is endowed with an equivariant Hermitian metric and vector bundlemaps between Q -bundles are assumed to be metric-preserving ( i.e. isometries). Homotopy classification of “Quaternionic” vector bundles.
The assumption that the basespace X is compact allows us to extend usual homotopy properties valid for complex vector bundles tothe category of Q -bundles. Given two involutive spaces ( X , τ ) and ( X , τ ) we say that a continuousmap ϕ : X → X is equivariant if and only if ϕ ◦ τ = τ ◦ ϕ . An equivariant homotopy betweenequivariant maps ϕ and ϕ is a continuous map F : [0 , × X → X such that ϕ t ( · ) : = F ( t , · ) is equi-variant for all t ∈ [0 , X , τ ) and ( X , τ ) will be denoted by [ X , X ] Z . Theorem 2.12 (Homotopy property) . Let ( X , τ ) and ( X , τ ) be two involutive spaces with X verifies(at least) condition 0 of Definition 1.1. Let ( E , Θ ) be a Q -bundle of rank m over ( X , τ ) and F :[0 , × X → X an equivariant homotopy between the equivariant maps ϕ and ϕ . Then, the pullbackbundles ϕ ∗ t E → X have an induced Q -structure for all t ∈ [0 , and ϕ ∗ E ≃ ϕ ∗ E in Vec m Q ( X , τ ) .sketch of. This theorem can be proved by a suitable equivariant generalization of [AB, Proposition1.3]. The only new point is the Q -structure on ϕ ∗ t E → X . By definition ϕ ∗ t E | x : = { x } × E | ϕ t ( x ) forall x ∈ X . Hence, the equivariance of ϕ t implies ϕ ∗ t E | τ ( x ) = { τ ( x ) } × E | τ ( ϕ t ( x )) . Then, the pullbackconstruction induces a Q -structure on ϕ ∗ t E by the map Θ ∗ t that acts anti -linearly between the fibers ϕ ∗ t E | x and ϕ ∗ t E | τ ( x ) as the product τ × Θ . (cid:4) LASSIFICATION OF “QUATERNIONIC”
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Theorem 2.12 is the starting point for a homotopy classification of Q -bundles. Complex and realvector bundles are classified by the set of homotopy equivalent maps from the base space to the com-plex or real Grassmann manifold, respectively [MS]. A similar result holds true also for “Real”-bundles provided that the Grassmann manifold is endowed with a suitable involution and the homo-topy equivalence is restricted to equivariant maps [Ed] (see also [DG1, Section 4.4]). In this sectionwe provide a similar result for Q -bundles of even rank.We recall that the Grassmann manifold is defined as G m ( C ∞ ) : = ∞ [ n = m G m ( C n ) , where, for each pair m n , G m ( C n ) ≃ U ( n ) / (cid:0) U ( m ) × U ( n − m ) (cid:1) is the set of m -dimensional (com-plex) subspaces of C n . The space G m ( C n ) can be endowed with the structure of a finite CW-complex,making it into a closed (i.e.compact without boundary) manifold of (real) dimension 2 m ( n − m ). Theinclusions C n ⊂ C n + ⊂ . . . yield inclusions G m ( C n ) ⊂ G m ( C n + ) ⊂ . . . and one can equip G m ( C ∞ )with the direct limit topology. The resulting space G m ( C ∞ ) has the structure of an infinite CW-complexwhich is, in particular, paracompact and path-connected. Following [LLM] or [BHH], in case of evendimension one can endow G m ( C ∞ ) with an involution of quaternionic-type in the following way: let Σ = h v , v , . . . v m − , v m i C be any 2 m -plane in G m ( C n ) generated by the basis { v , v , . . . v m − , v m } and define ρ ( Σ ) ∈ G m ( C n ) as the 2 m -plane spanned by h Q v , Q v , . . . , Q v m − , Q v m i C where v j is the complex conjugate of v j and Q is the 2 n × n matrix (2.1). Clearly, the definition of ρ ( Σ )does not depend on the choice of a particular basis and one can immediately check that the map ρ : G m ( C n ) → G m ( C n ) is an involution that makes the pair ( G m ( C n ) , ρ ) into an involutive space.Since all the inclusions G m ( C n ) ֒ → G m ( C n + ) ֒ → . . . are equivariant, the involution extends tothe infinite Grassmann manifold in such a way that ˆ G m ( C ∞ ) ≡ ( G m ( C ∞ ) , ̺ ) becomes an involu-tive space. Let Σ = ρ ( Σ ) be a fixed point of ˆ G m ( C ∞ ). Since ρ acts on vectors as a quaternionicstructure one has Σ ≃ H m (cf.Remark 2.1). More precisely, if Σ is ρ -invariant we can find a ba-sis of h v , v , . . . v m − , v m i C made by quaternionic pairs (v k − , v k : = Q v k − ), k = , . . . , m whichleads to Σ = h v , v , . . . v m − , v m − i H (one has v k = jv k − with respect to the left quaternionicmultiplication). Let G m ( H n ) ≃ S p ( n ) / (cid:0) S p ( m ) × S p ( n − m ) (cid:1) be the set of m -dimensional quater-nionic hyperplanes passing through the origin of H n . As for the complex case we can define the quatrnionic Grassmann manifold as the inductive limit G m ( H ∞ ) : = S ∞ n = m G m ( H n ). This space hasthe structure of an infinite CW-complex. In particular it is paracompact and path-connected. If Σ is a fixed point of ˆ G m ( C ∞ ) under the involution ̺ then the map G m ( H ∞ ) → ˆ G m ( C ∞ ) given by h v , . . . , v m i H
7→ h v , Q v , . . . , v m , Q v m i C is an embedding of G m ( H ∞ ) onto the fixed point set ofˆ G m ( C ∞ ). This shows that G m ( H ∞ ) ≃ ˆ G m ( C ∞ ) ρ .Each manifold G m ( C n ) is the base space of a canonical rank m complex vector bundle π : T nm → G m ( C n ) where the total space T nm consists of all pairs ( Σ , v) with Σ ∈ G m ( C n ) and v any vector in Σ and the bundle projection is π ( Σ , v) = Σ . Now, when n tends to infinity, the same constructionleads to the tautological m -plane bundle π : T ∞ m → G m ( C ∞ ). This vector bundle is the universalobject which classifies complex vector bundles in the sense that any rank m complex vector bundle E → X can be realized, up to isomorphisms, as the pullback of T ∞ m with respect to a classifying map ϕ : X → G m ( C ∞ ), that is E ≃ ϕ ∗ T ∞ m . Since pullbacks of homotopic maps yield isomorphic vectorbundles ( homotopy property ), the isomorphism class of E only depends on the homotopy class of ϕ .This leads to the fundamental result Vec m C ( X ) ≃ [ X , G m ( C ∞ )] where in the right-hand side there is theset of homotopy equivalence classes of maps between X and G m ( C ∞ ). This classical result can beextended to the category of even rank “Quaternionic” vector bundles provided that the total space T ∞ m is endowed with a Q -structure compatible with the involution ρ on the Grassmann manifold G m ( C ∞ ).This can be done by means of the anti -linear map Ξ : T ∞ m → T ∞ m defined by Ξ : ( Σ , v) (cid:0) ρ ( Σ ) , Q v (cid:1) .The relation π ◦ Ξ = ρ ◦ π can be easily verified, therefore ( T ∞ m , Ξ ) is a Q -bundle over the involutive space ˆ G m ( C ∞ ) ≡ ( G m ( C ∞ ) , ρ ). This is the universal object for the homotopy classification of Q -bundles: Theorem 2.13 (Homotopy classification) . Let ( X , τ ) be an involutive space and assume that X verifies(at least) condition 0 of Definition 1.1. Each rank m Q -bundle ( E , Θ ) over ( X , τ ) can be obtained,up to isomorphisms, as a pullback E ≃ ϕ ∗ T ∞ m with respect to a map ϕ : X → ˆ G m ( C ∞ ) which isequivariant, ϕ ◦ τ = ρ ◦ ϕ . Moreover, the homotopy property implies that ( E , Θ ) depends only on theequivariant homotopy class of ϕ , i.e. one has the isomorphism Vec m Q ( X , τ ) ≃ [ X , ˆ G m ( C ∞ )] Z . sketch of. The proof of this theorem is a direct equivariant adaption of standard arguments for thecomplex case (cf.[MS, Hu]) and it is not new in the literature (e.g.[LLM, Theorem 11.2] or [BHH,Section 4.2]). Just for sake of completeness let us comment that Theorem 2.12 assures that eachequivariant pullback of the tautological vector bundle T ∞ m provides a Q -bundle and that homotopicequivalent pullbacks provide isomorphic Q -bundles. Therefore, one has only to show that each Q -bundle can be realized via an equivariant pullback. The easiest way to prove this point is to adaptequivariantly the construction in [Hu, Chapter 3, Section 5]. (cid:4) Remark 2.14.
If one consider the trivial involutive space ( X , Id X ), then equivariant maps ϕ : X → ˆ G m ( C ∞ ) are characterized by ϕ ( x ) = ρ ( ϕ ( x )) for all x ∈ X . Since the fixed point set of G m ( C ∞ )is parameterized by G m ( H ∞ ) one has that [ X , ˆ G m ( C ∞ )] Z ≃ [ X , G m ( H ∞ )]. However, G m ( H ∞ ) is theclassifying space for quaternionic vector bundles [Hu, Chapter 8, Theorem 6.1] hence Vec m Q ( X , Id X ) ≃ Vec m H ( X ) in agreement with Proposition 2.4. ◭ The homotopy classification provided by Theorem 2.13 can be directly used to classify Q -bundles overthe involutive space ˜ S ≡ ( S , τ ). Proposition 2.15.
Vec m Q ( S , τ ) = . Proof.
In view of Theorem 2.13 it is enough to show that [ ˜ S , ˆ G m ( C ∞ )] Z reduces to the equivarianthomotopy class of the constant map. This fact is a consequence of a general result called Z -homotopyreduction [DG1, Lemma 4.26] which is based on the Z -skeleton decomposition of ˜ S . The involutivespace ˜ S has two fixed cells of dimension 0 and one free cell of dimension 1 (cf.[DG1, Example4.20]). Moreover, π ( G m ( C ∞ )) ≃ π ( G m ( C ∞ )) ≃ G m ( C ∞ ) ρ ≃ G m ( H ∞ ) alsoimplies π ( G m ( C ∞ ) ρ ) ≃
0. These data are su ffi cient for the application of the Z -homotopy reductionLemma. (cid:4) Stable range condition.
In topology, spaces which are homotopy equivalent to CW-complexesare very important. A similar notion can be extended to Z -spaces like ˜ S d ≡ ( S d , τ ) or ˜ T d ≡ ( T d , τ ).These spaces have the structure of a CW-complex with respect to a skeleton decomposition made bycells of various dimension that carry a Z -action. Such Z -cells can be only of two types: They are fixed if the action of Z is trivial or are free if they have no fixed points. We refer to [DG1, Section4.5] for a precise definition of the notion of Z -CW-complex (see also [Ma, AP]). Moreover, the Z -CW-complex structure of ˜ S d and ˜ T d is explicitly described in [DG1, Example 4.20 and Example4.20], respectively. We point out that this construction is modelled after the usual definition of CW-complex, replacing the “point” by “ Z -point”. For this reason (almost) all topological and homologicalproperties of CW-complexes have their “natural” counterparts in the equivariant setting.This notion of Z -CW-complex plays an important role in the proof of the next result which is theequivariant generalization of [Hu, Chapter 2, Theorem 7.1] to the category of even rank Q -bundles. Proposition 2.16 (existence of a global Q -pair of section) . Let ( X , τ ) be an involutive space such thatX has a finite Z -CW-complex decomposition with fixed cells only in dimension 0. Let us denote withd the dimension of X ( i.e. the maximal dimension of the cells in the decomposition of X). Let ( E , Θ ) be a Q -vector bundle over ( X , τ ) with fiber of rank m. If d m − there exists a pair of sections ( s , s ) ∈ Γ ( E ) which is a global Q -pair in the sense of Definition 2.6. LASSIFICATION OF “QUATERNIONIC”
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Proof.
The zero section s ( x ) = ∈ E x for all x ∈ X is Θ -invariant since Θ is an anti-linear isomor-phism in each fiber. Let E × ⊂ E be the subbundle of nonzero vectors. The fibers E × x are all isomorphicto ( C m ) × : = C m \ { } which is a 2(2 m − m −
1) homotopy groupsvanish identically). The anti-involution Θ endows E × with a Q -structure over ( X , τ ). A Q -pair ofsections ( s , s : = τ Θ ( s )) of E × can be seen as an everywhere-nonzero (i.e.global) Q -pair for the Q -vector bundle ( E , Θ ). We will show that if d m − Q -pair of sections of E × always exists.We prove the claim by induction on the dimension of the skeleton. This is the case for X which is afinite collection of fixed points { x j } j = ,..., N and conjugated pairs { x j , τ ( x j ) } j = ,..., ˜ N . In this case a global Q -pair can be defined as in the proof of Proposition 2.9. On the fixed points x j = τ ( x j ) one sets apair of vectors ( s ′ ( x j ) , s ′ ( x j )) ⊂ E × x , with s ′ ( x j ) : = τ Θ ( s ′ )( x j ). This pair is automatically independent.For each free pairs { x j , τ ( x j ) } one starts with a section s ′ : = ( s ′ ( x j ) , s ′ ( τ ( x j ))) ∈ E × x × E × τ ( x ) and, asusual, one defines s ′ : = τ Θ ( s ′ ). In this way ( s ′ , s ′ ) is a Q -pair of sections for E × x ∪ E × τ ( x ) . Assume nowthat the claim is true for the Z -CW-subcomplex X j − of dimension j − j d . By theinductive hypothesis we have a Q -pair of sections ( s ′ , s ′ ) of the restricted bundle E × | X j − . Let Y ⊂ X be a free j -cell of X with equivariant attaching map φ : Z × D j → Y ⊂ X . The pullback bundle φ ∗ ( E × ) → Z × D j has a Q -structure since φ is equivariant and it is locally Q -trivial. The Q -pair( s ′ , s ′ ) defines a Q -pair ( σ ′ , σ ′ ) on φ ∗ ( E × ) | Z × ∂ D j by σ ′ j : = s ′ j ◦ φ .Since D j is contractible we know from Theorem 2.12 that φ ∗ ( E × ) | Z × D j is Q -isomorphic to ( Z × D j ) × ( C m ) × (endowed with the standard trivial anti-involution Θ , even if the particular form of theinvolution is not important for the rest of the proof). Then, the Q -pair ( σ ′ , σ ′ ) defined on Z × ∂ D j canbe identified with a pair of independent equivariant maps Z × ∂ D j → ( C m ) × . Because j − ≤ m − π j − (( C m ) × ) ≃
0, the restriction of the map induced by σ ′ to { } × ∂ D j → ( C m ) × prolongs to amap { } × D j → ( C m ) × . At this point σ ′ can be seen as a map Z × ∂ D j → ( C m / h σ ′ i C ) × ≃ ( C m − ) × .Since j − ≤ m −
2) and π j − (( C m − ) × ) ≃
0, also the restriction of the map induced by σ ′ to { } × ∂ D j → ( C m ) × prolongs to a map { } × D j → ( C m ) × . Moreover, these two prolonged mapsare independent by construction. Using the equivariant constraints σ ′ ( − , x ) : = ( Θ σ )(1 , x ) and σ ′ ( − , x ) : = − ( Θ σ )(1 , x ) one obtains a Q -pair of maps Z × D j → ( C m ) × . This prolonged mapsyields a Q -pair of sections ( σ , σ ) of φ ∗ ( E × ). Using the natural morphism ˆ φ : φ ∗ ( E × ) → E × over φ (defined by the pullback construction) we have a unique Q -pair of sections ( s Y , s Y ) of E × | Y defined byˆ φ ◦ σ j = s Yj ◦ φ such that s Yj = s ′ j on X j − ∩ Y . Now, one defines a global Q -pair ( s , s ) of E × | X j − ∪ Y by the requirements that s j | X j − ≡ s ′ j and s j | Y ≡ s Yj for the free j -cell Y . By the weak topology propertyof CW-complex, the s j ’s are also continuous. This argument applies to every other free j -cell, and theclaim is true on X j and eventually on X d = X . (cid:4) Remark 2.17.
In principle the condition that the involutive space ( X , τ ) must have fixed cells onlyin dimension 0 should be removed since fibers over fixed points x = τ ( x ) are isomorphic to H m (cf.Proposition 2.4) and E × x ≃ H m \ { } is 2(2 m − ◭ The next theorem provides the stable range decomposition for “Quaterninic” vector bundles.
Theorem 2.18 (Stable range) . Let ( X , τ ) be an involutive space such that X has a finite Z -CW-complexdecomposition of dimension d with fixed cells only in dimension 0. Each rank m Q -vector bundle ( E , Θ ) over ( X , τ ) such that d m − splits as E ≃ E ⊕ ( X × C m − σ ) ) (2.3) where E is a Q -vector bundle over ( X , τ ) , X × C m − σ ) → X is the trivial product Q -bundle over ( X , τ ) and σ : = [ d + ] (here [ x ] denotes the integer part of x ∈ R ).Proof. By Proposition 2.16 there is a global Q -pair of sections ( s , s ) ⊂ Γ ( E ). This sections deter-mines a monomorphism f : X × C → E given by f ( x , ( a , a )) : = a s ( x ) + a s ( x ). This monomor-phism is equivariant, i.e. f ( τ ( x ) , ( − a , a )) = − a s ( τ ( x )) + a s ( τ ( x )) = Θ ( a s ( x ) + a s ( x )). Let E ′ be the cokernel of f in E , namely E ′ is the quotient of E by the relation: p ∼ p ′ if p and p ′ are in thesame fiber of E and if p − p ′ ∈ Im( f ). The map E ′ → X is a vector bundle of rank 2( m −
1) (cf.[Hu,Chapter 3, Corollary 8.3]) which inherits a Q -structure from E and the equivariance of f . Since X is compact, by [Hu, Chapter 3, Theorem 9.6] there is an isomorphism of Q -bundles between E and E ′ ⊕ ( X × C ). If d m − − E ′ and by iterating this procedureone gets eventually (2.3). (cid:4) Corollary 2.19.
Let ( X , τ ) be an involutive space such that X has a finite Z -CW-complex decomposi-tion of dimension d with fixed cells only in dimension 0. Then Vec m Q ( X , τ ) ≃ Vec Q ( X , τ ) ∀ m ∈ N .
3. T he FKMM- invariant
In an unpublished work M. Furuta, Y. Kametani, H. Matsue, and N. Minami proposed a topologicalinvariant capable to classify “Quaternionic” vector bundles over certain involutive spaces, providedthat certain conditions are met [FKMM]. Interestingly, this object was originally introduced to classify Q -bundles on ˜ T . We present in this section a more general and slightly di ff erent definition for thisinvariant recognizing, of course, that the original ideas contained in [FKMM] has been of inspirationto us.3.1. A short reminder of the equivariant Borel cohomology.
The proper cohomology theory forthe analysis of vector bundles in the category of spaces with involution is the equivariant cohomolgy introduced by A. Borel in [Bo]. This cohomology plays an important role for the classification of“Real” vector bundles [DG1] and we will show that it is also relevant for the study of “Quaternionic”vector bundles. A short self-consistent summary of this cohomology theory can be found in [DG1,Section 5.1]. For an introduction to the subject we refer to [Hs, Chapter 3] and [AP, Chapter 1].Since we need this tool we briefly recall the main steps of the Borel construction for the equivariantcohomology. The homotopy quotient of an involutive space ( X , τ ) is the orbit space X ∼ τ : = X × ˆ S ∞ / ( τ × ϑ ) . Here ϑ is the antipodal map on the infinite sphere S ∞ (cf.[DG1, Example 4.1]) and ˆ S ∞ is used for thepair ( S ∞ , ϑ ). The product space X × S ∞ (forgetting for a moment the Z -action) has the same homotopytype of X since S ∞ is contractible. Moreover, since ϑ is a free involution, also the composed involution τ × ϑ is free, independently of τ . Let R be any commutative ring (e.g., R , Z , Z , . . . ). The equivariantcohomology ring of ( X , τ ) with coe ffi cients in R is defined as H • Z ( X , R ) : = H • ( X ∼ τ , R ) . More precisely, each equivariant cohomology group H j Z ( X , R ) is given by the singular cohomologygroup H j ( X ∼ τ , R ) of the homotopy quotient X ∼ τ with coe ffi cients in R and the ring structure is given,as usual, by the cup product. As the coe ffi cients of the usual singular cohomology are generalizedto local coe ffi cients (see e.g.[Hat, Section 3.H] or [DK, Section 5]), the coe ffi cients of the Borel’sequivariant cohomology are also generalized to local coe ffi cients. Given an involutive space ( X , τ ) letus consider the fundamental group π ( X ∼ τ ) and the associated group ring Z [ π ( X ∼ τ )]. Each module Z over the group Z [ π ( X ∼ τ )] is, by definition, a local system on X ∼ τ . Using this local system one defines,as usual, the equivariant cohomology with local coe ffi cients in Z : H • Z ( X , Z ) : = H • ( X ∼ τ , Z ) . We are particularly interested in modules Z whose underlying groups are identifiable with Z . For eachinvolutive space ( X , τ ), there always exists a particular family of local systems Z ( m ) labelled by m ∈ Z .Here Z ( m ) ≃ X × Z denotes the Z -equivariant local system on ( X , τ ) made equivariant by the Z -action( x , l ) ( τ ( x ) , ( − m l ). Because the module structure depends only on the parity of m , we consideronly the Z -modules Z (0) and Z (1). Since Z (0) corresponds to the case of the trivial action of π ( X ∼ τ )on Z one has H k Z ( X , Z (0)) ≃ H k Z ( X , Z ) [DK, Section 5.2]. LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 17
Let us recall two important group isomorphisms involving the first two equivariant cohomologygroups. Let ( X , τ ) be an involutive space, then H Z ( X , Z (1)) ≃ [ X , U (1)] Z , H Z ( X , Z (1)) ≃ Vec R ( X , τ ) . (3.1)The first isomorphism [Go, Proposition A.2] says that the first equivariant cohomology group is iso-morphic to the set of Z -homotopy classes of equivariant maps ϕ : X → U (1) where the involution on U (1) is induced by the complex conjugation, i.e. ϕ ( τ ( x )) = ϕ ( x ). The second isomorphism is due toB. Kahn [Kah] and expresses the equivalence between the Picard group of “Real” line bundles (in thesense of [At1, DG1]) over ( X , τ ) and the second equivariant cohomology group of this space. A moremodern proof of this result can be found in [Go, Corollary A.5].The fixed point subset X τ ⊂ X is closed and τ -invariant and the inclusion ı : X τ ֒ → X extends to aninclusion ı : X τ ∼ τ ֒ → X ∼ τ of the respective homotopy quotients. The relative equivariant cohomologycan be defined as usual by the identification H • Z ( X | X τ , Z ) : = H • ( X ∼ τ | X τ ∼ τ , Z )and one has a related long exact sequence in cohomology . . . −→ H j Z ( X | X τ , Z ) −→ H j Z ( X , Z ) r −→ H j Z ( X τ , Z ) −→ H j + Z ( X | X τ , Z ) −→ . . . where the map r : = ı ∗ restricts cochains on X to cochains on X τ . The j -th cokernel of r is by definitionCoker j ( X | X τ , Z ) : = H j Z ( X τ , Z ) / r (cid:0) H j Z ( X , Z ) (cid:1) . Lemma 3.1.
Let ( X , τ ) be an involutive space such that X τ , ∅ . Then [ X τ , U (1)] Z / [ X , U (1)] Z ≃ Coker ( X | X τ , Z (1)) (3.2) where the group action of [ f ] ∈ [ X , U (1)] Z on [ g ] ∈ [ X τ , U (1)] Z is given by multiplication andrestriction, namely [ f ] : [ g ] [ f | X τ g ] . Moreover, if H Z ( X , Z (1)) = then [ X τ , U (1)] Z / [ X , U (1)] Z ≃ H Z ( X | X τ , Z (1)) . sketch of. The first isomorphism (3.2) is a direct consequence of the isomorphism (3.1) proved in [Go,Proposition A.2]. Under the extra assumption H Z ( X , Z (1)) = H Z ( X , Z (1)) r −→ H Z ( X τ , Z (1)) s −→ H Z ( X | X τ , Z (1)) −→ . Since, Ker( s ) = Im( r ) and Im( s ) = H Z ( X | X τ , Z (1)) one deduces from the homomorphism theoremIm( s ) ≃ H Z ( X τ , Z (1)) / Ker( s )the required isomorphism Coker ( X | X τ , Z (1)) ≃ H Z ( X | X τ , Z (1)). (cid:4) In many situations of interest the cokernel in (3.2) has a very simple form. In the cases of theTR-spheres ˜ S d described in Definition 1.4 one has thatCoker (cid:0) ˜ S d | ( ˜ S d ) τ , Z (1) (cid:1) ≃ H Z (cid:0) ˜ S d | ( ˜ S d ) τ , Z (1) (cid:1) ≃ Z ∀ d > . (3.3)The first isomorphism is a consequence of H Z ( ˜ S d , Z (1)) = T d the evaluation of the cokernel dependson the dimension according to the formulaCoker (cid:0) ˜ T d | ( ˜ T d ) τ , Z (1) (cid:1) ≃ H Z (cid:0) ˜ T d | ( ˜ T d ) τ , Z (1) (cid:1) ≃ Z d − ( d + ∀ d > Z ≡ { } . Again the first isomorphism follows from H Z ( ˜ T d , Z (1)) = The determinant construction.
In order to define the FKMM-invariant we need the notion of determinat line bundle associated with a (complex) vector bundle. Let V a complex vector space ofdimension n . The determinant of V is by definition det( V ) : = V n V where the symbol V n denotesthe top exterior power of V (i.e.the skew-symmetrized n -th tensor power of V ). This is a complexvector space of dimension one. If W is a second vector space of same dimension n and T : V → W isa linear map then there is a naturally associated map det( T ) : det( V ) → det( W ) which in the specialcase V = W coincides with the multiplication by the determinant of the endomorphism T . Thisdeterminant construction is a functor from the category of vector spaces to itself and by a standardargument [Hu, Chapter 5, Section 6] it induces a functor on the category of complex vector bundlesover an arbitrary space X . Given a rank n complex vector bundle E → X , one defined the associateddeterminant line bundle det( E ) → X as the rank 1 complex vector bundle with fiber descriptiondet( E ) x = det( E x ) x ∈ X . (3.5)If { s , . . . , s n } is a local trivializing frame for E over the open set U ⊂ X then det( E ) is trivialized overthe same open set U by the section s ∧ . . . ∧ s n . For each map ϕ : X → Y one has the isomorphismdet( ϕ ∗ ( E )) ≃ ϕ ∗ (det( E )) which is a special case of the compatibility between pullback and tensorproduct operations. Finally, if E = E ⊕ E in the sense of the Whitney sum then det( E ) = det( E ) ⊗ det( E ).If ( E , Θ ) is a rank n Q -bundle over the involutive space ( X , τ ) then the associated determinant linebundle det( E ) inherits an involutive structure given by the map det( Θ ) which acts anti -linearly betweenthe fibers det( E ) x and det( E ) τ ( x ) according to det( Θ )( p ∧ . . . ∧ p n ) = Θ ( p ) ∧ . . . ∧ Θ ( p n ). Clearlydet( Θ ) is a fiber preserving map which coincides with the multiplication by ( − n . Hence: Lemma 3.2.
Let ( E , Θ ) be a rank n Q -bundle over ( X , τ ) and (det( E ) , det( Θ )) the associated determi-nant line bundle endowed with the involutive structure det( Θ ) .(i) If n = m then (det( E ) , det( Θ )) is a “Real” line bundle over ( X , τ ) ;(ii) If n = m + then (det( E ) , det( Θ )) is a “Quaternionic” line bundle over ( X , τ ) . We recall once more that the adjective “Real” is used in the sense of [At1, DG1].
Remark 3.3 (Metric, line bundle, circle bundle) . Let ( E , Θ ) be a Q -bundle over ( X , τ ) of even degree.According to Assumption 2.11, E carries an equivariant Hermitian metric m that fixes a unique Her-mitian metric m det on det( E ) which is equivariant with respect to the induced R -structure det( Θ ). Moreexplicitly, if ( p i , q i ) ∈ E | x × E | x , i = , . . . , m then, m det ( p ∧ . . . ∧ p m , q ∧ . . . ∧ q m ) : = m Y i = m ( p i , q i ) . The R -line bundle (det( E ) , det( Θ )) endowed with the equivariant Hermitian metric m det is R -trivial ifand only if there exists an isometric R -isomorphism with X × C , or equivalently, if and only if thereexists a global R -section s : X → det( E ) of unit length (cf.[DG1, Theorem 4.8]). Let us introduce the circle bundle S (det( E )) : = { p ∈ det( E ) | m det ( p , p ) = } . Then, the R -triviality of det( E ) is equivalentto the existence of an R -section for the circle bundle S (det( E )) → X . ◭ Corollary 3.4.
Let ( E , Θ ) be a rank n Q -bundle over ( X , τ ) and (det( E ) , det( Θ )) the associated deter-minant line bundle endowed with the involutive structure det( Θ ) . If X τ , ∅ and H Z ( X , Z (1)) = thenthere exists a global trivializing map h det : det( E ) −→ X × C which is equivariant in the sense that h det ◦ det( Θ ) = Υ ◦ h det where Υ is the standard R -structure onthe product bundle X × C defined by Υ ( x , v) : = ( τ ( x ) , v) .Proof. Since X τ , ∅ Proposition 2.2 assures that ( E , Θ ) has even rank and Lemma 3.2 implies that(det( E ) , det( Θ )) is an R -line bundle. Therefore, the condition H Z ( X , Z (1)) = R -trivialityof (det( E ) , det( Θ )) as showed by (3.1). This means the existence of a global R -trivialization h det . (cid:4) LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 19
The restriction of the map h det to X τ provides a trivialization for the restricted line bundle det( E ) | X τ and, as a consequence of the fiber description (3.5), one has h det | X τ : det( E | X τ ) −→ X τ × C . (3.6) Lemma 3.5.
Let ( E , Θ ) be a Q -bundle over a space X with trivial involution τ = Id X . Then, the associ-ated determinant line bundle det( E ) is a trivial R -bundle which admits a unique canonical trivializingmap det X : det( E ) −→ X × C (3.7) such that det X ◦ det( Θ ) = Υ ◦ det X . Moreover, the map det X can be chosen to be metric-preservingand it leads to a unique canonical R -section s X : X → S (det( E )) defined bys X ( x ) : = det − X ( x , , ∀ x ∈ X . Proof.
Since X has trivial involution E has even rank (Proposition 2.2) and (det( E ) , det( Θ )) is an R -line bundle (Lemma 3.2). Let { U α } be a cover of X associated with a system of local trivializations h α : E | U α → U α × C m such that h α ◦ Θ = Θ ◦ h α where Θ ( x , v) = ( x , Q v) is the standard trivial Q -structure on the product bundle U α × C m (the matrix Q is defined in (2.1)). Such a trivializationexists in view of Proposition 2.9. Associated with each h α there is a local trivialization det( h α ) :det( E ) | U α → U α × C for the restricted determinant line bundle det( E | U α ) = det( E ) | U α . The map h α isusually not unique: a choice for h α is equivalent to a choice of a global Q -frame (cf.Definition 2.6) { s ( α )1 , s ( α )2 , s ( α )3 , s ( α )4 . . . , s ( α )2 m − , s ( α )2 m } ⊂ Γ ( E | U α ) such that s ( α ) j ( x ) : = h − α ( x , e j ). Di ff erent Q -frames leadto di ff erent trivializations h α and each pair of Q -frames is related by a gauge transformation f α . Ifone assumes that E is endowed with an invariant Hermitian metric (cf.Proposition2.10) each Q -framecan be chosen to be orthonormal and so f α : U α → U (2 m ). The Q -structure and the fact that X hastrivial involution imply that f α ( x ) = − Q f α ( x ) Q for all x ∈ U α , namely f α : U α → U (2 m ) µ ⊂ S U (2 m )(cf.Remark 2.1). The trivialization det( h α ) is uniquely specified by the “Real” section s ( α ) : = s ( α )1 ∧ s ( α )2 ∧ . . . ∧ s ( α )2 m − ∧ s ( α )2 m . If h α and h ′ α are two di ff erent trivializations for E | U α related by the gaugetransformation f α then det( h α ) and det( h ′ α ) are related by det( f α ) =
1, namely det( h α ) = det( h ′ α ). Inthis sense the local trivializations of det( E ) are canonical and we write det U α : det( E ) | U α → U α × C for the unique trivializing map.The topology of the vector bundle E is uniquely determined by the set of transition functions g αβ : U α ∩ U β → U (2 m ) associated with the maps h α ◦ h − β (which are well defined on U α ∩ U β ). Theequivariance of the maps h α implies g αβ : U α ∩ U β → U (2 m ) µ . The determinant line bundle det( E ) iscompletely specified by the set of transition functions det( g αβ ) : U α ∩ U β → U (1). Since det( g αβ ) = α and β the local canonical trivializations det U α glue together to give rise to the unique globalcanonical trivialization (3.7). (cid:4) If X τ , ∅ the restricted vector bundle E | X τ → X τ can be seen as a Q -bundle over a space with trivialinvolution and Lemma 3.5 provides the canonical trivializationdet X τ : det( E | X τ ) −→ X τ × C (3.8)for the restricted determinant line bundle det( E | X τ ) which is “a priori” di ff erent from (3.6).3.3. Construction of the FKMM-invariant.
In this section we construct the
FKMM-invariant asso-ciated to a “Quaternionic” vector bundle ( E , Θ ) over an involutive space ( X , τ ). Although a more gen-eral approach is possible [DG2] (cf.also Remark 3.10) we decided, for pedagogical reasons, to presenthere a construction which is specific for a particular (albeit su ffi ciently large) class of Q -bundles. Definition 3.6 ( Q -bundles of FKMM-type) . A Q -bundle ( E , Θ ) over the involutive space ( X , τ ) isof FKMM-type if X τ , ∅ and if the associated “Real” determinant line bundle (det( E ) , det( Θ )) is R -trivial. The property to be of FKMM-type is an isomorphism invariant and we use the notation Vec m FKMM ( X , τ ) ⊆ Vec m Q ( X , τ ) for the set of equivalence classes of FKMM Q -bundles of rank m. For certain involutive spaces ( X , τ ) all possible Q -bundles are of FKMM-type. Proposition 3.7.
Let ( X , τ ) be an FKMM-space in the sense of Definition 1.1. Then Vec m FKMM ( X , τ ) = Vec m Q ( X , τ ) ∀ m ∈ N . Proof.
Since X τ , ∅ the admissible Q -bundles have even rank (cf.Proposition 2.2). This implies thatthe associated determinant line bundles carry an R -structure (cf.Lemma 3.2). Finally, the condition H Z ( X , Z (1)) = R -triviality of each R -line bundleover ( X , τ ). (cid:4) Let ( E , Θ ) be a Q -bundle of FKMM-type over the involutive space ( X , τ ) and consider the restricteddeterminant line bundle det( E | X τ ) → X τ . This line bundle is R -trivial and according to Lemma 3.5 itadmits a canonical trivialization (3.8). On the other hand, the full determinant line bundle det( E ) → X is R -trivial by assumption and so (as in Corollary 3.4) there exists a global trivialization h det : det( E ) → X × C which restricts to a trivialization for det( E | X τ ) as in (3.6). If one fixes an equivariant Hermitianmetric on E the maps det X τ and h det | X τ can be chosen to be isometries with respect to the standardHermitian metric on the product bundle. This implies that the di ff erence h det | X τ ◦ det − X τ : X τ × C −→ X τ × C identifies a map ω E : X τ −→ U (1)such that ( h det | X τ ◦ det − X τ )( x , λ ) = ( x , ω E ( x ) λ ) for all ( x , λ ) ∈ X τ × C . The equivariance property( h det | X τ ◦ det − X τ ) ◦ Υ = Υ ◦ ( h det | X τ ◦ det − X τ ) implies that ω E is equivariant with respect to the involutionon U (1) given by the complex conjugation, i.e. ω E ( τ ( x )) = ω E ( x ). Since ω E is defined on the fixedpoint set X τ and the invariant subset of U (1) is {− , + } one has that ω E : X τ −→ {− , + } ≃ Z , (3.9)namely ω E ∈ Map( X τ , Z ) ≃ [ X τ , U (1)] Z . Considering that the canonical trivialization det X τ isunique, the construction of ω E only depends on the choice of h det . This freedom is equivalent tothe choice of a global equivariant gauge transform f : X → U (1) that a ff ects ω E by multiplication andrestriction (as in Lemma 3.1). Moreover, only the homotopy class [ f ] ∈ [ X , U (1)] Z is relevant. Definition 3.8 (FKMM-invariant, [FKMM]) . To each Q -bundle ( E , Θ ) of FKMM-type is associatedthe class κ ( E ) : = [ ω E ] ∈ [ X τ , U (1)] Z / [ X , U (1)] Z . We say that κ ( E ) is the FKMM-invariant of the Q -bundle ( E , Θ ) . Remark 3.9.
The FKMM-invariant can be introduced also from the point of view of sections. Sincedet( E ) is R -trivial by assumption, the set of R -sections of the circle bundle S (det( E )) is non-empty andany two of such R -sections are related by the multiplication by a Z -equivariant map u : X → U (1).Each R -section t : X → S (det( E )) restricts to a section t | X τ of S (det( E | X τ )). We can compare thissection with the unique canonical section s X τ constructed in Lemma 3.5. The di ff erence between them t | X τ = ω E · s X τ , is specified by a Z -equivariant map ω E : X τ → U (1) that coincides with the oneintroduced in Definition 3.8. This equivalent description helps us to understand the meaning of theFKMM-invariant. Indeed, each equivariant section of the “Real” circle bundle S (det( E )) defines, byrestriction, an equivariant section over X τ . On the other hand, according to the topology of X τ (forinstance when there are several disconnected components) there may exist equivariant sections over X τ that are not obtainable as the restriction of global equivariant sections over X . In a certain sense, itis exactly this redundancy which is measured by the FKMM-invariant. ◭ Remark 3.10 (A more general definition of the FKMM-invariant) . The-FKMM invariant can be for-mulated in a more general setting. The key observation is that the cohomology group H Z ( X | X τ , Z (1))can be realized as isomorphism classes of pairs consisting of a “Real” line bundles on ( X , τ ) and a LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 21 nowhere vanishing section on X τ . Then, in view of Lemma 3.5, a generalized version of the FKMM-invariant can be defined for every Q -bundle ( E , Θ ) on every involutive space ( X , τ ) as the element in H Z ( X | X τ , Z (1)) represented by the pair (det( E ) , s X τ ). The details of this construction will be given ina future work [DG3]. ◭ The main properties of the FKMM-invariant are listed in the following theorem.
Theorem 3.11.
The FKMM-invariant is well defined in the sense that if ( X , τ ) is an involutive spacesuch that X τ , ∅ and ( E , Θ ) and ( E , Θ ) are two isomorphic Q -bundles of FKMM-type over ( X , τ ) then κ ( E ) = κ ( E ) . Moreover:(i) The FKMM-invariant is natural , meaning that if ( E , Θ ) is a Q -bundle of FKMM-type over ( X , τ X ) and f : ( Y , τ Y ) → ( X , τ X ) is an equivariant map such that f ( Y ) ∩ X τ X , ∅ then κ ( f ∗ ( E )) = f ∗ ( κ ( E )) ;(ii) If ( E , Θ ) is Q -trivial then κ ( E ) = + ;(iii) If ( E , Θ ) and ( E , Θ ) are two Q -bundles of FKMM-type over ( X , τ ) then κ ( E ⊕ E ) = κ ( E ) · κ ( E ) . Finally, if ( X , τ ) is a FKMM-space then(iv) κ ( E ) ∈ H Z ( X | X τ , Z (1)) . Proof.
If ( E , Θ ) ≃ ( E , Θ ) as Q -bundles then the argument of the proof of Lemma 3.5 shows thatthe two canonical trivializations for det( E | X τ ) and det( E | X τ ) coincide (up to a suitable identification).Then ω E and ω E may di ff er only for the multiplication by a gauge transformation [ f ] ∈ [ X , U (1)] Z which connects the two global trivializations of the isomorphic R -line bundles det( E ) ≃ det( E ). Thisimplies κ ( E ) = κ ( E ) by construction.(i) The condition f ( Y ) ∩ X τ X , ∅ implies Y τ Y , ∅ . Moreover, the global R -trivialization h det :det( E ) → X × C induces the global R -trivialization f ∗ ( h det ) : det( f ∗ ( E )) → Y × C for det( f ∗ ( E )) ≃ f ∗ (det( E )). Then, also f ∗ ( E ) is a Q -bundle of FKMM-type and κ ( f ∗ ( E )) is well defined. The relations f ∗ ( h det ) = h det ◦ det( ˆ f ) and det Y τ Y = det X τ X ◦ det( ˆ f ) imply κ ( f ∗ ( E )) = f ∗ ( κ ( E )) (here ˆ f denotes thecanonical morphism between total spaces induced by f ).(ii) In this case ω E is the constant map on X τ with value + E ⊕ E ) ≃ det( E ) ⊗ det( E ).(iv) This is a consequence of Lemma 3.1. (cid:4)
4. C lassification in dimension d m Q ( S d , τ ) and Vec m Q ( T d , τ ) of isomorphism classesof “Quaternionic” vector bundles over the involutive spaces described in Definition 1.4 in the lowdimensional regime d
3. Since we already know the classification for d = d = ,
3. The case d = T d and ˜ S d . These are exactly theFKMM-spaces of Definition 1.1. When ( X , τ ) is a FKMM-space of “low dimension”, namely with free Z -cells of dimension not exceeding d =
3, Corollary 2.19 and Theorem 3.11 imply that the followingfacts hold true: • Vec m Q ( X , τ ) ≃ Vec Q ( X , τ ) for all m ∈ N ; • The FKMM-invariant associated with an element [ E ] ∈ Vec m Q ( X , τ ) coincides with the FKMM-invariant associated with its non-trivial part [ E ′ ] ∈ Vec Q ( X , τ ); • The FKMM-invariant provides a map κ : Vec Q ( X , τ ) → H Z ( X | X τ , Z (1)).The main result we want to prove is that the map κ is in e ff ect an isomorphism in many cases of interestlike ˜ T d or ˜ S d . Injectivity of the FKMM-invariant.
The first step of our analysis is to show that the map κ : Vec m Q ( X , τ ) −→ H Z ( X | X τ , Z (1)) (4.1)is injective if ( X , τ ) is an FKMM-space. We start with an important (and quite general) technical result. Lemma 4.1 ([FKMM]) . Let ( E , Θ ) and ( E , Θ ) be rank 2 “Quaternionic” vector bundles of FKMM-type over the involutive space ( X , τ ) . Assume also the equality κ ( E ) = κ ( E ) of the respective FKMM-invariants and the existence of a Q -isomorphism F : E | X τ → E | X τ . Then:(i) There exists an R -isomorphism f : det( E ) → det( E ) such that f | X τ = det( F ) ;(ii) The set SU ( E , E , f ) : = G x ∈ X ( G x : E | x → E | x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) vector bundle isomorphismsuch that det( G x ) = f | x ) defines a locally trivial fiber bundle over X with typical fiber S U (2) ;(iii) There is a natural involution on SU ( E , E , f ) covering τ . Moreover, on the fixed points x ∈ X τ this involution is identifiable with the involutive map µ : U
7→ −
QUQ on S U (2) defined inRemark 2.1;(iv) The set of equivariant sections of SU ( E , E , f ) is in bijection with the set of Q -isomorphismsG : E → E such that det( G ) = f .Proof. (i) By Lemma 3.5 we know that there exist two unique canonical R -sections s ( j ) X τ : X τ → det( E j ) | X τ , j = , F ) ◦ s (1) X τ = s (2) X τ . Accordingto Remark 3.9, the FKMM-invariant κ ( E j ) is representable as a Z -equivariant map ω E j : X τ → U (1)such that ω E j · s ( j ) X τ = t ( j ) | X τ where t ( j ) : X → det( E j ) is any arbitrarily chosen global R -section. Theassumption κ ( E ) = κ ( E ) implies the existence of a Z -equivariant map u : X → U (1) such that ω E = u | X τ · ω E . Moreover, since the R -line bundles det( E j ) → X are both trivial there is an R -isomorphism g : det( E ) → det( E ) such that g ◦ t (1) = t (2) . Then, by combining these facts onededuces the existence of an R -isomorphism f : det( E ) → det( E ) given by f ( p ) = u ( x ) g ( p ) for all p ∈ det( E ) | x such that f | X τ ◦ s (1) X τ = s (2) X τ . The uniqueness of the canonical sections implies f | X τ = det( F ).(ii) Let Hom( E , E ) → X be the homomorphism vector bundle (cf.[Hu, Chapter 5, Section 6]).The inclusion SU ( E , E , f ) ⊂ Hom( E , E ) provides a topology for SU ( E , E , f ). Moreover, if U isa trivializing neighborhood such that Hom( E , E ) | U ≃ U × U (2) (we are assuming that a Hermitianmetric has been fixed), then SU ( E , E , f ) | U ≃ U × S U (2) due to the requirement det G = f in thedefinition of SU ( E , E , f ).(iii) A natural involution ˆ Θ on SU ( E , E , f ) which relates τ -conjugated fibers is given by the col-lection of maps SU ( E , E , f ) | x ∋ G x
7→ − Θ | x ◦ G x ◦ Θ | τ ( x ) ∈ SU ( E , E , f ) | τ ( x ) .(iv) Follows from (ii) and (iii). (cid:4) Though SU ( E , E , f ) → X is a fiber bundle with typical fiber S U (2), it is not a principal S U (2)-bundle.This object plays an essential role in the proof of one of our main results: of Theorem 1.2. Due to the low dimension assumption (cf.Corollary 2.19) and the properties of theFKMM-invariant (cf.Theorem 3.11) it is enough to prove the claim only for the case of rank 2 Q -bundles (i.e. m = E , Θ ) and ( E , Θ ) be rank 2 Q -bundles over the FKMM-space ( X , τ ). (We notice that as aconsequence of Proposition 3.7 the two Q -bundles are of FKMM-type. Since X τ is a finite collectionof points, the restricted bundles E j | X τ , j = , Q -trivial and so we can set a Q -isomorphism F : E | X τ → E | X τ induced by these trivializations. If κ ( E ) = κ ( E ), Lemma 4.1 allows us to introducethe fiber bundle SU ( E , E , f ) → X . To complete the proof of the injectivity, it is enough to provethat there exists a global Z -equivariant section of SU ( E , E , f ). This fact can be proved exactly as inProposition 2.16 by using the fact that the fibers of SU ( E , E , f ) are 2-connected as a consequence of π k (S U (2)) = k = , , LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 23
Let us describe first the group structure of Vec Q ( X , τ ). For isomorphism classes [ E ] , [ E ] ∈ Vec Q ( X , τ ),we define the addition by [ E ] + [ E ] : = [ E ], where E is a rank 2 Q -bundle such that E ⊕ E ≃ E ⊕ ( X × C ) and X × C stands for the trivial product Q -bundle. The existence of E is guaranteed byTheorem 2.18. The class [ X × C ] plays the role of the neutral element and, in view of Theorem 3.11,the FKMM-invariant acts as a (multiplicative) morphism κ ([ E ]) = κ ([ E ]) · κ ([ E ]). This morphismis well defined since the value of the FKMM-invariant depends only on the equivalence class of a Q -bundle and κ ( E ) = κ ( E ⊕ ( X × C )) = κ ( E ⊕ E ) = κ ( E ) · κ ( E ) . Moreover, if E ′ is a second rank 2 Q -bundle such that E ⊕ E ≃ E ′ ⊕ ( X × C ) the above computationshows that κ ( E ) = κ ( E ′ ) and the injectivity of κ implies that E ≃ E ′ are in the same equivalenceclass. To define the inverse element in Vec Q ( X , τ ) we observe that under the hypothesis above foreach Q -bundle ( E , Θ ) there exists a Q -bundle ( E ′ , Θ ′ ) such that E ⊕ E ′ ≃ X × C . In this case theinjectivity of κ allows us to define [ E ′ ] = − [ E ]. The construction of E ′ is quite explicit: The complexvector bundle E → X is trivial (cf.Proposition 4.3) and so it admits a global frame { t , t } ∈ Γ ( E )subjected to a Θ -action of type (4.2). Let E ′ → X be the rank 2 Q -bundle generated by the globalframe t ′ j : = τ Θ ( t j ) = Θ ◦ t j ◦ τ , j = , Q -structure induced by Θ . The sum E ⊕ E ′ is Q -trivial since the collection { s = t , s = t ′ , s = t , s = t ′ } provides a global Q -frame.Finally, a group structure on Vec m Q ( X , τ ) can be induced from the group structure of Vec Q ( X , τ ) andthe isomorphism in Corollary 2.19. (cid:4) Remark 4.2.
Under the hypothesis of Theorem 1.2 the FKMM-invariant su ffi ces to establish the non-triviality of a given “Quaternionic” vector bundle. Hence, as a consequence of Theorem 2.7, we deducethat the FKMM-invariant can be interpreted as the (first) topological obstruction for the existence ofa global Q -frame. This is exactly the point of view explored in [GP] in the particular case of theinvolutive space ˜ T . ◭ General structure of low-dimensional Q -bundles. The low dimension assumption d Proposition 4.3.
Let ( X , τ ) be an involutive space such that X verifies (at least) condition 0 of Defini-tion 1.1 and has cells of dimension not bigger than d = . Assume also X τ , ∅ . If ( E , Θ ) is a (evenrank) Q -bundle over ( X , τ ) of FKMM-type, then the underlying vector bundle E → X is trivial in thecategory of complex vector bundles.sketch of.
Due to the low dimension assumption E → X is trivial (as a complex vector bundle) if andonly if its first Chern class c ( E ) ∈ H ( X , Z ) is trivial (cf.[DG1, Section 3.3]). Moreover, one of themain properties of the determinant construction is c ( E ) = c (det( E )). Since ( E , Θ ) is of FKMM-type,it follows that det( E ) → X is an R -trivial line bundle and so the first “Real” Chern class is trivial,˜ c (det( E )) = c (det( E )) = c (det( E )) = (cid:4) As a consequence of this result each Q -bundle ( E , Θ ) of FKMM-type over a base space of dimensionnot bigger than 3 can be endowed with a global frame of sections { t , . . . , t m } ⊂ Γ ( E ). However,this frame is not in general a Q -frame in the sense of Definition 2.6 and the possibility to deform { t , . . . , t m } into a Q -frame is linked to the Q -triviality of ( E , Θ ) as a “Quaternionic” vector bundle.The Q -structure acts on the frame { t , . . . , t m } via the map τ Θ (cf.Section 2.2) τ Θ ( t i )( x ) = ( Θ ◦ t i )( τ ( x )) = m X j = w ji ( τ ( x )) t j ( x ) i = , . . . , m , x ∈ X , (4.2)where w ji ( x ) ∈ C are components of a matrix-valued map w : X → Mat m ( C ). The relation τ Θ = − Idimplies m X k = w ki ( τ ( x )) w jk ( x ) = − δ i , j , ∀ x ∈ X . (4.3) Without loss of generality, one can assume that ( E , Θ ) is endowed with an equivariant Hermitian metric m (cf.Proposition 2.10) with respect to which the frame { t , . . . , t m } is orthonormal. In this case the Q -structure of ( E , Θ ) is encoded in a map w : X → U (2 m ) with components w ji ( x ) : = m (cid:0) t j ( τ ( x )) , τ Θ ( t i )( τ ( x )) (cid:1) = m (cid:0) t j ( τ ( x )) , Θ ◦ t i ( x ) (cid:1) (4.4)and the relation (4.3) reads w ( τ ( x )) = − w t ( x ) , ∀ x ∈ X (4.5)where w t is the transpose of the matrix w . In particular, (4.5) implies the skew -symmetric relation w ( x ) = − w t ( x ) on the fixed point set x ∈ X τ .If one uses the (orthonormal) frame { t , . . . , t m } as a basis for the trivialization of the vector bundleone ends with the identification E ≃ X × C m and the Q -structure Θ : X × C m → X × C m is fixed by Θ : ( x , v) −→ ( τ ( x ) , w ( x ) v) . (4.6)Therefore, in view of Proposition 4.3, the most general “Quaternionic” vector bundle of FKMM-typeover an involutive base space ( X , τ ) of dimension not bigger than 3 is represented by a product X × C m endowed with a map (4.6) associated with a matrix-valued continuous function w : X → U (2 m ) whichverifies (4.5). The topology of such a Q -bundle is entirely contained in the map w . For instance thequestion about the Q -triviality can be rephrased in the equivalent question about the existence of a map u : X → U (2 m ) such that u ∗ ( x ) · w ( τ ( x )) · u ( τ ( x )) = Q for all x ∈ X , where the matrix Q is the onedefined in (2.1).Also the FKMM-invariant can be reconstructed from w . This is linked to the fact that the induced“Real” structure on the determinant line bundle det( E ) ≃ X × C is specified by the mapdet( Θ ) : ( x , λ ) −→ ( τ ( x ) , det( w )( x ) λ ) . (4.7)where det( w ) : X → U (1) inherits from (4.5) the property det( w )( x ) = det( w )( τ ( x )) for all x ∈ X .Moreover, since (det( E ) , det( Θ )) is R -trivial by assumption there exists a map q w : X → U (1) suchthat det( w )( x ) = q w ( x ) q w ( τ ( x )) for all x ∈ X . The choice of q w is not unique. In fact, if ε : X → U (1)is a Z -equivariant map ε ( τ ( x )) = ε ( x ) then q ′ w ( x ) : = ε ( x ) q ( x ) also verifies det( w )( x ) = q ′ w ( x ) q ′ w ( τ ( x )).Moreover, all the possible choices for q w are related by a gauge transformation of this type. On thefixed points x ∈ X τ the matrix w ( x ) is skew-symmetric and so det( w )( x ) = Pf[ w ( x )] where the symbolPf[ A ] is used for the Pfa ffi an of the skew-symmetric matrix A (cf.[MS, Appendix C, Lemma 9]).Therefore, the map w : X → U (2 m ) and the notion of Pfa ffi an fix a well defined map Pf w : X τ → U (1)given by Pf w ( x ) : = Pf[ w ( x )]. Observing that on X τ the maps q w and Pf w can di ff er only by a sign, wecan set a sign map d w : X τ → {± } by d w ( x ) : = q w ( x )Pf w ( x ) x ∈ X τ . (4.8)By construction, the map d w is defined only up to the choice of a gauge transformation ε and so it canbe identified with a representative for a class [ d w ] in the cokernel [ X τ , U (1)] Z / [ X , U (1)] Z . Proposition 4.4 (FKMM-invariant via the sign map) . Let ( X , τ ) be an involutive space such that Xverifies (at least) condition 0 of Definition 1.1 and has cells of dimension not bigger than d = .Assume also X τ , ∅ . Let E ≃ X × C m endowed with a Q -structure Θ of type (4.6) associated to a mapw : X → U (2 m ) which verifies (4.5) . Then κ ( E ) = [ d w ] where the sign map d w is the one defined by (4.8) .Proof. Let us consider the section s X τ : X τ → X τ × C defined by s X τ ( x ) : = ( x , Pf w ( x )). It is notdi ffi cult to check that this section is “Real”. Indeed(det( Θ ) ◦ s X τ ◦ τ )( x ) = det( Θ )( x , Pf w ( x )) = ( x , det( w )( x ) Pf w ( x ) − ) = s X τ ( x ) . LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 25
This section agrees with the canonical section for the restricted R -bundle X τ × C described in Lemma3.5. On the other side, the map t : X → X × C given by t ( x ) : = ( x , q w ( x )) provides a global “Real”-section since(det( Θ ) ◦ t ◦ τ )( x ) = det( Θ )( τ ( x ) , q w ( τ ( x ))) = ( x , det( w )( x ) q w ( τ ( x )) − ) = t ( x ) . Evidently, the di ff erence between the canonical section s X τ and the restricted section t | X τ is expressedexactly by the sign map d w . This implies, in view of Remark 3.9, that d w provides a representative forthe FKMM-invariant of ( E , Θ ), namely κ ( E ) = [ d w ]. (cid:4) Remark 4.5 (The sign map by Fu, Kane & Mele) . The sign map (4.8) was introduced for the first timeby L. Fu, C. L. Kane and E. J. Mele in a series of works concerning “Quaternionic” vector bundlesover TR-tori ˜ T d with d = d = Q -bundleis investigated through the properties of a matrix-valued map w defined similarly to (4.4) and a relatedtopological invariant that agrees (morally) with our sign map (4.8). However, there is a di ff erence:the map q w (that expresses the R -triviality of the determinat line bundle) is replaced by a less naturaland more ambiguous branched function √ det( w ). Let us point out that, albeit apparently similar, ourdefinition of the sign map is much more general and it applies to all Q -bundles of FKMM-type overbase spaces of dimension not bigger that 3 (which in turns implies that the underlying complex vectorbundles are trivial). ◭ Classification for TR-spheres.
This section is devoted to the justification of the following fact:
Proposition 4.6.
Let ˜ S d ≡ ( S d , τ ) be the TR-sphere of Definition 1.4. Then, for all m ∈ N there is agroup isomorphism Vec m Q ( S d , τ ) κ ≃ H Z (cid:0) ˜ S d | ( ˜ S d ) τ , Z (1) (cid:1) ≃ Z if d = , which is established by the FMMM-invariant κ .Proof. Since the second isomorphism has already been established in (3.3), it remains only to justifythe first isomorphism. Theorem 1.2 already assures the injectivity of the group morphism κ . Hence, weneed only to show that both Vec m Q ( S , τ ) and Vec m Q ( S , τ ) have a non-trivial element. Corollary 2.19assures that it is enough to show the existence of non-trivial Q -bundle of rank 2. A specific realizationof these non-trivial elements is presented below. (cid:4) Let us start with the two-sphere ˜ S endowed with the TR involution τ ( k , k , k ) = ( k , − k , − k ).Following the arguments in Section 4.2 we can represent any rank two “Quaternionic” vector bundle( E , Θ ) over ˜ S by a product bundle E ≡ S × C on which the action of Θ is specified by a map w : S → U (2) such that w ( k , − k , − k ) = − w t ( k , k , k ) for all k = ( k , k , k ) ∈ S according to therelation (4.5). One possible realization for w is w ( k , k , k ) : = k + i k + k − k k − i k ! . (4.9)We can compute the FKMM-invariant of this Q -bundle by using the sign map as in Proposition 4.4.Since det( w ) is constantly 1 we can chose q w ( k ) = k ∈ S . On the other side, on the fixedpoints k ± : = ( ± , ,
0) a simple computation shows that Pf w ( k ± ) = ±
1. Therefore, the sign map d w ( k ± ) = ± S ) τ , U (1)] Z / [ ˜ S , U (1)] Z ≃ Z .Equivalently, κ ( E ) = [ d w ] can be identified with the non trivial element − ∈ Z showing that the Q -bundle ( E , Θ ) associated with w is non-trivial.For the three-sphere ˜ S with involution τ ( k , k , k , k ) = ( k , − k , − k , − k ) the argument is similar.A non-trivial Q -structure over E ≡ S × C can be induced by the map w : S → U (2) w ( k , k , k , k ) : = k + i k i k + k i k − k k − i k ! . (4.10)It is straightforward to check that this map verifies (4.5). As before, we can compute the FKMM-invariant by means of the sign map as in Proposition 4.4. Again det( w )( k ) = q w ( k ) =
1. Moreover, on the fixed points k ± : = ( ± , , ,
0) one checks Pf w ( k ± ) = ±
1. Therefore, d w ( k ± ) = ± S ) τ , U (1)] Z / [ ˜ S , U (1)] Z ≃ Z showing that the associated Q -bundle can not be trivial.For later use in Section 5, we derive here a consequence of Proposition 4.6. To state it we needthe sphere ˆ S ≡ ( S , ϑ ) endowed with the antipodal action ϑ : k
7→ − k . We know that the group[ ˆ S , U (1)] Z ≃ H Z ( ˆ S , Z (1)) ≃ Z is generated by the constant maps with values {± } ⊂ U (1) [DG1]. Corollary 4.7.
Let ˆ S be the sphere endowed with the antipodal action k
7→ − k. Then the determinant det : U (2) → U (1) induces an isomorphism of groups det : [ ˆ S , U (2)] Z −→ [ ˆ S , U (1)] Z ≃ Z where the space U (1) is endowed with the involution given by the complex conjugation and U (2) withthe involution µ described in Remark 2.1.Proof. Let H ± : = { ( k , k , k , k ) ∈ ˜ S | ± k ≥ } . The intersection H + ∩ H − is exactly ˆ S . Because each H ± is equivariantly contractible, the clutching construction [At2, Lemma 1.4.9] adapted to Q -bundlesleads to a bijection Vec Q ( ˆ S ) ≃ [ ˆ S , U (2)] Z . Hence, Proposition 4.6 implies that [ ˆ S , U (2)] Z (cid:27) Z ,at least abstractly. To complete the proof, it su ffi ces to verify that there exists a Z -equivariant map ϕ : ˆ S → U (2) such that det ϕ : ˆ S → U (1) is non-trivial in [ ˆ S , U (1)] Z ≃ Z . An example is providedby ϕ ( k , k , k ) : = i k + i k − k k k − i k ! , whose determinant is evidently the constant map with value − (cid:4) Connection with the Fu-Kane-Mele invariant.
In this section we will give a deeper look to theclassification of Q -bundles over su ffi ciently “nice” involutive spaces of dimension d =
2. With this wemean that:
Assumption 4.8.
Let ( X , τ ) be an involutive space such that:0. X is a closed (compact without boundary) and oriented -dimensional manifold;1. The fixed point set X τ , ∅ consists of a finite number of points;2. The involution τ : X → X is smooth and preserves the orientation.
We point out that both the involutive spaces ˜ S and ˜ T fulfill the previous hypothesis. Moreover,spaces of this type share with ˜ S and ˜ T the following general features: Proposition 4.9.
Let ( X , τ ) be an involutive space which verifies Assumption 4.8, thenH Z ( X , Z (1)) ≃ , H Z ( X | X τ , Z (1)) ≃ Z . Moreover, the number of fixed points is even, i.e. X τ = { x , . . . , x n } . We postpone the technical proof of this result to Appendix A. Here, we are mainly interested in thefollowing consequences of Proposition 4.9: if the space ( X , τ ) verifies Assumption 4.8 then it is a bonafide FKMM-space and “Quaternionic” vector bundles over it are classified by the FKMM-invariantwhich takes values in Z .Let L → X be a complex line bundle over an involutive space ( X , τ ). It is well known that thetopology of L is fully specified by the first Chern class c ( L ) ∈ H ( X , Z ). Moreover, if ( X , τ ) is as inAssumption 4.8, then c ( L ) is completely specified by the integer C ( L ) : = h c ( L ); [ X ] i ∈ Z called (first) Chern number . In the last equation the brackets h· ; ·i denote the pairing between cohomologyclasses in H ( X , Z ) with the generator of the homology [ X ] ∈ H ( X ) usually called the fundamentalclass . This pairing can be also understood (when possible) as the integration of the de Rham formwhich represents c ( L ) over X . Let L be the conjugate line bundle of L , namely L and L agree as(rank 2) real vector bundles but they have opposite complex structures. It’s well known that c ( L ) = LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 27 − c ( L ) [MS, Lemma 14.9]. The involution τ : X → X can be used to define the pullback line bundle τ ∗ ( L ). Since τ is an orientation-preserving involution it induces a map of degrees 1 in homology,hence c ( τ ∗ ( L )) = τ ∗ c ( L ) = c ( L ). Given a line bundle L over ( X , τ ) we can build (via theWhitney sum) the rank 2 complex vector bundle E L : = τ ∗ ( L ) ⊕ L . (4.11)The first Chern class of E L vanishes identically since c ( E L ) = c ( τ ∗ ( L )) + c ( L ) = c ( L ) − c ( L ).This agrees with the fact that E L admits a standard Q -structure Θ which can be defined as follow:first of all notice that E L | x ≃ L | τ ( x ) ⊕ L | x , then each point p ∈ E L | x has the form p = ( l , l ) with l ∈ L | τ ( x ) and l ∈ L | x and the bar denotes the inversion of the complex structure (i.e.the complexconjugation) in each fiber L | x . Accordingly, we can define the anti-linear anti-involution Θ between E L | x and E L | τ ( x ) by Θ : ( l , l ) ( − l , l ). Lemma 4.10.
Let ( X , τ ) be an involutive space which verifies Assumption 4.8 and L → X a complexline bundle with (first) Chern number C = . Consider the rank 2 “Quaternionic” vector bundle ( E L , Θ ) associated with L by the construction (4.11) . For each fixed point x j ∈ X τ , j = , . . . , n, letus define a map φ j : X τ → {± } by φ j ( x i ) : = − δ i , j . Then, κ ( E L ) = [ φ ] = . . . = [ φ n ] ∈ [ X τ , U (1)] Z / [ X , U (1)] Z ≃ Z , namely all these maps φ j are representatives for the FKMM-invariant of E L . Moreover, κ ( E L ) coin-cides with the non-trivial element − ∈ Z showing that the Q -bundle ( E L , Θ ) is non-trivial.Proof. Let X τ : = { x , . . . , x n } be the fixed point set of ( X , τ ). As a consequence of the so-called sliceTheorem [Hs, Chapter I, Section 3] a neighborhood of each x j ∈ X τ can be identified with an opensubset around the origin of R endowed with the involution given by the reflection x
7→ − x . Let D j ⊂ X be a disk (under this identification) around x j . More precisely, D j is an invariant set τ ( D j ) = D j inwhich x j is the only fixed point; it is closed, Z -contractible and with boundary ∂ D j ≃ S . Moreover,without loss of generality, we can choose su ffi ciently small disks in such a way that D i ∩ D j = ∅ if i , j . Let us set D : = S ni = D i and X ′ : = X \ Int( D ). By construction X ′ is a manifold with boundary ∂ D ≃ S ni = S i on which the involution τ acts freely.Since the action τ : ∂ D j → ∂ D j is free we can fix an isomorphism ˜ ψ j : ∂ D j ≃ S ∋ θ e i θ ∈ U (1)which is antipodal -equivariant, i.e. ˜ ψ j ◦ τ = e i π ˜ ψ j = − ˜ ψ j . Each of these isomorphisms defines a map ψ j : ∂ D ≃ S ∋ θ e i θ ∈ U (1) given by ψ j | ∂ D j = ˜ ψ j and ψ j | ∂ D i ≡ i , j . Finally, we can setmaps Ψ j : D → U (2) by Ψ j ( x ) : = ψ j ( τ ( x )) 00 ψ j ( x ) ! which verify the equivariance condition Ψ j ◦ τ = µ ◦ Ψ j where the involution µ on U (2) has beendefined in Remark 2.1. It follows that det( Ψ j )( x ) = − x ∈ ∂ D j and det( Ψ j )( x ) = + x ∈ D \ ∂ D j .Associated with each Ψ j we can construct a Q -bundle E j : = ( X ′ × C ) ∪ Ψ j ( D × C )by means of the equivariant version of the clutching construction [At2, Lemma 1.4.9] based on theequivariant map Ψ j . By construction E j turns out to be isomorphic to the Q -bundle E L j : = τ ∗ ( L j ) ⊕ L j where the line bundle L j = ( X ′ × C ) ∪ ψ j ( D × C ) is realized with the clutching construction basedon the map ψ j . Let c ( L j ) be the Chern class of the line bundle L j → X . Since the clutching around x j (and only around this point) is done with a phase-function of the type D j ≃ S ∋ θ → e i θ ∈ U (1) itfollows that the first Chern number obtained from the integration of c ( L j ) is C j =
1. In particular, itfollows that L j ≃ L for each j = , . . . , n . This also implies E j ≃ E L for each j = , . . . , n .To finish the proof we need only to show that the FKMM-invariant of E j is represented by φ j . Tosee this we choose an equivariant nowhere vanishing section of t : X → det( E j ). Sections of this typeare in one-to-one correspondence with a pair of equivariant maps u X ′ : X ′ → U (1) and u D : D → U (1)such that u X ′ = det( Ψ j ) · u D on X ′ ∩ D . As a specific choice, we can take u X ′ ≡ u D ( x ) = − x ∈ ∂ D j and u D ( x ) = + x ∈ D \ ∂ D j . This choice shows that φ j expresses the di ff erence betweenthe restricted section t | X τ and the (constant) canonical section s X τ associated with det( E j | X τ ). (cid:4) Corollary 4.11.
Let ( X , τ ) be an involutive space which verifies Assumption 4.8. Then Vec m Q ( X , τ ) ≃ Z .Proof. It follows from the injectivity of κ proved in Theorem 1.2. (cid:4) Corollary 4.12.
Let ( X , τ ) be an involutive space which verifies Assumption 4.8 and L → X a complexline bundle with (first) Chern number C ∈ Z . Consider the rank 2 “Quaternionic” vector bundle ( E L , Θ ) associated with L by the construction (4.11) . The topology of E L is completely specified bythe parity of C through the formula κ ( E L ) : = ( − C which relates the (image of the) FKMM-invariant κ ( E ) ∈ Z with the Chern class of L .Proof. We can repeat the same proof of Lemma 4.10 with respect to a generalized isomorphism ˜ ψ j : ∂ D j ≃ S ∋ θ e i C θ ∈ U (1) such that ˜ ψ j ◦ τ = e i C π ˜ ψ j = ( − C ˜ ψ j . With this choice the line bundle L j = ( X ′ × C ) ∪ ψ j ( D × C ) has Chern number C and the FKMM-invariant of E L ≃ E L j ≃ E j isrepresented by a function φ j : X τ → {± } such that φ j ( x j ) = ( − C and φ j ( x i ) = i , j . (cid:4) Remark 4.13.
Corollary 4.12 can be considered as the abstract version of the justification of theQuantum Spin Hall E ff ect given in [KM1] (cf.also [FK, eq. (3.26)]). The Chern numbers associatedwith the two line bundles which define E L are opposite in sign and, therefore define opposite travelingcurrents. Since these currents carry opposite spins they sum up and produce a non trivial e ff ect whichis quantified by the parity of the absolute value of the Chern number carried by each band. ◭ A less obvious interesting consequence of Lemma 4.10 is explored in the following proposition.
Proposition 4.14.
Let ( X , τ ) be an involutive space which verifies Assumption 4.8. The isomorphism [ X τ , U (1)] Z / [ X , U (1)] Z ≃ Z (4.12) is induced by the map Π : [ X τ , U (1)] Z → Z , [ ω ] Π n Y i = ω ( x i ) . Proof.
The isomorphism (4.12) is a consequence of the identification (3.2) and Proposition 4.9. Letˆ Φ : [ X τ , U (1)] Z / [ X , U (1)] Z → Z be such an isomorphism and Φ : [ X τ , U (1)] Z → Z a morphismwhich generates ˆ Φ . Evidently, the action of [ X , U (1)] Z on [ X τ , U (1)] Z by restriction is given byelements which are in the kernel of Φ . Since [ X τ , U (1)] Z ≃ Map( X τ , Z ) ≃ Z n , where 2 n is thenumber of fixed points in X τ , one has that Φ can be uniquely represented as a map Φ : Z n → Z , Φ ( ǫ , . . . , ǫ n ) : = n Y i = ( ǫ i ) σ i where for ω ∈ Map( X τ , Z ) one defines ǫ i : = ω ( x i ) ∈ {± } and σ i ∈ { , } . As a consequence ofLemma 4.10 one has that the value of Φ ([ φ i ]) = Φ (1 , . . . , , − , , . . . , ∈ Z has to be independentof i = , . . . , n or, equivalently, has to be independent of the position of the unique negative entry − , . . . , , − , , . . . , σ = . . . = σ n . Finally, the requirement that ˆ Φ has to be an isomorphism fixes σ i = i = , . . . , n . (cid:4) The content of Proposition 4.14 is quite relevant since, in combination with Proposition 4.4, it providesa complete justification, as well as a generalization, of the Fu-Kane-Mele formula for the classificationof “Quaternionic” vector bundles over the TR-torus ˜ T [FK, FKM]. LASSIFICATION OF “QUATERNIONIC”
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Theorem 4.15 (Fu-Kane-Mele formula) . Let ( X , τ ) be an involutive space which verifies Assumption4.8. The topology of a “Quaternionic” vector bundle ( E , Θ ) over ( X , τ ) is completely specified by the(image of the) FKMM-invariant κ ( E ) ∈ Z given by the Fu-Kane-Mele formula κ ( E ) : = n Y i = d w ( x i ) (4.13) where x i ∈ X τ are the fixed points of X and d w ∈ Map( X τ , Z ) is the sign map defined by (4.8) .Proof. Corollary 4.11 says that the topology of an element of Vec m Q ( X , τ ) ≃ Z is completely specifiedby FKMM-invariant κ ( E ) ∈ [ X τ , U (1)] Z / [ X , U (1)] Z . In view of Proposition 4.4, the class κ ( E ) isrepresented by the sign map d w and the isomorphism Π : [ X τ , U (1)] Z / [ X , U (1)] Z → Z described inProposition 4.14 applied to κ ( E ) = [ d w ] gives rise to formula (4.13). (cid:4) We point out that we made a slight abuse of notation in equation (4.13) where more correctly we shouldwrite Π ( κ ( E )) instead κ ( E ). Nevertheless, since there is no risk of confusion, we prefer to write theFu-Kane-Mele formula in the simplest and more evocative form (4.13).4.5. Classification for TR-tori.
Let us start with the case d = Proposition 4.16.
Let ˜ T ≡ ( T , τ ) be the TR-involutive space described in Definition 1.4. Then, forall m ∈ N there is a group isomorphism Vec m Q ( T , τ ) κ ≃ H Z (cid:0) ˜ T | ( ˜ T ) τ , Z (1) (cid:1) ≃ Z which is provided by the FMMM-invariant κ . Although this result is only a special case of Corollary 4.11, we find instructive to show an explicitrealization of a non-trivial rank 2 Q -bundle over T . The existence of a non-trivial element and theinjectivity of κ (cf.Theorem 1.2) provide a complete justification of Proposition 4.16 along the sameline of the proof of Proposition 4.6. A direct way to produce a non trivial Q -bundle of rank 2 is to startwith a line bundle over T with (first) Chern number C =
1. Then, the construction (4.11) providesthe required result in view of Corollary 4.12. A di ff erent way is to start with E ′ ≡ T × C and tointroduce a Q -structure Θ by means of a map w ′ : T → U (2) which verifies the relation (4.5). Asusual we parametrize points of T ≃ R / (2 π Z ) with pairs z : = ( θ , θ ) ∈ [ − π, π ] . With this choicethe involution τ acts as τ ( θ , θ ) = ( − θ , − θ ) and the four (distinct) fixed points are z : = (0 , z : = (0 , π ), z : = ( π,
0) and z : = ( π, π ). A simple way to introduce a non-trivial Q -structure on E ′ is to construct an equivariant map π : ˜ T → ˜ S such that π ( z ) = π ( z ) = π ( z ) = ( + , ,
0) and π ( z ) = ( − , ,
0) and to identify E ′ with the pullback π ∗ E where E ≡ S × C is the non-trivial Q -bundle over ˜ S with Q -structure w given by (4.9). In this case the Q -structure on E ′ is simplygiven by w ′ = π ∗ w = w ◦ π and we can compute the FKMM-invariant of this Q -bundle by using theFu-Kane-Mele formula (4.13), i.e. κ ( E ′ ) : = Y i = d w ′ ( z i ) = Y i = d w ( π ( z i )) = − . Remark 4.17 (Smash product construction) . A concrete realization of an equivariant map π : ˜ T ∋ ( θ , θ ) ( k , k , k ) ∈ ˜ S which verifies the properties required above is given by k ( θ , θ ) : = + cos( θ ) + cos( θ ) − θ ) cos( θ )9 − cos( θ ) − cos( θ ) − θ ) cos( θ ) k ( θ , θ ) : = θ )(1 − cos( θ ))9 − cos( θ ) − cos( θ ) − θ ) cos( θ ) k ( θ , θ ) : = θ )(1 − cos( θ ))9 − cos( θ ) − cos( θ ) − θ ) cos( θ ) . (4.14) The equivariance of π is evident from k ( − θ , − θ ) = k ( θ , θ ) and k j ( − θ , − θ ) = − k j ( θ , θ ), j = , θ , θ ) , (0 , = z and onthe three fixed points z , z , z one verifies that k ( z ) = k ( z ) = ( + , ,
0) and k ( z ) = ( − , , z withvalue k ( z ) = ( + , , T : = { ( θ , , (0 , θ ) | θ , θ ∈ [ − π, π ] } ⊂ T be the one-dimensionalsubcomplex of T consisting of two copies of S joined together on the fixed point z = (0 , T = S ∨ S is a wedge sum of two circles. From (4.14) it followsthat π ( T ) = ( + , , π corresponds to the (equivariant) projection π : T −→ T / ( S ∨ S ) : = S ∧ S ≃ S where the symbol ∧ denotes the smash product of two topological spaces [Hat, Chapter 0]. ◭ The case d = Proposition 4.18.
Let ˜ T ≡ ( T , τ ) be the TR-involutive space described in Definition 1.4. Then, forall m ∈ N there is a group isomorphism Vec m Q ( T , τ ) κ ≃ H Z (cid:0) ˜ T | ( ˜ T ) τ , Z (1) (cid:1) ≃ Z which is provided by the FMMM-invariant κ .Proof. Theorem 1.2 establishes the injectivity of the group morphism κ and we know from (3.4) theisomorphisms H Z (cid:0) ˜ T | ( ˜ T ) τ , Z (1) (cid:1) ≃ H Z (( ˜ T ) τ , Z (1)) / r (cid:0) H Z ( ˜ T , Z (1)) (cid:1) ˆ Φ ≃ Z . We can realize ˆ Φ starting from a morphism Φ : H Z (( ˜ T ) τ , Z (1)) → Z which acts trivially on r (cid:0) H Z ( ˜ T , Z (1)) (cid:1) . Let us build such a morphism Φ . The fixed point set ( ˜ T ) τ ≃ ( ˜ S ) τ × ( ˜ S ) τ × ( ˜ S ) τ has eight distinct points which can be labelled with eight vectors v k : = ( v k , v k , v k ) ∈ Z ; explicitlyv = ( + , + , + , v = ( + , + , − , v = ( + , − , + , v = ( − , + , + , v = ( + , − , − , v = ( − , + , − , v = ( − , − , + , v = ( − , − , − . (4.15)The presence of eight fixed points implies H Z (( ˜ T ) τ , Z (1)) ≃ Z [DG1, eq. (5.7)] and with the helpof the recursive relations [DG1, eq. (5.9)] one obtains H Z (cid:0) ˜ T , Z (1) (cid:1) ≃ Z ⊕ Z . The isomorphism H Z (cid:0) ˜ T , Z (1) (cid:1) ≃ [ ˜ T , U (1)] Z shows that the Z -summand is generated by the constant map ǫ : ˜ T →− Z -summand is spanned by the three (canonical) projections π j : ˜ T → ˜ S ≃ U (1).More precisely, the equality ˜ T = ˜ S × ˜ S × ˜ S allows us to label each k ∈ ˜ T as k : = ( z , z , z ) with z j ∈ U (1). Let z + = + U (1) ≃ ˜ S (with respect to the involutiongiven by the complex conjugation). With this notation the projections π j act as˜ T ∋ ( z , z , z ) π −→ ( z , + , + ≡ z ∈ ˜ S , ˜ T ∋ ( z , z , z ) π −→ ( + , z , + ≡ z ∈ ˜ S , ˜ T ∋ ( z , z , z ) π −→ ( + , + , z ) ≡ z ∈ ˜ S . The map r : H Z ( ˜ T , Z (1)) → H Z (( ˜ T ) τ , Z (1)) ≃ Z coincides with the evaluations on the fixedpoint set, hence we have that the image r (cid:0) H Z ( ˜ T , Z (1)) (cid:1) is generated by the four linearly independentvectors φ : = r ( ǫ ) = ( − , . . . , −
1) and φ j : = r ( π j ) : = ( π j (v ) , . . . , π j (v )), j = , , ϕ ∈ H Z (( ˜ T ) τ , Z (1)) be represented as a map ϕ : { v , . . . , v } → Z and consider the morphism Φ : H Z (( ˜ T ) τ , Z (1)) → Z defined by Φ : ϕ ( Φ ( ϕ ) , Φ ( ϕ ) , Φ ( ϕ ) , Φ ( ϕ )) Φ ( ϕ ) : = ϕ (v ) ϕ (v ) ϕ (v ) ϕ (v ) , Φ ( ϕ ) : = ϕ (v ) ϕ (v ) ϕ (v ) ϕ (v ) , Φ ( ϕ ) : = ϕ (v ) ϕ (v ) ϕ (v ) ϕ (v ) , Φ ( ϕ ) : = ϕ (v ) ϕ (v ) ϕ (v ) ϕ (v ) . LASSIFICATION OF “QUATERNIONIC”
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By construction Φ : φ j ( + , + , + , +
1) for all j = , . . . ,
3, hence r (cid:0) H Z ( ˜ T , Z (1)) (cid:1) is in the kernelof Φ . Moreover the images of ψ : = ( − , + , + , − , + , + , + , + , ψ : = ( − , + , − , + , + , + , + , + ,ψ : = ( − , − , + , + , + , + , + , + , ψ : = ( + , + , + , + , + , + , + , − , under the map Φ are Φ ( ψ ) = ( − , − , + , + , Φ ( ψ ) = ( − , + , − , + , Φ ( ψ ) = ( − , + , + , − , Φ ( ψ ) = ( − , + , + , + , showing that the morphism Φ induces the isomorphism ˆ Φ .In order to finish the proof, we need only to show the existence of Q -bundles E j over ˜ T such that κ ( E j ) is represented by ψ j for j = , , . . . ,
3. Let E → ˜ T be a non-trivial Q -bundle classified by κ ( E ) = [ d E ] where the sign function d E : ( ˜ T ) τ → Z takes values d E (w ) = − d E (w s ) = + s = , , = ( + , + , w = ( + , − , w = ( − , + , w = ( − , − , of the set ( ˜ T ) τ ≃ ( ˜ S ) τ × ( ˜ S ) τ . Let us consider the three projections˜ T ∋ ( z , z , z ) π −→ ( + , z , z ) ≡ ( z , z ) ∈ ˜ T , ˜ T ∋ ( z , z , z ) π −→ ( z , + , z ) ≡ ( z , z ) ∈ ˜ T , ˜ T ∋ ( z , z , z ) π −→ ( z , z , + ≡ ( z , z ) ∈ ˜ T . The Q -bundle π ∗ E → ˜ T has the FKMM-invariant κ ( π ∗ E ) which is represented by the map π ∗ d E : = d E ◦ π . Observing that π (v ) = π (v ) = w , π (v ) = π (v ) = w ,π (v ) = π (v ) = w , π (v ) = π (v ) = w , one verifies that π ∗ d E = ψ and so π ∗ E is a representative for a Q -bundle E with FKMM-invariant[ ψ ]. In the same way, one checks that π ∗ d E = ψ and π ∗ d E = ψ in such a way that π ∗ E is arepresentative for a Q -bundle E with FKMM-invariant [ ψ ] and π ∗ E is a representative for a Q -bundle E with FKMM-invariant [ ψ ]. To construct the Q -bundle E classified by [ ψ ] we consider theprojection π : ˜ T → ˜ S defined by the standard smash product construction [Hat, Chapter 0]˜ S ≃ ˜ T / T which produces the sphere ˜ S from the torus ˜ T collapsing the subcomplex T ⊂ ˜ T to the fixedpoint k ∗ ≡ ( + , + , + ∈ T (this is the same construction described in Remark 4.17 for the case ofa two-torus). More precisely, T : = π ( ˜ T ) ∪ π ( ˜ T ) ∪ π ( ˜ T ) is a collection of three 2-tori suchthat their common intersection is the fixed point k ∗ and each two of them intersect along a circle. Thisconstruction is compatible with the definition of the involution τ . Let p ± : = ( ± , , ,
0) be the twofixed points of ˜ S and E ′ → ˜ S the non-trivial Q -bundle classified by a sign map d E ′ ( p ± ) = ±
1. Byconstruction π (v j ) = p + for all j = , . . . , π (v ) = p − and so π ∗ d E ′ : = d E ′ ◦ π coincides with ψ . Then, π ∗ E ′ is a model for E . (cid:4) Remark 4.19 (Week and strong invariants) . According to the Proposition 4.18, each “Quaternionic”vector bundle ( E , Θ ) over ˜ T is specified by a quadruple ( κ ( E ) , κ ( E ) , κ ( E ) , κ ( E )) ∈ Z which pro-vides a representative for the FKMM-invariant κ ( E ). The proof of Proposition 4.18, together withthe group structure of Vec m Q ( T , τ ) described in Theorem 1.2, gives us also a recipe to compute thesenumbers using the sign function d E : ( ˜ T ) τ → Z associated with the Q -bundle E . Let us use theconvention (4.15) for the fixed point of ˜ T . The first three invariants κ ( E ) : = Y j ∈{ , , , } d E (v j ) , κ ( E ) : = Y j ∈{ , , , } d E (v j ) , κ ( E ) : = Y j ∈{ , , , } d E (v j ) can be understood as follows: if one considers, for instance, the restricted bundle E | π ( ˜ T ) over the two-dimensional torus π ( ˜ T ) ⊂ ˜ T one has that κ ( E ) coincides with the FKMM-invariant of E | π ( ˜ T ) .Similarly, κ ( E ) and κ ( E ) are the FKMM-invariants of the restrictions E | π ( ˜ T ) and E | π ( ˜ T ) , respec-tively. These three numbers are called week invariants in the jargon of [FKM] (cf.equation (2), inparticular) since they express only two-dimensional properties of the system. From a topological pointof view, these three invariants describe the obstruction to extending a Q -frame from the (equivariant)1-skeleton of ˜ T to its 2-skeleton. The fourth invariant κ ( E ) : = Y j = d E (v j )is called strong [FKM, eq. (1)] since it expresses a genuine three-dimensional property of the system.This number can be understood as follows : if the weak invariants are trivial the Q -bundle E is triv-ial when restricted to the subcomplex T . Hence E is Q -isomorphic to a Q -bundle over the sphere˜ S ≃ ˜ T / T [At2, Lemma 1.47] with FKMM-invariant κ ( E ). Then, the strong invariant provides theobstruction to extending a Q -frame from the subcomplex T to the full torus ˜ T . ◭
5. C lassification in dimension d = S and ˜ T . Weprove some general preliminary results under the following rather restrictive hypothesis: Assumption 5.1.
Let ( X , τ ) be an involutive space such that:0. X is a closed (compact without boundary) and oriented -dimensional manifold;1. The fixed point set X τ = { x , . . . , x N } consists of a finite number of points with N > ;2. The involution τ : X → X is smooth and preserves the orientation;3. H Z ( X , Z (1)) = ;4. There exists a fixed point x ∗ ∈ X τ such that the space X ∗ : = X \ { x ∗ } is Z -homotopic to a Z -CW complex whose cells are of dimension less than 4. First of all we notice that under the above conditions ( X , τ ) turns out to be an FKMM-space as inDefinition 1.1. Second, the reduced space X ∗ = X \ { x ∗ } is also an FKMM-space which verifies theconditions of Theorem 1.2, and so the map κ : Vec m Q ( X ∗ , τ ) −→ H Z ( X ∗ | X τ ∗ , Z (1)) ≃ [ X τ ∗ , U (1)] Z / [ X ∗ , U (1)] Z is injective. The fact that X ∗ is an FKMM-space depends on the following lemma: Lemma 5.2.
Under Assumption 5.1 H Z ( X ∗ , Z (1)) = . sketch of. As a consequence of the so-called slice Theorem [Hs, Chapter I, Section 3] a neighborhoodof x ∗ can be identified with an open subset around the origin of R endowed with the involution givenby the reflection x
7→ − x . Let D be the unit (closed) ball under this identification. Then D is aninvariant set in which x ∗ is the only fixed point. The space X ′ : = X \ Int( D ) is Z -homotopy equivalentto X ∗ . Moreover, D is Z -contractible and so with the help of the Meyer-Vietoris exact sequence for { D , X ′ } one can prove the isomorphism H Z ( X ∗ , Z (1)) ≃ H Z ( X , Z (1)) which concludes the proof. (cid:4) The following technical lemma will provide us important information which turn out to be equiv-alent to the classification of Q -bundles over ˜ S . We recall, following the same notation of Corollary4.7, that ˆ S ≡ ( S , ϑ ) is the three-sphere endowed with the antipodal action ϑ : k
7→ − k . Moreover, weneed also the well-known isomorphism [ ˆ S , U (2)] ≃ π ( U (2)) ≃ Z given by the topological degree . Lemma 5.3.
Let ˆ S be the sphere endowed with the antipodal action k
7→ − k and [ ˆ S , U (2)] Z theset of Z homotopy equivalent maps with respect to the involution µ on U (2) described in Remark LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 33 [ ˆ S , U (2)] Z → [ ˆ S , U (2)] defined by “forgetting” the involutive structures is anisomorphism. In particular, this leads to a group isomorphism deg : [ ˆ S , U (2)] Z −→ Z induced by the topological degree. The proof of this Lemma is based on the following idea: The determinant det : U (2) → U (1)induces a homomorphismdet : [ ˆ S , U (2)] Z → [ ˆ S , U (1)] Z ≃ H Z ( ˆ S , Z (1)) ≃ Z , (5.1)where H Z ( ˆ S , Z (1)) ≃ Z has been proved in [DG1]. This group is generated by the constant mapswith values {± } ⊂ U (1). One has the group morphism(det , deg) : [ ˆ S , U (2)] Z −→ Z ⊕ Z (5.2)which associates to each [ ϕ ] ∈ [ ˆ S , U (2)] Z the pair ( ǫ, n ) ∈ Z ⊕ Z where ǫ = [det ϕ ] and n = deg ϕ .This morphism turns out to be injective and so it defines an isomorphism between [ ˆ S , U (2)] Z and asubgroup of Z ⊕ Z described by[ ˆ S , U (2)] Z (det , deg) ≃ { ( ǫ, n ) ∈ Z ⊕ Z | ǫ = ( − n } ≃ Z . (5.3)From (5.3) it results evident that the degree completely classifies classes in [ ˆ S , U (2)] Z . of Lemma 5.3. Let us start by proving that (5.2) is injective. To this end, we recall the Z -CW decom-position of ˆ S (cf.[DG1, Section 4.5]): ˆ S = ˜ e ∪ ˜ e ∪ ˜ e ∪ ˜ e , where ˜ e d = Z × e d = e + d ⊔ e − d is a free Z -cell, on which the involution acts by exchanging the usual d -dimensional cells e ± d . Let X = ˜ e ∪ ˜ e ∪ ˜ e be the 2-skeleton of ˆ S , which provides also the Z -CWdecomposition of the sphere ˆ S with the antipodal free involution. Now, let ϕ : ˆ S → U (2) be a Z -equivariant map such that ǫ = [det ϕ ] = S , U (1)] Z ≃ Z and n = deg ϕ = S , U (2)] ≃ Z .The first assumption leads to [det ϕ | X ] = X , U (1)] Z , hence Corollary 4.7 assures the existence ofan equivariant homotopy between ϕ | X and the constant map at ∈ U (2). By applying the equivarianthomotopy extension property [Ma] to the subcomplex X ⊂ ˆ S , we get an equivariant homotopyfrom ϕ to an equivariant map ϕ ′ : ˆ S → U (2) such that ϕ ′ | X = . Then, the restriction ϕ ′ | e + isidentifiable with an element [ ϕ ′ | e + ] ∈ π ( U (2)). The involution on ˆ S preserves the orientation, andthe involution µ on U (2) induces the identity on π ( U (2)). As a result, [ ϕ ′ | e + ] = [ ϕ ′ | e − ] holds trueand [ ϕ ] ∈ π ( U (2)) is expressed as [ ϕ ] = ϕ ′ | e + ]. Because π ( U (2)) ≃ Z has no torsion [ ϕ ] = ϕ =
0) implies [ ϕ ′ | e + ] = π ( U (2)). Then, there exists a homotopyfrom ϕ ′ | e + to the constant map at . By means of the involution on ˆ S , this homotopy extends toan equivariant homotopy from ϕ ′ | ˜ e to the constant map at . Again by the equivariant homotopyextension property, we get an equivariant homotopy from ϕ ′ to the constant map at . In summary,we proved that if (det , deg) : [ ϕ ] (1 ,
0) then [ ϕ ] is Z -homotopy equivalent to the constant map at , namely the injectivity of (det , deg). It remains to prove that the image of (det , deg) agrees with thesubgroup (5.3). We point out that from the previous argument it follows that there is no equivariantmap ϕ : ˆ S → U (2) such that ǫ = n =
1. In fact we already proved that ǫ = [det ϕ ] = ϕ is of even degree, i.e. n ∈ Z . Finally, the equivariant map ϕ : ˆ S → U (2) defined by ϕ ( k , k , k , k ) : = i k + i k − k + i k k + i k k − i k ! clearly has ǫ = det ϕ = − n = deg ϕ = (cid:4) We are now in position to prove Theorem 1.6 under Assumption 5.1. Before going through thetechnical part of the proof of this theorem, let us point out that Theorem 1.6 is true under the hypothesisas stated, which is less restrictive than Assumption 5.1. The generalized proof makes strong use of the obstruction theory and this implies a considerable increasing of technical complications. Since the aimof this work is to provide a classification in the case of the spaces ˜ S and ˜ T (which verify Assumption5.1) we opted here for a simplified proof, leaving the more general case for a future work [DG3]. Theorem 5.4 (Injective group homomorphism: d = . Let ( X , τ ) be as in Assumption 5.1. Then, theFKMM-invariant κ and the second Chern class c define a map ( κ, c ) : Vec m Q ( X , τ ) −→ H Z ( X | X τ , Z (1)) ⊕ H ( X , Z ) m ∈ N (5.4) that is injective. Moreover, Vec m Q ( X , τ ) can be endowed with a group structure in such a way that thepair ( κ, c ) sets an injective group homomorphism.Proof. The stable range condition implies Vec m Q ( X , τ ) ≃ Vec Q ( X , τ ) (cf.Corollary 2.19). Moreover, if E ≃ E ⊕ ( X × C m − ) as Q -bundles we obtain from Theorem 3.11 (iii) that κ ( E ) = κ ( E ) and fromthe usual properties of Chern classes that c ( E ) = c ( E ). Then, without loss of generality, we canconsider only the case m = X ′ : = X \ Int( D ) where D is an invariant ball around thefixed point x ∗ ∈ X τ . The space X ′ is Z -homotopy equivalent to X ∗ and so, by assumption, it isequivalent to a Z -CW complex whose cells are of dimension less or equal to 3. By assumptionalso ( X ′ ) τ , ∅ . The inclusion map ı : X ′ ֒ → X induces by the naturality of the FKMM-invariant theequalities κ ( E i | X ′ ) = κ ( ı ∗ E i ) = ı ∗ κ ( E i ), i = ,
2. Then the coincidence of the FKMM-invariants of E and E , immediately implies κ ( E | X ′ ) = κ ( E | X ′ ) and Theorem 1.2 assures the existence of a Q -isomorphism f ′ : E | X ′ → E | X ′ . Since D is Z -contractible there are also two Q -isomorphisms g i : E i | D → D × C , i = ,
2. Let Y : = ∂ X ′ = ∂ D . This space is Z -homotopic to a three-sphere ˆ S endowed with theantipodal involution. The composition of Q -isomorphisms Y × C g − | Y −→ E | Y f ′ | Y −→ E | Y g | Y −→ Y × C can be expressed by a mapping ( x , v) ( x , θ ( x ) · Q · v) where θ : Y → U (2) is a µ -equivariant map inthe sense of Remark 2.1, i.e. θ ( τ ( x )) = − Q θ ( x ) Q . In view of the homotopy property for “Quaternionic”vector bundles (cf.Theorem 2.12), the map f ′ : E | X ′ → E | X ′ extends to an isomorphism f : E → E if and only if the Z -homotopy class [ θ ] ∈ [ Y , U (2)] Z = [ ˆ S , U (2)] Z is in the image of [ D , U (2)] Z under the restriction morphism. But we know that D is Z -contractible, hence [ D , U (2)] Z ≃ Q -isomorphism which extends f ′ if and only if [ θ ] is trivial, or equivalently if and onlyif deg( θ ) = c ( E ) = c ( E )implies deg( θ ) = X ′ has the structure of a CW-complex of dimension less or equal to 3. Then, anycomplex vector bundle over X ′ with associated trivial determinant line bundle is automatically trivialin the complex category (cf.Proposition 4.3). This implies the existence of an isomorphism of complexvector bundles h : E | X ′ → X ′ × C and we can set a second isomorphism h : E | X ′ → X ′ × C by h : = h ◦ ( f ′ ) − . For each i = , Y × C g − i | Y −→ E i | Y h i | Y −→ Y × C can be expressed by a mapping ( x , v) ( x , ϕ i ( x ) · v) where ϕ i : Y → U (2). This is nothing but aclutching function of the underlying complex vector bundle E i . In view of the Meyer-Vietoris exactsequence H ( X ′ , Z ) ⊕ H ( D , Z ) −→ H ( Y , Z ) −→ H ( X , Z ) −→ D ≃ {∗} , Y = X ′ ∩ D and X = X ′ ∪ D ) we obtain that H ( Y , Z ) ≃ H ( X , Z ) which impliesthat the degree of ϕ i coincides with the second Chern class of E i under the choice of an orientationon D compatible with X . By construction, one readily sees θ = ϕ ◦ ϕ − which implies deg( θ ) = deg( ϕ ) − deg( ϕ ) = c ( E ) − c ( E ). (cid:4) The case of ˜ S . If we remove from the TR-sphere ˜ S one of the two fixed points k ± : = ( ± , , , Z -homotopic to R endowed with the involution given by the reflection LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 35 around the origin x
7→ − x . This space is Z -contractible to a point and so Assumption 5.1 is verified.This implies that Theorem 1.6 applies to ˜ S , although the use of this result is not strictly necessary forthe classification of Vec m Q ( S , τ ). of Theorem 1.7 (i). One has the following isomorphismsVec m Q ( S , τ ) ≃ Vec Q ( S , τ ) ≃ [ ˆ S , U (2)] Z ≃ Z where the first is a consequence of Corollary 2.19, the second follows from an equivariant adaptationof the standard clutching construction [At2, Lemma 1.4.9] and the last one has been proved in Lemma5.3. To finish the proof, one has only to recall that if [ E ϕ ] ∈ Vec Q ( S , τ ) is the Q -bundle associated with[ ϕ ] ∈ [ ˆ S , U (2)] Z then C = h c ( E ϕ ) , [ S ] i = deg ϕ = n and κ ( E ϕ ) = [det ϕ ] = ǫ ∈ {± } . While the firstis a classical identity in the theory of complex vector bundles, the second deserves some comments.Let H ± : = { ( k , k , k , k , k ) ∈ ˜ S | ± k ≥ } with intersection H + ∩ H − = ˆ S . Then, by construction E ϕ ≃ ( H + × C ) ∪ ϕ ( H − × C )The set ( ˜ S ) τ = { ( ± , , , , } has two fixed points and we can set the identity map on ( ˜ S ) τ × C asthe trivialization of E ϕ | ( ˜ S ) τ and the induced canonical trivialization det ( ˜ S ) τ : det( E ϕ | ( ˜ S ) τ ) → ( ˜ S ) τ × C which obviously agrees with the identity map on ( ˜ S ) τ × C . On the other hand,det( E ϕ ) ≃ ( H + × C ) ∪ det ϕ ( H − × C )is obtained by gluing together trivial “Real” line bundles over H ± by means of the clutching functiondet ϕ . Then, a global trivialization h det : det( E ϕ ) → ˜ S × C can be realized as a pair of R -isomorphisms f ± : H ± × C → H ± × C such that f + | ˆ S = det ϕ · f − | ˆ S . Due to equation (5.1), we can assume thatdet ϕ = ± f + | ˆ S = ± f − | ˆ S , accordingly. This di ff erence in signis exactly the FKMM-invariant. (cid:4) The case of ˜ T . Let us briefly justify why the TR-torus ˜ T verifies Assumption 5.1. Under theidentifications ˜ T ≃ ˜ S × . . . × ˜ S and ˜ S ≃ U (1) (endowed with the complex conjugation) we canrepresent each point of the torus by a quadruple ( z , z , z , z ) with z j ∈ U (1), and the fixed points arelabelled by choosing z j ∈ {± } . Let k ∗ : = ( + , + , + , + k ∗ from ˜ T one easily seesthat the resulting space is Z -homotopy equivalent to the union of four TR-tori ˜ T j : = { ( z , z , z , z ) ∈ ˜ T | z j = − } with common intersection at the fixed point ( − , − , − , − T \ { k ∗ } ≃ T : = ˜ T ∪ . . . ∪ ˜ T We notice that T inherits the structure of a Z -CW complex from that of each summand ˜ T j and thedimension of the Z -cells does not exceed 3. Then, we showed that Assumption 5.1 is verified andTheorem 1.6 applies to ˜ T . of Theorem 1.7 (ii). We provide here only a sketch of the proof that, in spirit, is very close to the proofof Proposition 4.18. The interested reader can possibly complete the details following the schemebelow. • The injectivity follows from Theorem 1.6, the isomorphism H Z (cid:0) ˜ T | ( ˜ T ) τ , Z (1) (cid:1) ≃ Z inequation (3.4) and the isomorphism H ( T , Z ) ≃ Z induced by the pairing with the fundamen-tal class [ T ] ∈ H ( T ). • To prove the isomorphism one follows the same strategy of the proof of Proposition 4.18;More precisely one shows the existence of elements in Vec m Q ( T , τ ) (but as usual m = Z ⊕ Z . • Let ǫ = ( ǫ , . . . , ǫ , ǫ ) be the image of κ ( E ) in Z . One can realize the first six components ǫ , . . . , ǫ by the pullback of the (unique) non trivial element of Vec Q ( T , τ ) by means of thesix projections π i j : ˜ T → ˜ T , i , j = , . . . ,
4, onto the six sub-tori of dimension 2. We refer tothese components as ultra-week . • The next four components ǫ , . . . , ǫ are given by the pullback of the non trivial element ofVec Q ( T , τ ) described by the strong FKMM-invariant ( − , + , + , +
1) by means of the fourprojections π i : ˜ T → ˜ T , i = , . . . ,
4, onto the four sub-tori of dimension 3. These are the week components of ǫ . • These 10 components ( ǫ , . . . , ǫ ) generate a subgroup of Z . Moreover, by construction, allthe Q -bundles associated with these invariants have trivial second Chern classes. Then, weprovided the generators for the first summand Z . • Let π : ˜ T → ˜ T / T ≃ ˜ S be the projection obtained according to the usual collapsingconstruction . The pullback of the non trivial element in Vec m Q ( S , τ ) provides a Q -bundle over˜ T with second Chern number C = ǫ , . . . , ǫ , ǫ ) = (0 , ..., , (cid:4)
6. U niversality of the
FKMM- invariant
In this section we provide a “universal” interpretation of the FKMM-invariant. More precisely, weshow that the FKMM-invariant associated to a “Quaternionic” vector bundle can be defined functori-ally from the universal
FKMM-invariant of the classifying “Quaternionic” vector bundle described inSection 2.4. In this sense the FKMM-invariant is a bona fide characteristic class. We point out thatthe construction of the universal FKMM-invariant requires a generalization and an improvement of theoriginal idea in [FKMM].6.1.
Universal FKMM-invariant.
Theorem 2.13 says that each rank 2 m Q -bundle ( E , Θ ) over theinvolutive space ( X , τ ) is obtained (up to isomorphisms) as the pullback of the tautological 2 m -plane Q -bundle ( T ∞ m , Ξ ) over the involutive Grassmannian ˆ G m ( C ∞ ) ≡ ( G m ( C ∞ ) , ρ ).The determinat construction described in Section 3.2 applies also to the tautological Q -bundle T ∞ m and it defines a line bundle π : det( T ∞ m ) −→ ˆ G m ( C ∞ ) (6.1)which is endowed with a “Real” structure det( Ξ ) : det( T ∞ m ) → det( T ∞ m ) in agreement with Lemma3.2. Moreover, we recall that S (det( T ∞ m )) ⊂ det( T ∞ m ) denotes the circle bundle according to thenotation introduced in Remark 3.3. The following characterization will play an important role in thesequel. Lemma 6.1.
Let ( X , τ ) be an involutive space such that X verifies (at least) condition 0 of Definition1.1 and ϕ : X → ˆ G m ( C ∞ ) a Z -equivariant map. Then:(i) Global “Real” sections of the circle bundle S (det( ϕ ∗ T ∞ m )) are in one-to-one correspondencewith Z -equivariant maps ˜ ϕ : X → S (det( T ∞ m )) which make the following diagram (cid:0) S (det( T ∞ m )) , det( Ξ ) (cid:1) ˜ ϕ ( X , τ ) ϕ ✲ ✲ ˆ G m ( C ∞ ) π ❄ ( π ◦ ˜ ϕ = ϕ ) (6.2) commutative;(ii) The determinant line bundle det( ϕ ∗ T ∞ m ) associated with ϕ ∗ T ∞ m is R -trivial if and only if thereexists a Z -equivariant map ˜ ϕ : X → S (det( T ∞ m )) such that the diagram (6.2) is commutative;Proof. (i) Since the determinat construction is functorial one has the natural isomorphism det( ϕ ∗ T ∞ m ) ≃ ϕ ∗ det( T ∞ m ). Let ˆ ϕ : ϕ ∗ det( T ∞ m ) → det( T ∞ m ) be the standard morphism associated with the pullbackconstruction; we recall that ˆ ϕ establishes fiberwise (metric-preserving) isomorphisms ϕ ∗ det( T ∞ m ) | x ≃ det( T ∞ m ) | ϕ ( x ) . If s : X → S ( ϕ ∗ det( T ∞ m )) is a global R -section then ˜ ϕ : = ˆ ϕ ◦ s is a Z -equivariant map LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 37 such that π ◦ ˜ ϕ = ϕ . Vice versa, if one has a Z -equivariant map ˜ ϕ : X → S (det( T ∞ m )) which verifies π ◦ ˜ ϕ = ϕ then s ( x ) : = ˆ ϕ | − ϕ ( x ) ◦ ˜ ϕ ( x ) defines a global “Real” section of S ( ϕ ∗ det( T ∞ m )). (ii) This is aconsequence of the fact that the R -triviality of an R -line bundle (endowed with an equivariant metric)is equivalent to the existence of a global “Real” section of the associated circle bundle (cf.Remark3.3). (cid:4) Remark 6.2.
It is rather evident that if ˜ ϕ and ˜ ϕ are Z -equivariant maps which make the diagram(6.2) commutative then ˜ ϕ = u · ˜ ϕ for some Z -equivariant map u : X → U (1). ◭ In order to define a universal
FKMM-invariant we need to apply the pullback construction to thetautological 2 m -plane Q -bundle ( T ∞ m , Ξ ) with respect to the map (6.1) understood as an equivariantmap between involutive base spaces. More precisely, we need to use the following diagram( π ∗ T ∞ m , π ∗ Ξ ) ˆ π ✲ ( T ∞ m , Ξ ) (cid:0) S (det( T ∞ m )) , det( Ξ ) (cid:1) π ′′ ❄ π ✲ ˆ G m ( C ∞ ) π ′ ❄ . (6.3)Just for sake of completeness, we recall that the projection π ′′ is associated with the projection π ′ bythe pullback construction while the maps π ′ and π are related by the determinat construction, namely π = det( π ′ ) if one prefers the functorial notation. The peculiar structure of the Q -bundle ( π ∗ T ∞ m , π ∗ Ξ )is compatible with the construction of an FKMM-invariant. First of all, we notice that the fixed pointset of the base space (cid:0) S (det( T ∞ m )) , det( Ξ ) (cid:1) is non-empty; More precisely we will prove in Lemma 6.8that S (det( T ∞ m )) Ξ : = { x ∈ S (det( T ∞ m )) | det( Ξ )( x ) = x } ≃ G m ( H ∞ ) ⊔ G m ( H ∞ ) (6.4)which shows that S (det( T ∞ m )) Ξ decomposes as the disjoint union of two identical components that arepath-connected π ( G m ( H ∞ )) =
0. The second important ingredient is contained in the next result.
Lemma 6.3.
The rank m “Quaternionic” vector bundle ( π ∗ T ∞ m , π ∗ Ξ ) over the involutive space (cid:0) S (det( T ∞ m )) , det( Ξ ) (cid:1) is of FKMM-type according to Definition 3.6.Proof. The functoriality of the determinant construction implies that det( π ∗ T ∞ m ) ≃ π ∗ det( T ∞ m ). Tocomplete the proof we only need to show that π ∗ det( T ∞ m ) is R -trivial and we will show this fact in twodi ff erent ways.The first proof uses Lemma 6.1 (ii). Let us consider the diagram (6.2) with (cid:0) S (det( T ∞ m )) , det( Ξ ) (cid:1) instead ( X , τ ) and ϕ = π . The identity map Id : det( T ∞ m ) → det( T ∞ m ) provides a realization for theequivariant map ˜ ϕ , hence det( π ∗ T ∞ m ) is R -trivial.The second proof is more direct since it is based on the construction of an explicit trivialization.The pullback construction applied to diagram (6.3) shows that π ∗ det( T ∞ m ) = n ( x , y ) ∈ S (det( T ∞ m )) × det( T ∞ m ) | π ( x ) = π ( y ) o . Hence, the diagonal sections diag : S (det( T ∞ m )) −→ π ∗ det( T ∞ m ) , s diag ( x ) : = ( x , x ) (6.5)provides a global equivariant section and defines a global R -trivialization h diag : π ∗ det( T ∞ m ) −→ S (det( T ∞ m )) × C , h diag ( x , y ) : = ( x , ϕ diag ( x ))where the equivariant map ϕ diag : det( T ∞ m ) → U (1) is specified by the condition y = ϕ diag ( x ) · x . (cid:4) Remark 6.4 (Diagonal section) . Let us point out that the diagonal section s diag in (6.5) can be seen asa global R -section for the circle bundle S ( π ∗ det( T ∞ m )) → S (det( T ∞ m )). Moreover, it is evident that therelation s diag ( x ) = h − ( x , ∀ x ∈ S (det( T ∞ m )) holds true. ◭ Owing to Lemma 6.3, the following definition is well posed.
Definition 6.5 (Universal FKMM-invariant) . The universal FKMM-invariant K univ is the FKMM-invariant of the rank m “Quaternionic” vector bundle ( π ∗ T ∞ m , π ∗ Ξ ) defined over the involutive space (cid:0) S (det( T ∞ m )) , det( Ξ ) (cid:1) , i.e. K univ : = κ ( π ∗ T ∞ m ) ∈ [ S (det( T ∞ m )) Ξ , U (1)] Z / [ S (det( T ∞ m )) , U (1)] Z . Naturality.
The FKMM-invariant K univ is a universal characteristic class for Vec m FKMM ( X , τ ) inthe sense that it acts naturally with respect to the homotopy classification of Theorem 2.12. Theorem 6.6 (Naturality of K univ ) . Let ( E , Θ ) be a Q -bundle of FKMM-type ( cf. Definition 3.6) overthe involutive space ( X , τ ) . Let ϕ : X → ˆ G m ( C ∞ ) be the Z -equivariant map which classifies ( E , Θ ) (up to isomorphisms). Then κ ( E ) = ˜ ϕ ∗ ( K univ ) (6.6) where the Z -equivariant map ˜ ϕ : X → S (det( T ∞ m )) verifies π ◦ ˜ ϕ = ϕ according to Diagram (6.2) .Moreover, equality (6.6) is well defined in the sense that if ˜ ϕ , ˜ ϕ are two Z -equivariant maps suchthat π ◦ ˜ ϕ j = ϕ , j = , then ˜ ϕ ∗ ( K univ ) = ˜ ϕ ∗ ( K univ ) .Proof. To establish equation (6.6) is quite easy, indeed κ ( E ) = κ ( ϕ ∗ T ∞ m ) = κ ( ˜ ϕ ∗ ◦ π ∗ T ∞ m ) = ˜ ϕ ∗ κ ( π ∗ T ∞ m ) = ˜ ϕ ∗ ( K univ ) . In particular, in the first equality we used the invariance of κ under isomorphisms and in the thirdequality we used the naturality of κ under pullbacks (cf.Theorem 3.11). Of course, the above relationalso implies ˜ ϕ ∗ ( K univ ) = κ ( E ) = ˜ ϕ ∗ ( K univ ). (cid:4) Remark 6.7.
The “well-posedness” of definition (6.6) can also be established by a direct argument.By definition K univ = κ ( π ∗ T ∞ m ) is represented by a Z -equivariant map ω univ : S (det( T ∞ m )) Ξ → U (1)induced by the diagonal section s diag restricted to the fixed point set S (det( T ∞ m )) Ξ (cf.Remark 3.9).As argued in Remark 6.2, the two pullback sections ˜ ϕ ∗ j ( s diag ) : = s diag ◦ ˜ ϕ j , j = , Z -equivariant map u : X → U (1). The pullback classes ˜ ϕ ∗ j ( K univ ) are, byconstruction, represented by the maps ˜ ϕ ∗ j ( ω univ ) : = ω univ ◦ ˜ ϕ j | X τ induced by the restricted sections˜ ϕ ∗ j ( s diag ) | X τ and the di ff erence between these two maps is exactly the multiplication by u | X τ . However,this ambiguity is removed by the quotient with respect to the action of [ X , U (1)] Z which appears inthe definition of the FKMM-invariant. ◭ Characterization of the universal FKMM-invariant.
In this section we provide a useful char-acterization of the universal invariant K univ . First of all we need an analysis of the fixed point set of thespace S (det( T ∞ m )) under the involution given by the “Real” structure det( Ξ ). Lemma 6.8.
The following topological isomorphism S (det( T ∞ m )) Ξ ≃ G m ( C ∞ ) ρ ⊔ G m ( C ∞ ) ρ holds true.Proof. By Lemma 3.5 the restriction det( T ∞ m ) | G m ( C ∞ ) ρ → G m ( C ∞ ) ρ is R -trivial and admits a canoni-cal (metric-preserving) trivializationdet G ρ : det( T ∞ m ) | G m ( C ∞ ) ρ → G m ( C ∞ ) ρ × C with related canonical section s G ρ : G m ( C ∞ ) ρ −→ S (det( T ∞ m )) | G m ( C ∞ ) ρ ≃ G m ( C ∞ ) ρ × U (1) (6.7)defined by s G ρ ( x ) = det − G ρ ( x , π : det( T ∞ m ) → G m ( C ∞ ) is equivariantit follows that S (det( T ∞ m )) Ξ ⊂ π − ( G m ( C ∞ ) ρ ) ∩ S (det( T ∞ m )) = S (det( T ∞ m )) | G m ( C ∞ ) ρ . On the other side LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 39 the inclusion det( T ∞ m ) | G m ( C ∞ ) ρ ⊂ det( T ∞ m ) restricted at level of sphere-bundle and the isomorphismgiven by det G ρ provide S (det( T ∞ m )) Ξ = (cid:0) S (det( T ∞ m )) | G m ( C ∞ ) ρ ) Ξ ≃ G m ( C ∞ ) ρ × {± } . (6.8) (cid:4) Equation (6.4) follows from Lemma 6.8 and the isomorphism G m ( C ∞ ) ρ ≃ G m ( H ∞ ) discussed inSection 2.4. Theorem 6.9.
The following isomorphism [det( T ∞ m ) Ξ , U (1)] Z / [det( T ∞ m ) , U (1)] Z ≃ Z (6.9) holds true and under this identification K univ = − corresponds to the non trivial element of Z .Proof. First of all we notice that it is enough to show the isomorphism (6.9) in order to conclude also K univ = −
1. Indeed, this is a consequence of the naturality property proved in Theorem 6.6 and theexistence of non-trivial Q -bundles (e.g.Vec m Q ( ˜ S ) ≃ Z ).As a consequence of the first isomorphism in (3.1), the proof of (6.9) is equivalent to showCoker (cid:0) det( T ∞ m ) | det( T ∞ m ) Ξ , Z (1) (cid:1) ≃ Z . Due to Lemma 6.8 we know that det( T ∞ m ) has two connected components and so we only need toprove H (det( T ∞ m ) , Z ) = GysinsequenceH k ( G m ( C ∞ ) , Z ) π ∗ −→ H k (det( T ∞ m ) , Z ) ı k −→ H k − ( G m ( C ∞ ) , Z ) ∪ c −→ H k + ( G m ( C ∞ ) , Z ) −→ . . . for the complex line bundle π : det( T ∞ m ) → G m ( C ∞ ). The map on the third arrow is justified bythe equality c (det( T ∞ m )) = c ( T ∞ m ) and by the fact that c j ( T ∞ m ) : = c j ∈ H j ( G m ( C ∞ ) , Z ) are, bydefinition, the generators of the cohomology ring H • ( G m ( C ∞ ) , Z ) ≃ Z [ c , . . . , c m ]. When k = ∪ c is an isomorphism, hence ı =
0. Since H ( G m ( C ∞ ) , Z ) =
0, this implies H (det( T ∞ m ) , Z ) = (cid:4) Although not necessary, we find instructive to compute K univ = − s diag (6.5) on the fixedpoint set S (det( T ∞ m )) Ξ ≃ G m ( C ∞ ) ρ × {± } . The last isomorphism says that we can identify each pointin S (det( T ∞ m )) Ξ with a pair ( x , ǫ ) ∈ G m ( C ∞ ) ρ × {± } and in view of the fact that s diag takes values inthe circle bundle S ( π ∗ det( T ∞ m )) (cf.Remark 6.4) we can write S (det( T ∞ m )) Ξ ∋ ( x , ǫ ) s diag −→ (cid:0) ( x , ǫ ) , ( x , ǫ ) (cid:1) ∈ S ( π ∗ det( T ∞ m )) . (6.10)On the other side the canonical equivariant section s G ρ (6.7) can be represented as a map G m ( C ∞ ) ρ ∋ x s G ρ −→ ( x , ∈ S (det( T ∞ m )) Ξ in view of the equality (6.8). The pullback section π ∗ s G ρ : S (det( T ∞ m ) | G m ( C ∞ ) ρ −→ π ∗ det( T ∞ m )defined by the diagram (6.3) is still isometric, and when restricted to S (det( T ∞ m )) Ξ (cf.equation 6.8) itleads to S (det( T ∞ m )) Ξ ∋ ( x , ǫ ) π ∗ s G ρ −→ (cid:0) ( x , ǫ ) , ( x , (cid:1) ∈ S ( π ∗ det( T ∞ m )) . (6.11)A comparison between the diagonal section (6.10) and the canonical section (6.11) shows that thedi ff erence between the two is a map ω univ : S (det( T ∞ m )) Ξ → U (1) such that ω univ ( x , ǫ ) = ǫ . This map,which represents the universal FKMM-invariant, provides a representative for the non-trivial elementin (6.9). A ppendix A. C ondition for a Z - value FKMM- invariant
By construction, the FKMM-invariant κ ( E ) associated with a Q -bundle ( E , Θ ) over ( X , τ ) takesvalues in the cokernel Coker (cid:0) X | X τ , Z (1) (cid:1) : = H Z (cid:0) X τ , Z (1) (cid:1) / r (cid:0) H Z ( X , Z (1)) (cid:1) . This fact follows from a comparison between Definition 3.8 and Lemma 3.1. It is interesting to haveconditions on ( X , τ ) which assure that Coker j (cid:0) X | X τ , Z (1) (cid:1) reduces to the simplest (non-trivial) abeliangroup Z . Proposition A.1.
Let ( X , τ ) be an involutive space and assume that: (a) X verifies condition 0 of Definition 1.1; (b) X τ , ∅ and consists of a finite number N of path-connected components;Under these assumptions a necessary condition for Coker (cid:0) X | X τ , Z (1) (cid:1) ≃ Z is: (c.1) H ( X , Z ) ≃ Z b and N b + .and a su ffi cient condition is: (c.2) b = and N = .Proof. First of all, let us recall the exact sequence (cf.Proposition 2.3 in [Go]) H Z ( X , Z (1)) −→ H ( X , Z ) −→ H Z ( X , Z ) −→ H Z ( X , Z (1)) −→ H ( X , Z ) −→ . . . Since X is connected and has at least one fixed point, the exact sequence above reduces to0 −→ ˜ H Z ( X , Z (1)) −→ ˜ H ( X , Z ) = H ( X , Z ) ≃ Z b where ˜ H j is the standard notation for the reduced cohomology groups. Therefore, ˜ H Z ( X , Z (1)) is asubgroup of Z b and so ˜ H Z ( X , Z (1)) ≃ Z b − for some b − b . Moreover, H Z ( X τ , Z (1)) ≃ Z N with a Z summand for each connected component. This is evident from the isomorphism H Z ( X τ , Z (1)) ≃ Map( X τ , Z ) which is a consequence of the first of (3.1). Let us now estimate the size of H Z ( X , Z (1)) ≃ Z ⊕ Z b − in H Z ( X τ , Z (1)) ≃ Z N under the restriction map r . Evidently, r (cid:0) H Z ( X , Z (1)) (cid:1) ≃ Z a for someinteger 0 a N . On the other hand, the direct summand H Z ( {∗} , Z (1)) ≃ Z in H Z ( X , Z (1)) ≃ [ X , U (1)] Z consists of constant maps X → ±
1. Hence the image of this Z summand under r is anon-trivial subgroup Z in H Z ( X τ , Z (1)). The remaining summand Z b − in H Z ( X , Z (1)) has an imageunder r which can be at most isomorphic to Z b − . Therefore, one has the inequality a + b − . Thecondition Coker (cid:0) X | X τ , Z (1) (cid:1) ≃ Z is equivalent to a = N − N b − + b +
2. In the case N = b = H Z ( X , Z (1)) ≃ Z and H Z ( X τ , Z (1)) ≃ Z .Both these groups are represented by constant maps with values ± H Z ( X , Z (1)) on H Z ( X τ , Z (1)) is diagonal. This implies that Coker (cid:0) X | X τ , Z (1) (cid:1) ≃ Z . (cid:4) ( X , τ ) N b N b + S S d ( d >
2) 2 0 yes˜ T T d ( d >
3) 2 d d noT able A.1.
According to Preposition A.1 all the TR-spheres ˜ S d with d > Z . In the case of TR-tori only ˜ T fulfills the necessarycondition (c.1) but not the su ffi cient condition (c.2). A formula for the cokernel of ˜ T d can be explicitly derived. LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 41
Proposition A.2.
Let ˜ T d = ( T d , τ ) the involutive torus described in Definition 1.4, then Coker (cid:0) ˜ T d | ( ˜ T d ) τ , Z (1) (cid:1) ≃ Z d − ( d + ∀ d > . sketch of. One starts with H Z (cid:0) ˜ T d , Z (1) (cid:1) ≃ Z ⊕ Z d and H Z (cid:0) ( ˜ T d ) τ , Z (1) (cid:1) ≃ Z d (e.g.one can use therecursive relations [DG1, eq. (5.9)]). Let 0 a d be an integer such that r (cid:0) H Z (cid:0) ˜ T d , Z (1) (cid:1)(cid:1) ≃ Z a where r is the restriction map in cohomology induced by the inclusion ı : ( ˜ T d ) τ ֒ → ˜ T d . To concludethe proof is enough to show that a = d +
1. Since Z is a field, we can think of Z ⊕ . . . ⊕ Z as avector space over Z . We recall that the Z -summand in H Z (cid:0) ˜ T d , Z (1) (cid:1) is generated by the constant map ǫ : ˜ T d → − Z d -summand by the d canonical projections π j : ˜ T d = ˜ T × . . . × ˜ T → ˜ T ≃ U (1).The set ( ˜ T ) τ contains two fixed points which can be parametrized with ±
1. This leads to a bijectionbetween ( ˜ T d ) τ and Z d . Let us fix an order for the 2 d points in Z d , i.e. Z d = { v , . . . , v d } wherev k : = ( v k , . . . , v dk ) with v jk ∈ {± } , ( k = , . . . , d , j = , . . . , d ). The map r , which coincides withthe evaluations on the fixed points, sends ǫ to a vector r ( ǫ ) ∈ Z d represented as r ( ǫ ) = ( − , . . . , − d vectors r ( π j ) ∈ Z d are given by r ( π j ) : = ( π j (v ) , . . . , π j (v d )) with π j (v k ) : = v jk . Thelinear independence of { r ( ǫ ) , r ( π j ) , . . . , r ( π d ) } can be checked by the help of the Gauss elimination andthis shows that a = d + (cid:4) The proof of Proposition 4.9.
Let ( X , τ ) be an involutive space which verifies Assumption 4.8 and X τ = { x , . . . , x N } the fixed point set for some integer N >
0. Let us fix some notation: For each x j ∈ X τ let D j ⊂ X be a disk (closed, contractible set with boundary ∂ D j ≃ S ) which contains x j .We can choose su ffi ciently small disks D j such that D i ∩ D j = ∅ if i , j and τ ( D j ) = D j (this is aconsequence of the slice Theorem [Hs, Chapter I, Section 3] as in the proof of Lemma 4.10). Let usset D : = S Ni = D i and X ′ : = X \ Int( D ). By construction X ′ is a manifold with boundary ∂ D ≃ S Ni = S i on which the involution τ acts freely. The orbit space X /τ has boundary ∂ ( X /τ ) = ( ∂ X ) /τ .As a first step, let us prove that N = n for some integer n >
0. From the assumptions it followsthat a choice of a Riemannian metric on X makes it into a Riemann surface (a complex manifold ofdimension 1) without boundary. By means of the average construction we can assume without lossof generality that the metric is τ -invariant. Because τ : X → X preserves the orientation, we canthink of it as a holomorphic map. Then, the quotient X /τ also gives rise to a Riemann surface, and theprojection π : X → X /τ is holomorphic. In particular π is a ramified double covering with N branchingpoints x , . . . , x N . The Riemann-Hurwicz formula [GH, Chapter 2] tells us4 g ( X /τ ) = g ( X ) + − N , where g denotes the genus of the related Riemann surface, hence N = n has to be even. The Riemannsurface X ′ has genus g = g ( X ) and 2 n boundary components, hence H k ( X ′ , Z ) ≃ Z if k = Z g + n ) − if k =
10 if k > . (A.1)The Riemann surface X ′ /τ is obtained by removing 2 n disks around the branching points, hence g ( X ′ /τ ) = ( g + − n ) and it has 2 n boundary components. This implies H k ( X ′ /τ, Z ) ≃ Z if k = Z g + n if k =
10 if k > . (A.2)Now, we can prove that H Z ( X , Z (1)) =
0. We start with the exact sequence (cf.Proposition 2.3 in[Go]) H k ( X ′ /τ, Z ) ≃ H k Z ( X ′ , Z ) → H k ( X ′ , Z ) → H k Z ( X ′ , Z (1)) → H k + Z ( X ′ , Z ) ≃ H k + ( X ′ /τ, Z ) where we used the fact that the action of τ on X ′ is free. From (A.2) it follows that H k Z ( X ′ , Z (1)) ≃ H k ( X ′ , Z ) for all k > H k Z ( X ′ , Z (1)) = k >
2. Next, we use theMeyer-Vietoris sequence for { X ′ , D } , i.e. H Z ( X ′ , Z (1)) ⊕ H Z ( D , Z (1)) ∆ → H Z ( X ′ ∩ D , Z (1)) → H Z ( X , Z (1)) → H Z ( X ′ , Z (1)) ⊕ H Z ( D , Z (1))where the di ff erence homomorphism ∆ is given by ∆ ( a X ′ , a D ) = − a X ′ | X ′ ∩ D + a D | X ′ ∩ D for a X ′ ∈ H Z ( X ′ , Z (1)) ≃ [ X ′ , U (1)] Z and a D ∈ H Z ( D , Z (1)) ≃ [ D , U (1)] Z (here we used the isomorphism(3.1)). Since each connected component D j of D is equivariantly contractible H Z ( D , Z (1)) ≃ Z n consists of locally constant function with values in {± } . On the other side, the action of τ is free on X ′ ∩ D ≃ F nj = S and ( X ′ ∩ D ) /τ ≃ F nj = S since τ acts isomorphically to the antipodal map on eachcomponent. This yields H k Z ( X ′ ∩ D , Z (1)) ≃ H k ( S , Z (1)) n ≃ Z n if k =
10 if k , . Since also H Z ( X ′ ∩ D , Z (1)) can be represented by locally constant functions from X ′ ∩ D to {± } itfollows that the map H Z ( D , Z (1)) → H Z ( X ′ ∩ D , Z (1)) is surjective. Moreover, H Z ( X ′ , Z (1)) ≃ H Z ( D , Z (1)) ≃ L nj = H Z ( {∗} , Z (1)) ≃ H Z ( X , Z (1)) = H Z ( X | X τ , Z (1)) ≃ Z . By the homotopy axiom and the excisionaxiom for the Borel cohomology theory, we get isomorphisms H Z ( X | X τ , Z (1)) ≃ H Z ( X | D , Z (1)) ≃ H Z ( X ′ | ∂ X ′ , Z (1)) . Let us consider the exact sequence [Go, Proposition 2.3] Z ≃ H Z ( X ′ | ∂ X ′ , Z ) f → H ( X ′ | ∂ X ′ , Z ) → H Z ( X ′ | ∂ X ′ , Z (1)) → H Z ( X ′ | ∂ X ′ , Z ) = f which “forgets” the Z -action. We recall that τ acts freely on X ′ and f can beidentified with the pullback by the projection π : X ′ → X ′ /τ . Then, after an application of the Poincar´eduality and the universal coe ffi cient theorem to (A.2) we get H k Z ( X ′ | ∂ X ′ , Z ) ≃ H k ( X ′ /τ | ∂ ( X ′ /τ ) , Z ) ≃ Z g + n if k = Z if k =
20 if k , , . In the same way, one deduces from (A.1) that H ( X ′ | ∂ X ′ , Z ) ≃ Z . Because the projection π : X ′ → X ′ /τ is a honest double covering, f is a two-to-one map. This allows us to conclude that H Z ( X ′ | ∂ X ′ , Z (1)) ≃ Z .A ppendix B. S patial parity and quaternionic vector bundles
Let us introduce the spatial parity operator ˆ I defined by ( ˆ I ψ )( x ) = ψ ( − x ) for vectors ψ in L ( R d , d x ) ⊗ C L (continuous case) or in ℓ ( Z d ) ⊗ C L (periodic case). A Hamiltonian ˆ H is parity-invariant if ˆ I ˆ H ˆ I = ˆ H .If ˆ H posses also a -TR symmetry ˆ Θ then the combination ˆ Θ ′ : = ˆ I ˆ Θ provides an anti-linear fiber-preserving map of the associated Bloch-bundle. If it happens that the two symmetries are indepen-dently broken but the combination ˆ Θ ′ survives as a fundamentally symmetry, then Proposition 2.4assures that the topological phase of ˆ H is classified by a quaternionic vector bundle.The classification of quaternionic vector bundles in dimension d = , . . . , KSp -theory (see Table C.2 & Table C.3) one obtainsVec m H ( T d ) ≃ Vec m H ( S d ) ≃ ( d = , , Z if d = ∀ m ∈ N . LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 43 A ppendix C. A n overview to KQ - theory According to [Du, Se] we denote with KQ ( X , τ ) the Grothendieck group of Q -vector bundles overthe involutive space ( X , τ ). We refer also to [LM, Section 3.6] for pedagogical description of the KQ -theory and its relation with the Atiyah’s KR -theory. By a restriction to the fixed point set X τ (hereafterassumed non empty) one has a homomorphism KQ ( X , τ ) → KQ ( X τ , Id X ) ≃ KSp ( X τ ), where KSp denotes the K -theory for vector bundles with quaternionic fiber (see e.g.[Hu]).The reduced group g KQ ( X , τ ) is the kernel of the homomorphism KQ ( X , τ ) → KQ ( {∗} ) where ∗ ∈ X is a τ -invariant base point. When X is compact one has the usual relation KQ ( X , τ ) ≃ g KQ ( X , τ ) ⊕ KQ ( {∗} ) ≃ g KQ ( X , τ ) ⊕ Z (C.1)where we used KQ ( {∗} ) ≃ KSp ( {∗} ) ≃ Z . Moreover, the isomorphism˜ KR ( X , τ ) ≃ Vec R ( X , τ ) : = [ m ∈ N Vec m R ( X , τ ) (C.2)establishes the fact that ˜ KQ ( X , τ ) provides the description for Q -bundles in the stable regime (i.e.whenthe rank of the fiber is assumed to be su ffi ciently large).As for the KR -theory also the KQ -theory can be endowed with a grading structure as follows: firstof all one introduces the groups KQ j ( X , τ ) : = KQ ( X × D , j ; X × S , j , τ × ϑ ) KQ − j ( X , τ ) : = KQ ( X × D j , ; X × S j , , τ ) j = , , , , . . . where D p , q and S p , q are the unit ball and unit sphere in the space R p , q : = R p ⊕ i R q made involutiveby the complex conjugation (cf.[DG1, Exampe 4.2]). The relative group KQ ( X ; Y , τ ) of an involutivespace ( X , τ ) with respect to a τ -invariant subset Y ⊂ X is defined as g KQ ( X / Y , τ ) and corresponds to theGrothendieck group of Q -bundles over X which vanish on Y . The negative groups KQ − j agree withthe usual suspension groups since the spaces D j , and S j , are invariant. The positive groups KQ j are“twisted” suspension groups since D , j and S , j are endowed with the Z -action of the antipodal map ϑ . With respect to this grading the KQ groups are 8-periodic, i.e. KQ j ( X , τ ) ≃ KQ j + ( X , τ ) , j ∈ Z . If X has fixed points one can extend the isomorphism (C.1) for negative groups: KQ − j ( X , τ ) ≃ g KQ − j ( X , τ ) ⊕ KQ − j ( {∗} ) , j = , , , , . . . (C.3)where KQ − j ( {∗} ) ≃ KSp − j ( {∗} ) for all j . j = j = j = j = j = j = j = j = KQ − j ( {∗} ) Z Z Z Z able C.1.
The table is calculated using KQ − j ( {∗} ) ≃ KSp − j ( {∗} ) ≃ g KO ( S j + ) [Kar, Theorem 5.19,Chapter III]. The Bott periodicity implies KQ − j ( {∗} ) ≃ KQ − j − ( {∗} ). Finally, the following connection with KR -theory KQ j ( X , τ ) ≃ KR j ± ( X , τ ) , g KQ j ( X , τ ) ≃ g KR j ± ( X , τ ) j ∈ Z (C.4)has been proved in [Du].We can use equation (C.4) to compute the KQ -theory for the involutive sphere ˜ S d ≡ ( S d , τ ) bymeans of the KR -theory. Indeed, one has that g KQ ( ˜ S d ) ≃ g KR ( ˜ S d ) ≃ g KR ( ˜ S d + ) mod . d = d = d = d = d = d = d = d = e K ( S d ) 0 Z Z Z Z g KQ ( ˜ S d ) 0 Z Z Z Z g KSp ( S d ) 0 0 0 Z Z Z Z T able C.2.
The reduced KQ groups are computed with the help of the formula (C.5) and the TableB.3 in [DG1]. The reduced K groups for complex and quaternionic vector bundles are computed in[Hu, Chapter 9, Corollary 5.2]. We recall the relation g KO ( S d ) ≃ g KSp ( S d + ) (mod. 8) which relates the K -theories of real and quaternionic vector bundles. In order to compute the KQ groups for TR-tori we can adapt the formula [DG1, eq. (B.7)] with thehelp of (C.4) in order to obtain KQ − j ( ˜ T d ) ≃ d M n = (cid:16) KQ − ( j − n ) ( {∗} ) (cid:17) ⊕ ( dn ) . (C.6) d = d = d = d = d = d = d = d = e K ( T d ) 0 Z Z Z Z Z Z Z g KQ ( ˜ T d ) 0 Z Z Z ⊕ Z Z ⊕ Z Z ⊕ Z Z ⊕ Z Z ⊕ Z g KSp ( T d ) 0 0 0 Z Z ⊕ Z Z ⊕ Z Z ⊕ Z Z ⊕ Z T able C.3.
The groups g KQ ( ˜ T d ) are obtained from equation (C.6) and the isomorphism (C.1).In the quaternionic case a recursive formula can be derived from the isomorphism KSp − j ( S × Y ) ≃ KSp − ( j + ( Y ) ⊕ KSp − j ( Y ) . R eferences [AB] Atiyah, M. F.; Bott, R.: Ontheperiodicitytheoremforcomplexvectorbundles. Acta Math. , 229-247 (1964)[AP] Allday, C.; Puppe, V.: Cohomological Methods in Transformation Groups . Cambridge University Press, Cam-bridge, 1993[ASV] Avila, J. C.; Schulz-Baldes, H.; Villegas-Blas, C.: Topological Invariants of Edge States for Periodic Two-DimensionalModels. Math. Phys. Anal. Geom. , 137-170 (2013)[At1] Atiyah, M. F.: K -theoryandreality. Quart. J. Math. Oxford Ser. (2) , 367-386 (1966)[At2] Atiyah, M. F.: K-theory . W. A. Benjamin, New York, 1967[AZ] Altland, A.; Zirnbauer, M.: Non-standard symmetry classes in mesoscopic normal-superconducting hybridstructures, Phys. Rev.
B 55 , 1142-1161 (1997)[BHH] Biswas, I.; Huisman, J.; Hurtubise J.: The moduli space of stable vector bundles over a real algebraic curve .Math. Ann. , 201-233 (2010)[Bo] Borel, A.:
Seminar on transformation groups , with contributions by G. Bredon, E. E. Floyd, D. Montgomery,R. Palais. Annals of Mathematics Studies , Princeton University Press, Princeton, 1960[DG1] De Nittis, G.; Gomi, K.: Classificationof“Real”Bloch-bundles: Topological Quantum Systemsoftype AI . J.Geom. Phys. , 303-338 (2014)[DG2] De Nittis, G.; Gomi, K.: Di ff erential geometric invariants for time-reversal symmetric Bloch-bundles: the“Real”case. . E-print arXiv:1502.01232 (2015)[DG3] De Nittis, G.; Gomi, K.: Di ff erential geometric invariants for time-reversal symmetric Bloch-bundles: the“Quaternionic”case. . In preparation[DK] Davis, J. F.; Kirk, P.: Lecture Notes in Algebraic Topology . AMS, Providence, 2001[DL] De Nittis, G.; Lein, M.: Topologicalpolarizationingraphene-like systems. J. Phys. A , 385001 (2013)[DSLF] Dos Santos, P. F.; Lima-Filho, P.: Quaternionic algebraic cycles and reality. Trans. Amer. Math. Soc. ,4701-4736 (2004)[Du] Dupont, J. L.: SymplecticBundlesand KR -Theory. Math. Scand. , 27-30 (1969) LASSIFICATION OF “QUATERNIONIC”
BLOCH-BUNDLES 45 [Ed] Edelson, A. L.: RealVectorBundles andSpaceswithFreeInvolutions. Trans. Amer. Math. Soc. , 179-188(1971)[EMV] Essin, A. M.; Moore, J. E.; Vanderbilt, D. Magnetoelectric PolarizabilityandAxionElectrodynamics inCrys-tallineInsulators. Phys. Rev. Lett. , 146805 (2009)[FK] Fu, L.; Kane, C. L.: Timereversalpolarizationanda Z adiabaticspinpump. Phys. Rev. B , 195312 (2006)[FKM] Fu, L.; Kane, C. L.; Mele, E. J.: TopologicalInsulatorsinThreeDimensions. Phys. Rev. Lett. , 106803 (2007)[FKMM] Furuta, M.; Kametani, Y.; Matsue, H.; Minami, N.: Stable-homotopy Seiberg-Witten invariants and Pin bor-disms. UTMS Preprint Series 2000, UTMS 2000-46. (2000)[GH] Gri ffi ths, P.; Harris, J.: Principles of algebraic geometry . Wiley, New York, 1978[Go] Gomi, K.: AvariantofK-theoryandtopologicalT-dualityforRealcirclebundles. Commun. Math. Phys. ,923-975 (2015)[GP] Graf, G. M.; Porta, M.: Bulk-Edge Correspondence for Two-Dimensional Topological Insulators. Commun.Math. Phys. , 851-895 (2013)[Hat] Hatcher, A.:
Algebraic Topology . Cambridge University Press, Cambridge, 2002[HPB] Hughes, T. L.; Prodan, E.; Bernevig, B. A.: Inversion-symmetric topological insulators. Phys. Rev. B ,245132 (2011)[Hs] Hsiang, W. Y.: Cohomology Theory of Topological Transformation Groups . Springer-Verlag, Berlin, 1975[Hu] Husemoller, D.:
Fibre bundles . Springer-Verlag, New York, 1994[Kah] Kahn, B.: Construction de classes de Chern ´equivariantes pour un fibr´e vectoriel R´eel. Comm. Algebra. ,695-711 (1987)[Kar] Karoubi, M.: K-Theory. An Introduction . Springer-Verlag, New York, 1978.[KM1] Kane, C. L.; Mele, E. J.: QuantumSpinHallE ff ectinGraphene. Phys. Rev. Lett. , 226801 (2005)[KM2] Kane, C. L.; Mele, E. J.: Z Topological OrderandtheQuantum SpinHallE ff ect. Phys. Rev. Lett. , 146802(2005)[Ki] Kitaev, A.: Periodictablefortopologicalinsulatorsandsuperconductors. AIP Conf. Proc. , 22-30 (2009)[LM] Luke, G.; Mishchenko, A. S.: Vector Bundles and Their Applications . Kluwer Academic Publishers, 1998[LLM] Lawson Jr., H. B.; Lima-Filho, P.; Michelsohn, M.-L.: Algebraiccyclesandtheclassicalgroups.PartII:Quater-nioniccycles. Geometry & Topology , 1187-1220 (2005)[LY] Lin, H.; Yau, S.-T.: Onexoticsphere fibrations,topologicalphases, andedge statesinphysical systems. Int. J.Mod. Phys. B , 1350107 (2013)[Ma] Matumoto, T.: On G -CWcomplexesandatheoremofJ.H.C.Whitehead. J. Fac. Sci. Univ. Tokyo , 363-374(1971)[MB] Moore, J. E.; Balents, L.: Topological invariants of time-reversal-invariant band structures. Phys. Rev. B ,121306 (2007)[MHZ] Maciejko, J.; Hughes, T. L.; Zhang, S.-C.: The Quantum Spin HallE ff ect. Annu. Rev. Condens. Matter Phys. , 31-53 (2011)[MS] Milnor, J.; Stashe ff , J. D.: Characteristic Classes . Princeton University Press, 1974[Ro] Roy, R.: Topological phases and the quantum spin Hall e ff ect in three dimensions. Phys. Rev. B , 195322(2009)[Se] Seymour, R. M.: TheRealK-TheoryofLieGroupsandHomogeneousSpaces. Quart. J. Math. Oxford , 7-30(1973)[SRFL] Schnyder, A. P.; Ryu, S.; Furusaki, A.; Ludwig, A. W. W.: Classification of topological insulators and super-conductorsinthreespatialdimensions. Phys. Rev. B 78 , 195125 (2008)[Va] Vaisman, I.: Exoticcharacteristicclassesofquaternionic bundles. Israel J. Math. , 46-58 (1990)(De Nittis) D epartment M athematik , U niversit ¨ at E rlangen -N¨ urnberg , G ermany E-mail address : [email protected] (Gomi) D epartment of M athematical S ciences , S hinshu U niversity , N agano , J apan E-mail address ::