The cohomological nature of the Fu-Kane-Mele invariant
aa r X i v : . [ m a t h - ph ] O c t THE COHOMOLOGICAL NATURE OF THE FU-KANE-MELE INVARIANT
GIUSEPPE DE NITTIS AND KIYONORI GOMIA bstract . In this paper we generalize the definition of the FKMM-invariant introduced in [DG2] forthe case of “Quaternionic” vector bundles over involutive base spaces endowed with free involution orwith a non-finite fixed-point set. In [DG2] it has already be shown how the FKMM-invariant providesa cohomological description of the Fu-Kane-Mele index used to classify topological insulators in classAII. It follows that the FKMM-invariant described in this paper provides a cohomological generaliza-tion of the Fu-Kane-Mele index which is applicable to the classification of protected phases for othertype of topological quantum systems (TQS) which are not necesarily related to models for topologicalinsulators (e.g. the two-dimensional models of adiabatically perturbed systems discussed in [GR]). Asa byproduct we provide the complete classification of “Quaternionic” vector bundles over a big class oflow dimensional involutive spheres and tori. C ontents
1. Introduction 22. FKMM-invariant: Definitions and properties 82.1. Basic definitions 82.2. The determinant construction 92.3. A short reminder of the equivariant Borel cohomology 102.4. A geometric model for the relative equivariant cohomology 112.5. Generalized FKMM-invariant 122.6. Universal FKMM-invariant 143. “Quaternionic” line bundles and FKMM-invariant 163.1. The “Quaternionic” Picard torsor 173.2. Classification over involutive spheres 184. Topological classification of “Quaternionic” vector bundles 214.1. Stable rank condition 224.2. Injectivity in low dimension 234.3. The question of the surjectivity of the FKMM-invariant 264.4. Classification over low dimensional involutive spheres 274.5. Classification over low dimensional involutive tori 295. Application to prototype models of topological quantum systems 33Appendix A. About the equivariant cohomology of the Classifying space 37Appendix B. About the equivariant cohomology of involutive spheres 41B.1. TR-involution 41B.2. Antipodal involution 42B.3. More general involutions 44Appendix C. About the equivariant cohomology of involutive tori 45C.1. The free-involution cases 46C.2. The cases with non-empty fixed point sets 46References 50
MSC2010
Primary: 57R22; Secondary: 55N25, 53C80, 19L64.
Keywords. “Quaternionic” vector bundle, FKMM-invariant, Fu-Kane-Mele index, spectral bundle, Topologicalinsulators.
1. I ntroduction
In its simplest incarnation, a
Topological Quantum System (TQS) is a continuous matrix-valued map X ∋ x H ( x ) ∈ Mat( C d ) (1.1)defined on a topological space X sometimes called Brillouin zone . Although the precise definition ofTQS requires some more ingredients and can be stated in a more general form [DG1, DG2, DG3], onecan certainly state that the most relevant feature of these systems is the nature of the spectrum whichis made by continuous (energy) bands. It is exactly the family of eigenprojectors emerging from thispeculiar band structure which may encode information that are of topological nature. The study andthe classification of the topological properties of TQS have recently risen to the level of “hot topic”in mathematical physics because its connection with the study of topological insulators in condensedmatter (we refer to the two reviews [HK] and [AF] for a modern overview about topological insulatorsand an updated bibliography of the most relevant publications on the subject). However, systems like(1.1) are ubiquitous in the mathematical physics and are not necessarily linked with problem comingfrom condensed matter.For the sake of simplicity we consider here a very specific and simple realization of a TQS: H ( x ) : = N X j = F j ( x ) Σ j . (1.2)The { Σ , . . . , Σ N } ∈ Mat( C N ) define a (non-degenerate) irreducible representation of the complexCli ff ord algebra Cl C (2 N +
1) and the real-valued functions F j : X → R , with j = , , . . . , N , areassumed to be continuous. Under the gap conditionQ ( x ) : = N X j = F j ( x ) > x H ( x ) with a complexvector bundle E → X of rank 2 N − , called spectral bundle . For the details of this identification werefer to Section 5 as well as to [DL1, Section IV] or [DG1, Section 2] (the specific model (1.2) isconsidered in [DL2, Section 4]). The considerable consequence of the duality between gapped TQS’sand spectral bundles is that one can classify the possible topological phases of the TQS by means ofthe elements of the set Vec N − C ( X ) given by the isomorphism classes of all rank 2 N − vector bundlesover X . Therefore, in this general setting, the classification problem for the topological phases of aTQS like (1.2) can be traced back to a classic problem in topology which has been elegantly solved in[Pe]. For instance, in the (physically relevant) case of a low dimensional base space X , the completeclassification of the topological phases of (1.2) is provided byVec N − C ( X ) c ≃ H (cid:0) X , Z (cid:1) , dim( X ) X and the map c isthe first Chern class .The problem of the classification of the topological phases becomes more interesting, and challeng-ing, when the TQS is constrained by the presence of certain symmetries or pseudo-symmetries . Amongthe latter, the time-reversal symmetry (TRS) attracted recently a considerable interest in mathematicalphysics community. A system like (1.2) is said to be time-reversal symmetric if there is an involution τ : X → X on the base space and an anti -unitary map Θ such that Θ H ( x ) Θ ∗ = H (cid:0) τ ( x ) (cid:1) , ∀ x ∈ X Θ = ǫ N ǫ = ± . (1.5) Spectral bundles obtained as a result of a Bloch-Floquet transform, are sometimes called
Bloch bundle [PST, Pa2]
COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 3
The case ǫ = + E turns out to be equipped with an additional structure named “Real” by M. F. Atiyah in [At1].Therefore, in the presence of an even TRS the classification problem of the topological phases isreduced to the study of the set Vec N − R ( X , τ ) of isomorphism classes of rank 2 N − vector bundles over X endowed with a “Real” structure. This problem has been analyzed and solved in [DG1]. In particular,in the low dimensional case one has thatVec N − R ( X , τ ) c R ≃ H Z (cid:0) X , Z (1) (cid:1) , dim( X ) . (1.6)Here, in the right-hand side there is the equivariant Borel cohomology with local coe ffi cient system Z (1) of the involutive space ( X , τ ) (see Section 2.3 and references therein for more details) and theisomorphism is given by first “Real” Chern class c R introduced for the first time by B. Kahn in [Kah](see also [DG1, Section 5.6]). Note the close similarity between the equations (1.4) and (1.6).The case ǫ = − E acquires an additional structure called “Quaternionic” (or symplectic [Du]) and the topologicalphases of a TQS with an odd TRS turn out to be labelled by the set Vec N − Q ( X , τ ) of isomorphismclasses of rank 2 N − vector bundles over X with “Quaternionic” structure.The study of systems with an odd TRS is more interesting, and for several reasons also harder, thanthe case of an even TRS. Historically, the fame of these fermionic systems begins with the seminalpapers [KM, FKM] by L. Fu, C. L. Kane and E. J. Mele. The central result of these works is the inter-pretation of a physical phenomenon called Quantum Spin Hall E ff ect as the evidence of a non-trivialtopology for TQS constrained by an odd TRS. Specifically, the papers [KM, FKM] are concernedabout the study of systems like (1.2) (with N =
2) where the base space is a torus of dimension 2 or3 endowed with a “time-reversal” involution. These involutive
TR-tori are the quotient spaces ( R / Z ) d endowed with the quotient involution defined on R by x → − x (we denote these spaces with T , d , according to a general notation which will be clarified in equation (1.10)). The distinctive aspect of thespaces T , d , is the existence of a fixed point set formed by 2 d isolated points. The latter plays a crucialrole in the classification scheme proposed in [KM, FKM] where the di ff erent topological phases aredistinguished by the signs that a particular function d E defined by the spectral bundle (essentially theinverse of a normalized pfa ffi an) takes on the 2 d points fixed by the involution. These numbers areusually known as Fu-Kane-Mele indices .In the last years the problem of the topological classification of systems with an odd TRS has beendiscussed with several di ff erent approaches. As a matter of fact, many (if not almost all) of theseapproaches focus on the particular cases T , , and T , , with the aim of reproducing in di ff erentway the Z -invariants described by the Fu-Kane-Mele indices. From one hand there are classificationschemes based on K-theory and
KK-theory [Ki, FM, Th, Ke, Kub, BCR, PS] or equivariant homo-topy techniques [KG, KZ] which are extremely general. In the opposite side there are constructiveprocedures based on the interpretation of the topological phases as obstructions for the constructionof continuous time-reversal symmetric frames (see [GP] for the case T , , and [FMP, MCT] for thegeneralization to the case T , , ) or as spectral-flows [CPS, DS] or as index pairings [GS] or by meansof the Wess-Zumino amplitudes [CDFG, CDFGT, MT]. All these approaches, in our opinion, presentsome limitations. The K -theory is unable to distinguish the “spurious phases” (possibly) present out-side of the stable rank regime. The homotopy calculations are non-algorithmic and usually extremelyhard. The use of the KK -theory up to now is restricted only to the non-commutative version of theinvolutive Brillouin tori T , , and T , , . A similar consideration holds for the recipes for the “hand-made” construction of global continuous frames which are strongly dependent of the specific form ofthe involutive spaces T , , and T , , , and thus are di ffi cult to generalize to other spaces and higherdimensions. Finally, none of these approaches clearly identifies the invariant which labels the di ff erent G. DE NITTIS AND K. GOMI phases as a topological (characteristic) class , as it happens in the cases of systems with broken TRS(cf.eq. (1.4)) or with even TRS (cf.eq. (1.6)).To overcome these deficiencies we developed in [DG2] an alternative classification scheme basedon a cohomological description. By adapting an idea of M. Furuta, Y. Kametani, H. Matsue, and N.Minami [FKMM], we introduced a cohomological class κ called FKMM-invariant , that we have shownto be a characteristic class for the category of “Quaternionic” vector bundles. In the low dimensionalcase this invariant provides an injection
Vec N − Q ( X , τ ) κ −→ H Z (cid:0) X | X τ , Z (1) (cid:1) , dim( X ) . (1.7)The meaning of the cohomology group which appears in right-hand side can be guessed by looking atthe exact sequence (cf.Section 2.3) . . . −→ (cid:2) X τ , {± } (cid:3) −→ H Z (cid:0) X | X τ , Z (1) (cid:1) −→ Pic R (cid:0) X , τ (cid:1) r −→ Pic R (cid:0) X τ (cid:1) −→ . . . where [ X τ , {± } ] is the set of homotopy classes of maps from X τ into {± } , Pic R ( X , τ ) = Vec R ( X , τ ) isthe Picard group of the “Real” line bundles over ( X , τ ), Pic R ( X τ ) ≃ Pic R ( X τ , τ ) is the Picard group ofthe real line bundles over X τ and r is the restriction map.Despite its similarity with the equations (1.4) and (1.6), the equation (1.7) cannot be consideredcompletely satisfactory, at least as derived in [DG2, Theorem 1.1]. The most dramatic weakness ofthe (1.7) in its original form is that it has been proved only under the hypothesis of ( X , τ ) being anFKMM-space [DG2, Definition 1.1]. This is a strong restriction since: (i) it excludes involutive spaceswith fixed point sets X τ of dimension bigger than zero and involutive spaces with a free involution (i.e. X τ = Ø); (ii) it introduces by “brute force” the condition Pic R (cid:0) X , τ (cid:1) = H Z (cid:0) X | X τ , Z (1) (cid:1) ≃ (cid:2) X τ , {± } (cid:3) / (cid:2) X , U (1) (cid:3) Z (1.8)where [ X , U (1)] Z denotes the set of the homotopy classes of Z -equivariant maps from ( X , τ ) into U (1) endowed with the involution given by the complex conjugation. Clearly, the isomorphism (1.8)fails in general for involutive spaces ( X , τ ) which are not of FKMM-type. On the other hand, (1.8)is also the principal ingredient to link the FKMM-invariant κ with the (strong) Fu-Kane-Mele index[DG2, Theorem 4.2] through the well-known formula κ ( E ) = Y x ∈ X τ d E ( x ) , ( X , τ ) an FKMM-space . (1.9)There is also a second, more delicate, reason which makes (1.7) weaker than (1.4) or (1.6). In fact(1.7) provides, in general, only an injection while the other two are isomorphisms. This topic will bediscussed in Section 4.3.The main result of this paper is the extension of the range of validity of (1.7) to involutive spaces( X , τ ) in full generality. More precisely, we introduce a generalized version of the FKMM-invariantwhich extends the “old” invariant constructed in [DG2] to involutive spaces which are not necessary ofFKMM-type. This generalization has a pay-o ff : The (1.7) becomes meaningful (and valid) for spaceswith fixed point set of any co-dimension. This is, in our opinion, a big step forward in the theoryof the classification of topological phases for systems with odd TRS. In fact, motivated by the (1.9),we can interpret κ as the extension of the Fu-Kane-Mele indices when X τ is not simply a collection ofisolated points. Also the case X τ = Ø can be handled inside our formalism: Not only is the generalized
FKMM-invariant well defined for even-rank “Quaternonic” vector bundles over base spaces with freeinvolution, but we can also define a FKMM-invariant for odd-rank “Quaternonic” vector bundles andclassify them. The expression topological (characteristic) class is used here to denote an element which can be defined for eachelement in the (topological) category of “Quaternionic” vector bundles and which is natural with respect to the pullback inthis category.
COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 5
Before discussing some interesting consequences of the (1.7) in its generalized meaning, let usspend few words about the importance of considering spaces di ff erent form the TR-tori T , d , . If fromone hand the spaces T , d , emerge naturally as quasi-momentum spaces (a.k.a. Brillouin zone) for d -dimensional periodic electronic systems, on the other hand TQSs of type (1.2) are ubiquitous inmathematical physics and are not necessarily related to model in condensed matter (see e.g. the richmonographs [BMKNZ, CJ]). Just to give few examples let us mention that TQSs can be used to modelsystems subjected to cyclic adiabatic processes in classical and quantum mechanics [Pa1, Be], or in thedescription of the magnetic monopole [Di, YA] and the Aharonov-Bohm e ff ect [AB] or in the moleculardynamics in the context of the Born-Oppenheimer approximation [Te]. Usually, the space X plays therole of a configuration space for parameters which describe an adiabatic action of external fields ona system governed by the instantaneous Hamiltonian H ( x ). The phenomenology of these systems isenriched by the presence of certain symmetries like a TRS as in (1.5). Models of adiabatic topologicalsystems of this type have been recently investigated in [CDFG, CDFGT, GR]. In particular, in thesecond of these works [GR] the authors consider the adiabatically perturbed dynamics of a classicrigid rotor and a classical particle on a ring . In the first case the classical phase space turns out to bea two-dimensional sphere endowed with an antipodal involution induced by the TRS (we use S , forthis space). In the second case the phase space is a two dimensional torus T = S × S where the TRSinduces a time-reversal involution only on one of its factors (we denote this space by T , , ). The mainresult of [GR] consists in the classification of “Quaternionic” vector bundles over the involutive spaces S , and T , , and it is based on the analysis of the obstruction for the “handmade” construction ofa global frame. As already commented above, this technique is hard (and tricky) to extend to higherdimensions and other involutive spaces. Conversely, the classification provided by the map (1.7) turnsout to be extremely e ff ective and versatile. In fact it is algorithmic! As a matter of fact the formula(1.7) allows us to classify “Quaternionic” vector bundles over a big class of involutive spheres and toriup to dimension three extending, in this way, the results in [GR]. The cost for such a classificationamounts to just a little more than the (algorithmic) calculation of the cohomology group in (1.7). Wepostpone a synthesis of the results of this classification at the end of this introductory section. As afinal remark let us mention that recently the classification technique based on the FKMM-invariant hasbeen successfully used in [TSG] in order to classify Weyl semimetals with TRS.The map (1.7) can be used also to obtain information about the possibility of non trivial topologicalstates for systems of type (1.2) independently of the details of the functions F j . For instance, the studyof the case N = Theorem 1.1 (Necessary condition for non-trivial phases) . Consider a topological quantum systemof type 1.2 with N = and endowed with an odd TRS of type (1.5) . A necessary condition for theexistence of non-trivial phases is that F and at least two of the functions { F , F , F , F } must be not identically zero. For sake of precision, one has to refer the statement of the theorem above to the definitions and no-tations introduced in Section 5. Note that this result is just a (partial) summary of the content ofProposition 5.2 and Proposition 5.3.This paper is organized as follows: In
Section 2 we construct the generalized version of the FKMM-invariant and we prove that this is a characteristic class for the category of “Quaternionic” vectorbundles. Moreover, we show that it reduces to the “old version” of the FKMM-invariant describedin [DG2], and so to the Fu-Kane-Mele indices in the case of spaces with a discrete number of fixedpoints.
Section 3 is devoted to the study of “Quaternionic” line bundles. In
Section 4 we use the “new”FKMM-invariant to classify “Quaternionic” vector bundles of even and odd rank over low dimensionalinvolutive spaces. We also discuss the classification over involutive spheres and tori.
Section 5 isdevoted to the study of TQS of type (1.2) with N =
2. Finally,
Appendices A, B and C contain all thecohomology computations needed in the paper. These calculations are technical, but also of generalvalidity and may also be useful in other areas of the mathematical research. G. DE NITTIS AND K. GOMI
Acknowledgements.
GD’s research is supported by the grant
Iniciaci´on en Investigaci´on 2015 - N o Resum´e of the classification over involutive spheres and tori.
Let us fix some notations: The involutive sphere of type S p , q : = ( S p + q − , θ p , q ) is defined as the sphere of dimension d : = p + q − S d : = n ( k , k , . . . , k d ) ∈ R d + | k + k + . . . + k d = o endowed with the involution θ p , q ( k , k , . . . , k p − | {z } first p coord. , k p , k p + , . . . , k p + q − | {z } last q coord. ) : = ( k , k , . . . , k p − | {z } first p coord. , − k p , − k p + , . . . , − k p + q − | {z } last q coord. )The cases p = q = S d + , coincides with the sphere S d endowed with the trivial involution fixing all points. On the oppo-site side one has the space S , d + which is the sphere S d endowed with the antipodal (or free ) involution .The space S , d (denoted with the symbol ˜ S d in [DG1, DG2, DG3]) coincides with the “Brillouin zone”of condensed matter systems invariant under continuous translations and subjected to a TRS . For thisreason one often refers to θ , d as a TR-involution . For more details, the reader can refer to [DG1,Section 2]. The classification of “Quaternionic” vector bundles over these spaces in low dimension issummarized in Table 1.1. We notice that the case S , studied in [GR] is also included.With multiple products of involutive one-dimensional spheres one can define several types of invo-lutive tori . By using as building blocks the three involutive space S , , S , and S , one can define theinvolutive torus of type ( a , b , c ) by T a , b , c : = S , × . . . × S , | {z } a − times × S , × . . . × S , | {z } b − times × S , × . . . × S , | {z } c − times a , b , c ∈ N ∪ { } . (1.10)Topologically this is a torus of dimension d = a + b + c endowed with the natural product involution.The space T d , , coincides with the torus T d endowed with the trivial involution fixing all points. Thespace T , d , (denoted with the symbol ˜ T d in [DG1, DG2, DG3]) coincides with the “Brillouin zone”of condensed matter systems invariant under Z d -translations and subjected to a TRS (for more details,we refer to [DG1, Section 2]). Notice that c , d there are ( d + d +
2) possible combinations for a + b + c = d but only 2 d + T a , b , c ≃ T a + c − , b , c > . (1.11)The classification in the case of tori with fixed point ( c =
0) up to dimension three is described in Table1.2. Note that also the case T , , discussed in [GR] is included.The case with free involution (the equivalence (1.11) allows to consider only the case c =
1) issummarized in the Table 1.3.
Remark 1.2 (Flip involution) . Over T = S × S there is another interesting involution which isessentially di ff erent from the involutions described by (1.10). Consider the flip map γ : ( k , k ′ ) ( k ′ , k ) and the associated involutive space T : = ( T , γ ). This space has a non-empty fixed pointset ( T ) γ ≃ S given by the diagonal subset of points ( k , k ) and therefore it admits only even rank“Quaternionic” vector bundles. We anticipate thatVec m Q (cid:0) T (cid:1) c ≃ Z (1.12)although we will study this case in a separated paper [DG6] due to its relevance with the physics of asystem of two identical one-dimensional particles. Anyway, the justification of (1.12), which is quitelaborious, requires the computation of H Z ( T | ( T ) γ , Z (1)) and the study of the morphism whichconnects this group with H ( T , Z ) in a long exact sequence of type (2.8). Let us point out that over COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 7 p + q q = q = q = q = q = m + Q ( S , q ) Ø 0 0 2 Z + m Q ( S , q ) Ø 0 0 2 Z m Q ( S , q ) 0 0 Z Z . . . Vec m Q ( S , q ) 0 2 Z . . . Vec m Q ( S , q ) 0 0 . . . Vec m Q ( S , q ) 0 . . . T able Complete classification of “Quaternionic” vector bundles over involutive spheres of type S p , q in low dimension d = p + q −
3. The symbol Ø (empty set) denotes the impossibility ofconstructing “Quaternionic” vector bundles. The symbol 0 denotes the existence of a unique elementwhich can be identified with the trivial product bundle in the even rank case. In the odd rank case, onlypossible over the free-involution spheres S , q , the notion of trivial bundle is not well defined, and evenwhen there is only one representative (e.g.the cases q = ,
2) this is not given by a product bundlewith a product “Quaternionic” action. The classification by integers (even, odd) is attained by lookingat the first Chern class of the underlying complex vector bundle. The Z classification is given by theFKMM-invariant. The results summarized in this table are discussed in detail in Section 4.4. T there are only five inequivalent (non-trivial) involutions which are listed, for instance, in pg. 164of [Sa]. In this paper the inequivalent involutions are denoted with r j and a simple inspection shows r , r , r and r correspond to the involutive tori T , , , T , , , T , , and T , , , respectively. The map r can be related with T after some manipulation. In conclusion the content of Table 1.2 and Table1.3 along with (1.12) provide a classification of “Quaternionic” vector bundles over all possible two-dimensional involutive tori. The inequivalent involutions for the three-dimensional torus are classifiedin [KT]. ◭ a + b c = a = a = a = a = m Q ( T a , , ) Ø 0 0 0Vec m Q ( T a , , ) 0 2 Z (2 Z ) . . . Vec m Q ( T a , , ) Z Z ⊕ (2 Z ) . . . Vec m Q ( T a , , ) Z . . . T able Complete classification of “Quaternionic” vector bundles over involutive tori of type T a , b , in low dimension d = a + b
3. The existence of fixed points implies that only even rank “Quaternionic”vector bundles are admissible. The classification by (even) integers is attained by looking at the firstChern class of the underlying complex vector bundle. The Z classification is given by the FKMM-invariant. The results summarized in this table are discussed in detail in Section 4.5. G. DE NITTIS AND K. GOMI a + b c = a = a = a = m Q ( T a , , ) 0 Z Z Vec m Q ( T a , , ) 2 Z Z ⊕ (2 Z ) . . . Vec m Q ( T a , , ) (2 Z ) . . . . . . T able Complete classification of “Quaternionic” vector bundles over involutive tori of type T a , b , in low dimension d = a + b +
3. The absence of fixed points allows for “Quaternionic” vector bundlesof every rank (here m ∈ Z ). The classification by (even) integers is given by the first Chern class of theunderlying complex vector bundle. The Z classification is given by the FKMM-invariant. The resultssummarized in this table are discussed in detail in Section 4.5.
2. FKMM- invariant : D efinitions and properties
At a topological level, isomorphism classes of “Quaternionic” vector bundles can be classified bythe
FKMM-invariant [FKMM, DG2]. In its original form the FKMM-invariant can be introducedonly for a subfamily of “Quaternionic” vector bundles provided that certain conditions are met. Theaim of this section is to drop this unpleasant restriction by introducing a more general version of theFKMM-invariant.2.1.
Basic definitions.
In this section we recall some basic facts about the category of “Quaternionic”vector bundles and we refer to [DG2] for a more systematic presentation.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 called an involutive space . The fixed point set of the ( X , τ ) is by definition X τ : = { x ∈ X | τ ( x ) = x } . Henceforth, we will assume that: Assumption 2.1.
X is a topological space which admits the structure of a Z -CW-complex. The dimension d of X is the maximal dimension of its cells. 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 dimensions whichcarry a Z -action. For a precise definition of the notion of Z -CW-complex, the reader can referto [DG1, Section 4.5] or [Mat, AP]. Assumption (2.1) allows the space X to be made by severaldisconnected component. However, in the case of multiple components, we will tacitly assume thatvector bundles built over X possess fibers of constant rank on the whole X . Let us recall that a spacewith a CW-complex structure is automatically Hausdor ff and paracompact and it is compact exactlywhen it is made by a finite number of cells [Hat]. Almost all the examples considered in this paperwill concern with spaces with a finite CW-complex structure.We are now in position to introduce the principal object of interest in this work. Definition 2.2 (“Quaternionic” vector bundles [Du, DG2]) . A “Quaternionic” vector bundle, or Q -bundle, over ( X , τ ) is a complex vector bundle π : E → X endowed with a 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 − , 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 [DG2, Proposition 2.5]. COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 9
A vector bundle morphism f between two vector bundles π : E → X and π ′ : E ′ → X over thesame base space is a continuous map f : E → E ′ which is fiber preserving in the sense that π = π ′ ◦ f and that restricts to a linear map on each fiber f | x : E x → E ′ x . Complex vector bundles over X togetherwith vector bundle morphisms define a category and the symbol Vec m C ( X ) is used to denote the setof isomorphism classes of vector bundles of rank m . Also Q -bundles define a category with respectto Q -morphisms . A Q -morphism f between two “Quaternionic” vector bundles ( E , Θ ) and ( E ′ , Θ ′ )over the same involutive space ( X , τ ) is a vector bundle morphism commuting with the “Quaternionic”structures, i.e. f ◦ Θ = Θ ′ ◦ f . The set of isomorphisms classes of Q -bundles of rank m over ( X , τ )will be denoted by Vec m Q ( X , τ ). Remark 2.3 (“Real” vector bundles) . By changing condition ( Q ) of Definition 2.2 with( R ) Θ acts fiberwise as the multiplication by , namely Θ | E x = E x one ends in the category of “Real” (or R ) vector bundles . Isomorphism classes of rank m R -bundlesover the involutive space ( X , τ ) are denoted by Vec m R ( X , τ ). For more details we refer to [DG1]. ◭ Consider the fiber E x ≃ C m over a fixed point x ∈ X τ . In this case the restriction Θ | E x ≡ J defines an anti -linear map J : E x → E x such that J = − E x . Then, fibers over fixed points areendowed with a quaternionic structure in the sense of [DG2, Remark 2.1]. This fact has an importantconsequence: if X τ , Ø then every “Quaternionic” vector bundle over ( X , τ ) has necessarily even rank[DG2, Proposition 2.1]. However, odd rank Q -bundles are possible in the case of a free involution,i.e.when X τ = Ø (cf.Section 3).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 given by Q : = −
11 0 ! ⊗ m = −
11 0 . . . . . . −
11 0 . (2.1)A Q -bundle is said to be Q -trivial if it is isomorphic to the product Q -bundle in the category of Q -bundles.2.2. The determinant construction.
Let V be a complex vector space of dimension m . The deter-minant of V is by definition det( V ) : = V m V where the symbol V m denotes the top exterior power of V (i.e.the skew-symmetrized m -th tensor power of V ). This is a complex vector space of dimensionone. If W is a second vector space of the same dimension m and T : V → W is a linear map then thereis a naturally associated map det( T ) : det( V ) → det( W ) which in the special case V = W coincideswith the multiplication by the determinant of the endomorphism T . This determinant constructionis a functor from the category of vector spaces to itself and, by a standard argument [Hus, Chapter5, Section 6], induces a functor on the category of complex vector bundles over an arbitrary space X .More precisely, for each rank m complex vector bundle E → X , the associated determinant line bundle det( E ) → X is the rank 1 complex vector bundle with fibersdet( E ) x = det( E x ) x ∈ X . (2.2)Each local frame of sections { s , . . . , s m } of E over the open set U ⊂ X induces the section s ∧ . . . ∧ s m of det( E ) which fixes a trivialization over U . Given a map ϕ : X → Y one can prove the isomorphismdet( ϕ ∗ ( E )) ≃ ϕ ∗ (det( E )) which is a special case of the compatibility between pullback and tensorproduct. Finally, if E = E ⊕ E in the sense of the Whitney sum then det( E ) = det( E ) ⊗ det( E ).Let ( E , Θ ) be a rank m “Quaternionic” vector bundle over ( X , τ ). The associated determinant linebundle det( E ) inherits an involutive structure given by the map det( Θ ) which acts anti -linearly between the fibers det( E ) x and det( E ) τ ( x ) according to det( Θ )( p ∧ . . . ∧ p m ) = Θ ( p ) ∧ . . . ∧ Θ ( p m ). Clearlydet( Θ ) is a fiber preserving map which coincides with the multiplication by ( − m . Hence, in view ofRemark 2.3 one has the following result. Lemma 2.4.
Let ( E , Θ ) be a rank m “Quaternionic” vector bundle over ( X , τ ) . The associated deter-minant line bundle det( E ) endowed with the involutive structure det( Θ ) is a “Real” line bundle if m iseven and a “Quaternionic” line bundle if m is odd. Let ( E , Θ ) be a rank m Q -bundle over ( X , τ ) endowed with an equivariant Hermitian metric m . Thesedata fix a unique Hermitian metric m det on det( E ) which is equivariant with respect to the structureinduced by det( Θ ). More explicitly, 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 line bundle (det( E ) , det( Θ )) endowed with m det is trivial if and only if there exists an isometricequivariant isomorphism (in the appropriate category) with X × C . Equivalently, if and only if thereexists a global equivariant section s : X → det( E ) of unit length (cf.[DG1, Theorem 4.8]). Let S (det( E )) : = { p ∈ det( E ) | m det ( p , p ) = } be the circle bundle underlying to (det( E ) , det( Θ )). Then the triviality of det( E ) can be rephrased asthe existence of a global equivariant section of S (det( E )) → X . The following result will play a crucialrole in our construction. Proposition 2.5 ([DG2, Lemma 3.3]) . Let ( E , Θ ) be a “Quaternionic” vector bundle over a space Xwith trivial involution τ = Id X . Then, the associated determinant line bundle det( E ) endowed with the“Real” structure det( Θ ) is R -trivial and admits a unique canonical R -section s : X → S (det( E )) . If X τ , Ø the restricted vector bundle E | X τ → X τ is a Q -bundle over a space with trivial involution.Proposition 2.5 assures that the restricted line bundle det( E | X τ ) is R -trivial with respect to the restricted“Real” structure det( Θ | X τ ) and admits a distinguished R -section s E : X τ → S (cid:0) det( E ) | X τ (cid:1) . (2.3)We will refer to s E as the canonical section associated to ( E , Θ ).2.3. A short reminder of the equivariant Borel cohomology.
The proper cohomology theory forthe study of vector bundles in the category of spaces with involution is the equivariant cohomolgyintroduced by A. Borel in [Bo]. This cohomology has been used for the topological classificationof “Real” vector bundles [DG1] and plays also a role in the classification of “Quaternionic” vectorbundles [DG2]. A short self-consistent summary of this cohomology theory can be found in [DG1,Section 5.1] and we refer to [Hs, Chapter 3] and [AP, Chapter 1] for a more complete introduction tothe subject.Since we need this tool we briefly recall the main steps of the Borel construction. The homotopyquotient of an involutive space ( X , τ ) is the orbit space X ∼ τ : = X × S ∞ / ( τ × θ ∞ ) . (2.4)Here θ ∞ is the antipodal map on the infinite sphere S ∞ (cf.[DG1, Example 4.1]) and S , ∞ is used asshort notation for the pair ( S ∞ , θ ∞ ). The product space X × S ∞ (forgetting for a moment the Z -action)has the same homotopy type 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 equivariant cohomology 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, COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 11 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 Borelequivariant cohomology are also generalized to local coe ffi cients. Given an involutive space ( X , τ ) onecan consider the homotopy 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 . Foreach involutive space ( X , τ ), there always exists a particular family of local systems Z ( j ) labelled by j ∈ Z . Here Z ( j ) ≃ X × Z denotes the Z -equivariant local system on ( X , τ ) made equivariant by the Z -action ( x , l ) ( τ ( x ) , ( − j l ). Given that the module structure depends only on the parity of j , onecan consider only the Z -modules Z (0) and Z (1). Noticing that Z (0) corresponds to the case of thetrivial action of π ( X ∼ τ ) on Z one has that H k Z ( X , Z (0)) ≃ H k Z ( X , Z ) [DK, Section 5.2].We recall the two important group isomorphisms H Z (cid:0) X , Z (1) (cid:1) ≃ (cid:2) X , U (1) (cid:3) Z ≃ (cid:2) X , S , (cid:3) Z H Z (cid:0) X , Z (1) (cid:1) ≃ Vec R (cid:0) X , τ (cid:1) ≡ Pic R (cid:0) X , τ (cid:1) (2.5)involving the first two equivariant cohomology groups. The first isomorphism [Go, Proposition A.2]says that the first equivariant cohomology group is isomorphic to the set of Z -homotopy classes ofequivariant maps ϕ : X → U (1) where the involution on U (1) is induced by the complex conjugation,i.e. ϕ ( τ ( x )) = ϕ ( x ). Of course there is a natural identification of this target involutive space with theinvolutive sphere S , . The second isomorphism is due to B. Kahn [Kah] and expresses the equivalencebetween the Picard group of “Real” line bundles (in the sense of [At1, DG1]) over ( X , τ ) and the secondequivariant cohomology group of this space. A more modern proof of this isomorphism can be foundin [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 ) . Consequently, one has the related long exact sequence in cohomology . . . −→ H j Z ( X | X τ , Z ) δ −→ H j Z ( X , Z ) r −→ H j Z ( X τ , Z ) δ −→ H j + Z ( X | X τ , Z ) −→ . . . (2.6)where the map r : = ı ∗ restricts cochains on X ∼ τ to cochains on X τ ∼ τ . The j -th cokernel of r is bydefinition Coker j ( X | X τ , Z ) : = H j Z ( X τ , Z ) / r (cid:0) H j Z ( X , Z ) (cid:1) . (2.7)Let us point out that with the same construction one can define relative cohomology theories H • Z ( X | Y , Z ) for each closed subset Y ⊆ X which is τ -invariant τ ( Y ) = Y . If Y = Ø then H j Z ( X | Ø , Z ) ≃ H j Z ( X , Z ) by definition. Therefore it is suitable to declare H j Z (Ø , Z ) = A geometric model for the relative equivariant cohomology.
In this section we provide ageometric model for the equivariant relative cohomology group H Z ( X | Y , Z (1)) which extends thesecond isomorphism in (2.5).Let ( X , τ ) be an involutive space and Y ⊆ X a closed τ -invariant subspace τ ( Y ) = Y (it is not requiredthat Y ⊆ X τ ). Consider pairs ( L , s ) consisting of: (a) a “Real” line bundle L → X with a given “Real”structure Θ and a Hermitian metric m ; (b) a (nowhere vanishing) “Real” section s : Y → S ( L | Y ) of thecircle bundle associated to the restriction L | Y → Y (or equivalently a trivialization h : L | Y → Y × C ).Two pairs ( L , s ) and ( L , s ) built over the same involutive base space ( X , τ ) and the same invariant subspace Y are said to be isomorphic if there is an R -isomorphism of line bundles f : L → L (preserving the Hermitian structure) such that f ◦ s = s . Definition 2.6 (Relative “Real” Picard group) . For a given involutive space ( X , τ ) and a closed sub-space Y ⊆ X such that τ ( Y ) = Y, we define
Pic R ( X | Y , τ ) to be the group of the isomorphism classes ofpairs ( L , s ) , with group structure given by the tensor product ( L , s ) ⊗ ( L , s ) ≃ ( L ⊗ L , s ⊗ s ) . The main result of this section is the following characterization.
Proposition 2.7.
There is a natural isomorphism of groups ˜ κ : Pic R ( X | Y , τ ) ≃ −→ H Z (cid:0) X | Y , Z (1) (cid:1) which reduces to the Kahn’s isomorphism (2.5) when Y = Ø .Proof (sketch of). The Kahn’s isomorphism H Z ( X , Z (1)) ≃ Pic R ( X , τ ) can be proved roughly in twosteps: The first step is to prove that Vec R ( X , τ ) is isomorphic to the sheaf cohomology H ( Z • × X , ˜ S • ) ofthe simplicial space Z • × X associated to the involutive space ( X , τ ) and this is carried out by realizingthe sheaf cohomology as a ˇCech cohomology; The second step is to identify the sheaf cohomology with H Z ( X , Z (1)). For the full argument we refer to [Go, Appendix A]. The same argument is replicableverbatim for the pair Y ⊆ X . Firstly, the use of the ˇCech cohomology allows to prove the groupisomorphism between Pic R ( X | Y , τ ) and the relative sheaf cohomology H ( Z • × X | Z • × Y , ˜ S • ). Secondly,this sheaf cohomology can be identified with the relative equivariant cohomology H ( X | Y , Z (1)) in thesame way as in [Go, Appendix A]. (cid:4) In the next sections we will exploit the result of Proposition 2.7 in the special case of Y ⊆ X τ .We recall that in this situation, as a consequence of Proposition 2.5, the choice of s is canonical andcorresponds to a trivialization of the restricted “Real” line bundle L | Y → Y .2.5. Generalized FKMM-invariant.
By combining the content of Proposition 2.5 with that of Propo-sition 2.7 we can introduce an intrinsic invariant for the category of “Quaternionic” vector bundles.
Definition 2.8 (FKMM-invariant for even rank Q -bundles) . Let ( E , Θ ) be an even rank “Quater-nionic” vector bundle over the involutive space ( X , τ ) and consider the pair (det( E ) , s E ) where det( E ) is the determinant line bundle associated to E endowed with the “Real” structure induced by det( Θ ) and s E is the canonical section described by (2.3) . The generalized FKMM-invariant of ( E , Θ ) is thecohomology class κ ( E , Θ ) ∈ H Z (cid:0) X | X τ , Z (1) (cid:1) given by κ ( E , Θ ) : = ˜ κ (cid:0) [(det( E ) , s E )] (cid:1) where [(det( E ) , s E )] ∈ Pic R ( X | X τ , τ ) is the isomorphism class of the pair (det( E ) , s E ) and ˜ κ is the groupisomorphism described in Proposition 2.7. The definition extends to the case X τ = Ø by interpretingthe class κ ( E , Θ ) as the (first) “Real” Chern class of det( E ) ∈ Pic R ( X , τ ) according to the Kahn’sisomorphism ( cf. Corollary 2.12 (2)).
The following properties are immediate consequence of the last definition.(1) Isomorphic “Quaternionic” vector bundles define the same FKMM-invariant;(2) The FKMM-invariant is natural under the pullback induced by equivariant maps;(3) If ( E , Θ ) is Q -trivial then κ ( E , Θ ) = additive with respect to the Whitney sum and the abelian structure of H Z (cid:0) X | X τ , Z (1) (cid:1) , namely κ ( E ⊕ E , Θ ⊕ Θ ) = κ ( E , Θ ) · κ ( E , Θ )for each pair of “Quaternionic” vector bundles ( E , Θ ) and ( E , Θ ) over the same involutivespace ( X , τ ). COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 13
Remark 2.9 (Comparison with the original definition of the FKMM-invariant) . The original definitionof the FKMM-invariant, firstly introduced in [FKMM] and then extensively used in [DG2], is stronglybased on two crucial assumptions: (a) X τ , Ø and (b) det( E ) has to be R -trivial (cf.[DG2, Definition3.1]). Under this assumption the FKMM-invariant can be defined as an element in the cokernel of therestriction map r : H Z ( X , Z (1)) → H Z ( X τ , Z (1)). Moreover, the isomorphism described in [DG2,Lemma 3.1] allows to describe the original FKMM-invariant as an equivariant map φ : X τ → U (1)which measures the di ff erence between the canonical section s E and a global “Real” section of det( E )(cf.[DG2, Definition 3.2 & Remark 3.2]). On the other hand, the generalized invariant introduced inDefinition 2.8 does not require any extra restrictions on the nature of ( X , τ ) and ( E , Θ ) and fulfills allthe structural properties of the original FKMM-invariant (compare the (1)-(4) above with the content of[DG2, Theorem 3.1]). Moreover, as soon as assumptions (a) and (b) are met, the generalized FKMM-invariant agrees with the original FKMM-invariant (see Proposition 2.10 and Corollary 2.11 below).This allows to state that Definition 2.8 really extends the notion of the FKMM-invariant. ◭ Let Coker ( X | X τ , Z (1)) be the cokernel described in (2.7) and [ X , U (1)] Z the set of Z -homotopyclasses of equivariant maps between the involutive space ( X , τ ) and the group U (1) endowed with theinvolution induced by the complex conjugation. Let us recall that the isomorphism (cid:2) X τ , U (1) (cid:3) Z / (cid:2) X , U (1) (cid:3) Z ≃ Coker ( X | X τ , Z (1))holds under the assumption X τ , Ø [DG2, Lemma 3.1].
Proposition 2.10.
Let ( E , Θ ) be an even rank “Quaternionic” vector bundle over the involutive space ( X , τ ) . Assume that: (a) X τ , Ø and (b) det( E ) is R -trivial. Then generalized FKMM-invariant κ ( E , Θ ) introduced in Definition 2.8 is the injective image of an element [ φ ] ∈ Coker ( X | X τ , Z (1)) .Proof. The homomorphism δ and δ in the long exact sequence . . . −→ H Z (cid:0) X , Z (1) (cid:1) r −→ H Z (cid:0) X τ , Z (1) (cid:1) δ −→ H Z (cid:0) X | X τ , Z (1) (cid:1) δ −→ H Z (cid:0) X , Z (1) (cid:1) −→ . . . (2.8)for the pair X τ ֒ → X can be interpreted as homomorphisms δ : (cid:2) X τ , U (1) (cid:3) Z −→ Pic R (cid:0) X | X τ , τ (cid:1) , δ : Pic R (cid:0) X | X τ , τ (cid:1) −→ Pic R (cid:0) X , τ (cid:1) in view of the isomorphism proved in Proposition 2.7 and the isomorphisms in (2.5), respectively.Moreover, the triviality of det( E ) implies that the pair (det( E ) , s E ) which enters into the definition ofthe FKMM-invariant is an element of the class [( X × U (1) , φ )] ∈ Pic R ( X | X τ , τ ) with φ : X τ → U (1) isa suitable equivariant map. Therefore δ turns out to be just the assignment of the class [( X × U (1) , φ )]to the map [ φ ] ∈ [ X τ , U (1)] Z and δ agrees with the map induced from ( L , s ) L . With thisinterpretation the result directly follows from the exact sequence. Finally, the injectivity is assured by0 −→ Coker ( X | X τ , Z (1)) δ −→ H Z (cid:0) X | X τ , Z (1) (cid:1) δ −→ H Z (cid:0) X , Z (1) (cid:1) −→ . . . which follows from (2.8) by general arguments. (cid:4) As it emerges from the proof, the element [ φ ] in the statement of Proposition 2.10 is representedby the canonical section s E modulo the action (multiplication and restriction) of an equivariant map s : X → U (1) which can be interpreted as a global trivialization of the trivial line bundle det( E ).Therefore [ φ ] is just the original FKMM-invariant as introduced in [FKMM] and [DG2, Definition3.2]. As a consequence, we can rephrase Proposition 2.10 by saying that: “ The generalized FKMM-invariant is just the image (under δ ) of the old FKMM-invariant when-ever the latter can be defined.”
According to [DG2, Definition 1.1] an
FKMM-space is an involutive space ( X , τ ) such that X τ is anon-empty finite set and H Z ( X , Z (1)) = Corollary 2.11.
Let ( E , Θ ) be an even rank “Quaternionic” vector bundle over the FKMM-space ( X , τ ) . Then, the generalized FKMM-invariant κ ( E , Θ ) introduced in Definition 2.8 is represented by amap φ : X τ → { + , − } .Proof. It has been proved in [DG2, Lemma 3.1] that the condition H Z ( X , Z (1)) = δ defines an isomorphisms between Coker ( X | X τ , Z (1)) and H Z (cid:0) X | X τ , Z (1) (cid:1) . Hence, κ ( E , Θ ) can beview as an element in Coker ( X | X τ , Z (1)) which, modulo equivalences, is represented by an equivari-ant function φ : X τ → U (1). Since, by definition, τ acts trivially on X τ , φ maps into the points of U (1)fixed by the complex conjugation. (cid:4) For the next result we need to recall that the Kahn’s isomorphism H Z ( X , Z (1)) ≃ Pic R (cid:0) X , τ (cid:1) isrealized by first “Real” Chern class c R (see [Kah] or [DG1, Section 5.2]). Corollary 2.12.
Let ( E , Θ ) be an even rank “Quaternionic” vector bundle over the involutive space ( X , τ ) and δ : H Z (cid:0) X | X τ , Z (1) (cid:1) −→ H Z (cid:0) X , Z (1) (cid:1) the homomorphism in the long exact sequence (2.8) for the pair X τ ֒ → X. Then: (1)
The homomorphism δ carries the FKMM-invariant κ ( E , Θ ) to the first “Real” Chern classc R (det( E )) of the “Real” line bundle det( E ) → X; (2) If in addition X τ = Ø , then the FKMM-invariant κ ( E , Θ ) agrees with c R (det( E )) .Proof. From the proof of Proposition 2.10 one knows that the δ can be interpreted as the homomor-phism Pic R ( X | X τ , τ ) → Pic R ( X , τ ) induced by ( L , s ) L . Therefore (1) follows from the fact thatthe Kahn’s isomorphism is induced by c R . (2) follows just by observing that the condition X τ = Ø,once inserted in the long exact sequence (2.8), forces δ to be an isomorphism. (cid:4) The original FKMM-invariant cannot be defined when X τ = Ø but, on the contrary, requires X τ , Øto be finite and the triviality of det( E ). On the other hand the generalized FKMM-invariant introducedin Definition 2.8 bypasses these restrictions and provides a way to define a topological invariant also inthe opposite case when X τ = Ø and det( E ) is non-trivial. In contrast to the definition of FKMM-space[DG2, Definition 1.1] we can call “anti-FKMM” an involutive space ( X , τ ) such that X τ = Ø and H Z ( X , Z (1)) ,
0. Corollary 2.11 says that the generalized FKMM-invariant agrees with the originalinvariant in the case of an FKMM-space and Corollary 2.12 (ii) shows that the generalized FKMM-invariant reduces to the first “Real” Chern class in the anti-FKMM case. In more general situationsthe classification of even rank “Quaternionic” vector bundles would require the generalized notion ofFKMM-invariant (in the form of Definition 2.8 ) which can be considered a sort of mixture of the twomentioned extreme cases.2.6.
Universal FKMM-invariant.
Even rank “Quaternionic” vector bundles can be classified bymaps with values in a classification space [DG2, Theorem 2.4]. This is the basic fact which allowsa description of the generalized FKMM-invariant as a characteristic class , namely as the image of aunique universal object under the pullback induced by the classification maps. This section provides aclarification and a simplification of some notions and results already discussed in [DG2, Section 6].Let us recall that the (even dimensional)
Grassmannian is defined as G m ( C ∞ ) : = ∞ [ n = m G m ( C n ) , where, for each pair 2 m n , G m ( C n ) ≃ U ( n ) / (cid:0) U (2 m ) × U ( n − m ) (cid:1) is the set of 2 m -dimensional(complex) subspaces of C n . The spaces G m ( C n ) are finite CW-complexes and the space G m ( C ∞ )inherits the direct limit topology given by the inclusions G m ( C n ) ֒ → G m ( C n + ) ֒ → . . . inducedby C n ⊂ C n + ⊂ . . . . Moreover, G m ( C ∞ ) can be endowed with an involution of quaternionic typein 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 bythe basis { v , v , . . . v m − , v m } and define a new point ρ ( Σ ) ∈ G m ( C n ) as the 2 m -plane spanned by COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 15 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). Notice that the definition of ρ ( Σ ) does not depend on the choice of a particular basis and 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 Grassmannian in such a way that ˆ G m ( C ∞ ) ≡ ( G m ( C ∞ ) , ρ ) becomes an involutive space.Let Σ = ρ ( Σ ) be a fixed point of ˆ G m ( C ∞ ). Since ρ acts on vectors as a quaternionic structure onehas Σ ≃ H m (cf.[DG2, Remark 2.1]). This fact implies that the fixed point set of ˆ G m ( C ∞ ) can beidentified with the quaternionic Grassmannian, namely ˆ G m ( C ∞ ) ρ ≃ G m ( H ∞ ) (see [DG2, Section 2.4]for more details).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 ) with total space T nm : = { ( Σ , v) ∈ ( G m ( C n ) × C n ) | v ∈ Σ } and bundle projection π ( Σ , v) = Σ .When n tends to infinity, the same construction leads to the tautological m -plane bundle π : T ∞ m → G m ( C ∞ ). The latter is the universal vector bundle which classifies complex vector bundles. In factany rank m complex vector bundle E → X is realized, up to isomorphisms, as a pullback E ≃ ϕ ∗ T ∞ m with respect to a classifying map ϕ : X → G m ( C ∞ ). Since pullbacks of homotopy equivalent mapsyield isomorphic vector bundles one gets the well known result Vec m C ( X ) ≃ [ X , G m ( C ∞ )]. This resultextends 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 . This is done by 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 assures that ( T ∞ m , Ξ ) isa Q -bundle over the involutive space ˆ G m ( C ∞ ). This tautological Q -bundle classifies “Quaternionic”vector bundles in the sense that Vec m Q ( X , τ ) ≃ [ X , ˆ G m ( C ∞ )] Z where in the right-hand side there is theset of Z -homotopy classes of equivariant maps between ( X , τ ) and ˆ G m ( C ∞ ) [DG2, Theorem 2.4].The construction of the generalized FKMM-invariant applies to the universal Q -bundle ( T ∞ m , Ξ ). Definition 2.13 (Universal FKMM-invariant) . The universal FKMM-invariant h univ ∈ H Z (cid:0) ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) is the (generalized) FKMM-invariant of the tautological Q -bundle ( T ∞ m , Ξ ) as constructed in Definition2.8. More precisely h univ : = κ ( T ∞ m , Ξ ) is the image of (cid:2)(cid:0) det( T ∞ m ) , s T ∞ m (cid:1)(cid:3) ∈ Vec R ( ˆ G m ( C ∞ ) | ˆ G m (cid:0) C ∞ ) ρ , ρ (cid:1) under the natural isomorphism ˜ κ described in Proposition 2.7. The naturality of the invariant κ implies the following important result. Theorem 2.14 (Universality) . Let ( E , Θ ) be a rank m “Quaternionic” vector bundle over the invo-lutive space ( X , τ ) classified by the equivariant map ϕ : X → ˆ G m ( C ∞ ) . The generalized FKMM-invariant of ( E , Θ ) verifies κ ( E , Θ ) = ϕ ∗ (cid:0) h univ (cid:1) where ϕ ∗ is the homomorphism in cohomology induced by ϕ . The identification of h univ as an element of H Z ( ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1)) requires a careful inves-tigation of the equivariant cohomology of ˆ G m ( C ∞ ). Some aspects of this analysis are quite technicaland are postponed in Appendix A. The main results are summarized below. Proposition 2.15 (Identification) . The following facts hold: (1)
The group H Z ( ˆ G m ( C ∞ ) , Z (1)) ≃ Z is generated by the (universal) first “Real” Chern class c R : = c R (cid:16) det (cid:0) T ∞ m (cid:1)(cid:17) of the “Real” line bundle det( T ∞ m ) → ˆ G m ( C ∞ ) associated to the tautological Q -bundle ( T ∞ m , Ξ ) . Moreover, under the map f : H Z ( ˆ G m ( C ∞ ) , Z (1)) → H ( G m ( C ∞ ) , Z ) which for-gets the Z - action the class c R is mapped in the first universal Chern class c ∈ H ( G m ( C ∞ ) , Z ) ; (2) The homomorphism δ : H Z (cid:0) ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) ≃ −→ H Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) in the long exact sequence (2.6) for the pair ˆ G m ( C ∞ ) ρ ֒ → ˆ G m ( C ∞ ) is, in fact, an isomor-phism. (3) The universal FKMM-invariant h univ is the generator of H Z ( ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1)) ≃ Z which is mapped by δ in the first “Real” Chern class c R , i.e. h univ = δ − (cid:0) c R (cid:1) . Proof.
The proof of (1) and (2) are given in Proposition A.3 and Proposition A.5, respectively. Item(3) follows from Corollary 2.12 (1). (cid:4)
Remark 2.16 (Comparison with the old definition of the universal FKMM-invariant) . The old def-inition of the universal FKMM-invariant [DG2, Definition 6.1] is based on an intricate constructionthat we briefly recall. The bundle map π : det( T ∞ m ) → ˆ G m ( C ∞ ) defines an equivariant map betweeninvolutive spaces π : (cid:16) S (cid:0) det( T ∞ m ) (cid:1) , det( Ξ ) (cid:17) −→ ˆ G m ( C ∞ ) . The pullback of the tautological Q -bundle ( T ∞ m , Ξ ) induced by π defines a Q -bundle ( π ∗ T ∞ m , π ∗ Ξ )over the involutive space ( S (det( T ∞ m )) , det( Ξ )). According to the old definition, the universal FKMM-invariant K univ is the old FKMM-invariant (here denoted with κ ) of ( π ∗ T ∞ m , π ∗ Ξ ). More precisely K univ : = κ ( π ∗ T ∞ m , π ∗ Ξ ) ∈ Coker (cid:16) S (cid:0) det( T ∞ m ) (cid:1) | S (cid:0) det( T ∞ m ) (cid:1) Ξ , Z (1) (cid:17) ≃ Z agrees with the non trivial element of Z [DG2, Theorem 6.2]. It is interesting to notice that the Q -line bundle ( π ∗ T ∞ m , π ∗ Ξ ) is of FKMM-type [DG2, Lemma 6.2] and consequently the description inProposition 2.10 applies. This means that the new FKMM-invariant κ ( π ∗ T ∞ m , π ∗ Ξ ) ∈ H Z (cid:16) S (cid:0) det( T ∞ m ) (cid:1) | S (cid:0) det( T ∞ m ) (cid:1) Ξ , Z (1) (cid:17) agree with the (non trivial) image of κ ( π ∗ T ∞ m , π ∗ Ξ ) under the injective map δ . Thus, one can replacein the definition of K univ above the old invariant κ with the new invariant κ . Since κ is natural (byconstruction), it follows that K univ : = κ ( π ∗ T ∞ m , π ∗ Ξ ) = π ∗ κ ( T ∞ m , Ξ ) = π ∗ ( h univ ) , namely the old version of the universal FKMM-invariant is just the pullback of the new universalFKMM-invariant h univ under the projection π . ◭
3. “Q uaternionic ” line bundles and FKMM- invariant
As a matter of fact “Quaternionic” vector bundles of odd rank can be defined only over involutivebase spaces ( X , τ ) with a free involution (meaning that X τ = Ø) [DG2, Proposition 2.1]. In this sectionwe investigate the rank one case providing a classification scheme for “Quaternionic” line bundles. Asa byproduct we provides the definition of FKMM-invariant for “Quaternionic” line bundles (Definition3.3 ) that will be used in Theorem 4.9 to provide a FKMM-invariant for for odd-rank “Quaternionic”vector bundles. As a matter of notational simplification we prefer to replace in the next the pompousnotation ( L , Θ ) with lighter symbols like L Q and L R for “Quaternionic” and “Real” line bundles,respectively. COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 17
The “Quaternionic” Picard torsor.
We denote by Pic R ( X , τ ) ≡ Vec R ( X , τ ) and Pic Q ( X , τ ) ≡ Vec Q ( X , τ ) the sets of isomorphism classes of “Real” and “Quaternionic” line bundles on ( X , τ ), re-spectively. The set Pic R ( X , τ ) gives rise to an abelian group under the tensor product of “Real” linebundles and it is known as the “Real” Picard group . On the other hand Pic Q ( X , τ ) does not possess agroup structure under the tensor product. Instead, under the essential assumption that Pic Q ( X , τ ) , Ø,we can use the tensor product to define a left group-action of Pic R ( X , τ ) on the set Pic Q ( X , τ ):Pic R ( X , τ ) × Pic Q ( X , τ ) −→ Pic Q ( X , τ )([ L R ] , [ L Q ]) [ L R ⊗ L Q ] . Let us introduce some terminology. For a group G , a (left) G -torsor , is a set T with a simply transitive(left) G -action such that the map φ : G × T −→ T × T ( g , t ) ( gt , t ) . is an isomorphism. In other words, the G -action on T has to be free, and T has to be identifiable with asingle G -orbit. We notice that one may think of a G -torsor as a principal G -bundle over a single point. Theorem 3.1 (“Quaternionic” Picard torsor) . Assume
Pic Q ( X , τ ) , Ø . Then Pic Q ( X , τ ) is a torsorunder the group-action induced by Pic R ( X , τ ) .Proof. Given a “Quaternionic” line bundle L Q let L ∗ Q be the associated dual bundle. The tensorproduct L Q ⊗ L ∗ Q provides a representative for the trivial element in Pic R ( X , τ ). This fact can bechecked, for instance, by looking at the transition functions. Given any pair L R , L ′ R ∈ Pic R ( X , τ ) anda L Q ∈ Pic Q ( X ) let assume that L R ⊗ L Q ≃ L ′ R ⊗ L Q . By tensorizing both sides with L ∗ Q oneobtains that L R ≃ L ′ R , namely Pic R ( X , τ ) acts freely on Pic Q ( X , τ ). On the other hand, for each pair L Q , L ′ Q ∈ Pic Q ( X , τ ) there is a “Real” line bundle L R : = L ′ Q ⊗ L ∗ Q such that L R ⊗ L Q ≃ L ′ Q . Thenthe action of Pic R ( X , τ ) is also transitive. (cid:4) Corollary 3.2 (Classification of Q -line bundles) . Let ( X , τ ) be an involutive space such that X τ = Ø and Pic Q ( X , τ ) , Ø . Then, Pic Q ( X , τ ) ≃ Pic R ( X , τ ) ≃ H Z (cid:0) X , Z (1) (cid:1) is a bijection of sets.Proof. The first bijection is just a consequence of Theorem 3.1 which assures that Pic Q ( X , τ ) can berealized as a single orbit under the left action of Pic R ( X , τ ). However, the form of the bijection is notnatural but depends on the choice of an initial element in Pic Q ( X , τ ). The second bijection (in fact agroup isomorphism) is proved in [Kah] or [Go, Corollary A.5]. (cid:4) Theorem 3.1 provides a way to extend the notion of (generalized) FKMM-invariant for “Quater-nionic” line bundles.
Definition 3.3 (FKMM-invariant for Q -line bundles) . Assume that
Pic Q ( X , τ ) , Ø and let us fix arbi-trarily a reference element [ L ref ] ∈ Pic Q ( X , τ ) . Given a L Q ∈ Pic Q ( X , τ ) let L R ∈ Pic R ( X , τ ) be theunique (up to isomorphisms) element such that [ L Q ] = [ L R ⊗ L ref ] . The FKMM-invariant of L Q ,normalized by L ref , is by definition κ (cid:0) L Q (cid:1) : = c R (cid:0) L R (cid:1) . Evidently the definition of the κ -invariant in the line bundle case su ff ers of the ambiguity due to thechoice of a reference “Quaternionic” line bundle L ref . However, this ambiguity can be understoodas the freedom to fix the normalization κ ( L ref ) =
0, a fact which is common in the theory of charac-teristic classes. From Definition 3.3 it results that κ is a complete classifying invariant for Pic Q ( X , τ ).Moreover, this definition is in accordance with the result in Corollary 2.12 (2).The structure of the “Quaternionic” line bundles can be investigated in a more direct way undersome additional assumptions. Consider a representative L in Pic Q ( X , τ ) such that the underlying line bundle is trivial in the complex category, i.e. L ≃ X × C . In this situation a “Quaternionic” structure Θ is fixed by a continuous map q : X → U (1) such that Θ : ( x , λ ) ( τ ( x ) , q ( x ) λ ). The “Quater-nionic” constraint requires q ( τ ( x )) q ( x ) = −
1, or equivalently q ( τ ( x )) = − q ( x ), for all x ∈ X . Evidently,this condition cannot be verified if the involution τ is not free. The identification U (1) ≃ S saysthat the “Quaternionic” structure Θ is specified by a Z -equivariant map q from the involutive space( X , τ ) into the one-dimensional sphere endowed with the antipodal free involution S , . Since equivari-ant homotopy deformations produce isomorphic “Quaternionic” vector bundles we can conclude thateach equivariant homotopy class in [ X , S , ] Z identifies a “Quaternionic” structure on the product linebundle L ≃ X × C . Finally, we notice that the “Quaternionic” structures induced by an equivariantmap q and its opposite − q are Q -isomorphic. In conclusion we obtained that “Quaternionic” struc-tures on a product line bundle are classified by [ X , S , ] Z / Z where the Z -action is induced by themultiplication by −
1. We are now in position to state the next result.
Proposition 3.4.
Let ( X , τ ) be an involutive space which verifies the following condition: (a) X τ = Ø ; (b) H ( X , Z ) has no torsion; (c) The pullback homomorphism τ ∗ : H ( X , Z ) → H ( X , Z ) acts as the identity.Then the following bijection Pic Q ( X , τ ) ≃ (cid:2) X , S , (cid:3) Z / Z holds true. The result is evidently valid under the stronger condition H ( X , Z ) = , which implies (b) and (c) .Proof. Condition (a) is necessary for the existence of “Quaternionic” line bundles. Let L be anelement in Pic Q ( X , τ ). Conditions (b) and (c) are su ffi cient to ensure that L ≃ X × C in the complexcategory. In fact the presence of a “Quaternionic” structure induces an isomorphism of complex vectorbundles τ ∗ L ≃ L where L is the conjugated complex line bundle obtained by reversing the complex structure in thefibers of L . This isomorphisms leads to the constraint τ ∗ c ( L ) = − c ( L ) (3.1)for the first Chern class of L . Condition (c) implies 2 c ( L ) = c ( L ) =
0. This fact forces L to be trivial (in the complex category) since complex line bundle arecompletely classified by the first Chern class. (cid:4) One interesting aspect of the theory of “Quaternionic” line bundles is that there is no natural candi-date for the definition of a trivial element in Pic Q ( X , τ )!3.2. Classification over involutive spheres.
We discuss in this section the classification of “Quater-nionic” line bundles over the involutive spheres of type S , d . Example 3.5 (The Dupont Q -line bundle over S , ) . The zero-dimensional sphere S , coincides withthe two-points space {− , + } endowed with the flip-action θ , : ±
7→ ∓
1. We can construct a“Quaternionic” line bundle over S , as follows: Let L : = {− , + } × C be the product line bundle anddefine Θ : L → L by Θ : (cid:0) ± , λ (cid:1) (cid:0) ∓ , ± λ (cid:1) , λ ∈ C . This example has been introduced for the fist time by J. L. Dupont in [Du]. In particular it provesthat, Pic Q ( S , ) , Ø. The next natural question is whether there are other elements in Pic Q ( S , )distinguished from the Dupont’s example. A way to answer this question is to use the characterizationof Corollary 3.2. To compute the group H Z ( S , , Z (1)) let us start with the ordinary cohomology H k (cid:0) {− , + } , Z (cid:1) ≃ Z if k =
00 if k , COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 19 and with the equivariant cohomology with fixed coe ffi cients H k Z (cid:0) S , , Z (cid:1) ≃ H k (cid:0) {∗} , Z (cid:1) ≃ ( Z if k =
00 if k , H Z ( S , , Z (cid:1) ≃ H Z ( S , , Z (1))and consequently Pic Q (cid:0) S , (cid:1) ≃ H Z (cid:0) S , , Z (1) (cid:1) = {− , + } is necessarily trivial. This means that every “Quater-nionic” line bundle over S , is built from the underlying complex line bundle L : = {− , + } × C . Any“Quaternionic” structure on L can be specified by a t ∈ [0 , π ) according to Θ t : (cid:0) ± , λ (cid:1) (cid:0) ∓ , ± e i t λ (cid:1) , λ ∈ C . On the other hand a Q -isomorphism between two “Quaternionic” structures Θ t and Θ t ′ is provided bya map f φ : (cid:0) ± , λ (cid:1) (cid:0) ± , φ ( ± λ (cid:1) , λ ∈ C subjected to the equivariant condition f φ ◦ Θ t = Θ t ′ ◦ f φ . For each pair t , t ′ ∈ [0 , π ) such a map is fixedby the prescription φ ( +
1) : = + φ ( −
1) : = e i ( t ′ − t ) . This proves directly that each “Quaternionic”line bundle over {− , + } is isomorphic to the Dupont’s example which corresponds to t = ◭ Example 3.6 ( Q -line bundles over S , d + with d > . Recall that S , d + agrees with S d as topologicalspace and H ( S d , Z ) = d >
3. Then, we can use the result in Proposition 3.4 to state thatPic Q (cid:0) S , d + (cid:1) ≃ (cid:2) S , d + , S , (cid:3) Z / Z , d > . However, an analysis of the equivariant homotopy shows that (cid:2) S , d + , S , (cid:3) Z = Ø , d > . (3.2)and this leads to the conclusion that it is not possible to build “Quaternionic” line bundle over involutivesphere S , d + of dimension bigger than two. As a matter of completeness, let us justify equation (3.2).The involutive space S , d + can be seen as the total space of a Z -sphere bundle S , d + → R P d andit is well known that the Stiefel-Whitney class of this bundle w ( S , d + ) is given by the non-trivialelement of H ( R P d , Z ) ≃ Z for all d >
1. Let us assume the existence of a Z -equivariant map q : S , d + → S , . Such a map would induce a map ˜ q : R P d → R P on the orbit spaces andan isomorphism S , d + ≃ ˜ q ∗ S , of bundles over R P d where ˜ q ∗ S , is the pullback of the Z -spherebundles S , → R P . By functoriality, this would imply w ( S , d + ) = w ( ˜ q ∗ S , ) = ˜ q ∗ w ( S , ) showingthat ˜ q ∗ w ( S , ) is not trivial. On the other hand one has that (cid:2) R P d , R P (cid:3) ≃ (cid:2) R P d , K ( Z , (cid:3) ≃ H (cid:0) R P d , Z (cid:1) = , d > R P has been identified with the Eilenberg-MacLane space K ( Z , q must be ho-motopy equivalent to the constant map and this contradicts the non-triviality of ˜ q ∗ w ( S , ). This con-tradiction shows that the set of Z -equivariant map from S , d + into S , must be empty in agreementwith (3.2). ◭ Example 3.7 ( Q -line bundles over S , ) . Let us start by proving that Pic Q ( S , ) , Ø. Since H ( S , Z ) = S , must be built over the underlying productbundle L : = S × C . Then, a (possible) “Quaternionic” structure Θ q : L → L must be specified bya map q : S → U (1) which fulfills the constraint q ( − k ) = − q ( k ) for all k ∈ S according to the recipe Θ q : (cid:0) k , λ (cid:1) (cid:0) − k , q ( k ) λ (cid:1) , ( k , λ ) ∈ S × C . A standard choice for such a map is the following: q : S −→ U (1)( k , k ) k + i k . (3.3)The existence of q assures that Pic Q ( S , ) is nonempty and so the classification of Corollary 3.2applies. The computation in Proposition B.2 givesPic Q (cid:0) S , (cid:1) = , namely there is only one isomorphism class of “Quaternionic” line bundles over S , with a represen-tative given by ( L , q ). It is possible to verify this claim in a more direct way. A Q -isomorphismbetween two “Quaternionic” structures Θ q and Θ q is specified by a map φ : S → U (1) such that f φ : (cid:0) k , λ (cid:1) (cid:0) k , φ ( k ) λ (cid:1) , ( k , λ ) ∈ S × C , and subjected to the equivariance condition f φ ◦ Θ q = Θ q ◦ f φ . This translates into q ( k ) = φ ( − k ) φ ( k ) q ( k ) k ∈ S . In a general way the map q can be related to q by a relation of the form q ( k ) = q ( k ) n e i 2 π χ ( k ) k ∈ S (3.4)for some n ∈ Z and χ : S → R . The “Quaternionic” constraints − = q ( − k ) q ( k ) and − = q ( − k ) q ( k ) impose the conditione i 2 π [ χ ( − k ) − χ ( k )] = ( − n + k ∈ S . If n were even ∆ χ ( k ) : = χ ( − k ) − χ ( k ) should take values in Z + . However, ∆ χ is continuous andso ∆ χ ( k ) should take a constant value N + for some N ∈ Z and for all k . But this is incompatiblewith the parity of ∆ χ which requires ∆ χ ( − k ) = − ∆ χ ( k ). Then n = p + ∆ χ musttake values in Z . Again, the continuity and the parity of ∆ χ imply that ∆ χ ( k ) =
0, or equivalently χ ( − k ) = χ ( k ) for all k . This allows to rewrite equation (3.4) in the form q ( k ) = q ( k ) − p e − i 2 π χ ( k ) q ( k ) = φ ( − k ) φ ( k ) q ( k ) k ∈ S with φ ( k ) : = ( − p q ( k ) − p e − i π χ ( k ) . The last equation shows that any “Quaternionic” structure Θ q is Q -isomorphic to the standard “Quaternionic” structure Θ q . ◭ Example 3.8 ( Q -line bundles over S , ) . Again, let us start by showing that Pic Q ( S , ) , Ø. Thiscase is richer than those described in Example 3.6 and Example 3.7 since H ( S , Z ) ≃ Z which inturn implies that for each ℓ ∈ Z there is an inequivalent complex line bundle L ℓ → S , specifiedby the Chern class ℓ . In principle each of these L ℓ can be used as the underlying line bundle for a“Quaternionic” structure. Equation (3.2) says that it is not possible to endow the trivial line bundle L : = S × C with a “Quaternionic” structure. Let us investigate the case ℓ =
1. A representative for L → S can be constructed from the family of Hopf projectionsP
Hopf ( k , k , k ) : = + X j = k j σ j + , k : = ( k , k , k ) ∈ S (3.5)according to L : = G k ∈ S Ran (cid:0) P Hopf ( k ) (cid:1) where σ , σ and σ are the Pauli matrices and Ran (cid:0) P Hopf ( k ) (cid:1) ⊂ C is the one-dimensional subspacespanned by P Hopf ( k ). Moreover, L has a “Quaternionic” structure induced by the anti-linear symme-try Θ P Hopf ( k , k , k ) Θ ∗ = P Hopf ( − k , − k , − k ) , Θ : = σ ◦ C COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 21 where C is the operator which implements the complex conjugation on C (cf.Section 5). The ex-istence of the “Quaternionic” line bundle ( L , Θ ) assures that Pic Q ( S , ) is non-empty and so theclassification of Corollary 3.2 applies. Proposition B.2 providesPic Q (cid:0) S , (cid:1) ≃ H (cid:0) S , , Z (1) (cid:1) ≃ Z , namely there are Z distinct isomorphism classes of Q -line bundles over S , . A look to the exactsequence [Go, Proposition 2.3.] H Z (cid:0) S , , Z (cid:1) ✲ H Z (cid:0) S , , Z (1) (cid:1) f ✲ H (cid:0) S , Z (cid:1) ✲ H Z (cid:0) S , , Z (cid:1) ✲ H Z (cid:0) S , , Z (1) (cid:1) = ≃ ≃ ≃ = Z Z Z f which forgets the Z -structure acts as the multiplication by 2. By combiningthis fact with the isomorphism c : Pic C ( S ) ≃ H (cid:0) S , Z (cid:1) induced by the first Chern class and theisomorphism c R : Pic R ( S , ) → H Z (cid:0) S , , Z (1) (cid:1) induced by the first “Real” Chern class one concludesthat there is an isomorphisms of groups Pic R (cid:0) S , (cid:1) ≃ −→ Z [ L R ] c ( L R )where L R is any “Real” line bundle over S , . The characterization of Theorem 3.1 which describesPic Q (cid:0) S , (cid:1) as a torsor over Pic R (cid:0) S , (cid:1) implies the bijection of sets given byPic Q (cid:0) S , (cid:1) ≃ −→ Z + L Q ] ≃ [ L R ⊗ L ] c ( L Q ) = c ( L R ) + c ( L ) . We point out that the “Quaternionic” line bundle L has been used as the reference element for theidentification of Pic Q ( S , ) as a Pic R ( S , )-orbit. Summarizing, one has:(1) A complex line bundle can support a unique (up to isomorphisms) “Real” structure over thetwo-dimensional sphere endowed with the free antipodal action if and only if its first Chernclass is even and di ff erent Chern classes distinguish between inequivalent “Real” line bundles; (2) A complex line bundle can support a unique (up to isomorphisms) “Quaternionic” structureover the two-dimensional sphere endowed with the free antipodal action if and only if its firstChern class is odd and di ff erent Chern classes distinguish between inequivalent “Quater-nionic” line bundles. Moreover, for each L Q ∈ Pic Q ( S , ) is specified by its FKMM-invariantgiven by κ ( L Q ) = c ( L Q ) − . constructed according to Definition 3.3. Let us mention that that results similar to (1) and (2) have been recently derived in [GR] with a dif-ferent technique based on the analysis of obstructions to the construction of trivializing frames. It isinteresting to note that our derivation of (1) and (2) turns out to be no more than an exercise onceTheorem 3.1 has been established. ◭
4. T opological classification of “Q uaternionic ” vector bundles In this section we analyze the role of the FKMM-invariant in the classification of the “Quaternionic”vector bundles. It turns out that the κ -invariant is an extremely e ffi cient tool to solve the classificationproblem in low dimensions. Just to fix the terminology (for the present work) we say that: Definition 4.1 (Low dimension) . A topological space X which verifies Assumption 2.1 is said to be low dimensional if d . Stable rank condition.
The stable rank condition for vector bundles expresses the pretty generalfact that the non trivial topology can be concentrated in a sub-vector bundle of minimal rank. Thisminimal value depends on the dimensionality of the base space and on the category of vector bundlesunder consideration. For complex (as well as real or quaternionic) vector bundles the stable rankcondition is a well-known result (see e.g.[Hus, Chapter 9, Theorem 1.2]). The proof of this result isbased on an “obstruction-type argument” which provides the explicit construction of a certain maximal number of global sections [Hus, Chapter 2, Theorem 7.1]. The latter argument can be generalized tovector bundles over spaces with involution by means of the notion of Z -CW-complex [Mat, AP] (seealso [DG1, Section 4.5]). A Z -CW-complex is a CW-complex made by cells of various dimension thatcarry a Z -action. These Z -cells can be only of two types: They are fixed if the action of Z is trivialor they are free if they have no fixed points. Since this construction is modelled after the usual definitionof CW-complex, just by replacing the “point” by “ Z -point”, (almost) all topological and homologicalproperties valid for CW-complexes have their “natural” counterparts in the equivariant setting. The useof this technique is essential for the determination of the stable rank condition in the case of “Real”vector bundles [DG1, Theorem 4.25] and even rank “Quaternionic” vector bundles [DG2, Theorem2.5]. In this section we discuss the generalization of the stable rank condition in the “Quaternionic”category in situation not covered by in [DG2, Theorem 2.5] (e.g. X τ of any codimension) and for oddrank vector bundles. We start with the even dimensional case. Theorem 4.2 (Stable condition: even rank) . Let ( X , τ ) be an involutive space such that X has a finite Z -CW-complex decomposition of dimension d. Each rank m “Quaternionic” vector bundle ( E , Θ ) over ( X , τ ) such that d m − splits as E ≃ E ⊕ ( X × C m − σ ) ) (4.1) where σ : = [ d + ] (here [ x ] denotes the integer part of x ∈ R ), E is a (possible) non-trivial “Quater-nionic” vector bundle of rank σ and the remainder is the trivial “Quaternionic” vector bundle ofrank m − σ ) . As a consequence, one has Vec m Q (cid:0) X , τ (cid:1) ≃ Vec σ Q (cid:0) X , τ (cid:1) ∀ m > d +
34 (4.2) and more specifically
Vec m Q (cid:0) X , τ (cid:1) = if d = , ∀ m ∈ N (4.3)Vec m Q (cid:0) X , τ (cid:1) ≃ Vec Q (cid:0) X , τ (cid:1) if d ∀ m ∈ N . (4.4) Proof.
The claim above generalizes that of [DG2, Theorem 2.5] because it includes also the cases: (a) X τ = Ø and, (b) X τ a Z -CW-complex of dimension bigger than zero. These extensions are subjectto the possibility of generalizing in the same direction the key construction described in the proof of[DG2, Proposition 2.7]. For the case (a) this generalization is trivial since we need only to assumethat the number of fixed points is zero. At the first step of the inductive argument one must dealonly with pairs of conjugated points { x j , τ ( x j ) } and one can construct Q -pairs of independent sections s , s over { x j , τ ( x j ) } just by choosing s ( x j ) = v , s ( τ x j ) = v , s ( x j ) = Θ (v ), s ( τ x j ) = Θ (v )with v , v ∈ C m \ { } and v , λ Θ (v ) for some λ ∈ C . At this point the rest of the argumentfollows exactly as in the proof of [DG2, Proposition 2.7]. The validity of the case (b) has been alreadyanticipated and justified in [DG2, Remark 2.4]. (cid:4) Remark 4.3 (Stable rank condition for the “Real” case) . The same argument used in the proof ofTheorem 4.2 can be applied to extend the stable rank condition for “Real” vector bundles [DG1, The-orem 4.25] to the case of a free action. In summary, under the conditions: (a) X τ = Ø or (b’) X τ a Z -CW-complex of dimension zero, one has thatVec m R (cid:0) X , τ (cid:1) ≃ Vec σ R ( X , τ ) ∀ m > d +
12 (4.5)
COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 23 where σ : = [ d ], and more specificallyVec m R (cid:0) X , τ (cid:1) = d = , ∀ m ∈ N (4.6)Vec m R (cid:0) X , τ (cid:1) ≃ Vec R (cid:0) X , τ (cid:1) if 2 d ∀ m ∈ N . (4.7)The situation turns out to be quite di ff erent when X τ is a Z -CW-complex of dimension bigger thanzero as pointed out in [DG1, Remark 4.24]. ◭ The odd rank case requires a slight di ff erent analysis and strongly depends on the existence of a“Quaternionic” line bundle. The following result provides the key argument. Lemma 4.4.
Let ( X , τ ) be an involutive space such that X τ = Ø and Pic Q ( X , τ ) , Ø . Then, one has abijection Vec m Q (cid:0) X , τ (cid:1) ≃ Vec m R (cid:0) X , τ (cid:1) ∀ m ∈ N . Moreover,
Vec m + Q (cid:0) X , τ (cid:1) = Ø ⇔ Pic Q (cid:0) X , τ (cid:1) = Ø . Proof.
The proof of the second claim is easy. If [ L Q ] ∈ Pic Q (cid:0) X , τ (cid:1) , Ø one has that L Q ⊕ ( X × C m )is a “Quaternionic” vector bundle of rank 2 m + m + Q ( X , τ ) , Ø. Conversely, if [( E , Θ )] ∈ Vec m + Q ( X , τ ) , Ø, then the line bundle det( E ) inherits a “Quaternionic” structure given by det( Θ )and so Pic Q ( X , τ ) , Ø. Let now assume that Pic Q (cid:0) X , τ (cid:1) , Ø and choose a representative L Q of someelement in Pic Q (cid:0) X , τ (cid:1) . The map L Q ⊗ : Vec m Q (cid:0) X , τ (cid:1) ≃ −→ Vec m R (cid:0) X , τ (cid:1) [ E Q ] [ L Q ⊗ E Q ]turns out to be a bijection of sets for each m ∈ N . This fact can be proved with an argument similarto that in the proof of Theorem 3.1 and is based on the observation that L Q ⊗ L ∗ Q is equivalent to thetrivial element in Pic R (cid:0) X , τ (cid:1) . (cid:4) Let us point out that the condition Vec m Q ( X , τ ) , Ø is always guaranteed by the existence of the trivial“Quaternionic” product bundle X × C m → X . The next result follows by combining Lemma 4.4 andthe stable rank condition for “Real” vector bundles discussed in Remark 4.3. Theorem 4.5 (Stable condition: odd rank) . Let ( X , τ ) be an involutive space such that X has a finite Z -CW-complex decomposition of dimension d and X τ = Ø . Assume also Pic Q ( X , τ ) , Ø . Then Vec m + Q (cid:0) X , τ (cid:1) ≃ Vec σ Q (cid:0) X , τ (cid:1) ∀ m > d − where σ : = [ d ] . In low dimensions this provides Vec m + Q (cid:0) X , τ (cid:1) ≃ Pic Q (cid:0) X , τ (cid:1) = if d = , ∀ m ∈ N (4.8)Vec m + Q (cid:0) X , τ (cid:1) ≃ Pic Q (cid:0) X , τ (cid:1) if d = , ∀ m ∈ N (4.9)Vec m + Q (cid:0) X , τ (cid:1) ≃ Vec Q (cid:0) X , τ (cid:1) if d = , ∀ m ∈ N . (4.10)We point out that the 0 in the (4.8) refers to the existence of a unique element which could also bedi ff erent from a (trivial) product vector bundle.4.2. Injectivity in low dimension.
The FKMM-invariant introduced in Definition 2.8 provides a map κ : Vec m Q (cid:0) X , τ (cid:1) −→ H Z (cid:0) X | X τ , Z (1) (cid:1) which can be used to classify “Quaternionic” vector bundles over ( X , τ ). A remarkable property isthat, under certain assumptions on ( X , τ ), κ provides a natural injection . A similar result has beenalready proved in [DG2, Theorem 1.1] under the hypothesis that: ( α ) ( X , τ ) is a low dimensionalspace according to Definition 4.1 ; and ( β ) ( X , τ ) is an FKMM-space [DG2, Definition 1.1]. While( α ) is indispensable (when d grows more invariants are needed to distinguish between inequivalent “Quaternionic” vector bundles, cf.[DG2, Theorem 1.3]), the assumption ( β ) can be weakened. Firstof all one can renounce to the requirement H Z ( X , Z (1)) = X τ , Ø of any codimension and the case X τ = Ø. The first step forthe study of the injectivity of κ consists in the following generalization (and simplification) of [DG2,Lemma 4.1]. Lemma 4.6.
Let ( X , τ ) be a space with involution such that X τ , Ø . Let ( E , Θ ) and ( E , Θ ) be rank2 “Quaternionic” vector bundles on ( X , τ ) such that: (a) There is an isomorphism Ψ : E | X τ → E | X τ of “Quaternionic” vector bundles over X τ ; (b) κ ( E , Θ ) = κ ( E , Θ ) in H Z ( X | X τ , Z (1)) .Then, there is an isomorphism of “Real” line bundles ψ : (det( E ) , det( Θ )) → (det( E ) , det( Θ )) on ( X , τ ) such that det( Ψ ) = ψ | X τ .Proof. Let j = ,
2. Recall that the two determinant “Real” line bundles (det( E j ) , det( Θ j )) associatedto ( E j , Θ j ) admit unique canonical R -sections s E j : X τ → S (det( E j ) | X τ ) for their restrictions on X τ (Proposition 2.5). Assumption (a) and the uniqueness of the canonical section assure that the R -isomorphism det( Ψ ) : det( E ) | X τ → det( E ) | X τ intertwines the canonical sections in the sense that s E = det( Ψ ) ◦ s E . According to Definition 2.8, the invariants κ ( E j , Θ j ) agree with the isomorphism classes of the pairs(det( E j ) , s E j ). Thus, by assumption (b), there is also an R -isomorphism ψ : det( E ) → det( E ) suchthat s E = ψ | X τ ◦ s E . The di ff erence between the isomorphisms det( Ψ ) and ψ | X τ is uniquely specified by a map u : X τ → U (1) according to the relation det( Ψ ) = ψ | X τ · u . This implies s E = det( Ψ ) ◦ s E = (cid:0) ψ | X τ · u (cid:1) ◦ s E = (cid:0) ψ | X τ ◦ s E (cid:1) · u = s E · u namely u is the constant map u ≡ X τ and so det( Ψ ) = ψ | X τ . (cid:4) Let us consider now a pair of rank 2 “Quaternionic” vector bundles ( E , Θ ) and ( E , Θ ) over ( X , τ )related by an R -isomorphism ψ : (det( E ) , det( Θ )) → (det( E ) , det( Θ )) of the respective determinantline bundles. The construction in [DG2, Lemma 4.1] allows us to define the locally trivial fiber bundle SU (cid:0) E , E , ψ (cid:1) : = G x ∈ X ( Υ x : E | x → E | x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) vector bundle isomorphismsuch that det( Υ x ) = ψ | x ) whose typical fiber is modeled out of S U (2). It has been proved in [DG2, Lemma 4.1] that:(1) There is a natural involution on SU ( E , E , ψ ) covering τ which can be identified with thestandard involution on S U (2) given by µ : u
7→ −
QuQ (see [DG2, Remark 2.1] for moredetails) on the fixed point set X τ ;(2) The set of equivariant sections of SU ( E , E , ψ ) is in bijection with the set of Q -isomorphisms Υ : E → E such that det( Υ ) = ψ .We are now in position to prove the following crucial result. Theorem 4.7 (Injectivity: even rank case) . Let ( X , τ ) be a low dimensional involutive space in thesense of Definition 4.1. Then: (1) If X τ , Ø the map κ : Vec m Q (cid:0) X , τ (cid:1) −→ H Z (cid:0) X | X τ , Z (1) (cid:1) , m ∈ N given by the FKMM-invariant provides a natural injective group homomorphism; COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 25 (2)
If X τ = Ø the map κ : Vec m Q (cid:0) X , τ (cid:1) −→ H Z (cid:0) X , Z (1) (cid:1) , m ∈ N given by the identification of κ with the first “Real” Chern class of the associated determinantline bundle ( cf. Corollary 2.12 (2)) provides a natural injective group homomorphism. Inaddition, this is an isomorphism if
Pic Q ( X , τ ) , Ø .Proof. For d = , m Q ( X , τ ) = d = , m Q ( X , τ ) ≃ Vec Q ( X , τ ) in equation (4.4) reduces the proof to the fact that κ isinjective on Vec Q (cid:0) X , τ (cid:1) .Let us start with the case (1). Let ( E , Θ ) be a “Quaternionic” vector bundle of rank 2 over ( X , τ ).The restriction E | X τ is isomorphic to a bundle of quaternionic lines over X τ [DG2, Proposition 2.2].Since the dimension of X τ is less than four one has that Vec H ( X τ ) = H -vector bundles [Hus, Chapter 9, Theorem 1.2]. Therefore E | X τ is Q -trivial. Let( E , Θ ) be a second “Quaternionic” vector bundle of rank 2 over ( X , τ ). Since also E | X τ turns out tobe Q -trivial for the same reason, there is an isomorphism of Q -bundles Ψ : E | X τ → E | X τ given by thecomposition of the respective trivializations. If κ ( E , Θ ) = κ ( E , Θ ) Lemma 4.6 assures that there isan R -isomorphism ψ : det( E ) → det( E ) such that det( Ψ ) = ψ | X τ . This allows to construct the fiberbundle SU ( E , E , ψ ) and Ψ provides an equivariant section of the restriction SU ( E , E , ψ ) | X τ . Now,the free Z -celles of X have dimensions less than 4 and π j (S U (2)) = j = , , U (2) is 2-connected). These facts are su ffi cient to build a global section Υ : X → SU ( E , E , ψ )which extends Ψ in the sense that Υ | X τ = Ψ . The explicit construction is realized following the samearguments in the proof of [DG2, Proposition 2.7]. The existence of Υ implies the Q -isomorphism( E , Θ ) ≃ ( E , Θ ) and this proves that κ is injective. The group structure on Vec m Q ( X , τ ), whichmakes κ a group homomorphism, has been described in full detail in the proof of [DG2, Theorem 1.1]and is based on the splitting (4.4) and on the additivity of the FKMM-invariant with respect to theWhitney sum (cf.property (4) in Section 2.5).The proof in the case (2) is quite similar. Let ( E , Θ ) and ( E , Θ ) be two “Quaternionic” vectorbundles of rank 2 over ( X , τ ) and assume that κ ( E , Θ ) = κ ( E , Θ ) in H Z ( X , Z (1)). This implies thatthere is an isomorphism of “Real” line bundles ψ : det( E ) → det( E ) which allows to build the fiberbundle SU ( E , E , ψ ). Since X has free Z -celles of dimension less than 4 and the typical fiber S U (2)is 2-connected it follows that SU ( E , E , ψ ) admits a global section which provides a Q -isomorphismbetween ( E , Θ ) and ( E , Θ ). The group structure can be introduced as in the case (i). The last claimis a consequence of Lemma 4.4. (cid:4) Corollary 4.8.
Let ( X , τ ) be a low dimensional involutive space in the sense of Definition 4.1. If theinvolution is trivial, X = X τ , it holds that Vec m Q (cid:0) X , τ (cid:1) = , m ∈ N . Proof.
It follows from the injectivity of κ proved in Theorem 4.7 (1) and the observation that H Z (cid:0) X | X , Z (1) (cid:1) = (cid:4) The odd rank case requires a base space with a free involution X τ = Ø and it is completely de-termined by the nature of Pic Q ( X , τ ). More precisely, when Pic Q ( X , τ (cid:1) = Ø is not possible to build“Quaternionic” vector bundles (cf.Lemma 4.4) and the κ -invariant simply cannot be defined. The casePic Q ( X , τ (cid:1) , Ø is described in the next result.
Theorem 4.9 (Injectivity: odd rank case) . Let ( X , τ ) be a low dimensional involutive space in the senseof Definition 4.1. Assume X τ = Ø and Pic Q ( X , τ ) , Ø . Then, there is a bijection κ : Vec m − Q (cid:0) X , τ (cid:1) ≃ −→ H Z (cid:0) X , Z (1) (cid:1) , m ∈ N and the form of κ depends on a normalization given by the choice a reference element [ L ref ] ∈ Pic Q ( X , τ ) . Proof.
By combining equation (4.9) with Corollary 3.2 one obtains the bijectionsVec m − Q (cid:0) X , τ (cid:1) ≃ Pic Q (cid:0) X , τ (cid:1) ≃ H Z (cid:0) X , Z (1) (cid:1) , m ∈ N both depending on the election of a reference element [ L ref ] ∈ Pic Q ( X , τ ). More precisely, one has theisomorphisms[ E Q ] ≃ [ L ref ] ⊗ [ X × C ] ⊕ (2 m − ≃ [ L ref ] ⊕ (2 m − ⊕ [ L ref ] d = , L ref ] ⊗ (cid:16) [ X × C ] ⊕ (2 m − ⊕ L R (cid:17) ≃ [ L ref ] ⊕ (2 m − ⊕ [ L ref ⊗ L R ] d = , E Q ] [ L ref ⊗ L R ] [ L R ] c R (cid:0) L R (cid:1) with the convention that L R is automatically trivial when d = ,
1. The value of the FKMM-invariantcan be fixed in accordance to Definition 3.3 by the prescription κ ( E Q ) = c R (cid:0) L R (cid:1) which implies the normalization κ ( L ⊕ m − ) = (cid:4) The question of the surjectivity of the FKMM-invariant.
Theorem 4.7 and Theorem 4.9 assurethe injectivity of the FKMM-invariant κ in the case of low dimensional base spaces ( d ff erent topological phases. On the other hand itcould be valuable to know whether the cohomology group where κ maps is a complete set of invariantsor if it is redundant. It turns out that, for a big class low dimensional spaces the FKMM-invariantprovides bijections. For instance, this is the case for the involutive spheres S p , q and the involutive tori T a , b , c , discussed in Section 4.4 and Section 4.5 respectively, or for oriented surfaces with finitely manyisolated fixed points [DG2, Corollary 4.2]. Therefore, one is legitimate to ask whether the surjectivityin low dimension is a general property of the FKMM-invariant or not!Let us start with two observation: ( α ) The FKMM-invariant fails to be surjective in dimension biggerthan three where other invariants like the second Chern class start to play a role in the classification(cf.[DG2, Theorem 1.4]); ( β ) The odd rank case (which requires X τ = Ø) is completely specified bythe nature of Pic Q ( X , τ ) as proved in Theorem 4.9. Therefore, the question of the surjectivity is relevantonly for even rank “Quaternionic” vector bundles over low dimensional involutive spaces. We pointout that the study of this question deserves a separated analysis for the cases X τ = Ø and X τ , Ø.The study of the surjectivity of κ goes beyond the scope of this work. However, we can antici-pate some results proved in [DG5]. Let ( X , τ ) be an involutive space of dimension d which fulfillsAssumption 2.1. Then:(1) When d = , κ -invariant provides a “trivial” isomorphism.(2) When d = κ -invariant provides isomorphismsVec m Q (cid:0) X , τ (cid:1) κ ≃ H Z ( X , Z (1)) if X τ = Ø H Z ( X | X τ , Z (1)) if X τ , Ø . (3) When d = κ -invariant fails to be surjective.The proof of (1) amounts to show that the condition d = , H Z ( X , Z (1)) = X τ = Ø and H Z ( X | X τ , Z (1)) = X τ , Ø. The proof of (2) is quite technical.The payo ff of (1) and (2) is that one can see the κ -invariant as the generalization of the isomorphisms(1.4) and (1.6) for the “Quaternionic” category up to dimension two. The claim (3) marks the di ff erencebetween the FKMM-invariant and the first (“Real”) Chern class.The proof of (3) is based on the construction of an explicit counterexample Let S : = { ( z , z ) ∈ C | | z | + | z | = } ⊂ C be the three dimensional sphere viewed as the unit sphere in C . Considerthe action of Z ≃ {± , ± i } on S given by ( z , z ) ( ρ z , ρ z ) for all ρ ∈ Z . The quotient space L (1; 4) : = S / Z is called (3-dimensional) lens space (cf.[BT, Example 18.5] or [Hat, Example 2.43] for more details).Since Z is preserved by the complex conjugation (with our convention) the involution on S ⊂ C induced by ( z , z ) ( z , z ) descends to an involution τ on L (1; 4). The involutive space ( L (1; 4) , τ )is a smooth (3-dimensional) manifold with a smooth involution, hence it admits a Z -CW-complexstructure [May, Theorem 3.6]. Moreover this space has a fixed point set of the type L (1; 4) τ ≃ S ⊔ S .One can construct the “Quaternionic” vector bundles over ( L (1; 4) , τ ) by the clutching construction providing Vec m Q (cid:0) L (1; 4) , τ (cid:1) = Z . (4.11)On the other hand the computation of the equivariant cohomology provides H Z (cid:0) L (1; 4) | L (1; 4) τ , Z (1) (cid:1) = Z . (4.12)The proofs of (4.11) and (4.12) are quite elaborate and technical and will be presented in [DG5] forthe slightly more general case of a lens space of type L (1; 2 q ) with q ∈ N . In any case, the importantmessage conveyed by (4.11) and (4.12) is that the map κ : Vec m Q (cid:0) L (1; 4) , τ (cid:1) −→ H Z (cid:0) L (1; 4) | L (1; 4) τ , Z (1) (cid:1) cannot be surjective, confirming the claim (3).4.4. Classification over low dimensional involutive spheres.
In this section we apply the results ofSection 4.1 and Section 4.2 to provide the complete classification of “Quaternionic” vector bundlesover the involutive spheres of type S p , q in low dimension d = p + q −
3. These results have beensummarized in Table 1.1. Since S , = Ø we conventionally writeVec k Q (cid:0) S , (cid:1) = Ø . • Spheres of type S p , . These are the spheres with the trivial involution which fixes all points. Corol-lary 4.8 immediately providesVec m Q (cid:0) S p , (cid:1) = , p , ∀ m ∈ N . • Spheres of type S , q . These are the spheres with free antipodal involution and are the only sphereswhich admit “Quaternionic” vector bundles of odd rank. The line bundle case has been already dis-cussed in Section 3.2.[q = q = S , is the two points set {− , + } endowed with the flip involution ±
7→ ∓ k Q (cid:0) S , (cid:1) = , ∀ k ∈ N . For k = m the unique element is given by the (trivial) “Quaternionic” product bundle over {− , + } while for k = m + m “Quaternionic” trivialbundle with the Dupont “Quaternionic” line bundle described in Example 3.5. Also for q = k Q (cid:0) S , (cid:1) = , ∀ k ∈ N . When k = m the result is justified by equation (4.3) and the unique element represented by the trivial“Quaternionic” product bundle. When k = m + m with the “Quaternionic” line bundle L described in Example3.7.[q =
3] - In the odd rank case equation (4.9) and the Example 3.8 provideVec m − Q (cid:0) S , (cid:1) ≃ Pic Q (cid:0) S , (cid:1) ≃ Z + , ∀ m ∈ N where the (second) identification is given by the first Chern class c . This identification requires areference element in Pic Q ( S , ) which can be chosen to be the line bundle L constructed in Example3.8. The proof of Theorem 4.9 shows that each [ E Q ] ∈ Vec m − Q ( S , ) can be uniquely represented by an element [ L R ] ∈ Pic R ( S , ) according to the factorization [ E Q ] ≃ [ L ] ⊕ (2 m − ⊕ [ L ⊗ L R ].Accordingly, one can fix the FKMM-invariant by the prescription κ (cid:0) E Q (cid:1) : = c R (cid:0) L R (cid:1) . The identification 2 c R ( L R ) ≃ c ( L R ) between the “Real” Chern class and the standard Chern class of L R discussed in Example 3.8 provides c (cid:0) E Q (cid:1) ≃ (2 m − c (cid:0) L (cid:1) + c (cid:0) L (cid:1) + c (cid:0) L R (cid:1) ≃ (2 m − c (cid:0) L (cid:1) + c R ( L R )Since c (cid:0) L (cid:1) = c R ( L R ) ∈ Z one obtains that the Chern class of any representative in Vec m − Q ( S , )is necessarily odd and it is su ffi cient for a complete characterization of the isomorphism class [ E Q ].Moreover, the FKMM-invariant takes the form κ (cid:0) E Q (cid:1) = c (cid:0) E Q (cid:1) − (2 m − , [ E Q ] ∈ Vec m − Q (cid:0) S , (cid:1) . In the even rank case one hasVec m Q (cid:0) S , (cid:1) ≃ Vec Q (cid:0) S , (cid:1) κ −→ H Z (cid:0) S , , Z (1) (cid:1) ≃ Z where the first isomorphism is justified by (4.4) and the FKMM-invariant provides an injective mapaccording to Theorem 4.7 (2). Moreover, the existence of L assures that κ is indeed an isomorphismMore precisely, for each [ E Q ] ∈ Vec m Q ( S , ) there exists a unique [ E R ] ∈ Vec m R ( S , ) such that [ E Q ] ≃ [ L ⊗ E R ]. The determinant construction provides det( E Q ) = L ⊗ m ⊗ det( E R ) and a straightforwardcomputation shows that c (cid:0) E Q (cid:1) ≃ c (cid:0) det( E Q ) (cid:1) ≃ m c (cid:0) L (cid:1) + c (cid:0) det( E R ) (cid:1) ≃ m c (cid:0) L (cid:1) + c R (cid:0) det( E R ) (cid:1) . Since c ( L ) = c R ( L R ) ∈ Z one concludes that the Chern class of any representative inVec m Q ( S , ) must be even. Moreover, by definition one has κ (cid:0) E Q (cid:1) : = c R (cid:0) det( E Q ) (cid:1) = c (cid:0) E Q (cid:1) , [ E Q ] ∈ Vec m Q (cid:0) S , (cid:1) . Summarizing, we proved that
Vec m Q (cid:0) S , (cid:1) c ≃ Z Vec m − Q (cid:0) S , (cid:1) c ≃ Z + , ∀ m ∈ N . (4.13)We point out that the first Chern class c turns out to be a complete invariant for the classification of Q -bundles of any rank over S , . The (4.13) recovers the results recently presented in [GR].[q =
4] - By combining Lemma 4.4 and Example 3.6 one concludesVec m − Q (cid:0) S , (cid:1) = Ø , ∀ m ∈ N for the odd rank case. For the even rank case one hasVec m Q (cid:0) S , (cid:1) ≃ Vec Q (cid:0) S , (cid:1) κ −→ H Z (cid:0) S , , Z (1) (cid:1) ≃ κ -invariant proved in Theorem 4.7(2) implies Vec m Q (cid:0) S , (cid:1) = , ∀ m ∈ N and the unique element is represented by the rank 2 m trivial Q -bundle. • Spheres of type S , q . These are the spheres with TR-involution (two distinguished fixed points)which have been studied in [DG2]. We recall that:Vec m Q (cid:0) S , (cid:1) ≃ Vec m Q (cid:0) S , (cid:1) = , Vec m Q (cid:0) S , (cid:1) ≃ Vec m Q (cid:0) S , (cid:1) ≃ Z . In the non-trivial cases S , and S , the Z -phase is discriminated by the FKMM-invariant. COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 29 • Spheres of type S , q . In this case the fixed point set ( S , q ) θ ≃ S is a circle and only even rank“Quaternionic” vector bundles are possible. Equation (4.4) and Theorem 4.7 (1) provideVec m Q (cid:0) S , q (cid:1) ≃ Vec Q (cid:0) S , q (cid:1) κ −→ H Z (cid:0) S , q | S , Z (1) (cid:1) q = , ∀ m ∈ N with κ being an injection.[q =
1] - Proposition B.3 providesVec Q (cid:0) S , (cid:1) κ −→ H Z (cid:0) S , | S , Z (1) (cid:1) ≃ Z Moreover, one has that the isomorphism H Z ( S , | S , Z (1)) ≃ Z is induced by the inclusion H Z (cid:0) S , | S , Z (1) (cid:1) δ ֒ → H Z (cid:0) S , , Z (1) (cid:1) f ≃ H (cid:0) S , Z (cid:1) c ≃ Z by means of the first Chern class of the underlying complex vector bundle. The surjectivity of κ can beproved as follow: Let E be the trivial element in Vec Q ( S , ) and L ∈ Pic R ( S , ) such that c ( L ) = R -line bundle with this property can be explicitly constructed as showed in the proof of PropositionB.3. Since c ( E ) = E n : = E ⊗ ( L ⊗ n ) has Chern class c (cid:0) E n (cid:1) = rk (cid:0) E (cid:1) c (cid:0) L ⊗ n (cid:1) = n c (cid:0) L (cid:1) = n . This fact shows that κ is also surjective. In particular, we proved the isomorphismVec m Q (cid:0) S , (cid:1) c ≃ Z which says that a vector bundle over S , can be endowed with a unique Q -structure if and only if itsfirst Chern class is even.[q =
2] - Equation (B.5) shows that H Z ( S , | S , Z (1)) =
0, hence the injectivity of κ immediatelyimplies that Vec m Q (cid:0) S , (cid:1) = m ∈ N . • Spheres of type S , q . The only case of interest in low dimension is S , . This space contains atwo-dimensional sphere as fixed point set ( S , ) θ ≃ S . Therefore, only the even rank case is relevant.Equation (4.4) and Theorem 4.7 (1) provideVec m Q (cid:0) S , (cid:1) ≃ Vec Q (cid:0) S , (cid:1) κ −→ H Z (cid:0) S , | S , Z (1) (cid:1) with κ being an injection. Since equation (B.6) shows that H Z ( S , | S , Z (1)) = m Q ( S , ) = m ∈ N .4.5. Classification over low dimensional involutive tori.
This section is devoted to the completeclassification of “Quaternionic” vector bundles over involutive tori of the type T a , b , c in low dimension a + b + c
3. These results have been summarized in Table 1.2 and Table 1.3. Since T , , = Ø weconventionally write Vec k Q (cid:0) T , , (cid:1) = Ø . - Cases with fixed points - Involutive tori of type T a , b , have non-empty fixed point sets. There are three di ff erent two-dimensionaltori and four di ff erent three-dimensional tori of this type. For all these cases only even rank Q -bundlesare possible due to the presence of fixed points. • The torus T a , , . Spaces of type T a , , are a -dimensional tori with the trivial involution which fixesall points. Corollary 4.8 immediately providesVec m Q (cid:0) T a , , (cid:1) = , a , ∀ m ∈ N . • The torus T , b , . The spaces T , b , are b -dimensional tori with the TR-involution (2 b distinguishedfixed points) which have been studied in [DG2]. We recall that:Vec m Q (cid:0) T , , (cid:1) ≃ Z , Vec m Q (cid:0) T , , (cid:1) ≃ Z Vec m Q (cid:0) T , , (cid:1) ≃ Z ⊕ Z . for all m ∈ N . The Z -phases are discriminated by the FKMM-invariant. • The torus T , , . In this case the fixed point set ( T , , ) τ ≃ S ⊔ S is given by a disjoint union oftwo circles. Equation (4.4) and Theorem 4.7 (1) provideVec m Q (cid:0) T , , (cid:1) ≃ Vec Q (cid:0) T , , (cid:1) κ −→ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) ≃ Z . with κ being an injection. The last isomorphism is proved in Proposition C.2 and is induced by thecomposition H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ ֒ → H Z (cid:0) T , , , Z (1) (cid:1) f −→ H (cid:0) T , Z (cid:1) c ≃ Z where the forgetting map f turns out to be surjective. The surjectivity of κ can be verified as follow:Let E be the trivial element in Vec Q ( T , , ) and L , ∈ Pic R ( T , , ) the non trivial R -line bundlewith c ( L , ) = c ( E ) = Q -bundle E n : = E ⊗ ( L ⊗ n , ) has Chern class c (cid:0) E n (cid:1) = rk (cid:0) E (cid:1) c (cid:0) L ⊗ n , (cid:1) = n c (cid:0) L , (cid:1) = n . This fact proves that κ is also surjective and in turn one has the isomorphismVec m Q (cid:0) T , , (cid:1) c ≃ Z , ∀ m ∈ N . In particular a vector bundle over T , , can be endowed with a unique Q -structure if and only if it hasan even Chern class. A similar result has been recently obtained in [GR]. • The torus T , , . The fixed point set ( T , , ) τ ≃ T ⊔ T consists of two disjoint copies of a two-dimensional torus. One hasVec m Q (cid:0) T , , (cid:1) ≃ Vec Q (cid:0) T , , (cid:1) κ −→ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) ≃ (2 Z ) where the first isomorphism is justified in (4.4) and Theorem 4.7 (1) assures that κ is an injection. Thelast isomorphism is proved in Proposition C.3 and is induced by the composition H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ ֒ → H Z (cid:0) T , , , Z (1) (cid:1) f −→ H (cid:0) T , Z (cid:1) c ≃ Z where the forgetting map f turns out to be bijective between the free part of H Z ( T , , , Z (1)) and a Z -summand in H ( T , Z ). Let E → T , , be the element of Vec m Q ( T , , ) identified by the Chern class c ( E ) = E in Vec m Q ( T , , ) under the two distinct projections T , , → T , , . These two vector bundles provide a basis for the bijectionVec m Q (cid:0) T , , (cid:1) c ≃ (2 Z ) , ∀ m ∈ N showing in turn that κ is bijective. In particular one has that a vector bundle over T , , can be endowedwith a unique Q − structure if and only if it has a Chern class of type (2 n , n ′ , ∈ Z . • The torus T , , . In this case the fixed point set ( T , , ) τ ≃ S ⊔ S ⊔ S ⊔ S is the disjoint unionof four circles. From (4.4) and Theorem 4.7 (1) one obtainsVec m Q (cid:0) T , , (cid:1) ≃ Vec Q (cid:0) T , , (cid:1) κ −→ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) ≃ Z ⊕ (2 Z ) with κ being an injection. The last isomorphism, proved in Proposition C.4, enters in the compositionof maps H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ −→ H Z (cid:0) T , , , Z (1) (cid:1) f −→ H (cid:0) T , Z (cid:1) c ≃ Z where δ acts injectively on the free part of H Z ( T , , | ( T , , ) τ , Z (1)) and the forgetting map f is abijection between the free part of H Z ( T , , , Z (1)) and a Z -summand in H ( T , Z ). It turns out that COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 31 the map κ is also surjective. In fact, the Z -summand in Vec m Q ( T , , ) is generated by the pullbackof Vec m Q ( T , , ) ≃ Z under the projection T , , → T , , . The two 2 Z -summands, instead, aregenerated by the pullback of Vec m Q ( T , , ) ≃ Z under the two distinct projections T , , → T , , . Inconclusion one obtains thatVec m Q (cid:0) T , , (cid:1) ≃ Z ⊕ (2 Z ) , ∀ m ∈ N . This implies that one can build a Q -structure on T , , if and only if the underlying complex vectorbundle has a first Chern class of type (2 n , n ′ , ∈ Z . Moreover, if this is the case, it is possible tobuild to inequivalent Q -structures. - Cases with free involution - As soon as c , T a , b , c turns out to be free. In this case the classification of“Quaternionic” vector bundles is simplified by two general results. The first of them concerns thepossibility to reduce situations with c > standard situation c = Proposition 4.10.
For each a , b ∈ N ∪ { } and c > there is an identification of Z -spaces T a , b , c ≃ T a + c − , b , . Proof.
Consider the identifications S ≡ U (1) and T d ≡ U (1) × . . . × U (1). By using the group structureof U (1) we can define maps f c : T d → T d of the type f c : ( z , . . . , z d − c + , . . . , z d − , z d | {z } last c coord. ) ( z , . . . , z d − c + z d , . . . , z d − z d | {z } c − , z d ) . Topologically these maps are homeomorphisms. Since S , is identifiable with U (1) endowed withtrivial involution z z and S , is identifiable with U (1) with antipodal involution z
7→ − z oneconcludes that f c is a Z -homeomorphism between T a , b , c and T a + c − , b , . (cid:4) The second result concerns involutive spaces with the product structure X × S , . Proposition 4.11.
For each involutive space ( X , τ ) the FKMM-invariant κ : Vec Q (cid:0) X × S , (cid:1) −→ H Z (cid:0) X × S , , Z (1) (cid:1) is surjective. Moreover there is a bijection Pic Q (cid:0) X × S , (cid:1) ≃ Pic R (cid:0) X × S , (cid:1) , Ø which preserves the first Chern class.Proof. Since the involution on X × S , is free the FKMM-invariant is given by the compositionVec Q (cid:0) X × S , (cid:1) det −→ Pic R (cid:0) X × S , (cid:1) c R −→ H Z (cid:0) X × S , , Z (1) (cid:1) (cf.Corollary 2.12 (2)). In order to show that Pic Q ( X × S , ) , Ø let us introduce a special Q -line bundleby mimicking the construction in Example 3.7. Consider the product line bundle X × S , × C → X × S , endowed with the “Quaternionic” structure Θ q : ( x , k , λ ) ( τ ( x ) , − k , q ( x , k ) λ ) ( x , k , λ ) ∈ X × S , × C where the map q : X × S , → U (1) must fulfill the constraint q ( τ ( x ) , − k ) = − q ( x , k ). A possible choiceis to fix q ( x , k ) = q ( k ) where q is the map described by (3.3). Let ˜ L ∈ Pic Q ( X × S , ) be this Q -linebundle. By construction c ( ˜ L ) = L : = ˜ L ⊗ ˜ L . This R -line bundle agrees with the trivial element in Pic R ( X × S , ). In fact theproduct structure of ˜ L is evident by construction. Moreover, the R -structure on ˜ L is induced by themap q which can be factorized as q ( k ) = φ ( − k ) φ ( k ) with φ : = i q . This shows that q is isomorphicthrough φ to the constant map 1. For each L R ∈ Pic R ( X × S , ) consider the rank two “Quaternionic”vector bundle over X × S , given by E Q : = (cid:0) L R ⊕ ( X × S , × C ) (cid:1) ⊗ ˜ L . The associated determinant line bundle verifiesdet (cid:0) E Q (cid:1) ≃ L R ⊗ ( X × S , × C ) ⊗ ˜ L ≃ L R where we used the triviality of ˜ L . This proves the surjectivity of κ . The last claim is a consequenceof Corollary 3.2 and the existence of ˜ L . (cid:4) Corollary 4.12.
Let ( X , τ ) be an involutive space that fulfills Assumption 2.1 and with dimension d .Then the FKMM-invariant provides a bijection Vec m Q (cid:0) X × S , (cid:1) κ ≃ H Z (cid:0) X × S , , Z (1) (cid:1) ≃ Pic R ( X × S , ) valid for each m ∈ N independently of its parity. In the odd rank case it is possible to normalize theFKMM-invariant κ in such a way that the bijection Vec m − Q (cid:0) X × S , (cid:1) ≃ Pic R ( X × S , ) preserves the Chern class of the underlying complex vector bundle.Proof. The condition about the dimension of X assures the validity of the stable rank condition (cf.eq.(4.4) and eq. (4.9)). In the even rank case the bijectivity of κ is consequence of Theorem 4.7 (2) andProposition 4.11. In the odd rank case the bijection follows from Theorem 4.9 and the existence of˜ L ∈ Pic Q ( X × S , ). The use of ˜ L as reference element for the construction of the bijection and thefact of c ( ˜ L ) = (cid:4) • The torus T , , . In view of Corollary 4.12 one has the bijectionVec m Q (cid:0) T , , (cid:1) κ ≃ H Z (cid:0) T , , , Z (1) (cid:1) ≃ Z ∀ m ∈ N . The computation of the cohomology group is provided in Proposition C.1. However, we can add moreinformation by using the exact sequence [Go, Proposition 2.3] H Z (cid:0) T , , , Z (1) (cid:1) f ✲ H (cid:0) T , Z (cid:1) ✲ H Z (cid:0) T , , , Z (cid:1) ✲ H Z (cid:0) T , , , Z (1) (cid:1) ≃ ≃ ≃ ≃ Z Z Z . Here we used H Z (cid:0) T , , , Z (cid:1) ≃ H Z (cid:0) S , , Z (cid:1) ⊕ H Z (cid:0) S , , Z (1) (cid:1) ≃ Z H Z (cid:0) T , , , Z (1) (cid:1) ≃ H Z (cid:0) S , , Z (1) (cid:1) ⊕ H Z (cid:0) S , , Z (cid:1) ≃ f acts as the multiplication by two and the composition with the first Chern classproduces the identificationVec m Q (cid:0) T , , (cid:1) κ ≃ H Z (cid:0) T , , , Z (1) (cid:1) c ≃ Z ∀ m ∈ N . We point out that the validity of this result in the odd rank case is a consequence of the existence of abijection which preserves the first Chern classes as claimed by Corollary 4.12. • The tori T , , ≃ T , , . Let us study the case T , , . Corollary 4.12 and Proposition C.1 implyVec m Q (cid:0) T , , (cid:1) κ ≃ H Z (cid:0) T , , , Z (1) (cid:1) ≃ Z ∀ m ∈ N . COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 33 • The torus T , , . From Corollary 4.12 and Proposition C.1 one obtainsVec m Q (cid:0) T , , (cid:1) κ ≃ H Z (cid:0) T , , , Z (1) (cid:1) ≃ Z ∀ m ∈ N . A more precise description can be obtained by looking at the exact sequence [Go, Proposition 2.3] H Z (cid:0) T , , , Z (1) (cid:1) f ✲ H (cid:0) T , Z (cid:1) ✲ H Z (cid:0) T , , , Z (cid:1) ✲ H Z (cid:0) T , , , Z (1) (cid:1) ≃ ≃ ≃ ≃ Z Z Z ⊕ Z Z . The cohomology groups can be computed by using the second isomorphism in (C.1) which gives H Z (cid:0) T , , , Z (cid:1) ≃ H Z (cid:0) T , , , Z (cid:1) ⊕ H Z (cid:0) S , , Z (1) (cid:1) ⊕ H Z (cid:0) S , , Z (cid:1) ≃ Z ⊕ Z H Z (cid:0) T , , , Z (1) (cid:1) ≃ H Z (cid:0) T , , , Z (1) (cid:1) ⊕ H Z (cid:0) T , , , Z (cid:1) ≃ Z . The exact sequence alone does not su ffi ce for the determination of the forgetting map f . However,we can notice that H Z ( T , , , Z (1)) ≃ Vec Q ( T , , ) contains two 2 Z -summands generated by thepullbacks of Vec Q ( T , , ) ≃ Z under the two distinct projections T , , → T , , . This fact impliesthat f must act as f : ( n , n ′ ) (2 n , n ′ ,
0) with n , n ′ ∈ Z . The composition with the first Chern classfinally produces the identificationVec m Q (cid:0) T , , (cid:1) c ≃ (2 Z ) ∀ m ∈ N . • The tori T , , ≃ T , , . Let us discuss the case T , , . Corollary 4.12 and Proposition C.1 implyVec m Q (cid:0) T , , (cid:1) κ ≃ H Z (cid:0) T , , , Z (1) (cid:1) ≃ Z ⊕ Z ∀ m ∈ N . However, a deeper analysis shows thatVec m Q (cid:0) T , , (cid:1) ≃ Z ⊕ (2 Z ) ∀ m ∈ N where the Z -summand is the result of the pullback of Vec m Q ( T , , ) ≃ Z under the projection T , , → T , , . In the same way, the two 2 Z -summands are generated by the pullbacks of Vec m Q ( T , , ) ≃ Z and Vec m Q ( T , , ) ≃ Z under the projections T , , → T , , and T , , → T , , , respectively. • The tori T , , ≃ T , , ≃ T , , . Let us focus on the case T , , . From Corollary 4.12 and Proposi-tion C.1 one obtains Vec Q (cid:0) T , , (cid:1) κ ≃ H Z (cid:0) T , , , Z (1) (cid:1) ≃ Z m ∈ N . The two Z -summands can be generated by the pullback of Vec m Q ( T , , ) ≃ Z under the two distinctprojections T , , → T , , .5. A pplication to prototype models of topological quantum systems Let { Σ , . . . , Σ } ∈ Mat( C ) be an irreducible representation of the Cli ff ord algebra C ℓ C (5) generatedby the rules Σ i Σ j + Σ j Σ i = δ i , j for all i , j = , , . . . ,
4. An explicit realization is given by Σ : = σ ⊗ σ Σ : = σ ⊗ σ Σ : = ⊗ σ Σ : = ⊗ σ Σ : = σ ⊗ σ where σ : = ! , σ : = − ii 0 ! , σ : = − ! are the Pauli matrices. With this special choice the following relations hold true: Σ ∗ j = Σ j , Σ j : = C Σ j C = ( − j Σ j , Σ Σ Σ Σ Σ = − . where C v = v is the complex conjugation on C . We need also the relations for the traceTr C ( Σ j ) = , Tr C ( Σ i Σ j ) = δ i , j . (5.1) Let J : = σ ⊗ and Θ : = C ◦ J = − J ◦ C . Evidently J ∗ = J = J − and Θ = − which in turnimplies Θ ∗ = J ◦ C . Then ( Θ Σ j Θ ∗ = − Σ j , j = , , , . Θ Σ Θ ∗ = +Σ Given an involutive space ( X , τ ) and five continuous functions F j : X → R , where j = , , . . . , , one can build a topological quantum system (in the sense of [DG1, Definition 1.1]) H : X → Mat( C )given by H ( x ) : = X j = F j ( x ) Σ j . (5.2)It should be noted that (5.2) provides only a convenient and simple toy model. In principle one canconsider more complicated Hamiltonians by adding terms proportional to the products Σ j . . . Σ j r andrelevant models of this type have been already discussed in the literature (e.g.the Kane-Mele model[KM]). However, in order to simplify the presentation, only the simpler case (5.2) will be consideredhere.By imposing the conditions F j (cid:0) τ ( x ) (cid:1) = − F j ( x ) , j = , , , , F (cid:0) τ ( x ) (cid:1) = + F ( x ) x ∈ X (5.3)one can verify that Θ H ( x ) Θ ∗ = H (cid:0) τ ( x ) (cid:1) , x ∈ X . (5.4)Equation (5.4) says that x H ( x ) defines a topological quantum system with an odd time reversalsymmetry in the sense of [DG1, Definition 1.2]. According to a commonly used notation, the class ofthese systems is denoted by the label class AII (see e.g.[AZ, SRFL]).A simple computation provides H ( x ) = Q ( x ) , Q ( x ) : = X j = F j ( x ) and, after assuming the zero gap condition Q >
0, one can define two projection-valued maps P ± : X → Mat( C ) given by P ± ( x ) : = ± X j = F j ( x ) √ Q ( x ) Σ j . (5.5)The projection relations P ± ( x ) = P ± ( x ), P ± ( x ) P ∓ ( x ) = P + ( x ) ⊕ P − ( x ) = and the spectralrelations [ P ± ( x ) , H ( x )] = P ± ( x ) H ( x ) P ± ( x ) = ± √ Q ( x ) P ± ( x ) can be directly checked. The con-ditions Tr C P ± ( x ) = P ± : X → Mat( C ) defines a continuous family of twodimensional planes in C , namely a rank two complex vector bundle over X . Moreover, under thecondition (5.3) which assures Θ P ± ( x ) Θ ∗ = P ± (cid:0) τ ( x ) (cid:1) , x ∈ X , (5.6)one obtains that the vector bundles E ± → X associated to P ± ( x ) are indeed “Quaternionic”. Formore details about the construction of the vector bundles E ± we refer to [DG1, Section 2 & Section6]. Define f j ( x ) : = F j ( x ) / √ Q ( x ). The family of these functions f j : X → R is subjected to thepointwise constraint P j = f j ( x ) =
1. Therefore, the vector valued map φ : = ( f , f , . . . , f ) provides acontinuous map φ : X → S . This map identifies uniquely the projections P ± ( x ) through the formula(5.5). COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 35
The last observation suggests the introduction of the
Hopf vector bundle E Hopf → S defined by theprojection-valued map P Hopf : S → Mat( C ) given by P Hopf ( k , k , . . . , k ) : = + X j = k j Σ j . (5.7)The involution Θ induces a “Quaternionic” structure on E Hopf provided that the base space S is en-dowed with the involution ( k , k , k , k , k ) ( − k , − k , k , − k , − k ). This space, up to a reorderingof the indices, agrees with the TR sphere S , described in Section 1 (see also [DG1, Example 4.2]).Then, ( E Hopf , Θ ) is an element of Vec m Q ( S , ). This set has been classified in [DG2, Theorem 1.4 (i)]and we know that Vec Q ( S , ) ≃ Z is completely classified by the second Chern class c . Moreover, theFKMM-invariant κ of elements in Vec m Q ( S , ) takes values in Z and is fixed by the parity of c . Thesecond Chern class of E Hopf is c ( E Hopf ) = κ ( E Hopf ) = ( − c ( E Hopf ) = − . Therefore, the Hopf vector bundle E Hopf provides an example of non-trivial Q -bundle over S , with anon-trivial FKMM-invariant.A comparison between (5.7) and (5.5) shows that the vector bundles E ± → X can be obtained from E Hopf → S by a pullback with respect a map φ : X → S which amounts to the replacement of ± f j ( x )with k j in (5.7). This means that all possible vector bundle generated by a topological quantum systemof type (5.2) can be labelled by continuous maps φ : X → S . Moreover, since homotopy equivalentmaps generate (via pullback) isomorphic vector bundles we obtain that the possible topological phasesof the system (5.2) can be labelled by the set [ X , S ] of classes of homotopy equivalence functions.However, if we want to respect the odd time reversal symmetry we need to take into account theconstraints (5.3). These are compatible with the involution θ , since θ , (cid:0) f ( x ) , f ( x ) , f ( x ) , f ( x ) , f ( x ) (cid:1) = (cid:0) − f ( x ) , − f ( x ) , f ( x ) , − f ( x ) , − f ( x ) (cid:1) = (cid:0) f ( τ ( x )) , f ( τ ( x )) , f ( τ ( x )) , f ( τ ( x )) , f ( τ ( x )) (cid:1) and one deduces that the map φ is indeed Z -equivariant from ( X , τ ) to S , . Therefore, the topology ofa systems (5.2) with an odd time reversal symmetry imposed by (5.3) is encoded in the set [ X , S , ] Z of classes of Z -homotopy equivalence of equivariant functions. In summary we showed that: Proposition 5.1 (Su ffi cient condition for non-trivial phases) . Each class AII topological quantum sys-tem given by a gapped
Hamiltonian of type (5.2) subjected to the constraints (5.3) is obtained as thepullback of the Hopf Q -bundle E Hopf with respect to a map [ φ ] ∈ [ X , S , ] Z . In this sense [ X , S , ] Z provides a (generally non injective) complete set of labels for the topological phases of these systems.Finally, by naturality, the FKMM-invariant for these systems is obtained as κ (cid:0) φ ∗ E Hopf (cid:1) = φ ∗ (cid:0) κ ( E Hopf ) (cid:1) = φ ∗ (cid:0) − and is non-trivial whenever the equivariant map φ : X → S , induces a homomorphism φ ∗ : H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) −→ H Z (cid:0) X | X τ , Z (1) (cid:1) which is injective . In Appendix B we computed H Z (cid:0) S , d | ( S , d ) θ , Z (1) (cid:1) ≃ ( d = , Z if d > . (5.8)Since the condition f j ≡ f j ’s vanishes then only asub-vector bundle of the Hopf Q -bundle E Hopf is responsible for the pullback. To be more precise, letus introduce the embedding maps ı : S , d − → S , d which identify S , d − with the subset of S , d fixed by the condition k j = θ , d (according to the above notation this means j , S , ı −→ S , ı −→ . . . ı −→ S , d ı −→ S , d + . . . and a related sequence of restricted Hopf Q -bundle E ( iv − n )Hopf : = ı n ∗ E Hopf ∈ Vec Q (cid:0) S , − n (cid:1) where we used the short notation ı n : = ı ◦ . . . ◦ ı and the convention E ( iv )Hopf ≡ E Hopf . We know from theclassification established in [DG2] thatVec Q (cid:0) S , (cid:1) ≃ Vec Q (cid:0) S , (cid:1) = E ( i )Hopf and E (0)Hopf are both trivial. On the other hand we know thatVec Q (cid:0) S , (cid:1) ≃ Vec Q (cid:0) S , (cid:1) = Z are completely classified by the FKMM-invariant and by naturality κ (cid:0) E ( iii )Hopf (cid:1) = ı ∗ κ (cid:0) E Hopf (cid:1) = ı ∗ ( − , κ (cid:0) E ( ii )Hopf (cid:1) = ı ∗ κ (cid:0) E Hopf (cid:1) = ı ∗ ( − . Since the maps ı : S , d − → S , d induce isomorphisms (see Lemma B.1) ı ∗ : H Z (cid:0) S , d | ( S , d ) θ , Z (1) (cid:1) ≃ −→ H Z (cid:0) S , d − | ( S , d − ) θ , Z (1) (cid:1) , d > E ( iii )Hopf ∈ Vec Q ( S , ) , E ( ii )Hopf ∈ Vec Q ( S , )are non-trivial representatives of the respective classes. These considerations have implications in theanalysis of topological quantum systems of type (5.2). Proposition 5.2 (The case F . . Consider class AII topological quantum systems given by gapped
Hamiltonians of type (5.2) subjected to the constraints (5.3) and F . . A necessary condition toproduce a non-trivial topological phases is that at least two of the functions { F , F , F , F } have tobe not identically zero. If this is the case, the set [ X , S , − n ] Z provides a (generally non injective)complete set of labels for the topological phases on these systems. Here n = , , represents thenumber of functions in { F , F , F , F } which are identically zero. Finally, by naturality, the FKMM-invariant of these systems is given by κ (cid:0) φ ∗ E ( iv − n )Hopf (cid:1) = φ ∗ (cid:0) κ ( E ( iv − n )Hopf ) (cid:1) = φ ∗ (cid:0) − , n = , , and is non-trivial whenever the equivariant map φ : X → S , − n induces a homomorphism φ ∗ : H Z (cid:0) S , − n | ( S , − n ) θ , Z (1) (cid:1) −→ H Z (cid:0) X | X τ , Z (1) (cid:1) which is injective . It should be emphasized that the calculation of H Z ( S , − n | ( S , − n ) θ , Z (1)) is quite easy since the co-homology classes can be represented in terms of sign maps defined on the fixed-point set ( S , − n ) θ . Atthis point the verification of the injectivity of φ ∗ is not a di ffi cult problem provided that the equivariantcohomology of X can be computed. The latter, however, in general may be a di ffi cult problem!Let us now discuss the implications of the condition F ≡
0. In this case the vector bundles E ± → X are generated by the pullback with respect to a map φ : X → S , ⊂ S , where S , is the subspacefixed by the constraint k =
0. In particular S , is the three dimensional sphere with antipodal freeinvolution. The equivariant embedding η : S , → S , produces by pullback the restricted Hopf Q -bundle ˆ E Hopf : = η ∗ E Hopf ∈ Vec Q (cid:0) S , (cid:1) = . Since the vector bundles E ± → X subjected to the condition F ≡ E Hopf which is trivial one obtains:
COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 37
Proposition 5.3 (The case F ≡ . Consider class AII topological quantum systems given by gapped
Hamiltonians of type (5.2) subjected to the constraints (5.3) and F ≡ . These systems possesses onlythe trivial phase. A ppendix A. A bout the equivariant cohomology of the C lassifying space The cohomology of G m ( C ∞ ) is the polynomial ring H • (cid:0) G m ( C ∞ ) , Z (cid:1) ≃ Z [ c , . . . , c m ] (A.1)generated by the universal Chern classes c k ∈ H k (cid:0) G m ( C ∞ ) , Z (cid:1) [MS, Theorem 14.5]. The involution ρ on the space G m ( C ∞ ) described in Section 2.6 induces a homomorphism ρ ∗ on each cohomologygroup H k ( G m ( C ∞ ) , Z ). ρ ∗ acts on the universal Chern classes as follows: Lemma A.1.
It holds that ρ ∗ ( c k ) = ( − k c k , k = , , . . . , m . Proof.
Since c k is the k -th Chern class of the tautological vector bundle T ∞ m → G m ( C ∞ ) it followsby naturality that ρ ∗ ( c k ) is the k -th Chern class of the pullback vector bundle ρ ∗ ( T ∞ m ). The presence ofthe Q -structure implies that the fibers of T ∞ m and ρ ∗ ( T ∞ m ) relative to a given point of the base spaceare related by an anti-linear transform. The result follows from the same argument as in [MS, Lemma14.9]. (cid:4) The equivariant cohomology of a fixed point {∗} plays the role of the coe ffi cient system for the equi-variant cohomology of an equivariant space. We recall that (cf.[Go, Section 2.3] and [DG1, Section5.1]) H • Z (cid:0) {∗} , Z (cid:1) = H • (cid:0) R P ∞ , Z (cid:1) ≃ Z [ t ] / (2 t ) (A.2)where the generator t ∈ H (cid:0) R P ∞ , Z (cid:1) ≃ Z obeys 2 t = H • Z (cid:0) {∗} , Z (1) (cid:1) = H • (cid:0) R P ∞ , Z (1) (cid:1) ≃ t / Z [ t ] / (2 t / , t ) (A.3)where also t / ∈ H (cid:0) R P ∞ , Z (1) (cid:1) ≃ Z is subjected to 2 t / =
0. The relation t / ∪ t / = t given bythe cup product provides the ring structure H • Z (cid:0) {∗} , Z (cid:1) ⊕ H • Z (cid:0) {∗} , Z (1) (cid:1) ≃ Z [ t / ] / (2 t / ) . (A.4)The generators t / , t and t / : = t / ∪ t ∈ H Z (cid:0) {∗} , Z (1) (cid:1) play an important role in the description ofthe (low dimensional) equivariant cohomology of the involutive space ˆ G m ( C ∞ ) ≡ ( G m ( C ∞ ) , ρ ). Lemma A.2.
The cohomology groups of the involutive space ˆ G m ( C ∞ ) up to the degree are sum-marized in the Table A.1. Moreover, the generators of H Z ( ˆ G m ( C ∞ ) , Z (1)) , H Z ( ˆ G m ( C ∞ ) , Z ) andH Z ( ˆ G m ( C ∞ ) , Z (1)) can be identified with t / , t and t / , respectively. Similarly, the generator of the Z -summand in the group H Z ( ˆ G m ( C ∞ ) , Z ) can be identified with t : = t ∪ t .k = k = k = k = k = H k (cid:0) G m ( C ∞ ) , Z (cid:1) Z Z Z H k Z (cid:0) ˆ G m ( C ∞ ) , Z (cid:1) Z Z Z Z ⊕ Z H k Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) Z Z Z Z T able A.1.
Proof.
The ordinary (non-equivariant) cohomology is described by (A.1). The computation of theequivariant cohomology requires the use of the Leray-Serre spectral sequence. For more details werefer to [DG3, Appendix D] and references therein. We know that there is a converging spectralsequence E p , q ⇒ H p + q (cid:0) ˆ G m ( C ∞ ) , Z ( j ) (cid:1) which in page 2 is given by the group cohomology of Z , E p , q : = H p group (cid:0) Z ; H q ( G m ( C ∞ ) , Z ) ⊗ Z Z ( j ) (cid:1) , where j = , ffi cient systems H q ( G m ( C ∞ ) , Z ) ⊗ Z Z ( j ) are regarded as Z -modules withrespect to the involution ρ ∗ on H q ( G m ( C ∞ ) , Z ) described in Lemma A.1 and the action n ( − j n on Z ( j ). The page 2 is concentrated in the first quadrant, meaning that E p , q = p < q <
0. Inthe case j = E p , q are showed in the following table: q = q = Z Z Z q = q = Z Z Z q = q = Z Z Z E p , q ( j = p = p = p = p = p = p = E p , i + = H q ( G m ( C ∞ ) , Z )is concentrated in even degree. This implies that the di ff erentials d : E p , q → E p + , q − are trivial sothat E p , q ≃ E p , q . Moreover, the di ff erentials d : E p , → E p + , , p = , , , . . . have to be trivial since E p , must survive inside the direct summand E p , ≃ H p Z ( {∗} , Z (1)) of thedecomposition H p (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) ≃ H p Z (cid:0) {∗} , Z (1) (cid:1) ⊕ ˜ H p (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) (A.5)where ˜ H • denotes the relative cohomology with respect to a fixed point. In fact, if d were nontrivial, some of the E p + , would be killed producing a contradiction. For instance, suppose that d : E , → E , is non trivial. Then, E , = E , ≃ E , ≃ . . . ≃ E , ∞ =
0. However, from ageneral property of the spectral sequence E , ∞ injects into H ( ˆ G m ( C ∞ ) , Z (1)). Consider the followingcommutative diagram E p , ( {∗} ) ≃ H Z (cid:0) {∗} , Z (1) (cid:1) ≃ Z E , s ✲ ∗ ✛ H (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) ∗ ✲ (A.6)where s is the injection induced by the spectral sequence and ∗ is the homomorphism induced bythe collapsing (equivariant) map : ˆ G m ( C ∞ ) → {∗} . Also ∗ is injective due to the decomposition COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 39 (A.5). The map ∗ can be seen as the homomorphism induced by between the spectral sequenceassociated to the equivariant cohomology of the fixed point E p , ( {∗} ) ≃ E p , ∞ ( {∗} ) ≃ H • Z ( {∗} , Z (1)) andthe spectral sequence for the cohomology of ˆ G m ( C ∞ ). Since s ◦ ∗ = ∗ is injective also ∗ has tobe injective, proving that E , ∞ ,
0. This contradiction shows that d is trivial. The triviality of d implies that E p , q ≃ E p , q for all 0 p + q
4. Moreover, all the di ff erentials d r : E p , qr → E p + r , q − r + r are automatically trivial for 0 p + q r >
5, so that one concludes E p , q ≃ E p , q ∞ in the range0 p + q
4. In this situation the extension problem0 −→ F p + H p + q (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) −→ F p H p + q (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) −→ E p , q ∞ −→ p + q p + q = n there is at leastone pair which gives a non-trivial E p , q . In conclusion one finds that H (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) ≃ E , ∞ ≃ H (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) ≃ E , ∞ ≃ Z H (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) ≃ E , ∞ ≃ Z H (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) ≃ E , ∞ ≃ Z H (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) ≃ E , ∞ ≃ Z . The case j = q = q = Z Z Z q = q = Z Z Z q = q = Z Z Z E p , q ( j = p = p = p = p = p = p = E p , q ≃ E p , q and one can show that the maps d : E p , → E p + , are trivial for all p since E p , ≃ H p Z ( {∗} , Z ) has to survive to contribute as a direct summand of H p Z ( ˆ G m ( C ∞ ) , Z ) dueto the presence of a fixed point. Therefore, one can determine E p , q ∞ for all p + q
3. For p + q = d : E , → E , is trivial or not. However, inany event, the kernel of this map is Z and this is enough to compute E p , q ∞ also for all p + q =
4. Since E , must survive as the contribution from H Z ( {∗} , Z ), the extension problem for H Z ( ˆ G m ( C ∞ ) , Z )is trivial, and the computation is completed.Finally we notice that the identification with the generators t / , t , t / and t is clear from the directsummand argument. (cid:4) The last result has important consequences on the description of the cohomology of ˆ G m ( C ∞ ). Proposition A.3.
The group H Z ( ˆ G m ( C ∞ ) , Z (1)) ≃ Z is generated by the (universal) first “Real”Chern class c R : = c R (cid:16) det (cid:0) T ∞ m (cid:1)(cid:17) of the “Real” line bundle det( T ∞ m ) → ˆ G m ( C ∞ ) associated to the tautological Q -bundle. Moreover,under the map f : H Z ( ˆ G m ( C ∞ ) , Z (1)) → H ( G m ( C ∞ ) , Z ) which forgets the Z -action the class c R is mapped in the first universal Chern class c .Proof. The exact sequence [Go, Proposition 2.3] H Z (cid:0) ˆ G m ( C ∞ ) , Z (cid:1) δ −→ H Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) f −→ H (cid:0) G m ( C ∞ ) , Z (cid:1) g −→ H Z (cid:0) ˆ G m ( C ∞ ) , Z (cid:1) . . . and H Z (cid:0) ˆ G m ( C ∞ ) , Z (cid:1) = f is injective. Moreover, g −→ H Z (cid:0) ˆ G m ( C ∞ ) , Z (cid:1) δ ′ −→ H Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) −→ H (cid:0) G m ( C ∞ ) , Z (cid:1) . . . , along with H (cid:0) G m ( C ∞ ) , Z (cid:1) = H Z (cid:0) ˆ G m ( C ∞ ) , Z (cid:1) ≃ Z ≃ H Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) , implies that δ ′ is an isomorphism. Therefore g = f : H Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) ≃ −→ H (cid:0) G m ( C ∞ ) , Z (cid:1) is an isomorphism. The general behavior of the “Real” Chern classes under the forgetting map is f : c R (cid:16) det (cid:0) T ∞ m (cid:1)(cid:17) c (cid:16) det (cid:0) T ∞ m (cid:1)(cid:17) . Finally, the equality c (det( T ∞ m )) = c ( T ∞ m ) and the definition of universal Chern class c = c ( T ∞ m )conclude the proof. (cid:4) Since the fixed point set ˆ G m ( C ∞ ) ρ can be identified with G m ( H ∞ ) one has that H • (cid:0) ˆ G m ( C ∞ ) ρ , Z (cid:1) ≃ H • (cid:0) G m ( H ∞ ) , Z (cid:1) . (A.7)The latter is given by the polynomial ring H • (cid:0) G m ( H ∞ ) , Z (cid:1) ≃ Z [ q , . . . , q m ] (A.8)generated by the universal (symplectic) Pontryagin classes q k ∈ H k ( G m ( H ∞ ) , Z ) [Hus, Chapter 17]. Lemma A.4.
It holds thatH • Z (cid:0) ˆ G m ( C ∞ ) ρ , Z (cid:1) ⊕ H • Z (cid:0) ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) ≃ Z [ t / , q , . . . , q m ] / (2 t / ) where the class t / agrees with the generator of H Z (cid:0) {∗} , Z (1) (cid:1) ≃ Z and q k ∈ H k ( ˆ G m ( C ∞ ) ρ , Z ) arethe symplectic Pontryagin classes (up to the identification (A.7) ).Proof. The involution on ˆ G m ( C ∞ ) ρ ≃ G m ( H ∞ ) is trivial, hence H • Z (cid:0) ˆ G m ( C ∞ ) ρ , Z ( j ) (cid:1) ≃ H • (cid:0) G m ( H ∞ ) × R P ∞ , Z ( j ) (cid:1) , j = , H • ( G m ( H ∞ )) imply that H • Z (cid:0) ˆ G m ( C ∞ ) ρ , Z ( j ) (cid:1) ≃ H • Z (cid:0) {∗} , Z ( j ) (cid:1) ⊗ H • (cid:0) G m ( H ∞ ) , Z (cid:1) More precisely, a comparison with (A.2) and (A.3) provides that H • Z (cid:0) ˆ G m ( C ∞ ) ρ , Z ( j ) (cid:1) ≃ Z [ t , q , . . . , q m ] / (2 t ) if j = t / Z [ t , q , . . . , q m ] / (2 t / , t ) if j = t / ∈ H Z (cid:0) {∗} , Z (1) (cid:1) , t ∈ H (cid:0) {∗} , Z (cid:1) ≃ Z and q k ∈ H k ( ˆ G m ( C ∞ ) ρ , Z ) are the symplecticPontryagin classes. The final result is a consequence of (A.4). (cid:4) The low order cohomology of ˆ G m ( C ∞ ) ρ is summarized in the following table:The relative equivariant cohomology of the pair ˆ G m ( C ∞ ) ρ ֒ → ˆ G m ( C ∞ ) can be conputed with thehelp of the long exact sequence (2.6). COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 41 k = k = k = k = k = H k (cid:0) ˆ G m ( C ∞ ) ρ , Z (cid:1) Z Z H k Z (cid:0) ˆ G m ( C ∞ ) ρ , Z (cid:1) Z Z Z ⊕ Z H k Z (cid:0) ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) Z Z Proposition A.5.
It holds that k = k = k = k = k = H k Z (cid:0) ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) Z Z and the map δ : H Z (cid:0) ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) ≃ −→ H Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) is an isomorphism.Proof. The restriction map ˆ G m ( C ∞ ) ρ ֒ → ˆ G m ( C ∞ ) induces homomorphisms r : H k Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) −→ H k Z (cid:0) ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) . The analysis in Lemma A.2 and Lemma A.4 shows that the homomorphisms in degree k = k = t / and t / , respectively. To concludethe proof one needs to inspect the long exact sequence (2.6) for the relative equivariant cohomology H k Z ( ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1)). The degree k = k = δ −→ H Z (cid:0) ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) δ −→ Z r −→ ≃ Z δ −→ and this shows that H Z ( ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1)) =
0. In degree k = δ −→ H Z (cid:0) ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) δ −→ H Z (cid:0) ˆ G m ( C ∞ ) , Z (1) (cid:1) r −→ H Z ( ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1)) and H Z ( ˆ G m ( C ∞ ) , Z (1)) ≃ Z .In degree k = δ −→ H Z (cid:0) ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) δ −→ Z r −→ ≃ Z δ −→ which implies H Z ( ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1)) =
0. Finally, in degree k = δ −→ H Z (cid:0) ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1) (cid:1) δ −→ Z r −→ H Z ( ˆ G m ( C ∞ ) | ˆ G m ( C ∞ ) ρ , Z (1)) ≃ Z . (cid:4) A ppendix B. A bout the equivariant cohomology of involutive spheres
B.1.
TR-involution.
Let S , d : = ( S d , θ , d ) be the d -dimensional sphere with TR-involution θ , d :( k , k , . . . , k d ) ( k , − k , . . . , − k d ) (cf.[DG1, Example 4.2]). The equivariant cohomology groups H k Z ( S , d , Z (1)) have been computed in [DG1, Section 5.4]: H even Z (cid:0) S , d , Z (1) (cid:1) ≃ H odd Z (cid:0) S , d , Z (1) (cid:1) ≃ Z ⊕ Z if k = d Z if k < d Z ⊕ Z if k > d . (B.1) Moreover, since ( S , d ) θ = {− , + } fixed consists of two fixed points (in each dimension), one has that H k Z (cid:0) ( S , d ) θ , Z (1) (cid:1) ≃ H k Z (cid:0) {∗} , Z (1) (cid:1) ⊕ H k Z (cid:0) {∗} , Z (1) (cid:1) ≃ ( Z ⊕ Z if k is odd0 if k is even . The groups H Z (cid:0) S , d | ( S , d ) θ , Z (1) (cid:1) for d > H Z (cid:0) S , d | ( S , d ) θ , Z (1) (cid:1) ≃ Z , d > . When d = S , = {− , + } fixed = ( S , d ) θ immediately implies H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) = . The case d = −→ H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) δ −→ Z ⊕ Z r −→ Z ⊕ Z δ −→ H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) −→ . The same argument as in the proof of [DG2, Proposition A.1] shows that r acts bijectively on the Z summand. On the other hand each homomorphism Z → Z has Z as kernel. Thus H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) = Z . Finally, since the Z -summand in Z ⊕ Z is diagonally embedded into Z ⊕ Z and the image of the Z -summand is { (1 , − , ( − , } , one concludes that r is surjective, and so H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) = . Let ı : S , d − → S , d be the embedding maps which identify S , d − with the subset of S , d defined bythe constraint k d = Lemma B.1.
The homomorphisms ı ∗ : H Z (cid:0) S , d | ( S , d ) θ , Z (1) (cid:1) −→ H Z (cid:0) S , d − | ( S , d − ) θ , Z (1) (cid:1) induced by ı : S , d − → S , d are isomorphisms for all d > .Proof. For d > H Z ( S , d , Z (1)) ≃ Z is given by the summand H Z (cid:0) {∗} , Z (1) (cid:1) where the fixed point ∗ ∈ S , d is preserved by the embedding. Therefore, ı ∗ : H Z ( S , d , Z (1)) → H Z ( S , d − , Z (1)) is an isomorphism. Also the maps ı ∗ : H Z (( S , d ) θ , Z (1)) → H Z (( S , d − ) θ , Z (1))are isomorphisms since the fixed point set {− , + } fixed is preserved by the embedding. The naturalityof the exact sequence for pairs concludes the proof. (cid:4) B.2.
Antipodal involution.
Let S , d + : = ( S d , θ , d + ) be the d -dimensional sphere with (free) antipo-dal involution θ , d + : ( k , k , . . . , k d ) ( − k , − k , . . . , − k d ) (cf.[DG1, Example 4.1]). We want tocompute the equivariant cohomology of S , d + . The ordinary integer cohomology of the sphere isgiven by H k (cid:0) S d , Z (cid:1) ≃ ( Z if k = , d k , , d . The equivariant cohomology with integer coe ffi cients can be computed by observing that θ , d + actsfreely with orbit space S , d + /θ , d + ≃ R P d . Therefore H k Z (cid:0) S , d + , Z (cid:1) ≃ H k (cid:0) R P d , Z (cid:1) ≃ Z if k = Z if k even 0 < k < d Z if k = d odd Z if k = d even0 otherwiseby construction. The equivariant cohomology with Z (1) coe ffi cients can be investigated by the help ofthe Gysin exact sequence . Proposition B.2.
Let S , d + be the sphere endowed with the free antipodal involution. Then: COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 43 (1)
In odd dimension, d = n + :H k Z (cid:0) S , n + , Z (1) (cid:1) ≃ ( Z if k odd k n + otherwise . (2) In even dimension, d = n:H k Z (cid:0) S , n + , Z (1) (cid:1) ≃ Z if k odd k < n Z if k = n otherwise . Proof. (1) Let π : R d + → {∗} be the Z -equivariant real vector bundle with non-trivial Z -action givenby θ , d + : ( k , . . . , k d ) ( − k , . . . , − k d ). The d -dimensional antipodal sphere S , d + coincides withthe sphere-bundle of π : R d + → {∗} , i.e. S , d + = S ( R d + ). The use the Gysin exact sequence requiresthe specification of the equivariant orientability and the equivariant Euler class of π : R d + → {∗} . If d is odd, π : R d + → {∗} is equivariant orientable since its first Z -equivariant Stiefel-Whiteny classvanishes, i.e. w Z ( R d + ) =
0. This can be proved by observing that w Z ( R d + ) ∈ H Z ( {∗} , Z ) ≃ H ( R P ∞ , Z ) ≃ Hom Z ( Z , Z ) ≃ Z agrees with the homomorphism induced by det( θ , d + ) : Z → Z and det( θ , d + ) = + d is odd. According to the proof of [Go, Proposition 2.6], the Z -equivariant Euler class χ Z ( R ) of π : R → {∗} can be identified with the class t / which generates the ring (A.4), so that χ Z ( R d + ) = χ Z ( R ) d + = t ( d + / ∈ H d + Z ( {∗} , Z ) . Now, we are ready to use the Gysin sequence for the π : S , d + = S ( R d + ) → {∗} : H k − d − Z (cid:0) {∗} , Z (1) (cid:1) χ ∪ −→ H k Z (cid:0) {∗} , Z (1) (cid:1) π ∗ −→ H k Z (cid:0) S , d + , Z (1) (cid:1) π ∗ −→ H k − d Z (cid:0) {∗} , Z (1) (cid:1) χ ∪ −→ . . . where χ ∪ denotes the cup product with χ Z ( R d + ). The map π ∗ turns out to be an isomorphism when k d . Moreover, χ ∪ is an isomorphism for k > d + H k Z ( S , d + , Z (1)) = k > d + d is even the main di ff erence is that the Z -vector bundle π : R d + → {∗} is not equivariantlyorientable since w Z ( R d + ) ≃ det( θ , d + ) = −
1. This implies that the relevant Gysin sequence reads H k − d − Z (cid:0) {∗} , Z (cid:1) χ ∪ −→ H k Z (cid:0) {∗} , Z (1) (cid:1) π ∗ −→ H k Z (cid:0) S , d + , Z (1) (cid:1) π ∗ −→ H k − d Z (cid:0) {∗} , Z (cid:1) χ ∪ −→ . . . and, as in the odd case, one deduces that π ∗ is an isomorphism for k d and H k Z ( S , d + , Z (1)) = k > d + (cid:4) Proposition B.2 produces the following table in low dimension: H k Z (cid:0) S , d + , Z (1) (cid:1) k = k = k = k = k = k = k = . . . d = Z . . . d = Z Z . . . d = Z Z . . . d = Z Z Z . . . d = Z Z Z . . . B.3.
More general involutions.
Let S p , q be an involutive sphere of type ( p , q ) according to the nota-tion introduced in Section 1. There is an identification of Z -spaces S p , q ≃ S , ∧ . . . ∧ S , | {z } ( p − ∧ S , ∧ . . . ∧ S , | {z } q -times , p > ∧ denotes the smash product of topological spaces [Hat, Chapter 0]. Moreover, the equivariant suspension isomorphism [Go, Section 2.4] provides˜ H k Z (cid:0) S p , q , Z ( j ) (cid:1) ≃ ˜ H k − q Z (cid:0) S p , , Z ( j − q ) (cid:1) ≃ H k − q − ( p − Z (cid:0) {∗} , Z ( j − q ) (cid:1) where ˜ H k Z is the reduced cohomology. Finally, let us recall that in presence of a fixed point {∗} ∈ X τ the exact sequence (2.6) can be extended to the reduced cohomology theory and one obtains . . . −→ H k Z (cid:0) X | X τ , Z (cid:1) δ −→ ˜ H k Z (cid:0) X , Z (cid:1) r −→ ˜ H k Z (cid:0) X τ , Z (cid:1) δ −→ H k + Z (cid:0) X | X τ , Z (cid:1) −→ . . . (B.2)With this information one can compute the equivariant cohomology of S , . The fixed point set( S , ) θ ≃ S is a circle and the suspension isomorphism says˜ H k Z (cid:0) S , , Z (1) (cid:1) ≃ H k − Z (cid:0) {∗} , Z (cid:1) ≃ H k − (cid:0) R P ∞ , Z (cid:1) ˜ H k Z (cid:0) ( S , ) θ , Z (1) (cid:1) ≃ H k − Z (cid:0) {∗} , Z (1) (cid:1) ≃ H k − (cid:0) R P ∞ , Z (1) (cid:1) . The values of H k Z (cid:0) {∗} , Z ( j ) (cid:1) are listed in (A.2) and (A.3). From (B.2) one extracts the following exactsequence 0 ✲ H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) δ ✲ ˜ H Z (cid:0) S , , Z (1) (cid:1) r ✲ ˜ H Z (cid:0) ( S , ) θ , Z (1) (cid:1) ≃ ≃ ≃ ? Z Z (B.3)which can be used to prove the following result: Proposition B.3.
The map r in (B.3) is surjective and consequently there is an isomorphism of groupsH Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) ≃ Z which is induced by the compositionH Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) δ ֒ → H Z (cid:0) S , , Z (1) (cid:1) f ≃ H (cid:0) S , Z (cid:1) c ≃ Z with f the map that forgets the Z -action and c the first Chern class.Proof. To understand the homomorphism r in (B.3) one identifies the bases of these cohomologygroups by “Real” line bundles according to the Kahn’s isomorphism (2.5). The sequence of groupisomorphisms ˜ H Z (cid:0) ( S , ) θ , Z (1) (cid:1) ≃ H Z (cid:0) S , , Z (1) (cid:1) ≃ Pic R (cid:0) S (cid:1) ≃ Z (B.4)tells that the non-trivial element of ˜ H Z (( S , ) θ , Z (1)) can be identified with the M¨obius bundle on S or, equivalently, with the “Real” line bundle L M = S × C with “Real” structure ( k , λ ) ( k , φ ( k ) λ )where φ : S → U (1) is the map given by φ ( k ) : = k + i k . The first isomorphism in (B.4) can beverified directly by looking at the construction of the reduced cohomology groups (see e.g.[DG1, eq.5.10]). The second isomorphism is justified by the fact that ( S , ) θ coincides with S endowed with thetrivial involution and by [DG1, Proposition 4.5]. In a similar way one has the group isomorphisms˜ H Z (cid:0) S , , Z (1) (cid:1) ≃ H Z (cid:0) S , , Z (1) (cid:1) ≃ Pic R (cid:0) S , (cid:1) . The isomorphism Pic R ( S , ) ≃ Z can be realized explicitely. Consider the map f : H Z ( S , , Z (1)) → H ( S , Z ) which forgets the Z -action. Since H Z ( S , , Z ) = H Z ( S , , Z ) = f is a bijection. A non trivial representative for a generator of H ( S , Z ) ≃ Pic C ( S ) is provided by the line bundle L → S described by the family of Hopf projections k COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 45 P Hopf ( k ) in equation (3.5). We know that L has Chern class c ( L ) =
1. Moreover, L can beendowed with a “Real” structure over S , induced by C P
Hopf ( k , k , k ) C = P Hopf ( k , − k , k )where C implements the complex conjugation on C (cf.Section 5). Set the “Real” line bundle ( L , C )as a representative for a generator of H Z ( S , , Z (1)). The inclusion ı : S , → S , , realized by ı :( k , k ) ( k , , k ), can be used to define the restricted (pullback) line bundle ı ∗ L ∈ H Z ( S , , Z (1)).The latter is equivalently described by the family of “restricted” Hopf projections S ≃ R / π Z ∋ ω ˜ P Hopf ( ω ) : = + cos( ω ) sin( ω )sin( ω ) 1 − cos( ω ) ! through the parameterization k ≡ sin( ω ) and k ≡ cos( ω ). There is a continuous section of normalizedeigenvectors R / π Z ∋ ω s ( ω ) : = e − i ω cos (cid:16) ω (cid:17) sin (cid:16) ω (cid:17) . We notice that the pre-factor e − i ω is need to assure the continuity of the section. One can use s to fixa global trivialization S × C ∋ ( ω, λ ) ψ λ s ( ω ) ∈ ı ∗ L . Since the map ψ is compatible with the “Real” structure of L M it follows that ı ∗ L ≃ L M as “Real”line bundles. This fact immediately implies that the restriction map r in (B.3) is surjective. (cid:4) The case S , can be studied along the same lines. Again the fixed point set ( S , ) θ ≃ S is a circleand the suspension isomorphism provides˜ H k Z (cid:0) S , , Z (1) (cid:1) ≃ H k − Z (cid:0) {∗} , Z (1) (cid:1) ≃ H k − (cid:0) R P ∞ , Z (1) (cid:1) ˜ H k Z (cid:0) ( S , ) θ , Z (1) (cid:1) ≃ H k − Z (cid:0) {∗} , Z (1) (cid:1) ≃ H k − (cid:0) R P ∞ , Z (1) (cid:1) . In this case the (B.2) immediately provides H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) = . (B.5)The fixed point set of S , is a two-dimensional sphere ( S , ) θ ≃ S . In this case the suspensionisomorphism provides ˜ H k Z (cid:0) S , , Z (1) (cid:1) ≃ H k − Z (cid:0) {∗} , Z (cid:1) ≃ H k − (cid:0) R P ∞ , Z (cid:1) ˜ H k Z (cid:0) ( S , ) θ , Z (1) (cid:1) ≃ H k − Z (cid:0) {∗} , Z (1) (cid:1) ≃ H k − (cid:0) R P ∞ , Z (1) (cid:1) and the exact sequence (B.2) implies H Z (cid:0) S , | ( S , ) θ , Z (1) (cid:1) = . (B.6)A ppendix C. A bout the equivariant cohomology of involutive tori
In this section we study the equivariant cohomology of involutive tori of type T a , b , c introduced inSection 1. Note that as soon as c > T a , b , c is free. An important computational toolis provided by the Gysin exact sequence [Go, Corollary 2.11] which establishes the isomorphisms ofgroups H k Z (cid:0) X × S , , Z ( j ) (cid:1) ≃ H k Z (cid:0) X , Z ( j ) (cid:1) ⊕ H k − Z (cid:0) X , Z ( j ) (cid:1) H k Z (cid:0) X × S , , Z ( j ) (cid:1) ≃ H k Z (cid:0) X , Z ( j ) (cid:1) ⊕ H k − Z (cid:0) X , Z ( j − (cid:1) . (C.1) C.1.
The free-involution cases.
When c > T a , b , c has a free involution. Due to Proposition4.10 we can focus our attention on the case c = Proposition C.1.
For each a , b ∈ N ∪ { } there is a group isomorphismH Z (cid:0) T a , b , , Z (1) (cid:1) ≃ Z a ⊕ Z ( a + b . Proof.
An iterated use of the Gysin exact sequences (C.1) provides H Z (cid:0) T a , b , , Z (1) (cid:1) ≃ H Z (cid:0) T , b , , Z (1) (cid:1) ⊕ H Z (cid:0) T , b , , Z (1) (cid:1) ⊕ a ⊕ H Z (cid:0) T , b , , Z (1) (cid:1) ⊕ ( a )where H − j Z ( X , Z (1)) ≡ H Z (cid:0) T , b , , Z (1) (cid:1) ≃ H Z (cid:0) S , , Z (1) (cid:1) ⊕ H Z (cid:0) S , , Z (cid:1) ⊕ b ⊕ H Z (cid:0) S , , Z (1) (cid:1) ⊕ ( b ) H Z (cid:0) T , b , , Z (1) (cid:1) ≃ H Z (cid:0) S , , Z (1) (cid:1) ⊕ H Z (cid:0) S , , Z (cid:1) ⊕ b H Z (cid:0) T , b , , Z (1) (cid:1) ≃ H Z (cid:0) S , , Z (1) (cid:1) due to the equality T , , = S , . The cohomology of S , has been computed in Section B.2. (cid:4) In low dimension one has: H Z ( T a , b , , Z (1)) b = b = b = a = Z Z a = Z Z ⊕ Z Z ⊕ Z a = Z Z ⊕ Z Z ⊕ Z C.2.
The cases with non-empty fixed point sets.
The condition c = T a , b , has fixedpoints. Let us start with the two-dimensional torus T , , which has a fixed point set given by thedisjoint union of two circles, ( T , , ) τ ≃ S , ⊔ S , . A repeated use of the Gysin exact sequences (C.1)provides H k Z (cid:0) T , , , Z ( j ) (cid:1) ≃ H k Z (cid:0) S , , Z ( j ) (cid:1) ⊕ H k − Z (cid:0) S , , Z ( j − (cid:1) ≃ H k Z (cid:0) {∗} , Z ( j ) (cid:1) ⊕ H k − Z (cid:0) {∗} , Z ( j ) (cid:1) ⊕ H k − Z (cid:0) {∗} , Z ( j − (cid:1) ⊕ H k − Z (cid:0) {∗} , Z ( j − (cid:1) . The equivariant cohomology of the fixed point set is computed by H k Z (cid:0) ( T , , ) τ , Z ( j ) (cid:1) ≃ H k Z (cid:0) S , , Z ( j ) (cid:1) ⊕ H k Z (cid:0) S , , Z ( j ) (cid:1) ≃ H k Z (cid:0) {∗} , Z ( j ) (cid:1) ⊕ ⊕ H k − Z (cid:0) {∗} , Z ( j ) (cid:1) ⊕ . These computations allow to extract from (2.6) the exact sequence Z ✲ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ ✲ H Z (cid:0) T , , , Z (1) (cid:1) r ✲ H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) . ≃ ≃ ≃ ? Z ⊕ Z Z (C.2) Proposition C.2.
The map r in (C.2) is surjective and consequently there is an isomorphism of groupsH Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) ≃ Z (C.3) which is induced by the compositionH Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ ֒ → H Z (cid:0) T , , , Z (1) (cid:1) f −→ H (cid:0) T , Z (cid:1) c ≃ Z (C.4) where f is the map that forgets the Z -action and c is the first Chern class. COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 47
Proof.
The determination of H Z ( T , , | ( T , , ) τ , Z (1)) requires an accurate knowledge of the homo-morphism r in the exact sequence (C.2). Let us start with the r in degree 1. In this case we can use thegeometric identification (2.5) to write H Z (cid:0) T , , , Z (1) (cid:1) ≃ (cid:2) T , , , S , (cid:3) Z r −→ (cid:2) ( T , , ) τ , S , (cid:3) Z ≃ H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) . A basis for (cid:2) ( T , , ) τ , S , (cid:3) Z ≃ (cid:2) S , , S , (cid:3) Z ⊕ (cid:2) S , , S , (cid:3) Z ≃ Z is provided by two copies of the constant map g : S , → ( − , ∈ S , . A basis for (cid:2) T , , , S , (cid:3) Z ≃ (cid:2) S , , S , (cid:3) Z ≃ Z ⊕ Z (C.5)is given by the constant map ˜ g : S , × S , → ( − , ∈ S , for the Z -summand and by the projection π : S , × S , → S , for the Z -summand. Note that the first isomorphism in (C.5) is inducedby H Z ( T , , , Z (1)) ≃ H Z ( S , , Z (1)) which follows from the use of the first Gysin exact sequence(C.1). This analysis shows that the map r is surjective in degree 1 and so the exact sequence (C.2) canbe rewritten as 0 −→ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ −→ Z ⊕ Z r −→ Z . (C.6)In particular δ turns out to be an injection. The Kahn’s isomorphism provides an easy way to describethe basis of H Z (cid:0) T , , , Z (1) (cid:1) ≃ Pic R (cid:0) S , × S , (cid:1) ≃ Z ⊕ Z . The Z -summand in Pic R (cid:0) S , × S , (cid:1) is given by the pullback under the projection π : S , × S , → S , of the “Real” M¨obius bundle L M ∈ Pic R ( S , ) (cf.the proof of Proposition B.3). The constructionof the generator of the Z -summand requires a little of work. Let U (1) × R × C → U (1) × R be thetrivial complex line bundle over U (1) × R endowed with a Z -action α n : ( z , x , λ ) ( z , x + n , z n λ )for all n ∈ Z . Since the Z -action is free on the base space one can build the quotient line bundle L , : = ( U (1) × R × C ) /α over U (1) × ( R / Z ) ≃ T . This line bundle L , → T has Chern class c ( L , ) =
1. Moreover, it can be endowed with the “Real” structure Θ : [( z , x , λ )] [( z , − x , λ )]which converts L , into a non-trivial element of Pic R ( S , × S , ) with Chern class 1. This explicitconstruction proves, in particular, that the map f in (C.4) is surjective. A basis for H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) ≃ Pic R (cid:0) S , (cid:1) ⊕ Pic R (cid:0) S , (cid:1) ≃ Z is given by two copies of the M¨obius bundle L M . At this point it is enough to note that L , restrictsto the trivial “Real” bundle on S , × { } ⊂ T , , and to the M¨obius bundle on S , × { } ⊂ T , , . Thisfact means the map r in (C.6) is surjective, hence H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) ≃ Ker h r : H Z (cid:0) T , , , Z (1) (cid:1) → H Z (cid:0) ( T , , ) τ , Z (1) (cid:1)i ≃ Z . Finally, the injectivity of δ and the surjectivity of f assure that the isomorphism (C.3) is realized bythe composition of maps in (C.4). (cid:4) A similar result can be proved for the torus T , , . Let us start again with the exact sequence Z ✲ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ ✲ H Z (cid:0) T , , , Z (1) (cid:1) r ✲ H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) . ≃ ≃ ≃ ? Z ⊕ Z Z (C.7)The computation of the cohomology groups follows by a repeated use of the Gysin exact sequences(C.1) along with the fact that the fixed point set ( T , , ) τ ≃ T , , ⊔ T , , is the disjoint union of twotwo-dimensional tori. Proposition C.3.
The map r in (C.7) is surjective and consequently there is an isomorphism of groupsH Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) ≃ (2 Z ) (C.8) which is induced by the compositionH Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ ֒ → H Z (cid:0) T , , , Z (1) (cid:1) f −→ H (cid:0) T , Z (cid:1) c ≃ Z (C.9) where f is the map that forgets the Z -action and c is the first Chern class.Proof. The proof follows the same arguments as in Proposition C.2. One needs information about thehomomorphism r in degree 1 and 2. In degree 1 the use of the geometric identification (2.5) provides H Z (cid:0) T , , , Z (1) (cid:1) ≃ (cid:2) T , , , S , (cid:3) Z r −→ (cid:2) ( T , , ) τ , S , (cid:3) Z ≃ H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) . A basis for (cid:2) ( T , , ) τ , S , (cid:3) Z ≃ (cid:2) T , , , S , (cid:3) Z ⊕ (cid:2) T , , , S , (cid:3) Z ≃ Z is given by two copies of the constant map g : T , , → ( − , ∈ S , . A basis for (cid:2) T , , , S , (cid:3) Z ≃ (cid:2) S , , S , (cid:3) Z ≃ Z ⊕ Z (C.10)is given by the constant map ˜ g : T , , × S , → − ∈ S , for the Z -summand and by the projection π : T , , × S , → S , for the Z -summand. Notice that the first isomorphism in (C.10) is consequenceof H Z ( T , , , Z (1)) ≃ H Z ( S , , Z (1)) which follows from the use of the first Gysin exact sequence(C.1). This analysis says that the map r is surjective in degree 1 and the exact sequence (C.7) assumesthe short form 0 −→ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ −→ Z ⊕ Z r −→ Z . (C.11)In particular δ is an injection. In degree 2 we can use the Kahn’s isomorphism H Z (cid:0) T , , , Z (1) (cid:1) ≃ Pic R (cid:0) S , × S , × S , (cid:1) ≃ Z ⊕ Z . The two Z -summands are given by the pullback under the first and second projection π j : S , × S , × S , → S , of the “Real” M¨obius bundle L M ∈ Pic R ( S , ) (cf.the proof of Proposition B.3). The two Z -summands are given by the pullback under the two di ff erent projections ˜ π j : S , × S , × S , → S , × S , of the line bundle L , → S , × S , constructed in the proof of Proposition C.2. Thisexplicit construction also shows that the f in (C.9) restricts to a bijection from the free part Z of H Z ( T , , , Z (1)) to a direct summand Z in H ( T , Z ) ≃ Z . A basis for H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) ≃ Pic R (cid:0) T , , (cid:1) ⊕ Pic R (cid:0) T , , (cid:1) ≃ Z is given by two copies of π ∗ j L M for each j = ,
2. At this point the same argument in the proof ofProposition C.2 can be adapted to show that the map r in (C.7) is surjective, hence H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) ≃ Ker h r : H Z (cid:0) T , , , Z (1) (cid:1) → H Z (cid:0) ( T , , ) τ , Z (1) (cid:1)i ≃ (2 Z ) . Finally, the injectivity of δ and the properties of f assure that the isomorphism (C.3) is realized bythe sequence of maps in (C.9). (cid:4) Finally, let us investigate the involutive torus T , , . In this case the exact sequence reads Z ✲ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ ✲ H Z (cid:0) T , , , Z (1) (cid:1) r ✲ H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) . ≃ ≃ ≃ ? Z ⊕ Z Z (C.12)The computation of the cohomology groups follows from a repeated use of the Gysin exact sequences(C.1) along with the fact that ( T , , ) τ ≃ S , ⊔ S , ⊔ S , ⊔ S , . Proposition C.4.
There is an isomorphism of groupsH Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) ≃ Z ⊕ (2 Z ) (C.13) which is given by the direct sum of Coker h r : H Z (cid:0) T , , , Z (1) (cid:1) → H Z (cid:0) ( T , , ) τ , Z (1) (cid:1)i ≃ Z Ker h r : H Z (cid:0) T , , , Z (1) (cid:1) → H Z (cid:0) ( T , , ) τ , Z (1) (cid:1)i ≃ (2 Z ) . COHOMOLOGICAL GENERALIZATION OF THE FU-KANE-MELE INDEX 49
Moreover, in the sequence of mapsH Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ −→ H Z (cid:0) T , , , Z (1) (cid:1) f −→ H (cid:0) T , Z (cid:1) c ≃ Z (C.14) δ is an injection of the free part of H Z ( T , , | ( T , , ) τ , Z (1)) into the free part of H Z ( T , , , Z (1)) and f is a bijection between the free part of H Z ( T , , , Z (1)) and a direct Z -summand in H ( T , Z ) .Proof. Let us study the homomorphism r in degree 1 and 2. In degree 1 we have the geometricidentification H Z (cid:0) T , , , Z (1) (cid:1) ≃ (cid:2) T , , , S , (cid:3) Z r −→ (cid:2) ( T , , ) τ , S , (cid:3) Z ≃ H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) . A basis for (cid:2) ( T , , ) τ , S , (cid:3) Z ≃ (cid:2) S , , S , (cid:3) ⊕ Z ≃ Z is provided by four copies of the constant map g : S , → ( − , ∈ S , . A basis for (cid:2) T , , , S , (cid:3) Z ≃ H (cid:0) S , , Z (1) (cid:1) ⊕ H (cid:0) S , , Z (cid:1) ≃ Z ⊕ Z (C.15)is given by the constant map ˜ g : T , , → ( − , ∈ S , for the Z -summand (which comes from H ( S , , Z (1))) and by the two distinct projections S , × S , × S , → S , for the two Z -summands.This proves that Coker h r : H Z (cid:0) T , , , Z (1) (cid:1) → H Z (cid:0) ( T , , ) τ , Z (1) (cid:1)i ≃ Z and the exact sequence (C.12) assumes the short expression0 −→ Z −→ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) δ −→ Z ⊕ Z r −→ Z . (C.16)Before to describe the degree 2 let us note that the forgetting map f : H Z ( T , , , Z (1)) → H ( T , Z )induces a bijection from the free part Z of H Z ( T , , , Z (1)) to a Z -summand in H ( T , Z ) ≃ Z .In fact this last Z -summand turns out to be generated by the two projections S , × S , × S , → S , upon forgetting the Z -action. In degree 2 the Kahn’s isomorphism provides H Z (cid:0) T , , , Z (1) (cid:1) ≃ Pic R (cid:0) S , × S , × S , (cid:1) ≃ Z ⊕ Z . The Z -summand is generated by the pullback of the “Real” M¨obius bundle L M ∈ Pic R ( S , ) underthe projection S , × S , × S , → S , . In fact H Z ( S , , Z (1)) is contained in H Z ( T , , , Z (1)) as adirect summand by the splitting of the Gysin sequence (C.1). The two Z -summands are generated bythe pullbacks of L , ∈ Pic R ( T , , ) under the two distinct projections T , , → T , , . By an exactsequence argument one can show that the forgetting map f : H Z ( T , , , Z (1)) → H ( T , Z ) inducesan injection from the free part Z of H Z ( T , , , Z (1)) into H ( T , Z ) ≃ Z . Actually, this injectionis a bijection from the free part of H Z ( T , , , Z (1)) to a direct Z -summand in H ( T , Z ). In factthis Z -summand can be generated by the pullbacks of L , ∈ Pic R ( T , , ) under the two distinctprojections T , , → T , , upon forgetting the Z -action. A basis for H Z (cid:0) ( T , , ) τ , Z (1) (cid:1) ≃ Pic R (cid:0) S , (cid:1) ⊕ ≃ Z is given by four copies of the pullback of the line bundle L M under the four distinct projections( T , , ) τ → S , . This analysis shows thatKer h r : H Z (cid:0) T , , , Z (1) (cid:1) → H Z (cid:0) ( T , , ) τ , Z (1) (cid:1)i ≃ (2 Z ) . The conclusions above imply that the group H Z ( T , , | ( T , , ) τ , Z (1)) fits in the exact sequence0 −→ Z −→ H Z (cid:0) T , , | ( T , , ) τ , Z (1) (cid:1) −→ (2 Z ) −→ Z ) is a free group and this concludes the proof. (cid:4) R eferences [AB] Aharonov, Y.; Bohm, D.: Significanceofelectromagneticpotentialsinquantumtheory. Phy. Rev. , 485-491,(1959)[AF] Ando, Y.; Fu, L.: Topologicalcrystallineinsulators andtopological superconductors: fromconcepts tomateri-als. Annu. Rev. Cond. Matt. Phys. , 361-381 (2015)[AP] Allday, C.; Puppe, V.: Cohomological Methods in Transformation Groups . Cambridge University Press, Cam-bridge, 1993[At1] Atiyah, M. F.: K -theoryandreality. Quart. J. Math. Oxford Ser. (2) , 367-386 (1966)[AZ] Altland, A.; Zirnbauer, M.:: Non-standard symmetry classes in mesoscopic normal-superconducting hybridstructures. Phys. Rev.B , 1142-1161 (1997)[BCR] Bourne C.; Carey A.L.; Rennie A.: Anoncommutative frameworkfortopologicalinsulators. Rev. Math. Phys.28, 1650004, (2016)[Be] Berry M. V.: Quantal Phase Factors Accompanying Adiabatic Changes. Proc. Roy. Soc. Lond. A. , 45-57,(1984)[BMKNZ] B¨ohm, A.; and Mostafazadeh, A.; Koizumi, H.; Niu, Q.; Zwanziger, J.: The Geometric Phase in QuantumSystems . Springer-Verlag, Berlin, 2003[Bo] Borel, A.:
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