Geometric cycles, index theory and twisted K-homology
aa r X i v : . [ m a t h . K T ] J u l GEOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY
BAI-LING WANGA
BSTRACT . We study twisted
Spin c -manifolds over a paracompact Hausdorff space X with a twisting α : X → K ( Z , . We introduce the topological index and the analyticalindex on the bordism group of α -twisted Spin c -manifolds over ( X, α ) , taking values intopological twisted K-homology and analytical twisted K-homology respectively. The mainresult of this paper is to establish the equality between the topological index and the analyt-ical index for smooth manifolds. We also define a notion of geometric twisted K-homology,whose cycles are geometric cycles of ( X, α ) analogous to Baum-Douglas’s geometric cy-cles. As an application of our twisted index theorem, we discuss the twisted longitudinalindex theorem for a foliated manifold ( X, F ) with a twisting α : X → K ( Z , , whichgeneralizes the Connes-Skandalis index theorem for foliations and the Atiyah-Singer fami-lies index theorem to twisted cases. C ONTENTS
1. Introduction 22. Review of twisted K-theory 73. Twisted
Spin c -manifolds and analytical index 124. Twisted Spin c bordism and topological index 185. Topological index = Analytical index 296. Geometric cycles and geometric twisted K-homology 367. The twisted longitudinal index theorem for foliation 418. Final remarks 45References 49 Date : September 25, 2007.2000
Mathematics Subject Classification.
Key words and phrases.
Twisted
Spin c -manifolds, twisted K-homology, twisted index theorem.
1. I
NTRODUCTION
According to work of Baum and Douglas [10] [11], the Atiyah-Singer index theorem([6][7]) for a closed smooth manifold X can be formulated as in the following commutativetriangle K ( T ∗ X ) Index t y y ssssssssss Index a % % LLLLLLLLLL K t ( X ) µ ∼ = / / K a ( X ) , (1.1)whose arrows are all isomorphisms. Here K ( T ∗ X ) denotes the K-cohomology with com-pact supports of the cotangent bundle T ∗ X , corresponding to symbol classes of ellipticpseudo-differential operators on X . K t ( X ) is the topological K-homology constructed in[10], and K a ( X ) is the Kasparov’s analytical K-homology (see [31] and [27]) of the C ∗ -algebra C ( X ) of continuous complex-valued functions on X .The topological index and the analytical index can defined on the level of cycles. Thebasic cycles for K t ( X ) (resp. K t ( X ) ) are triples ( M, ι, E ) consisting of even-dimensional(resp. odd-dimensional) closed smooth manifolds M with a given Spin c structure on thetangent bundle of M together with a continuous map ι : M → X and a complex vectorbundle E over M . The equivalence relation on the set of all cycles is generated by thefollowing three steps (see [10] for details):(i) Bordism(ii) Direct sum and disjoint union(iii) Vector bundle modification.Addition in K tev/odd ( X ) is given by the disjoint union operation of topological cycles.A symbol class in K ( T ∗ X ) of an elliptic pseudo-differential operator D on X is repre-sented by σ ( D ) : π ∗ E −→ π ∗ E where π : T ∗ X → X is the projection, E and E are complex vector bundles over X .Choose a Riemannian metric on X , let S ( T ∗ X ⊕ R ) be the unit sphere bundle in T ∗ X ⊕ R ,equipped with the natural Spin c structure. Denote by φ the projection S ( T ∗ X ⊕ R ) → X .Let ˆ E be the complex vector bundle over S ( T ∗ X ⊕ R ) obtained by the clutch construction(See section 10 in [10]): as S ( T ∗ X ⊕ R ) consists of two copies of the unit ball bundle of T ∗ X glued together along the unit sphere bundle, one can use the symbol σ ( D ) to clutch π ∗ E and π ∗ E together along the unit sphere bundle S ( T ∗ X ) . The topological index Index t ([ σ ( D )]) is represented by the following topological cycle ( S ( T ∗ X ⊕ R ) , ˆ E, φ ) . The Kasparov’s analytical K-homology K aev/odd ( X ) , denoted KK ev/odd ( C ( X ) , C ) inthe literature, is generated by unitary equivalence classes of multi-graded Fredholm modules EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 3 over C ( X ) modulo operator homotopy relation [31]. Addition in KK ev/odd ( C ( X ) , C ) isdefined using a natural notion of direct sum of Fredholm modules, see [27] for details. Theanalytical index Index a ([ σ ( D )]) is defined in terms of Poincar´e duality (Cf. [32]) K ( T ∗ X ) ∼ = KK ( C , C c ( T ∗ X )) ∼ = KK ( C ( X ) , C ) (Kasparov’s Poincar´e duality) = K a ( X ) . On the level of cycles, an even dimensional topological cycle ( M, f, E ) defines a canon-ical element [ /D EM ] in K a ( M ) determined by the Dirac operator /D EM : C ∞ ( S + ⊗ E ) −→ C ∞ ( S − ⊗ E ) where S ± are the positive and negative spinor bundles (called reduced spinor bundles in[27]). Then the natural isomorphism µ : K t ( X ) −→ K a ( X ) is defined by the correspondence ( M, ι, E ) ι ∗ ([ /D EM ]) where ι ∗ : K a ( M ) −→ K a ( X ) is the covariant homomorphism induced by ι .The commutative triangle (1.1) has played an important role in the understanding ofthe Atiyah-Singer index theorem and its various generalizations such as the Baum-Connesconjecture in [9]. In this paper, we will generalize the Atiyah-Singer index theorem to theframework of twisted K-theory following ideas inspired from Baum-Douglas [10] [11] andBaum-Connes [9].In this paper, we aim to develop the index theorem in the framework of twisted K-theorywhich is a natural generalization of the Baum-Douglas commutative triangle (1.1). We needa notion of a twisting in complex K-theory, given by a continuous map α : X −→ K ( Z , where K ( Z , is an Eilenberg–MacLane space. We often choose a homotopy model of K ( Z , as the classifying space of the projective unitary group P U ( H ) of an infinitedimensional, complex and separable Hilbert space H , equipped with the norm topology.The norm topology could be too restrictive for some examples, one might have to use thecompact-open topology instead as discussed in [5].For any paracompact Hausdorff space X with a continuous map α : X → K ( Z , , thecorresponding principal K ( Z , -bundle over X will be denoted by P α . Then any base-point preserving action of K ( Z , on a spectrum defines an associated bundle of basedspectra. In this paper, we mainly consider two spectra, one is the complex K-theory spec-trum K = { Ω n K } , the other is the Spin c Thom spectrum
MSpin c = { MSpin c ( n ) } . The BAI-LING WANG corresponding bundle of based spectra over X will be denoted by P α ( K ) and P α ( MSpin c ) respectively.Twisted K-cohomology groups of ( X, α ) are defined to be π (cid:0) C c ( X, P α (Ω n K )) (cid:1) the homotopy classes of compactly supported sections of P α (Ω n K ) .Let K be the C ∗ -algebra of compact operators on the Hilbert space H , and P α ( K ) bethe associated bundle of compact operators corresponding to the P U ( H ) -action on K byconjugation. An equivalent definition of twisted K-theory of ( X, α ) is the algebraic K-cohomology groups of the continuous trace C ∗ -algebra over X of compactly supportedsections of P α ( K ) . The Bott periodicity of the K-theory spectrum implies that we onlyhave two twisted K-groups, denoted by K ( X, α ) and K ( X, α ) , or simply K ev/odd ( X, α ) . We will review twisted K-theory and its basic properties in sec-tion 2.We define topological twisted K-homology to be K tn ( X, α ) := lim −→ k →∞ [ S n +2 k , P α (Ω k K ) /X ] the stable homotopy groups of P α ( K ) /X .Due to Bott periodicity, we only have two different topological twisted K-homologygroups denoted by K tev/odd ( X, α ) . There is a notion of analytical twisted K-homologydefined as Kasparov’s analytical K-homology K aev/odd ( X, α ) := KK ev/odd ( C c ( X, P α ( K )) , C ) . Now we can state the main theorem of this paper, which should be thought of as the generalindex theorem in the framework of twisted K-theory.
Main Theorem (Cf. Theorem 5.1 and Remark 5.3)
Let X be a smooth manifold and π : T ∗ X → X be the projection , there is a natural isomorphism Φ : K tev/odd ( X, α ) −→ K aev/odd ( X, α ) , and there exist notions of the topological index and the analytical index on K ev/odd ( T ∗ X, α ◦ π ) EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 5 such that the following diagram K ev/odd ( T ∗ X, α ◦ π ) Index t v v lllllllllllll Index a ( ( RRRRRRRRRRRRR K tev/odd ( X, α ) ∼ =Φ / / K aev/odd ( X, α ) is commutative, and all arrows are isomorphisms. We remark that topological and analytical twisted K-homology groups are well-definedfor any paracompact Hausdorff space X with a continuous map α : X → K ( Z , . Theabove main theorem only holds for smooth manifolds, we believe that the isomorphism Φ : K tev/odd ( X, α ) −→ K aev/odd ( X, α ) , should be true for more general spaces such as paracompact Hausdorff spaces with thehomotopy type of finite CW complexes. We only establish this isomorphism for smoothmanifolds by applying the Poincar´e duality in twisted K-theory which requires differen-tial structures, see the proof of Theorem 5.1 for details. It would be interesting to havethis isomorphism for paracompact Hausdorff spaces with the homotopy type of finite CWcomplexes.To prove this main theorem, we introduce a notion of α -twisted Spin c manifolds overany paracompact Hausdorff space X with a continuous map α : X → K ( Z , in Section3, which consists of quadruples ( M, ν, ι, η ) where(1) M is a smooth, oriented and compact manifold together with a fixed classifyingmap of its stable normal bundle ν : M −→ BSO here
BSO = lim −→ k BSO ( k ) is the classifying space of stable normal bundle of M ;(2) ι : M → X is a continuous map;(3) η is an α -twisted Spin c structure on M , that is a homotopy commutative diagram(see Definition 3.1 for details) M ι (cid:15) (cid:15) ν / / BSO η v ~ v v v v vv v v v v W (cid:15) (cid:15) X α / / K ( Z , . A manifold M admits an α -twisted Spin c structure if and only if there exists a continuousmap ι : M → X such that ι ∗ ([ α ]) + W ( M ) = 0 in H ( M, Z ) . (This is known to physicists as the Freed-Witten anomaly cancellation con-dition for Type II D-branes (Cf. [26]) ). BAI-LING WANG
We then define an analytical index for each α -twisted Spin c manifold over X takingvalues in the analytical twisted K-homology K aev/odd ( X, α ) and establish its bordism in-variance.In Section 4, we study the geometric α -twisted bordism groups Ω Spin c ∗ ( X, α ) and estab-lish a generalized Pontrjagin-Thom isomorphism (Cf. Theorem 4.4) between our geomet-ric α -twisted bordism groups and the homotopy theoretic definition of α -twisted bordismgroups Ω Spin c n ( X, α ) ∼ = lim k →∞ π n + k ( P α ( MSpin c ( k )) /X ) . We also define a topological index on geometric α -twisted bordism groups. Then the maintheorem is proved in Section 5.In Section 6, we explain the notion of geometric cycles for any paracompact Hausdorffspace X with a continuous map α : X → K ( Z , . Geometric cycles in this sense arecalled ‘D-branes’ in string theory. These consist of an α -twisted Spin c manifold M over X together with an ordinary K-class [ E ] . Following the work of Baum-Douglas, we imposean equivalence relation, generated by(i) Direct sum and disjoint union(ii) Bordism(iii) Spin c vector bundle modificationon the set of all geometric cycles to obtain the geometric twisted K-homology K geoev/odd ( X, α ) .Then we establish the commutative diagram (Cf. Theorem 6.4) for a smooth manifold X with a twisting α : X → K ( Z , K tev/odd ( X, α ) Ψ w w nnnnnnnnnnn Φ ' ' PPPPPPPPPPPP K geoev/odd ( X, α ) ∼ = µ / / K aev/odd ( X, α ) (1.2)whose arrows are all isomorphisms. One consequence of this commutative diagram is thatevery twisted K-class in K ev/odd ( X, α ) can be realized by appropriate geometric cycles(Cf. Corollary 6.5).We remark that the commutative diagram (1.2) of isomorphisms should hold for generalparacompact Hausdorff spaces with the homotopy type of finite CW complexes. The re-striction to smooth manifolds is due to the fact that we only establish the isomorphism Φ inTheorem 5.1 for smooth manifolds. We expect that the equivalence of geometric, topologi-cal and analytical twisted K-homology exists for any finite CW complex. We will return tothis in a sequel paper ([13].In Section 7, we study the twisted longitudinal index theorem (Cf. Theorem 7.3) for afoliated manifold ( X, F ) with a twisting α : X → K ( Z , , and show that this twistedlongitudinal index theorem generalizes both the Atiyah-Singer families index theorem in EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 7 [8] and Mathai-Melrose-Singer index theorem for projective families of elliptic operatorsassociated to a torsion twisting in [35].In Section 8, we introduce a notion of twisted
Spin manifolds over a manifold X witha KO-twisting α : X → K ( Z , . A smooth manifold M admits an α -twisted Spin structure if and only if there exists a continuous map ι : M → X such that ι ∗ ([ α ]) + w ( M ) = 0 in H ( M, Z ) , here w ( X ) is the second Stiefel-Whitney class of T M . (This is the anomalycancellation condition for Type I D-branes (Cf. [45]) ). We also discuss a notion of twistedstring manifolds over a manifold X with a string twisting α : X → K ( Z , . A smoothmanifold M admits an α -twisted string structure if and only if there is a continuous map ι : M → X such that ι ∗ ([ α ]) + p ( M )2 = 0 in H ( M, Z ) , here p ( X ) is the first Pontrjagin class of T M . These notions could be usefulin the study of twisted elliptic cohomology.It would be interesting to establish a local index theorem in the framework of twisted K-theory in which differential twisted K-theory in [18] [29] will come into play. We will returnto these problems in subsequent work. Finally, we like to point that, except in Sections 5and 7 and Theorem 6.4 where X is smooth, X is assumed to be a paracompact Hausdorfftopological space throughout this paper.2. R EVIEW OF TWISTED K- THEORY
In this section, we briefly review some basic facts about twisted K-theory, the main ref-erence are [5] and [19] (See also [16] [25] [39]).Let H be an infinite dimensional, complex and separable Hilbert space. We shall con-sider locally trivial principal P U ( H ) -bundles over a paracompact Hausdorff topologicalspace X , the structure group P U ( H ) is equipped with the norm topology. The projec-tive unitary group P U ( H ) with the norm topology (Cf. [33]) has the homotopy type of anEilenberg-MacLane space K ( Z , . The classifying space of P U ( H ) , as a classifying spaceof principal P U ( H ) -bundle, is a K ( Z , . Thus, the set of isomorphism classes of principal P U ( H ) -bundles over X is canonically identified with (Proposition 2.1 in [5]) [ X, K ( Z , ∼ = H ( X, Z ) . A twisting of complex K -theory on X is given by a continuous map α : X → K ( Z , .For such a twisting, we can associate a canonical principal K ( Z , -bundle P α through thefollowing pull-back construction P α (cid:15) (cid:15) / / EK ( Z , (cid:15) (cid:15) X α / / K ( Z , . (2.1) BAI-LING WANG
Let K be the 0-th space of the complex K-spectrum, in this paper, we take K to be Fred ( H ) ,the space of Fredholm operators on H . There is a base-point preserving action of K ( Z , on the K-theory spectrum K ( Z , × K −→ K which is represented by the action of complex line bundles on ordinary K-groups. As weidentify K ( Z , with P U ( H ) and K with Fred ( H ) , the above base point preserving actionis given by the conjugation action P U ( H ) × Fred ( H ) −→ Fred ( H ) . (2.2)The action (2.2) defines an associated bundle of K-theory spectra over X . Denote P α ( K ) = P α × K ( Z , K the bundle of based spectra over X with fiber the K-theory spectrum, and { Ω nX P α ( K ) = P α × K ( Z , Ω n K } the fiber-wise iterated loop spaces. Definition 2.1.
The twisted K-groups of ( X, α ) are defined to be K − n ( X, α ) := π (cid:0) C c ( X, Ω nX P α ( K )) (cid:1) , the set of homotopy classes of compactly supported sections of the bundle of K-spectra.Due to Bott periodicity, we only have two different twisted K-groups K ( X, α ) and K ( X, α ) . Given a closed subspace A of X , then ( X, A ) is a pair of topological spaces,and we define relative twisted K-groups to be K ev/odd ( X, A ; α ) := K ev/odd ( X − A, α ) . Remark 2.2.
Given a pair of twistings α , α : X → K ( Z , , if η : X × [1 , → K ( Z , is a homotopy between α and α , written as in the following formation, X α ! ! α = = K ( Z , , η (cid:11) (cid:19) (cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31)(cid:31) then there is a canonical isomorphism P α ∼ = P α induced by η . This canonical isomor-phism determines a canonical isomorphism on twisted K-groups η ∗ : K ev/odd ( X, α ) ∼ = / / K ev/odd ( X, α ) , (2.3)This isomorphism η ∗ depends only on the homotopy class of η . The set of homotopy classesbetween α and α is labelled by [ X, K ( Z , . Note that the the first Chern class isomor-phism Vect ( X ) ∼ = [ X, K ( Z , ∼ = H ( X, Z ) where Vect ( X ) is the set of equivalence classes of complex line bundles on X . We remarkthat the isomorphisms induced by two different homotopies between α and α are related EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 9 through an action of complex line bundles. This observation will play an important role inthe local index theorem for twisted K-theory.
Remark 2.3.
Let K be the C ∗ -algebra of compact operators on H . The isomorphism P U ( H ) ∼ = Aut ( K ) via the conjugation action of the unitary group U ( H ) provides an ac-tion of K ( Z , on the C ∗ -algebra K . Hence, any K ( Z , -principal bundle P α definesa locally trivial bundle of compact operators, denoted by P α ( K ) = P α × K ( Z , K . Let C c ( X, P α ( K )) be the C ∗ -algebra of the compact supported sections of P α ( K ) . We remarkthat C c ( X, P α ( K ) is the (unique up to isomorphism) stable separable complex continuous-trace C ∗ -algebra over X with its Dixmier-Douday class [ α ] ∈ H ( X, Z ) , here we identifythe ˇCech cohomology of X with its singular cohomology (Cf, [39] and [38]). In [5] and[39], it was proved that twisted K-groups K ev/odd ( X, α ) are canonically isomorphic to al-gebraic K-groups of the stable continuous trace C ∗ -algebra C c ( X, P α ( K )) K ev/odd ( X, α ) ∼ = KK ev/odd (cid:0) C , C c ( X, P α ( K )) (cid:1) . (2.4)The twisted K-theory is a 2-periodic generalized cohomology theory : a contravariantfunctor on the category of pairs consisting a pair of topological spaces A ⊂ X with a twist-ing α : X → K ( Z , to the category of Z -graded abelian groups. Note that a morphismbetween two pairs ( X, α ) and ( Y, β ) is a continuous map f : X → Y such that β ◦ f = α .The twisted K-theory satisfies the following three axioms whose proofs are rather standardfor 2-periodic generalized cohomology theory.(I) ( The homotopy axiom ) If two morphisms f, g : (
Y, B ) → ( X, A ) are homotopicthrough a map η : ( Y × [0 , , B × [0 , → ( X, A ) , written in terms of thefollowing homotopy commutative diagram ( Y, B ) g (cid:15) (cid:15) f / / ( X, A ) η u } t t t t tt t t t t α (cid:15) (cid:15) ( X, A ) α / / K ( Z , , then we have the following commutative diagram K ev/odd ( X, A ; α ) g ∗ ) ) SSSSSSSSSSSSSS f ∗ u u kkkkkkkkkkkkkk K ev/odd ( Y, B ; α ◦ f ) η ∗ / / K ev/odd ( Y, B ; α ◦ g ) . Here η ∗ is the canonical isomorphism induced by the homotopy η . (II) ( The exact axiom ) For any pair ( X, A ) with a twisting α : X → K ( Z , , thereexists the following six-term exact sequence K ( X, A ; α ) / / K ( X, α ) / / K ( A, α | A ) (cid:15) (cid:15) K ( A, α | A ) O O K ( X, α ) o o K ( X, A ; α ) o o here α | A is the composition of the inclusion and α .(III) ( The excision axiom ) Let ( X, A ) be a pair of spaces and let U ⊂ A be a subspacesuch that the closure U is contained in the interior of A . Then the inclusion ι :( X − U, A − U ) → ( X, A ) induces, for all α : X → K ( Z , , an isomorphism K ev/odd ( X, A ; α ) −→ K ev/odd ( X − U, A − U ; α ◦ ι ) . In addition, twisted K-theory satisfies the following basic properties (see [5] [19] for de-tailed proofs).(IV) (
Multiplicative property ) Let α, β : X → K ( Z , be a pair of twistings on X .Denote by α + β the new twisting defined by the following map α + β : X ( α,β ) / / K ( Z , × K ( Z , m / / K ( Z , , (2.5) where m is defined as follows BP U ( H ) × BP U ( H ) ∼ = B ( P U ( H ) × P U ( H )) −→ BP U ( H ) , for a fixed isomorphism H ⊗ H ∼ = H . Then there is a canonical multiplication K ev/odd ( X, α ) × K ev/odd ( X, β ) −→ K ev/odd ( X, α + β ) , which defines a K ( X ) -module structure on twisted K-groups K ev/odd ( X, α ) .(V) ( Thom isomorphism ) Let π : E → X be an oriented real vector bundle of rank k over X with the classifying map denoted by ν E : X → BSO ( k ) , then there is acanonical isomorphism, for any twisting α : X → K ( Z , , K ev/odd ( X, α + ( W ◦ ν E )) ∼ = K ev/odd ( E, α ◦ π ) , (2.6) with the grading shifted by k ( mod .Here W : BSO ( k ) → K ( Z , is the classifying map of the principal K ( Z , -bundle BSpin c ( k ) → BSO ( k ) .(VI) ( The push-forward map ) For any differentiable map f : X → Y between twosmooth manifolds X and Y , let α : Y → K ( Z , be a twisting. Then there is acanonical push-forward homomorphism f ! : K ev/odd (cid:0) X, ( α ◦ f ) + ( W ◦ ν f ) (cid:1) −→ K ev/odd ( Y, α ) , (2.7) with the grading shifted by n mod (2) for n = dim ( X ) + dim ( Y ) . Here ν f is theclassifying map X −→ BSO ( n ) EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 11 corresponding to the bundle
T X ⊕ f ∗ T Y over X .(VII) ( Mayer-Vietoris sequence ) If X is covered by two open subsets U and U with atwisting α : X → K ( Z , , then there is a Mayer-Vietoris exact sequence K ( X, α ) / / K ( U ∩ U , α ) / / K ( U , α ) ⊕ K ( U , α ) (cid:15) (cid:15) K ( U , α ) ⊕ K ( U , α ) O O K ( U ∩ U , α ) o o K ( X, α ) o o where α , α and α are the restrictions of α to U , U and U ∩ U respectively. Remark 2.4. (1) Note that [ α + ( W ◦ ν E )] = [ α ] + W ( E ) , our Thom isomorphismagrees with the Thom isomorphism in [19] and [21], where the notation K ev/odd ( X, [ α ] + W ( E )) ∼ = K ev/odd ( E, π ∗ ([ α ])) is used.(2) The push-forward map constructed in [19] is established in the following form f ! : K ev/odd (cid:0) X, f ∗ [ α ] + W ( T X ⊕ f ∗ T Y ) (cid:1) −→ K ev/odd ( Y, [ α ]) , which is obtained by applying the Thom isomorphism and Bott periodicity as fol-lows.Choose an embedding i : X → R k . Then x ( f ( x ) , i ( x )) defines an em-bedding of X → Y × R k whose normal bundle N is identified with a tubularneighborhood of X . Let ν N : X → BSO be the classifying map of the normalbundle N , let ι : N → Y × R k be the inclusion map, and π : Y × R k → Y bethe projection. We use the following commutative diagram N ι / / (cid:15) (cid:15) Y × R kπ (cid:15) (cid:15) X ( f,i ) ; ; f / / Y α / / K ( Z , , to illustrate induced twistings α ◦ f, α ◦ π ◦ ι and α ◦ π on X, N and Y × R nk respectively. Notice the isomorphism, as bundles over X , N ⊕ T X ⊕ T X ∼ = T X ⊕ f ∗ T Y ⊕ R n and the canonical Spin c structure on T X ⊕ T X determines a canonical homotopybetween W ◦ ν N and W ◦ ν f , which in turn induces a canonical isomorphism K ev/odd (cid:0) X, ( α ◦ f ) + ( W ◦ ν f ) (cid:1) ∼ = K ev/odd (cid:0) X, ( α ◦ f ) + ( W ◦ ν N ) (cid:1) . Applying the Thom isomorphism (2.6), we have K ev/odd (cid:0) X, ( α ◦ f ) + ( W ◦ ν N ) (cid:1) ∼ = K ev/odd ( N, α ◦ π ◦ ι ) , with the grading shifted by n mod (2) for n = dim ( X ) + dim ( Y ) . The inclusionmap ι : N → Y × R k induces a natural push-forward map ι ! : K ev/odd ( N, α ◦ π ◦ ι ) → K ev/odd ( Y × R n , α ◦ π ) . The Bott periodicity gives a canonical isomorphism K ev/odd ( Y × R n , α ◦ π ) ∼ = K ev/odd ( Y, α ) . The composition of the above isomorphisms and the map ι ! gives rise to the canon-ical push-forward map (2.7).3. T WISTED
Spin c - MANIFOLDS AND ANALYTICAL INDEX
Definition 3.1.
Let ( X, α ) be a paracompact Hausdorff topological space with a twisting α . An α -twisted Spin c manifold over X is a quadruple ( M, ν, ι, η ) where(1) M is a smooth, oriented and compact manifold together with a fixed classifyingmap of its stable normal bundle ν : M −→ BSO here
BSO = lim −→ k BSO ( k ) is the classifying space of stable normal bundle of M ;(2) ι : M → X is a continuous map;(3) η is an α -twisted Spin c structure on M , that is a homotopy commutative diagram M ι (cid:15) (cid:15) ν / / BSO η v ~ v v v v vv v v v v W (cid:15) (cid:15) X α / / K ( Z , , where W is the classifying map of the principal K ( Z , -bundle BSpin c → BSO associated to the third integral Stiefel-Whitney class and η is a homotopy between W ◦ ν and α ◦ ι .Two α -twisted Spin c structures η and η ′ on M are called equivalent if there is a homotopybetween η and η ′ . Remark 3.2. (1) Definition of twisted
Spin c manifolds over X is previously givenby Douglas in [22] using Hopkins-Singer’s differential cochains developed in [29].Here in Definition 3.1, we define an α -twisted Spin c structure on M to be a homo-topy between W ◦ ν and α ◦ ι as it induces a canonical isomorphism η ∗ (3.5) whichwill play an important role in our definition of the analytical index. EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 13 (2) Let ( W, ν, ι, η ) be an α -twisted Spin c manifold with boundary over X , then thereis a natural α -twisted Spin c structure on the boundary ∂W with outer normal ori-entation, which is the restriction of the α -twisted Spin c structure on W : ∂W ι | ∂W (cid:15) (cid:15) ν | ∂W / / BSO η | ∂W v ~ u u u u uu u u u u W (cid:15) (cid:15) X α / / K ( Z , . (3.1)(3) Given an oriented real vector bundle E of rank k over a smooth manifold M , theclassifying map of E ν E : M −→ BSO ( k ) and the principal K ( Z , -bundle BSpin c ( k ) → BSO ( k ) define an associatedtwisting W ◦ ν E : M −→ BSO ( k ) −→ K ( Z , . Proposition 3.3.
Given a smooth, oriented and compact n-dimensional manifold M and aparacompact space X with a twisting α : X → K ( Z , , then (1) M admits an α -twisted Spin c structure if and only if there exists a continuous map ι : M → X such that ι ∗ ([ α ]) + W ( M ) = 0 (3.2) in H ( M, Z ) . Here W ( M ) is the third integral Stiefel-Whitney class, W ( M ) = β ( w ( M )) with β : H ( M, Z ) → H ( M, Z ) the Bockstein homomorphism and w ( M ) thesecond Stiefel-Whitney class of T M . (The condition (3.2) is the Freed-Witten anom-aly cancellation condition for Type II D-branes (Cf. [26] ).) (2) If ι ∗ ([ α ]) + W ( M ) = 0 , then the set of equivalence classes of α -twisted Spin c structures on M are in one-to-one correspondence with elements in H ( M, Z ) .Proof. If M admits an α -twisted Spin c structure, then W ◦ ν and α ◦ ι are homotopic asmaps from M to K ( Z , . This means that the third integral Stiefel-Whitney class of thestable normal bundle is equal to ι ∗ ([ α ]) . As M is compact, we can find an embedding i k : M n −→ R n + k for a sufficiently large k . Denote by ν ( i k ) the normal of i k , then we know that W ( ν ( i k )) = ι ∗ ([ α ]) , and ν ( i k ) ⊕ T M ∼ = i ∗ k ( T R n + k ) is a trivial bundle, which implies W ( M ) + W ( ν ( i k )) = 0 . Thus, ι ∗ ([ α ]) + W ( M ) = 0 . Conversely, if ι ∗ ([ α ]) + W ( M ) = 0 , then W ( ν ( i k )) agrees with ι ∗ ([ α ]) , hence, theclassifying map ν k : M → BSO ( k ) makes the following diagram homotopy commutativefor some homotopy η : M ι (cid:15) (cid:15) ν k / / BSO ( k ) η w (cid:127) w w w w ww w w w w W (cid:15) (cid:15) X α / / K ( Z , . This defines an α -twisted Spin c structure on M by letting k → ∞ .The set of equivalence classes of α -twisted Spin c structures on M corresponds to the setof homotopy classes of homotopies between W ◦ ν and α ◦ ι . The latter is an affine spaceover [Σ M, K ( Z , ∼ = [ M, K ( Z , here Σ denotes the suspension. As [ M, K ( Z , ∼ = H ( M, Z ) , so H ( M, Z ) acts freelyand transitively on the set of equivalence classes of α -twisted Spin c structures on M . (cid:3) Remark 3.4. (1) If the twisting α : X → K ( Z , is homotopic to the trivial map,then an α -twisted Spin c structure on M is equivalent to a Spin c structure on M .(2) Let τ X : X → BSO be a classifying map of the stable tangent bundle of X , then a W ◦ τ X -twisted Spin c structure on M is equivalent to a K-oriented map from M to X .(3) Let ( M, ν, ι, η ) be an α -twisted Spin c manifold over X . Any K-oriented map f : M ′ → M defines a canonical α -twisted Spin c structure on M ′ .Recall that for k ∈ { , , , · · · } and a separable C ∗ -algebra A , Kasparov’s K-homologygroup KK k ( A, C ) ∼ = KK ( A, Cliff ( C k )) is the abelian group generated by unitary equivalence classes of Cliff ( C k ) -graded Fred-holm modules over A modulo certain relations (See [27] for details). Then KK ev ( A, C ) and KK odd ( A, C ) denote the direct limits under the periodicity maps KK ev ( A, C ) = lim −→ k KK k ( A, C ) , and KK odd ( A, C ) = lim −→ k KK k +1 ( A, C ) . Definition 3.5.
Let X be a paracompact Hausdorff space with a twisting α : X → K ( Z , .Let P α ( K ) be the associated bundle of compact operators on X . Analytical twisted K-homology, denoted by K aev/odd ( X, α ) , is defined to be K aev/odd ( X, α ) := KK ev/odd (cid:0) C c ( X, P α ( K )) , C (cid:1) , the Kasparov’s Z -graded K-homology of the C ∗ -algebra C c ( X, P α ( K )) . Given a closedsubspace A of X , the relative twisted K-homology K aev/odd ( X, A ; α ) is defined to be KK ev/odd (cid:0) C c ( X − A, P α ( K )) , C (cid:1) . Analytical twisted K-homology is a 2-periodic generalized homology theory.
EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 15
We first discuss the relationship between the stable normal bundle of M and its stabletangent bundle, and apply it to study the corresponding twisted K-homology groups. Notethat the classifying space of SO ( k ) is given by the direct limit BSO ( k ) = lim −→ m Gr ( k, m + k ) where Gr ( k, m + k ) is the Grassmann manifold of oriented k -planes in R k + m . The classi-fying space of the stable special orthogonal group is lim −→ k BSO ( k ) , and will be denoted by BSO .The map I k,m : Gr ( k, m + k ) −→ Gr ( m, k + m ) of assigning to each oriented k -planein R k + m to its orthogonal m -plane induces a map I : BSO −→ BSO with I the identity map.For a compact n-dimensional manifold M n , the stable normal bundle is represented bythe normal bundle of an embedding i k : M n −→ R n + k for any sufficiently large k . Thenormal bundle ν ( i k ) of i k is the quotient of the pull-back of the tangent bundle T R n + k = R n + k × R n + k by the tangent bundle T M . Then the normal map ν k : M −→ Gr ( k, k + n ) and the tangent map τ k : M −→ Gr ( n, k + n ) are related to each other by τ k = I k,n ◦ ν k . So the classifying map for the stable normalbundle ν : M −→ BSO and the classifying map of the stable tangent bundle τ : M −→ BSO are related by τ = I ◦ ν . Thus, we have a natural isomorphism on the associated bundles ofcompact operators I ∗ : τ ∗ BSpin c ( K ) −→ ν ∗ BSpin c ( K ) . (3.3)This determines an isomorphism, denoted by I ∗ , on the corresponding twisted K-homologygroups I ∗ : K aev/odd ( M, W ◦ τ ) ∼ = K aev/odd ( M, W ◦ ν ) . (3.4) Remark 3.6.
Given an embedding i k : M → R n + k with the normal bundle N , the naturalisomorphism T M ⊕ N ⊕ N ∼ = R n + k ⊕ N and the canonical Spin c structure on N ⊕ N define a canonical homotopy between W ◦ τ and W ◦ ν . The isomorphism (3.4) is induced by this canonical homotopy. For a Riemannian manifold M , denote by Cliff ( T M ) the bundle of complex Clifford al-gebras of T M over M . As algebras of the sections, C ( M, Cliff ( T M )) is Morita equivalentto C ( M, τ ∗ BSpin c ( K )) . Hence, we have a canonical isomorphism K aev/odd ( M, W ◦ τ ) ∼ = KK ev/odd ( C ( M, Cliff ( M )) , C ) with the degree shift by dimM ( mod . Applying Kasparov’s Poincar´e duality (Cf. [32]) KK ev/odd ( C , C ( M )) ∼ = KK ev/odd ( C ( M, Cliff ( M )) , C ) , we obtain a canonical isomorphism P D : K ( M ) ∼ = K aev/odd ( M, W ◦ τ ) , with the degree shift by dimM ( mod . The fundamental class [ M ] ∈ K aev/odd ( M, W ◦ τ ) is the Poincar´e dual of the unit element in K ( M ) . Note that [ M ] ∈ K aev ( M, W ( M )) if M is even dimensional and [ M ] ∈ K aodd ( M, W ( M )) if M is odd dimensional. The capproduct ∩ : K aev/odd ( M, W ◦ τ ) ⊗ K ( M ) −→ K aev/odd ( M, W ◦ τ ) is defined by the Kasparov product. We remark that the cap product of the fundamentalclass [ M ] and [ E ] ∈ K ( M ) is given by [ M ] ∩ [ E ] = P D ([ E ]) . Given an α -twisted Spin c manifold ( M, ν, ι, η ) over X , the homotopy η induces anisomorphism ν ∗ BSpin c ∼ = ι ∗ P α as principal K ( Z , -bundles on M , hence defines anisomorphism ν ∗ BSpin c ( K ) η ∗ ∼ = / / ι ∗ P α ( K ) as bundles of C ∗ -algebras on M . This isomorphism determines a canonical isomorphismbetween the corresponding continuous trace C ∗ -algebras C ( M, ν ∗ BSpin c ( K )) ∼ = C ( M, ι ∗ P α ( K )) . Therefore, we have a canonical isomorphism η ∗ : K aev/odd ( M, W ◦ ν ) ∼ = K aev/odd ( M, α ◦ ι ) . (3.5)Notice that the natural push-forward map in analytic K-homology theory is ι ∗ : K aev/odd ( M, α ◦ ι ) −→ K aev/odd ( X, α ) . (3.6)We can introduce a notion of analytical index for any α -twisted Spin c manifold over X ,taking values in analytical twisted K-homology of ( X, α ) . Definition 3.7.
Given an α -twisted Spin c closed manifold ( M, ν, ι, η ) and [ E ] ∈ K ( M ) ,we define its analytical index Index a (( M, ν, ι, η ) , [ E ]) ∈ K aev/odd ( X, α ) EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 17 to be the image of the cap product [ M ] ∩ [ E ] ∈ K aev/odd ( M, W ◦ τ ) under the maps (3.4),(3.5) and (3.6): K aev/odd ( M, W ◦ τ ) ι ∗ ◦ η ∗ ◦ I ∗ / / K aev/odd ( X, α ) . The analytical index enjoys the following properties.
Proposition 3.8. (1)
The analytical index
Index a (( M, ν, ι, η ) , [ E ]) depends only onthe equivalence class of the α -twisted Spin c structure η . (2) (Disjoint union and Direct sum) Given a pair of α -twisted Spin c manifolds ( M , ν , ι , η ) and ( M , ν , ι , η ) , and [ E i ] ∈ K ( M i ) . Then Index a (cid:0) ( M , ν , ι , η ) ⊔ ( M , ν , ι , η ) , [ E ] ⊔ [ E ] (cid:1) = Index a (( M , ν , ι , η ) , [ E ]) + Index a (( M , ν , ι , η ) , [ E ]) . (3) (Bordism invariance) If ( W, ν, ι, η ) is an α -twisted Spin c manifold with boundaryover X and [ E ] ∈ K ( W ) , then Index a ( ∂W, ∂ν, ∂ι, ∂η ) , [ E | ∂W ]) = 0 . Proof.
The α -twisted Spin c structure η enters the definition of Index a (( M, ν, ι, η ) , [ E ]) only through η ∗ : K aev/odd ( M, W ◦ ν ) ∼ = K aev/odd ( M, α ◦ ι ) . This isomorphism depends only on the homotopy class of η . So Claim (1) is obvious.Claim (2) follows from the disjoint union and direct sum property of the fundamentalclasses and the cap product.To establish Claim (3), let ( W, ν, ι, η ) be an α -twisted Spin c manifold with boundaryover X , and denote its boundary by M = ∂W with the induced α -twisted Spin c structure ( ∂ν, ∂ι, ∂η ) . Let i : M → W be the boundary inclusion map. The exact sequence in topo-logical K-theory and analytical K-homology. are related through Poincar´e duality (assumethat W is odd dimensional) as in the following commutative diagram K ( W, M ) P D (cid:15) (cid:15) K ( M ) o o P D (cid:15) (cid:15) K ( W ) i ∗ o o P D (cid:15) (cid:15) K a ( W, W ◦ τ W ) K a ( M, W ◦ τ M ) i ∗ o o K a ( W, M ; W ◦ τ W ) , ∂ o o (3.7)where τ M and τ W are classifying maps of the stable tangent bundles of M and W re-spectively. One could get this K-homology exact sequence by applying the Kasparov KK-functor to the following short exact sequence of C ∗ -algebras: → C ( W, Cliff ( W )) −→ C ( W, Cliff ( W )) −→ C ( M, Cliff ( M )) → , where C ( W, Cliff ( W )) denotes the C ∗ -algebra of continuous sections of Cliff ( W ) van-ishing at the boundary M . The relative analytical K-homology KK ev/odd ( C ( W, Cliff ( W )) , C ) is isomorphic to K aev/odd ( W, M ; W ◦ τ ) , and hence isomorphic to K aev/odd ( W, M ; W ◦ ν ) under (3.4).From (3.7) that the Poincar´e dual of [ i ∗ E ] ∈ K ( M ) is mapped to zero in K a ( W, W ◦ τ W ) for [ E ] ∈ K ( W ) under the map i ∗ : i ∗ ◦ P D ◦ i ∗ ([ E ]) = i ∗ ◦ ∂ ◦ P D ([ E ]) = 0 . (3.8)Notice that Index a (cid:0) ( M, ∂ν, ∂ι, ∂η ) , [ E | M ] (cid:1) is image of the class P D ( i ∗ [ E ]) under thefollowing sequence of maps K a ( M, W ◦ τ M ) I ∗ / / K a ( M, W ◦ ∂ν ) ( ∂η ) ∗ / / K a ( M, α ◦ ∂ι ) ( ∂ι ) ∗ / / K a ( X, α ) , and the inclusion map i : M → W induces the following commutative diagram K a ( M, W ◦ τ M ) i ∗ (cid:15) (cid:15) I ∗ / / K a ( M, W ◦ ∂ν ) i ∗ (cid:15) (cid:15) ( ∂η ) ∗ / / K a ( M, α ◦ ∂ι ) i ∗ (cid:15) (cid:15) ( ∂ι ) ∗ ' ' PPPPPPPPPPPP K a ( W, W ◦ τ W ) I ∗ / / K a ( W, W ◦ ν ) η ∗ / / K a ( W, α ◦ ν ) ι ∗ / / K a ( X, α ) . Therefore, we conclude that
Index a (cid:0) ( M, ∂ν, ∂ι, ∂η ) , [ E | M ] (cid:1) = ( ∂ι ) ∗ ◦ ( ∂η ) ∗ ◦ I ∗ ◦ P D ( i ∗ [ E ]) (Definition 3.7) = ι ∗ ◦ η ∗ ◦ I ∗ ◦ i ∗ ◦ P D ( i ∗ [ E ]) (The above commutative diagram) = 0 (3.8) (cid:3) Remark 3.9.
Given an α -twisted Spin c structure η on ( M, ν, ι ) and a complex line bundle L over M , denote by c · [ η ] the action of the first Chern class c = c ( L ) ∈ H ( M, Z ) on the homotopy class of η , then the analytical index depends on the choice of equivalenceclasses of α -twisted Spin c structures through the following formulae Index a (( M, ν, ι, c · [ η ]) , [ E ]) = Index a (( M, ν, ι, [ η ]) , ([ L ] ⊗ [ E ])) .
4. T
WISTED
Spin c BORDISM AND TOPOLOGICAL INDEX
Given a manifold X with a twisting α : X → K ( Z , , α -twisted Spin c manifolds over X form a bordism category, called the α -twisted Spin c brodism over ( X, α ) , whose objectsare compact smooth manifolds over X with an α -twisted Spin c structure as in Definition EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 19 α -twisted Spin c manifolds ( M , ν , ι , η ) and ( M , ν , ι , η ) is a boundary preserving continuous map f : M → M and the following diagram M ι " " ν ! ! f ! ! CCCCCCCC M ι (cid:15) (cid:15) ν / / BSO η w (cid:127) v v v v vv v v v v W (cid:15) (cid:15) X α / / K ( Z , (4.1)is a homotopy commutative diagram such that(1) ν is homotopic to ν ◦ f through a continuous map ν : M × [0 , → BSO ;(2) ι ◦ f is homotopic to ι through continuous map ι : M × [0 , → X ;(3) the composition of homotopies ( α ◦ ι ) ∗ ( η ◦ ( f × Id )) ∗ ( W ◦ ν ) is homotopic to η .The boundary functor ∂ applied to an α -twisted Spin c manifold ( M, ν, ι, η ) is the manifold ∂M with outer normal orientation and the restriction of the α -twisted Spin c structure to M .Two α -twisted Spin c manifolds ( M , ν , ι , η ) and ( M , ν , ι , η ) are called isomor-phic if there exists a diffeomorphism f : M → M such that the diagram (4.1) is ahomotopy commutative diagram. Definition 4.1.
We say that an α -twisted Spin c manifold ( M, ν, ι, η ) is null-bordant ifthere exists an α -twisted Spin c manifold W whose boundary is ( M, ν, ι, η ) in the sense of(3.1). We define the α -twisted Spin c bordism group of X , denoted by Ω Spin c ( X, α ) , tobe the set of all isomorphism classes of closed α -twisted Spin c manifolds over X modulonull-bordism, with the sum given by the disjoint union.The subgroup of isomorphism classes of n-dimensional closed α -twisted Spin c mani-folds over X will be denoted Ω Spin c n ( X, α ) . Define Ω Spin c ev ( X, α ) = L k Ω Spin c k ( X, α )Ω Spin c odd ( X, α ) = L k Ω Spin c k +1 ( X, α ) . Proposition 4.2.
The analytical index defined in the previous section induces a homomor-phism
Index a : Ω Spin c ev/odd ( X, α ) −→ K aev/odd ( X, α ) (4.2) Proof.
Let ( M, ι, ν, η ) be α -twisted Spin c manifold over X , representing an element in the α -twisted Spin c bordism group Ω Spin c ev/odd ( X, α ) . Define Index a ( M, ι, ν, η ) =
Index a (( M, ι, ν, η ) , [ C ]) ∈ K aev/odd ( X, α ) , where C denotes the trivial line bundle over M , representing the unit element in K ( M ) .We need to show that for a pair of isomorphic objects ( M , ι , ν , η ) and ( M , ι , ν , η ) in the α -twisted Spin c bordism category over X , we have Index a ( M , ι , ν , η ) = Index a ( M , ι , ν , η ) . Let f be a diffeomorphism from M to M such that (4.1) is a homotopy commutativediagram. Let τ and τ be classifying maps of the stable tangent bundles of M and M respectively. The homotopy between ν and ν ◦ f implies that τ and τ ◦ f are homotopyequivalent. This defines a canonical Spin c structure on T M ⊕ f ∗ T M . Hence, there is acanonical Morita equivalence C ( M , Cliff ( M )) ∼ C ( M , f ∗ Cliff ( M )) . This Morita equivalence defines a canonical isomorphism K aev/odd ( M , W ◦ τ ) ∼ = K aev/odd ( M , W ◦ τ ◦ f ) . Recall that natural push-forward map in analytical K-homology is related to the K-theoreticalpush-forward map f ! in topological K-theory via the Poincar´e duality (PD): K ev/odd ( M ) ∼ = P D (cid:15) (cid:15) f ! / / K ev/odd ( M ) ∼ = P D (cid:15) (cid:15) K aev/odd ( M , W ◦ τ ) f ∗ / / K aev/odd ( M , W ◦ τ ) where the Poincar´e duality shifts the degree by the dimension of the underlying manifold.Applying the natural push-forward map in analytical K-homology, we obtain f ∗ : K aev/odd ( M , W ◦ τ ) ∼ = K aev/odd ( M , W ◦ τ ◦ f ) −→ K aev/odd ( M , W ◦ τ ) with the degree shifted by d ( f ) = dimM − dimM ( mod . The homotopy between W ◦ ν and W ◦ ν ◦ f defines a canonical homomorphism f ∗ : K aev/odd ( M , W ◦ ν ) ∼ = K aev/odd ( M , W ◦ ν ◦ f ) −→ K aev/odd ( M , W ◦ ν ) , such that the following diagram commutes K aev/odd ( M , W ◦ τ ) f ∗ (cid:15) (cid:15) I M ∗ / / K aev/odd ( M , W ◦ ν ) f ∗ (cid:15) (cid:15) K aev/odd ( M , W ◦ τ ) I M ∗ / / K aev/odd ( M , W ◦ ν ) . (4.3) EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 21
Similarly, the homotopy between ( α ◦ ι ) ∗ ( η ◦ ( f × Id )) ∗ ( W ◦ ν ) and η induces acommutative diagram K aev/odd ( M , W ◦ ν ) f ∗ (cid:15) (cid:15) ( η ) ∗ / / K aev/odd ( M , α ◦ ι ) f ∗ (cid:15) (cid:15) K aev/odd ( M , W ◦ ν ) ( η ) ∗ / / K aev/odd ( M , α ◦ ι ) . (4.4)The homotopy between α ◦ ι ◦ f and α ◦ ι induces the following commutative triangle K aev/odd ( M , α ◦ ι ) f ∗ (cid:15) (cid:15) ( ι ) ∗ ( ( RRRRRRRRRRRRR K aev/odd ( X, α ) .K aev/odd ( M , a ◦ ι ) ( ι ) ∗ lllllllllllll (4.5)These commutative diagrams (4.3), (4.4) and (4.5) imply that Index a ( M , ι , ν , η )= ( ι ) ∗ ◦ ( η ) ∗ ◦ I M ∗ ([ M ])= ( ι ) ∗ ◦ ( η ) ∗ ◦ I M ∗ ◦ f ∗ ([ M ])= ( ι ) ∗ ◦ ( η ) ∗ ◦ I M ∗ ([ M ])= Index a ( M , ι , ν , η ) . Now the cobordant invariance in Proposition 3.8 tells us that
Index a is a well-definedhomomorphism from Ω Spin c ev/odd ( X, α ) to K aev/odd ( X, α ) . (cid:3) We recall the construction of Thom spectrum of
Spin c bordism. Let ξ k be the universalbundle over BSO ( k ) . The pull-back bundle over BSpin c ( k ) is given by ˜ ξ k = ESpin c ( k ) × Spin c ( k ) R k . Denote by
MSpin c ( k ) the Thom space of ˜ ξ k . The inclusion map j k induces a pull-backdiagram j ∗ k ˜ ξ k +1 / / (cid:15) (cid:15) ˜ ξ k +1 (cid:15) (cid:15) BSpin c ( k ) j k / / BSpin c ( k + 1) with j ∗ k ˜ ξ k +1 ∼ = ˜ ξ k ⊕ R , where R denote the trivial real line bundle. Then the Thom spaceof j ∗ k ˜ ξ k +1 can be identified with Σ MSpin c ( k ) (the suspension of MSpin c ( k ) ). Thus wehave a sequence of continuous maps Th ( j k ) : Σ MSpin c ( k ) −→ MSpin c ( k + 1) , that means, { MSpin c ( k ) } k is the Thom spectrum associated to BSpin c = lim −→ k BSpin c ( k ) .As BSpin c ( k ) is a principal K ( Z , -bundle over BSO ( k ) , we have a base point pre-serving action of K ( Z , on the Thom spectrum { MSpin c ( k ) } , written as K ( Z , + ∧ MSpin c ( k ) = K ( Z , × MSpin c ( k ) K ( Z , × ∗ −→ MSpin c ( k ) which is compatible with the base-point action of K ( Z , on the K-theory spectrum K inthe sense that there exists a K ( Z , -equivariant map, called the index map Index : MSpin c −→ K . This K ( Z , -equivariant map has been constructed in [22] and [43]. Here we provide amore geometric construction. Write the principal BU (1) -bundle BSpin c (2 k ) as the fol-lowing pull-back bundle BSpin c (2 k ) / / (cid:15) (cid:15) EK ( Z , (cid:15) (cid:15) BSO (2 k ) W / / K ( Z , , which induces a natural P U ( H ) -action P U ( H ) × BSpin c (2 k ) −→ BSpin c (2 k ) . This action corresponds to the action of the set of complex line bundles on the set of
Spin c structures. The P U ( H ) -action on BSpin c (2 k ) can be lifted to a base point preservingaction of P U ( H ) on MSpin c (2 k ) P U ( H ) × MSpin c (2 k ) −→ MSpin c (2 k ) . Note that there is a fundamental Z -graded spinor bundle S + ⊕ S − over BSpin c (2 k ) seeTheorem C.9 in [14], which defines a canonical Thom class in K ( MSpin c (2 k )) . Thiscanonical Thom class determines a P U ( H ) -equivariant map Index : MSpin c (2 k ) −→ Fred ( H ) . Hence we have associated bundles of Thom spectra over X P α ( MSpin c ( k )) = P α × K ( Z , MSpin c ( k ) , and natural maps to the associated bundle of K-theory spectra Index : P α ( MSpin c ( k )) −→ P α ( K ) = P α × K ( Z , K . Remark 4.3.
The
Spin c bordism groups over a pointed space X , denoted by Ω Spin c ∗ ( X ) asin ([41]), can be identified as the stable homotopy groups of MSpin c ∧ X (the Pontrjagin-Thom isomorphism) Ω Spin c n ( X ) ∼ = π Sn ( MSpin c ∧ X ) := lim −→ k π n + k ( MSpin c ( k ) ∧ X ) . (4.6) EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 23
The index map
Index : MSpin c → K determines a natural transformation from the evenor odd dimensional Spin c bordism group of X to K-homology of X , which is called thetopological index: Ω Spin c ev/odd ( X ) → K tev/odd ( X ) . The following theorem is the twisted version of the Pontrjagin-Thom isomorphism (4.6).
Theorem 4.4.
The bordism group Ω Spin c n ( X, α ) of n-dimensional α -twisted Spin c mani-folds over X is isomorphic to the stable homotopy group Θ : Ω
Spin c n ( X, α ) ∼ = / / π S n (cid:0) P α ( MSpin c ) /X (cid:1) . Here we denote π S n (cid:0) P α ( MSpin c ) /X (cid:1) := lim k →∞ π n + k (cid:0) P α ( MSpin c ( k )) /X (cid:1) .Proof. The proof is modeled on the proof of the classical Pontrjagin-Thom isomorphism(Cf. [41])
Step 1.
Definition of the homomorphism Θ .Let σ be an element in Ω Spin c n ( X, α ) represented by a n -dimensional α -twisted Spin c manifold ( M, ι, ν, η ) over X . Let i k : M → R n + k be an embedding with the classifyingmap of the normal bundle denoted by ν k . Then we have the following pull-back diagram N ˜ ν k / / π (cid:15) (cid:15) ξ k (cid:15) (cid:15) M ν k / / BSO ( k ) . (4.7)Here the total space N of the normal bundle of i k can be thought of as a subspace of R n + k × R n + k . Under the addition map R n + k × R n + k → R n + k , for some sufficientlysmall ǫ > , the ǫ -neighborhood N ǫ of the zero section M × { } of N is an embedding ǫ | N ǫ : N ǫ → R n + k , whose restriction to the zero section M × { } is the embedding i k : M → R n + k .Consider S n + k as R n + k ∪{∞} (the one point compactification) so we have an embedding N ǫ → S n + k . Define c : S n + r → N ǫ /∂N ǫ by collapsing all points of S n + k outside and on the boundary of N ǫ to a point. Note that N ǫ /∂N ǫ is homeomorphic to the Thom space Th ( N ) of the normal bundle of i k , inducedby multiplication by ǫ − . Denote this homeomorphism by ǫ − : N ǫ /∂N ǫ −→ Th ( N ) . The pull-back diagram P W ◦ ν k (cid:15) (cid:15) / / EK ( Z , (cid:15) (cid:15) M W ◦ ν k / / K ( Z , , induces a homotopy pull-back P W ◦ ν k ( BSpin c ( k )) (cid:15) (cid:15) / / EK ( Z , BSpin c ( k )) (cid:15) (cid:15) M W ◦ ν k / / K ( Z , . As EK ( Z , is contractible, so EK ( Z , BSpin c ( k )) is homotopy equivalent to BSO ( k ) .This implies that the following diagram P W ◦ ν k ( BSpin c ( k )) (cid:15) (cid:15) / / BSO ( k ) (cid:15) (cid:15) M W ◦ ν k / / K ( Z , is a homotopy pull-back. Notice that the diagram M Id (cid:15) (cid:15) ν k / / BSO ( k ) W (cid:15) (cid:15) M W ◦ ν k / / K ( Z , is commutative. Thus there exists a unique map (up to homotopy) h : M → P W ◦ ν k ( BSpin c ( k )) such that the following diagram M Id ' ' ν k ( ( h ' ' OOOOOOOOOOOOO P W ◦ ν k ( BSpin c ( k )) (cid:15) (cid:15) / / BSO ( k ) r z m m m m m m m mm m m m m m m m W (cid:15) (cid:15) M W ◦ ν k / / K ( Z , . is homotopy commutative. Together with the pull-back diagram ˜ ξ k / / (cid:15) (cid:15) ξ k (cid:15) (cid:15) BSpin c ( k ) / / BSO ( k ) , we obtain a homotopy commutative diagram N (cid:15) (cid:15) / / P W ◦ ν k ( ˜ ξ k ) (cid:15) (cid:15) M / / P W ◦ ν k ( BSpin c ( k )) EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 25 which in turn determines a canonical map h ∗ : Th ( N ) −→ P W ◦ ν k ( MSpin c ( k )) /M. Notice that the α -twisted Spin c -structure on M defines a continuous map ι ∗ : P W ◦ ν k ( MSpin c ( k )) /M −→ P α ( MSpin c ( k )) /X. The composition ι ∗ ◦ h ∗ ◦ ǫ − ◦ c is a continuous map of pairs θ = θ ( M,ι,ν,η ) : ( S n + k , ∞ ) −→ ( P α ( MSpin c ( k )) /X, ∗ ) , here ∗ is the base point in P α ( MSpin c ( k )) /X , hence represents an element of lim k →∞ π n + k (cid:0) P α ( MSpin c ( k )) /X (cid:1) . The stable homotopy class of θ doesn’t depend on choices of i k , ν k , ǫ in the constructionfor sufficiently large k . Thus, we assign an element in lim k →∞ π n + k (cid:0) P α ( MSpin c ( k )) /X (cid:1) represented by θ to every closed α -twisted Spin c manifold ( M, ι, ν, η ) over X .Now we show that the stable homotopy class of θ depends only on the bordism class of M . Let W be an (n+1)-dimensional α -twisted Spin c manifold and j : ∂W → R n + k be anembedding for some sufficiently large k with the classifying map ν k for the normal bundle N ∂W ˜ ν k / / π (cid:15) (cid:15) ξ k (cid:15) (cid:15) ∂W ν k / / BSO ( k ) . Choose i : W → R n + k × [0 , to be an embedding agreeing with j × { } on ∂W , em-bedding a tubular neighborhood of ∂W orthogonally along j ( ∂W ) × { } , and with theimage missing R n + k × { } . The previous construction applied to the embedding i yields anull-homotopy of the map θ : ( S n + k , ∞ ) −→ ( P α ( MSpin c ( k )) /X, ∗ ) . Assigning the stable homotopy class of the map θ to each α -twisted Spin c bordism class,we have defined a map Θ : Ω
Spin c n ( X, α ) −→ π S n (cid:0) P α ( MSpin c ) /X (cid:1) . Step 2. Θ is a homomorphism.Given a pair of closed α -twisted Spin c manifolds ( M , ι , ν , η ) and ( M , ι , ν , η ) over X representing two classes in Ω Spin c n ( X, α ) . Then for a = 1 or , Θ([ M a , ι a , ν a , η a ]) is represented by a map θ a : ( S n + k , ∞ ) −→ ( P α ( MSpin c ( k )) /X, ∗ ) constructed as above.Choose an embedding i : M ⊔ M → R n + k such that the last coordinate is positive for M and negative for M . Taking small enough ǫ , the previous construction gives us a map ( S n + k , ∞ ) d / / S n + k ∧ S n + k θ + θ / / ( P α ( MSpin c ( k )) /X, ∗ ) , where d denotes the collapsing the equator of S n + k . This map represents the sum of thehomotopy classes of θ and θ . Hence, Θ([ M , ι , ν , η ] + [ M , ι , ν , η ]) = Θ([ M , ι , ν , η ]) + Θ([ M , ι , ν , η ]) . Step 3. Θ is a monomorphism.Let ( M, ι, ν, η ) be an α -twisted Spin c n -manifold such that Θ([
M, ι, ν, η ]) = 0 . Thenfor some large k , the above construction in Step 1 defines a continuous map θ = ι ∗ ◦ h ∗ ◦ ǫ − ◦ c : ( S n + k , ∞ ) −→ P α ( MSpin c ( k )) /X which is null-homotopic. As P α ( BSpin c ( k )) ⊂ P α ( MSpin c ( k )) /X and the fact that M is the zero section of N , and the map ι ∗ ◦ h ∗ : Th ( N ) → P α ( MSpin c ( k )) /X sending the zero section of N to P α ( BSpin c ( k )) . We have M = θ − (cid:0) P α ( BSpin c ( k )) (cid:1) . Note that the trivial map, denoted by θ , maps S n + k to the base point of P α ( MSpin c ( k )) /X ,so we know that θ − (cid:0) P α ( BSpin c ( k )) (cid:1) is a empty set. Now we can choose a homotopybetween θ and θ H : S n + k × [0 , −→ P α ( MSpin c ( k )) /X for some sufficiently large k , such that H is differentiable near and transversal to P α ( BSpin c ( k )) ⊂ P α ( MSpin c ( k )) /X. So W = H − (cid:0) P α ( BSpin c ( k )) (cid:1) is a submanifold of R n + k × [0 , with ∂W = M meeting R n + k × { } orthogonally along M . The map H | W sends W to P α ( BSpin c ( k )) , as P α ( BSpin c ( k )) is a fibration over X ,so we have a continuous map ι W : W → X .Note that the pull-back diagram P α (cid:15) (cid:15) / / EK ( Z , (cid:15) (cid:15) X α / / K ( Z , , EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 27 induces a homotopy pull-back P α ( BSpin c ( k )) (cid:15) (cid:15) / / EK ( Z , BSpin c k )) (cid:15) (cid:15) X α / / K ( Z , . As EK ( Z , is contractible, so the associated fiber bundle EK ( Z , BSpin c ( k )) is ho-motopy equivalent to BSO ( k ) . This implies that the following diagram P α ( BSpin c ( k )) (cid:15) (cid:15) / / BSO ( k ) (cid:15) (cid:15) X α / / K ( Z , is a homotopy pull-back. We see that the map H | W defines a homotopy commutative dia-gram W & & ' ' H | W & & MMMMMMMMMMM P α ( BSpin c ( k )) (cid:15) (cid:15) / / BSO ( k ) s { n n n n n n nn n n n n n n W (cid:15) (cid:15) X α / / K ( Z , . Hence, W admits an α -twisted Spin c structure such that the boundary inclusion M → W is a morphism in the α -twisted Spin c bordism category. This implies that ( M, ι, ν, η ) isnull-bordant, so [ M, ι, ν, η ] = 0 in Ω Spin c n ( X, α ) . Step 4. Θ is an epimorphism.Let θ : ( S n + k , ∞ ) −→ ( P α ( MSpin c ( k )) /X, ∗ ) , for a large k , represents an element in π S n (cid:0) P α ( MSpin c ) /X (cid:1) . As S n + k is compact, we may find a finite dimensional model for BSpin c ( k ) , so we maypretend that BSpin c ( k ) is finite dimensional. We can deform the map θ to a map h suchthat (1) h agrees with θ on an open set containing ∞ .(2) h is differentiable on the preimage of some open set containing P α ( BSpin c ( k )) and is transverse on P α ( BSpin c ( k )) .(3) Let M = h − ( P α ( BSpin c ( k ))) , then h is a normal bundle map from a tubularneighborhood of M in S n + k to P α ( MSpin c ( k )) /X . Then M is a smooth compact n-dimensional manifold with the following homotopy com-mutative diagram: M ι & & ν ' ' h | M & & MMMMMMMMMMM P α ( BSpin c ( k )) (cid:15) (cid:15) / / BSO ( k ) s { n n n n n n nn n n n n n n W (cid:15) (cid:15) X α / / K ( Z , . Therefore, M admits an α -twisted Spin c structure ( ι, ν, η ) . The above generalized Pontrjagin-Thom construction implies that Θ([
M, ι, ν, η ]) is the class represented by θ . (cid:3) The index map
Ind : MSpin c → K (the complex K-theory spectrum) induces a map ofbundles of spectra over X Ind : P α ( MSpin c ) −→ P α ( K ) . The stable homotopy group of P α ( K ) /X by definition is the twisted topological K-homologygroups K tev/odd ( X, α ) , due to the periodicity of K , we have K tev ( X, α ) = lim −→ k →∞ π k (cid:0) P α ( K ) /X (cid:1) and K todd ( X, α ) = lim −→ k →∞ π k +1 (cid:0) P α ( K ) /X (cid:1) . Here the direct limits are taken by the double suspension π n +2 k (cid:0) P α ( K ) /X (cid:1) −→ π n +2 k +2 (cid:0) P α ( S ∧ K ) /X (cid:1) and then followed by the standard map π n +2 k +2 (cid:0) P α ( S ∧ K ) /X (cid:1) b ∧ / / π n +2 k +2 (cid:0) P α ( K ∧ K ) /X (cid:1) m / / π n +2 k +2 (cid:0) P α ( K ) /X (cid:1) , where b : R → K represents the Bott generator in K ( R ) , m is the base point preservingmap inducing the ring structure on K-theory. Definition 4.5. (Topological index) There is a homomorphism, called the topological index
Index t : Ω Spin c ∗ ( X, α ) −→ K tev/odd ( X, α ) , (4.8)defined to be Ind ∗ ◦ Θ , the composition of Θ (as in Theorem 4.4) Θ : Ω
Spin c n ( X, α ) ∼ = / / π S n (cid:0) P α ( MSpin c ) /X (cid:1) . and the induced index transformation Ind ∗ : lim −→ k →∞ π n +2 k (cid:0) P α ( MSpin c (2 k )) /X (cid:1) → lim −→ k →∞ π n +2 k (cid:0) P α ( K ) /X (cid:1) EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 29
5. T
OPOLOGICAL INDEX = A
NALYTICAL INDEX
In this section,we will establish the main result of this paper. It should be thought of asthe generalized Atiyah-Singer index theorem for α -twisted Spin c manifolds over X with atwisting α : X → K ( Z , . In this section, we assume that X is a smooth manifold. Theorem 5.1.
There is a natural isomorphism
Φ : K tev/odd ( X, α ) −→ K aev/odd ( X, α ) suchthat the following diagram commutes Ω Spin c ev/odd ( X, α ) Index t w w ooooooooooo Index a ' ' OOOOOOOOOOOO K tev/odd ( X, α ) ∼ =Φ / / K aev/odd ( X, α ) , that is, given an closed α -twisted Spin c manifold ( M, ν, ι, η ) over X , we have Index a ( M, ν, ι, η ) =
Index t ( M, ν, ι, η ) under the isomorphism Φ . Remark 5.2. If α : X → K ( Z , is the trivial map, then we have following commutativediagram Ω Spin c ev/odd ( X ) Index t x x qqqqqqqqqq Index a & & MMMMMMMMMM K tev/odd ( X ) ∼ = / / K aev/odd ( X ) (5.1)where the isomorphism K tev/odd ( X ) ∼ = K aev/odd ( X ) follows from the work of Atiyah [2],Baum-Douglas [10] and Kasparov [31]. If X is a point, then the diagram (5.1) is the usualform of Atiyah-Singer index theorem for Spin c manifolds. Proof.
Notice that K tev/odd ( X, α ) and K aev/odd ( X, α ) are two generalized homology the-ories dual to the twisted K-theory. The twisted K-cohomology K ev/odd ( X, α ) is definedas K ev ( X, α ) = lim −→ k →∞ π (Γ( X, P α (Ω k K )) K odd ( X, α ) = lim −→ k →∞ π (Γ( X, P α (Ω k +1 K )) the homotopy classes of sections (with compact support if X is non-compact) of the asso-ciated bundle of K-theory spectra, and Ω k K is the iterated loop space of K . We will showthat there are natural isomorphisms from twisted K-homology (topological and analytical)to twisted K-cohomology with the twisting shifted by α α + ( W ◦ τ ) where τ : X → BSO is the classifying map of the stable tangent space and α + ( W ◦ τ ) denotes the map X → K ( Z , defined in (2.5), representing the class [ α ] + W ( X ) in H ( X, Z ) . Step 1.
There exists an isomorphism K tev/odd ( X, α ) ∼ = K ev/odd ( X, α + ( W ◦ τ )) with thedegree shifted by dimX ( mod .Assume X is n dimensional, choose an embedding i k : X → R n +2 k for some large k ,with its normal bundle π : N k → X identified as an ǫ -tubular neighborhood of X . Anytwo embeddings X → R n +2 k are homotopic through a regular homotopy for a sufficientlylarge k . Under the inclusion R n +2 k × ⊂ R n +2 k +2 , the Thom spaces of N k and N k +2 are related through the reduced suspension by S Th ( N k +2 ) = S ∧ Th ( N k ) . (5.2)By the Thom isomorphism ([19]), we have an isomorphism K ev/odd (cid:0) X, α + ( W ◦ τ ) (cid:1) ∼ = lim −→ k →∞ K ev/odd ( N k , α ◦ π ) , (5.3)where α ◦ π : N k → K ( Z , is the pull-back twisting on N k . There is a natural mapfrom K ev/odd ( N k , α ◦ π ) to K tev/odd ( X, α ) by considering S n +2 k as R n +2 k ∪ {∞} andthe following pull-back diagram P α ◦ π ( K ) / / (cid:15) (cid:15) P α ( K ) (cid:15) (cid:15) N k π / / X. Given an element of K ev ( N k , α ◦ π ) represented by a compactly supported section θ : N k → P α ◦ π ( K ) , then S n +2 k c / / Th ( N k ) θ / / P α ◦ π ( K ) /N k / / P α ( K ) /X representing an element in K tev/odd ( X, α ) . Replacing X by X × R , this construction givesa map from K odd ( N k , α ◦ π ) to K todd ( X, α ) . Recall that there is a homotopy equivalence K ∼ Ω K induced by the map S ∧ K b ∧ / / K ∧ K m / / K , EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 31 where b represents the Bott generator in K ( R ) , m is the base point preserving map in-ducing the ring structure on K-theory. Together with (5.2), we obtain S n +2 k c / / s (cid:15) (cid:15) Th ( N k ) / / s (cid:15) (cid:15) P α ( K ) /X s (cid:15) (cid:15) S ∧ S n +2 k (cid:15) (cid:15) ∧ c / / S ∧ Th ( N k ) (cid:15) (cid:15) / / P α ( S ∧ K ) /X m ◦ ( b ∧ (cid:15) (cid:15) S n +2 k +2 c / / Th ( N k +2 ) / / P α ( K ) /X (5.4)where s is the reduced suspension map by S . This implies that the stable homotopy equiv-alent class of sections defines the same element in K t ∗ ( X, α ) , with the degree given by n ( mod . Thus, we have a well-defined homomorphism Ψ t : K ev/odd ( X, α + ( W ◦ τ )) −→ K tev/odd ( X, α ) . (5.5)Conversely, for a sufficiently large k , let θ : ( S m +2 k , ∞ ) → ( P α ( K ) /X, ∗ ) representan element in K tev/odd ( X, α ) (depending on even or odd m ). We can lift this map to amap θ : S m +2 k → P α ( MSpin c (2 k )) /X . As in Step 4 of the proof of Theorem 4.4, θ can be deformed to a differentiable map h on the preimage of some open set containing P α ( BSpin c (2 k )) , is transverse to P α ( BSpin c (2 k )) and agrees with θ on an open setcontaining ∞ . Then M = h − ( P α ( BSpin c (2 k ))) ⊂ R m +2 k = S m +2 k − {∞} is a smooth compact manifold and admits a natural α -twisted Spin c structure M ι ' ' ν ' ' h | M ' ' NNNNNNNNNNNN P α ( BSpin c (2 k )) (cid:15) (cid:15) / / BSO s { n n n n n n n nn n n n n n n n W (cid:15) (cid:15) X α / / K ( Z , . (5.6)Therefore, we can assume that the map θ : ( S m +2 k , ∞ ) → ( P α ( K ) /X, ∗ ) comes from thefollowing commutative diagram R m +2 k θ / / P α ( K ) (cid:15) (cid:15) N ǫ j ; ; xxxxxxxxx π GGGGGGGGG M O O ι / / X, where N ǫ is the normal bundle of M in R n +2 k , identified as the ǫ -neighborhood of M in S m + k . In particular, the continuous map θ ◦ j : N ǫ → S m +2 k → P α ( K ) /X determines a compactly supported section of P α ◦ ι ◦ π ( K ) = ( ι ◦ π ) ∗ P α ( K ) .Choose an embedding ι k : M → R k , the α -twisted Spin c structure (5.6) on M over X induces a natural α ◦ π -twisted Spin c structure (5.6) on M over X × R k M ( ι,ι k ) (cid:15) (cid:15) ν / / BSO t | q q q q qq q q q q W (cid:15) (cid:15) X × R k α ◦ π / / K ( Z , such that ( ι, ι k ) is an embedding. Here π is the projection X × R k → X . Notice that K tev/odd ( X, α ) ∼ = K tev/odd ( X × R k , α ) . Therefore, without losing any generality, we may assume that ι : M → X is an embeddingand there is an embedding i k : X → R n +2 k . Denote by N X the normal bundle of theembedding i k , and N M the normal bundle of M in R n +2 k . We implicitly assume that anynormal bundle of an embedding is identified to a tubular neighborhood of the embedding.Then we have the following collapsing map Th ( N X ) −→ Th ( N M ) as N M is imbedded in N X with appropriate choices of tubular neighborhood.The map θ : ( S m +2 k , ∞ ) → ( P α ( K ) /X, ∗ ) is stable homotopic to ( S n +2 k , ∞ ) c / / ( Th ( N M ) , ∗ ) / / ( P α ( K ) /X, ∗ ) . (5.7)Hence, we obtain a map Th ( N X ) / / Th ( N M ) / / P α ( K ) /X , which gives a compactly supported section of P α ◦ π ( K ) where π denotes the projection N X → X . This section defines an element in K ev ( N X , α ◦ π ) , hence an element of K ev ( X, α + ( W ◦ τ )) under the isomorphism (5.3) and the diagram (5.4). It is straightfor-ward to show that this map from K tev/odd ( X, α ) to K ev/odd ( X, α + ( W ◦ τ )) is the inverseof Ψ t defined before.Hence, we have established the isomorphism Ψ t : K ev/odd ( X, α + ( W ◦ τ )) −→ K tev/odd ( X, α ) , (5.8)with the degree shifted by dimX ( mod . This is the Poincar´e duality in topological twistedK-theory. Step 2 . There is an isomorphism Ψ a : K aev/odd ( X, α ) ∼ = K ev/odd ( X, α + ( W ◦ τ )) withthe degree shifted by dimX ( mod . EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 33
Recall that for a twisting α : X → K ( Z , , there is an associated bundle of C ∗ al-gebras, denoted by P α ( K ) where K be the C ∗ -algebra of compact operators on an infi-nite dimensional, complex and separable Hilbert space H . Here we identify K ( Z , asthe projective unitary group P U ( H ) with the norm topology (See [5] for details). Thereis an equivalent definition of K ev/odd ( X, α ) in [39], using the continuous trace C ∗ alge-bra C c ( X, P α ( K )) , which consists of compactly supported sections of the bundle of C ∗ -algebras, P α ( K ) . Moreover, Atiyah-Segal established a canonical isomorphism in [5] be-tween K ev/odd ( X, α ) and the analytical K-theory of C c ( X, P α ( K )) . The latter K-theorycan be described as the Kasparov KK-theory KK (cid:0) C , C c ( X, P α ( K )) (cid:1) . There is an equivalent definition of K ev/odd ( X, α + ( W ◦ τ )) in [39], using the con-tinuous trace C ∗ algebras, which consists of compactly supported sections of the bundle of C ∗ -algebra, P α +( W ◦ τ ) ( K ) In [23] (see also [42]) , a natural isomorphism, called the Poincar´e duality in analyticaltwisted K-theory, KK (cid:0) C , C c ( X, P α ( K )) ˆ ⊗ C c ( X ) C c ( X, Cliff ( T X )) (cid:1) ∼ = KK (cid:0) C c ( X, P − α ( K )) , C (cid:1) is constructed using the Kasparov product with the weak dual-Dirac element associated to P α ( K ) , see Definition 1.11 and Theorem 1.13 in [23] for details.Note that there is a natural Morita equivalence C c ( X, P α ( K )) ˆ ⊗ C c ( X ) C c ( X, Cliff ( T X )) ∼ C c ( X, P α +( W ◦ τ ) ( K )) which induces a canonical isomorphism on their KK-groups. The isomorphism KK (cid:0) C c ( X, P − α ( K )) , C (cid:1) ∼ = KK (cid:0) C c ( X, P α ( K )) , C (cid:1) is obvious using the operator conjugation. So in our notation, the Poincar´e duality in ana-lytical twisted K-theory can be written in the following form Ψ a : K ev/odd ( X, α + ( W ◦ τ )) −→ K aev/odd ( X, α ) , (5.9)with the degree shifted by dimX ( mod coming from the shift of grading on the evev/odddimensional complex Clifford algebra.Applying the Poincar´e duality isomorphisms (5.8) and (5.9) in topological twisted K-theory and algebraic twisted K-theory, we have a natural isomorphism Φ : K tev/odd ( X, α ) Ψ a ◦ Ψ − t / / K aev/odd ( X, α ) , such that the following diagram commutes K ev/odd ( X, α + ( W ◦ τ )) Ψ t ∼ = u u kkkkkkkkkkkkkk Ψ a ∼ = ) ) SSSSSSSSSSSSSSS K tev/odd ( X, α ) ∼ =Φ / / K aev/odd ( X, α ) . (5.10) Step 3.
Show that
Index a = Φ ◦ Index t .Applying the suspension operation, we only need to prove the even case. Let ( M, ι, ν, η ) be a n dimensional closed α -twisted Spin c manifold over X , M ι (cid:15) (cid:15) ν / / BSO η v ~ v v v v vv v v v v W (cid:15) (cid:15) X α / / K ( Z , , representing an element in the α -twisted Spin c bordism group Ω Spin c n ( X, α ) .The analytical index of ( M, ι, ν, η ) , as defined Definition 3.7, is given by Index a ( M, ι, ν, η ) = ι ! ([ M ]) = ι ! ◦ P D ([ C ]) . (5.11)where P D : K ( M ) → K a ( M, W ◦ τ ) is the Poincar´e duality isomorphism with τ theclassifying map for the stable tangent bundle. The push-forward map ι ! in (5.11) is obtainedfrom the following sequence of maps K a ( M, W ◦ τ ) I ∗ / / K a ( M, W ◦ ν ) η ∗ / / K a ( M, α ◦ ι ) ι ∗ / / K a ( X, α ) . There is a natural push-forward map ι ! : Ω Spin c ev ( M, α ◦ ι ) → Ω Spin c ev ( X, α ) such that the following diagrams for the analytical index Ω Spin c ev ( M, α ◦ ι ) Index a (cid:15) (cid:15) ι ! / / Ω Spin c ev ( X, α ) Index a (cid:15) (cid:15) K a ( M, α ◦ ι ) ι ! / / K a ( X, α ) , and for the topological index Ω Spin c ev ( M, α ◦ ι ) Index t (cid:15) (cid:15) ι ! / / Ω Spin c ev ( X, α ) Index t (cid:15) (cid:15) K t ( M, α ◦ ι ) ι ! / / K t ( X, α ) , are commutative. EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 35 As ( M, Id, ν, Id ) is a natural α ◦ ι -twisted Spin c manifold over M , we only need toshow that ( I ∗ ) − ◦ Φ ◦ Index t ( M, Id, ν, Id ) = [ M ] = P D ( C ) (5.12)in K a ( M, W ◦ τ ) . We will show that the identity (5.12) follows from the Thom isomor-phism K ( M ) ∼ = K ( N M , W ◦ ν k ◦ π ) where we choose an embedding i k : M → R n + k with its normal bundle N M , π is theprojection N M → M and ν k : M → BSO ( k ) is the classifying map of the normal bundle N M . The image of [ C ] under the above Thom isomorphism is represented by the map θ M : ( S n + k , ∞ ) → ( Th ( N M ) , ∗ ) → ( P W ◦ ν k ◦ π ( K ) /N M , ∗ ) arising from the W ◦ ν k -twisted Spin c structure on M as in the following diagram Th ( N M ) / / P W ◦ ν k ( MSpin c ( k )) /M / / P W ◦ ν k ( K ) /MM (cid:31) ? O O Id ) ) SSSSSSSSSSSSSSSSSSSS / / P W ◦ ν k ( BSpin c ( k )) (cid:15) (cid:15) (cid:31) ? O O / / BSO ( k ) W (cid:15) (cid:15) M W ◦ ν k / / K ( Z , . This same diagram also defines the topological index of ( M, Id, ν, Id ) under the index map Index t : Ω Spin c ev ( M, W ◦ ν ) → K t ( M, W ◦ ν ) . Hence, we establish the following commutative diagram Ω Spin c ev/odd ( X, α ) Index t w w ooooooooooo Index a ' ' OOOOOOOOOOOO K tev/odd ( X, α ) ∼ =Φ / / K aev/odd ( X, α ) . (cid:3) Remark 5.3.
Let π X : T X → X be the projection. Applying the Thom isomorphism([19]), we obtain the following isomorphism K ev/odd ( T X, α ◦ π X ) ∼ = K ev/odd ( X, α + ( W ◦ τ )) . Hence, the above commutative diagram (5.10) becomes K ev/odd ( T X, α ◦ π X ) P D ∼ = u u lllllllllllll P D ∼ = ) ) RRRRRRRRRRRRRR K tev/odd ( X, α ) ∼ =Φ / / K aev/odd ( X, α ) , (5.13) which should be thought of as a generalized Atiyah-Singer index theorem. for α -twisted Spin c manifolds over X with a twisting α : X → K ( Z , . If α : X → K ( Z , is a trivialmap, the commutative diagram (5.13) becomes K ev/odd ( T X ) Index t w w ooooooooooo Index a ' ' OOOOOOOOOOO K tev/odd ( X ) ∼ =Φ / / K aev/odd ( X ) , which is the basic form for the Atiyah-Singer index theorem. The upper vertex representsthe symbols of elliptic pseudo-differential operators on X . Each of these index maps isessentially just the Poincar´e duality isomorphism between the K-cohomology of T ∗ X andthe two realizations of the K-homology K ( X ) . See [10] for more details.In particular, if α is the twisting associated to the classifying map W ◦ τ : X → K ( Z , of the stable tangent bundle, then we have the following twisted index theorem, given bythe following commutative diagram K ev/odd ( T X, W ◦ τ ◦ π ) Index t t t jjjjjjjjjjjjjjj Index a * * TTTTTTTTTTTTTTTT K tev/odd ( X, W ◦ τ ) ∼ =Φ / / K aev/odd ( X, W ◦ τ ) . This is a special case of Connes-Skandalis longitudinal index theorem for foliations . Wewill return to this issue later.6. G
EOMETRIC CYCLES AND GEOMETRIC TWISTED K- HOMOLOGY
Definition 6.1.
Let X be a paracompact Hausdorff space , and let α : X −→ K ( Z , be atwisting over X . A geometric cycle for ( X, α ) is a quintuple ( M, ι, ν, η, [ E ]) such that(1) M is a smooth closed manifold equipped with an α -twisted Spin c structure; M ι (cid:15) (cid:15) ν / / BSO η v ~ v v v v vv v v v v W (cid:15) (cid:15) X α / / K ( Z , , where ι : M → X is a continuous map, ν is a classifying map of the stable normalbundle, and η is a homotopy from W ◦ ν and α ◦ ι ;(2) [ E ] is a K-class in K ( M ) represented by a Z -graded vector bundle E over M .Two geometric cycles ( M , ι , ν , η , [ E ]) and ( M , ι, ν , η , [ E ]) are isomorphic if thereis an isomorphism f : ( M , ι , ν , η ) → ( M , ι , ν , η ) , as α -twisted Spin c manifoldsover X , such that f ! ([ E ]) = [ E ] .Let Γ( X, α ) be the collection of all geometric cycles for ( X, α ) . We now impose anequivalence relation ∼ on Γ( X, α ) , generated by the following three elementary relations: EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 37 (1)
Direct sum - disjoint union If ( M, ι, ν, η, [ E ]) and ( M, ι, ν, η, [ E ]) are two geometric cycles with the same α -twisted Spin c structure, then ( M, ι, ν, η, [ E ]) ∪ ( M, ι, ν, η, [ E ]) ∼ ( M, ι, ν, η, [ E ] + [ E ]) . (2) Bordism
Given two geometric cycles ( M , ι , ν , η , [ E ]) and ( M , ι , ν , η , [ E ]) , if thereexists a α -twisted Spin c manifold ( W, ι, ν, η ) and [ E ] ∈ K ( W ) such that ∂ ( W, ι, ν, η ) = − ( M , ι , ν , η ) ∪ ( M , ι , ν , η ) and ∂ ([ E ]) = [ E ] ∪ [ E ] . Here − ( M , ι , ν , η ) denotes the manifold M withthe opposite α -twisted Spin c structure.(3) Spin c vector bundle modification Suppose we are given a geometric cycle ( M, ι, ν, η, [ E ]) and a Spin c vector bundle V over M with even dimensional fibers. Denote by R the trivial rank one real vectorbundle. Choose a Riemannian metric on V ⊕ R , let ˆ M = S ( V ⊕ R ) be the sphere bundle of V ⊕ R . Then the vertical tangent bundle T v ( ˆ M ) of ˆ M admitsa natural Spin c structure with an associated Z -graded spinor bundle S + V ⊕ S − V .Denote by ρ : ˆ M → M the projection which is K-oriented. Then ( M, ι, ν, η, [ E ]) ∼ ( ˆ M , ι ◦ ρ, ν ◦ ρ, η ◦ ρ, [ ρ ∗ E ⊗ S + V ]) . Definition 6.2.
Denote by K geo ∗ ( X, α ) = Γ(
X, α ) / ∼ the geometric twisted K-homology.Addition is given by disjoint union - direct sum relation. Note that the equivalence relation ∼ preserves the parity of the dimension of the underlying α -twisted Spin c manifold. Let K geo ( X, α ) (resp. K geo ( X, α ) ) the subgroup of K geo ∗ ( X, α ) determined by all geometriccycles with even (resp. odd) dimensional α -twisted Spin c manifolds. Remark 6.3. (1) According to Proposition 3.3, M admits an α -twisted Spin c structureif and only if ι ∗ ([ α ]) + W ( M ) = 0 . (If ι is an embedding, this is the anomaly cancellation condition obtained by Freedand Witten in [26], ( M, ι, ν, η, [ E ]) is referred to by physicists as a D-brane ap-peared in Type IIB string theory, see [26] [44][30][15].)(2) Different definitions of topological twisted K-homology were proposed in [37] us-ing Spin c -manifolds and twisted bundles. It is not clear to the author if their defi-nition is equivalent to Definition 6.2.(3) If f : X → Y is a continuous map and α : X → K ( Z , is a twisting, then thereis a natural homomorphism of abelian groups f ∗ : K geoev/odd ( X, α ) −→ K geoev/odd ( Y, α ◦ f ) sending [ M, ι, ν, η, E ] to [ M, f ◦ ι, ν, η, E ] .Given a geometric cycle ( M, ι, ν, η, [ E ]) , the analytical index (as in Definition 3.7) de-termines an element µ ( M, ι, ν, η, [ E ]) = Index a ( M, ι, ν, η, [ E ])= ι ∗ ◦ η ∗ ◦ I ∗ ◦ P D ([ E ]) in K aev/odd ( X, α ) . Theorem 6.4.
The assignment ( M, ι, ν, η, [ E ]) → µ ( M, ι, ν, η, [ E ]) , called the assemblymap, defines a natural homomorphism µ : K geoev/odd ( X, α ) → K aev/odd ( X, α ) which is an isomorphism for any smooth manifold X with a twisting α : X → K ( Z , .Proof. Step 1.
We need to show that the correspondence is compatible with the three el-ementary equivalence relations, so the assembly map µ is well-defined. We only need todiscuss the even case.Proposition 3.8 ensures that Index a ( M, ι, ν, η, [ E ]) is compatible with the bordism re-lation and disjoint union - direct sum relation. We only need to check that the assembly mapis compatible with the relation of Spin c vector bundle modification.Suppose given a geometric cycle ( M, ι, ν, η, [ E ]) of even dimension and a Spin c vectorbundle V over M with even dimensional fibers. Then ( M, ι, ν, η, [ E ]) ∼ ( ˆ M , ι ◦ ρ, ν ◦ ρ, η ◦ ρ, [ ρ ∗ E ⊗ S + V ]) . where ˆ M = S ( V ⊕ R ) is the sphere bundle of V ⊕ R and ρ : ˆ M → M is the projection. Thevertical tangent bundle T v ( ˆ M ) of ˆ M admits a natural Spin c structure with an associated Z -graded spinor bundle S + V ⊕ S − V . The K-oriented map ρ induces a natural homomorphism(see [4]) ρ ! : K ( ˆ M ) −→ K ( M ) sending [ ρ ∗ E ⊗ S + V ] to [ E ] . This follows from the Atiyah-Singer index theorem for familiesof longitudinally elliptic differential operator associated to the Dirac operator on the round n -dimensional sphere. Applying the Poincar´e duality, we have the following commutativediagram K ( ˆ M ) ρ ! / / P D (cid:15) (cid:15) K ( M ) P D (cid:15) (cid:15) K a ( ˆ M , W ◦ τ ◦ ρ ) ρ ∗ / / K a ( M, W ◦ τ ) , which implies that P D ([ E ]) = ρ ∗ ◦ P D ([ ρ ∗ E ⊗ S + V ]) . Hence, we have µ ( M, ι, ν, η, [ E ]) = µ ( ˆ M , ι ◦ ρ, ν ◦ ρ, η ◦ ρ, [ ρ ∗ E ⊗ S + V ]) . EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 39
Step 2.
We establish the following commutative diagram K tev/odd ( X, α ) Ψ w w nnnnnnnnnnn Φ ∼ = ' ' PPPPPPPPPPPP K geoev/odd ( X, α ) µ / / K aev/odd ( X, α ) and show that Ψ is surjective. This implies that µ is an isomorphism.Firstly, we construct a natural map Ψ : K tev ( X, α ) → K geo ( X, α ) . Given an element of K t ( X, α ) represented by a map θ : ( S m +2 k , ∞ ) → ( P α ( K ) /X, ∗ ) for a sufficiently large k . We can lift this map to a map θ : S m +2 k → P α ( MSpin c (2 k )) /X .As in Step 4 of the proof of Theorem 4.4, θ can be deformed to a differentiable map h on thepreimage of some open set containing P α ( BSpin c (2 k )) , is transverse to P α ( BSpin c (2 k )) and agrees with θ on an open set containing ∞ . Then M = h − ( P α ( BSpin c (2 k ))) ⊂ R m +2 k = S m +2 k − {∞} admits a natural α -twisted Spin c structure M ι ' ' ν ' ' h | M ' ' NNNNNNNNNNNN P α ( BSpin c (2 k )) (cid:15) (cid:15) / / BSO s { n n n n n n n nn n n n n n n n W (cid:15) (cid:15) X α / / K ( Z , . A homotopy equivalence map gives rise to a bordant α -twisted Spin c manifold. Hence, wehave a geometric cycle ( M, ι, ν, η, [ C ]) , whose equivalence class doesn’t depend on variouschoices in the construction. This defines a map Ψ : K t ( X, α ) −→ K geo ( X, α ) . It is straightforward to show that Ψ is a homomorphism. Note that Φ = µ ◦ Ψ follows fromthe definition of Φ and Theorem 5.1.To show that Ψ is surjective, let ( M, ι, ν, η, [ E ]) be a geometric cycle. Then the α -twisted Spin c manifold ( M, ι, ν, η ) defines a bordism class in Ω Spin c ev ( X, α ) . The topological index Index t ( M, ι, ν, η ) ∈ K t ( X, α ) is represented by the canonical map θ : ( S m +2 k , ∞ ) → ( Th ( N M ) , ∗ ) → ( P α ( M Spin (2 k ) /X, ∗ ) → ( P α ( K ) /X, ∗ ) associated to the normal bundle π : N M → M of an embedding i k : M → R m +2 k asin Step 1 of the proof of Theorem 5.1. This map defines a compactly supported section of P α ◦ ι ◦ π ( K ) (a bundle of K-theory spectra over N M ). We also denote this section by θ . Thenthe homotopy class of the section θ defines a twisted K-class in K ( N M , α ◦ ι ◦ π ) , whichis mapped to [ C ] under the Thom isomorphism K ( N M , α ◦ ι ◦ π ) ∼ = K ( N M , W ◦ ν ◦ π ) ∼ = K ( M ) . Let σ : M → K be a map representing the K-class [ E ] . σ ◦ π is a section of thetrivial bundle K over N M . Define a new section of P α ◦ ι ◦ π ( K ) by applying the fiberwisemultiplication m : K ∧ K → K to ( θ, σ ◦ π ) . Then m ( θ, σ ◦ π ) is a compactly supportedsection of P α ◦ ι ◦ π ( K ) which determines a map, denoted by θ · σθ · σ : ( S m +2 k , ∞ ) → ( Th ( N M ) , ∗ ) → ( P α ◦ ι ◦ π ( K ) /N M , ∗ ) . The homotopy class of θ · σ as an element in K ( N M , α ◦ ι ◦ π ) , is uniquely determined bythe stable homotopy class of θ and the homotopy class of σ . Under the Thom isomorphism K ( N M , α ◦ ι ◦ π ) ∼ = K ( M ) , [ θ · σ ] is mapped to [ E ] . Hence, Ψ([ θ · σ ]) = [ M, ι, ν, η, [ E ]] . Therefore, Ψ is surjective. (cid:3) Corollary 6.5.
Given a twisting α : X → K ( Z , on a smooth manifold X , every twistedK-class in K ev/odd ( X, α ) is represented by a geometric cycle supported on an ( α + ( W ◦ τ )) -twisted closed Spin c -manifold M and an ordinary K-class [ E ] ∈ K ( M ) .Proof. We only need to prove the even case, the odd case can be obtained by the suspensionoperation. Assume that X is even dimensional and π : T X → X is the projection, then wehave the following isomorphisms K ( X, α ) ∼ = K ( T X, ( α ◦ π ) + ( W ◦ τ ◦ π )) (Thom isomorphism) ∼ = K a ( X, α + ( W ◦ τ )) (Remark 5.3) ∼ = K geo ( X, α + ( W ◦ τ )) (Theorem 6.4) From the definition of K geo ( X, α + ( W ◦ τ )) , we know that each element in K geo ( X, α +( W ◦ τ )) is represented by a geometric cycle ( M, ι, ν, η, [ E ]) for ( X, α + ( W ◦ τ )) , whichis a generalized D-brane supported on an ( α + ( W ◦ τ )) -twisted closed Spin c -manifold M and an ordinary K-class [ E ] ∈ K ( M ) . (cid:3) Remark 6.6.
Let Y be a closed subspace of X . A relative geometric cycle for ( X, Y ; α ) isa quintuple ( M, ι, ν, η, [ E ]) such that(1) M is a smooth manifold (possibly with boundary), equipped with an α -twisted Spin c structure ( M, ι, ν, η ) ;(2) if M has a non-empty boundary, then ι ( ∂M ) ⊂ Y ; EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 41 (3) [ E ] is a K-class in K ( M ) represented by a Z -graded vector bundle E over M , ora continuous map M → K .The relation ∼ generated by disjoint union - direct sum, bordism and Spin c vector bundlemodification is an equivalence relation. The collection of relative geometric cycles, modulothe equivalence relation is denoted by K geoev/odd ( X, Y ; α ) . Then we have the following commutative diagram whose arrows are all isomorphisms K tev/odd ( X, Y ; α ) Ψ v v mmmmmmmmmmmmm Φ ( ( RRRRRRRRRRRRR K geoev/odd ( X, Y ; α ) µ / / K aev/odd ( X, Y ; α ) .
7. T
HE TWISTED LONGITUDINAL INDEX THEOREM FOR FOLIATION
Given a C ∞ foliated manifold ( X, F ) , that is, F is an integrable sub-bundle of T X , let D be an elliptic differentiable operator along the leaves of the foliation. Denote by σ D thelongitudinal symbol of D , whose class in K ( F ∗ ) is denoted by [ σ D ] . In [20], Connes andSkandalis defined the topological index and the analytical index of D taking values in theK-theory of the foliation C ∗ -algebra C ∗ r ( X, F ) and established the equality between thetopological index and the analytical index of D . See [20] for more details. In this section,we will generalize the Connes-Skandalis longitudinal index theorem to a foliated manifold ( X, F ) with a twisting α : X → K ( Z , .Let N F = T X/F be the normal bundle to the leaves whose classifying map is denotedby ν F : X → BSO ( k ) . Here assume that F is rank k and X is n dimensional. We canequip X with a Riemannian metric such that we have a splitting T X = F ⊕ N F . Then the sphere bundle M = S ( F ∗ ⊕ R ) is a W ◦ ν F -twisted Spin c manifold over X . Let π be the projection M → X . To see this, we need to calculate the third Stiefel-Whitneyclass of M = S ( F ∗ ⊕ R ) from the following exact sequence of bundles over M → π ∗ ( F ⊕ R ) −→ T M ⊕ R −→ π ∗ T X → , (7.1)from which we have W ( T M ) = π ∗ W ( F ) + π ∗ W ( T X )= π ∗ W ( F ) + π ∗ ( W ( F ) + W ( N F ))= π ∗ W ( N F ) . Note that π ∗ W ( N F ) = [ W ◦ ν F ◦ π ] ∈ H ( M, Z ) . So S ( F ∗ ⊕ R ) admits a natural W ◦ ν F -twisted Spin c structure M π (cid:15) (cid:15) ν / / BSO η v ~ v v v v vv v v v v W (cid:15) (cid:15) X W ◦ ν F / / K ( Z , , where ν is the classifying map of the stable normal of M and η is a homotopy associated toa splitting of (7.1) as follows. Given a splitting of (7.1), the natural isomorphisms T M ⊕ R ∼ = π ∗ T X ⊕ π ∗ ( F ⊕ R ) ∼ = π ∗ ( F ⊕ N F ) ⊕ π ∗ ( F ⊕ R ) ∼ = π ∗ ( F ⊕ F ) ⊕ π ∗ N F ⊕ R and the canonical Spin c structure on π ∗ ( F ⊕ F ) define the homotopy between W ◦ τ and W ◦ ν F . Different choices of splittings of (7.1) gives rise to the same homotopy equivalenceclass, hence doesn’t change the twisted Spin c bordism class of M .Given an elliptic differentiable operator along the leaves of the foliation with longitudinalsymbol class [ σ D ] ∈ K ( F ∗ ) represented by a map σ D : π ∗ E → π ∗ E of a pair of vector bundles E and E over X such that σ D is an isomorphism away fromthe zero section of F ∗ . Applying the clutching construction as described in [10], M = S ( F ∗ ⊕ R ) consists of two copies of the unit ball bundle of F ∗ glued together by theidentity map of S ( F ∗ ) . We form a vector bundle over M by gluing π ∗ E and π ∗ E overeach copy of the unit ball bundle along S ( F ∗ ) by the symbol map σ D . Denote the resultingvector bundle by ˆ E . The quintuple ( M, π, ν, η, [ ˆ E ]) is a geometric cycle of ( X, W ◦ ν F ) .We define the topological index of [ σ ( D )] to be Index t ([ σ D ]) = [ M, π, ν, η, ˆ E ] ∈ K geo [ n + k ] ( X, W ◦ ν F ) where [ n + k ] denotes the mod 2 sum (even or odd if n + k is even or odd).The analytical index of [ σ D ] is defined through the following sequence of isomorphisms K ( F ∗ ) ∼ = K [ k ] ( X, W ( F )) ( Thom isomorphism ) ∼ = K a [ n + k ] ( X, W ( F ⊕ T X )) (
Poincar´e duality ) ∼ = K a [ n + k ] ( X, W ( N F )) ( F ⊕ T X ∼ = F ⊕ F ⊕ N F ) ∼ = K a [ n + k ] ( X, W ◦ ν F ) . The resulting element is denoted by
Index a ([ σ D ]) ∈ K a [ n + k ] ( X, W ◦ ν F ) . EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 43
Now we apply Theorems 5.1 and 6.4, and Remark 5.3 to obtain the following versionof the longitudinal index theorem for the foliated manifold ( X, F ) . This longitudinal in-dex theorem is equivalent to the Connes-Skandalis longitudinal index theorem through thenatural homomorphism K a [ n − k ] ( X, W ◦ ν F ) → K ( C ∗ r ( X, F )) . Theorem 7.1.
Given a C ∞ n-dimensional foliated manifold ( X, F ) of rank k , the longitu-dinal index theorem for ( X, F ) is given by the following commutative diagram K ( F ∗ ) Index t w w nnnnnnnnnnnn Index a ( ( PPPPPPPPPPPP K geo [ n − k ] ( X, W ◦ ν F ) µ ∼ = / / K a [ n − k ] ( X, W ◦ ν F ) , whose arrows are all isomorphisms. Remark 7.2. If ( X, F ) comes from a fibration π B : X → B such that the leaves are thefibers of π B , then F is given by the vertical tangent bundle T ( X/B ) and N F ∼ = π ∗ B T B .This isomorphism defines a canonical homotopy η realizing W ◦ ν F ∼ W ◦ τ B ◦ π B ,where τ B is the classifying map of the stable tangent bundle of B . The following homotopydiagram S ( F ∗ ⊕ R ) π (cid:15) (cid:15) ν / / BSO η q y l l l l l l l l ll l l l l l l l l W (cid:15) (cid:15) X π (cid:15) (cid:15) W ◦ ν F / / η w (cid:127) v v v v v vv v v v v v K ( Z , ,B W ◦ τ B kkkkkkkkkkkkkkkkk implies that ( S ( F ∗ ⊕ R ) , π B ◦ π, ν, η ∗ η , [ ˆ E ]) , where η ∗ η is the obvious homotopy joining η and η , is a geometric cycle of ( B, W ◦ τ B ) and ( π B ) ! ( S ( F ∗ ⊕ R ) , π, ν, η, [ ˆ E ]) = ( S ( F ∗ ⊕ R ) , π B ◦ π, ν, η ∗ η , [ ˆ E ]) . The commutative diagram K ( F ∗ ) Index t w w nnnnnnnnnnnn Index a ' ' PPPPPPPPPPPP K geo [ n − k ] ( X, W ◦ ν F ) ( π B ) ! (cid:15) (cid:15) µ / / K a [ n − k ] ( X, W ◦ ν F ) ( π B ) ! (cid:15) (cid:15) K geo [ n − k ] ( B, W ◦ τ B ) µ / / P D ' ' PPPPPPPPPPPP K a [ n − k ] ( B, W ◦ τ B ) P D w w nnnnnnnnnnnn K ( B ) becomes the Atiyah-Singer families index theorem in [8].In the presence of a twisting α : X → K ( Z , on a foliated manifold ( X, F ) , Theorems5.1 and 6.4, and Remark 5.3 give rise to the following twisted longitudinal index theorem. Theorem 7.3.
Given a C ∞ n-dimensional foliated manifold ( X, F ) of rank k and a twisting α : X → K ( Z , , let π : F ∗ → X be the projection. Then the twisted longitudinalindex theorem for the foliated manifold ( X, F ) with a twisting α is given by the followingcommutative diagram K ( F ∗ , α ◦ π ) Index t u u kkkkkkkkkkkkkkk Index a ) ) TTTTTTTTTTTTTTT K geo [ n − k ] ( X, α + ( W ◦ ν F )) µ ∼ = / / K a [ n − k ] ( X, α + ( W ◦ ν F )) , whose arrows are all isomorphisms. In particular, if ( X, F ) comes from a fibration π B : X → B and a twisting α ◦ π B on X comes from a twisting α on B , then we have thefollowing twisted version of the Atiyah-Singer families index theorem with notations fromRemark 7.2 K ( T ∗ ( X/B ) , α ◦ π B ◦ π ) ( π B ) ! ◦ Index t t t iiiiiiiiiiiiiiiii ( π B ) ! ◦ Index a * * UUUUUUUUUUUUUUUUU K geo [ n − k ] ( B, α + ( W ◦ τ B )) µ / / P D * * UUUUUUUUUUUUUUUUUU K a [ n − k ] ( B, α + ( W ◦ τ B )) P D t t iiiiiiiiiiiiiiiiii K ( B, α ) . In [35], Mathai-Melrose-Singer established the index theorem for projective families oflongitudinally elliptic operators associated to a fibration φ : Z → X and an Azumayabundle A α for α representing a torsion class in H ( X, Z ) .Given a local trivialization of A a for an open covering of X = ∪ i U i , according to [35], aprojective family of longitudinally elliptic operators is a collection of longitudinally ellipticpseudo-differential operators, acting on finite dimensional vector bundles of fixed rank overeach of the open sets { φ − ( U i ) } such that the compatibility condition over triple overlapsmay fail by a scalar factor. The symbol class of such a projective family of elliptic operatorsdetermines a class in K ( T ∗ ( Z/X ) , α ◦ φ ◦ π ) , where T ∗ ( Z/X ) is dual to the vertical tangent bundle of Z and π : T ∗ ( Z/X ) → Z is theprojection. Let n be the dimension of Z and k be the dimension of the fiber of φ . The Thom EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 45 isomorphism, Theorems 5.1 and 6.4 give rise to the following commutative diagram K ( T ∗ ( Z/X ) , α ◦ φ ◦ π ) φ ! ◦ Index t t t iiiiiiiiiiiiiiiii φ ! ◦ Index a * * UUUUUUUUUUUUUUUUU K geo [ n − k ] ( X, α + ( W ◦ τ )) µ / / P D * * UUUUUUUUUUUUUUUUUU K a [ n − k ] ( X, α + ( W ◦ τ )) P D ∼ = t t iiiiiiiiiiiiiiiiii K ( X, α ) here α +( W ◦ τ ) represents the class [ α ]+ W ( X ) ∈ H ( X, Z ) . Readers familiar with [35]will recognise that the above theorem is another way of writing the Mathai-Melrose-Singerindex theorem (Cf. Theorem 4 in [35]) for projective families of longitudinally ellipticoperators associated a fibration φ : Z → X and an Azumaya bundle A α for α representinga torsion class [ α ] ∈ H ( X, Z ) . 8. F INAL REMARKS
Let M be an oriented manifold with a map ν : M → BSO classifying its stable normalbundle. Given any fibration π : B → BSO , we can define a B -structure on M to be ahomotopy class of lifts ˜ ν of ν : B π (cid:15) (cid:15) M ˜ ν ; ; ν / / BSO . (8.1)When B is BSpin c , then a lift ˜ ν in (8.1) is a Spin c structure on its stable normal bundle.When B is BSpin , then a lift ˜ ν in (8.1) is a Spin structure on its stable normal bundle.Define
String = lim −→ k →∞ String ( k ) where String ( k ) is an infinite dimensional topological group constructed in [40]. There is amap String ( k ) → Spin ( k ) which induces an isomorphism π n ( String ( k )) ∼ = π n ( Spin ( k )) for all n except n = 3 when π ( String ( k )) = 0 and π ( Spin ( k )) ∼ = Z .Let M be a Spin manifold with a classifying map ν : M → BSpin for the
Spin structure on its stable normal bundle. A string structure on M is a lift ˜ ν of ν : BString π (cid:15) (cid:15) M ˜ ν : : vvvvv ν / / BSpin . We point out that an oriented manifold M admits a spin structure on its stable normalbundle if and only if its second Stiefel-Whitney class w ( M ) vanishes, and a spin manifold M admits a string structure on its stable normal bundle if p ( M )2 vanishes, where p ( M ) denotes the first Pontrjagin class of M (Cf. [34] [40]). If M is a string manifold, then M has a canonical orientation with respect to elliptic cohomology.The tower of Eilenberg-MacLane fibrations BString K ( Z , (cid:15) (cid:15) BSpin K ( Z , (cid:15) (cid:15) p / / K ( Z , BSO w / / K ( Z , gives rise to Thom spectra MString −→ MSpin −→ MSO , with corresponding bordism groups Ω String ∗ ( X ) −→ Ω Spin ∗ ( X ) −→ Ω SO ∗ ( X ) . Remark 8.1.
Given a paracompact space X , a continuous map α : X → K ( Z , is calleda KO-twisting, and a continuous map α : X → K ( Z , is called a string twisting. For aprincipal G -bundle P over X for a compact Lie group G equipped with a map BG → K ( Z , representing a degree 4 class in H ( BG, Z ) , there is a natural string twisting X −→ BG −→ K ( Z , . Given a string twisting α : X → K ( Z , , a universal Chern-Simons 2-gerbe was con-structed in [17].For any KO-twisting α , there is a corresponding notion of an α -twisted Spin manifoldover ( X, α ) . Definition 8.2.
Let ( X, α ) be a paracompact topological space with a twisting α : X → K ( Z , . An α -twisted Spin manifold over X is a quadruple ( M, ν, ι, η ) where(1) M is a smooth, oriented and compact manifold together with a fixed classifyingmap of its stable normal bundle ν : M −→ BSO . (2) ι : M → X is a continuous map;(3) η is an α -twisted Spin structure on M , that is a homotopy commutative diagram M ι (cid:15) (cid:15) ν / / BSO η v ~ u u u u uu u u u u w (cid:15) (cid:15) X α / / K ( Z , , EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 47 where w is the classifying map of the principal K ( Z , -bundle BSpin → BSO associated to the second Stiefel-Whitney class and η is a homotopy between w ◦ ν and α ◦ ι .Two α -twisted Spin structures η and η ′ on M are called equivalent if there is a homotopybetween η and η ′ . Remark 8.3.
Given a smooth, oriented and compact n-dimensional manifold M and aparacompact space X with a KO-twisting α : X → K ( Z , . (1) M admits an α -twisted Spin structure if and only if there exists a continuous map ι : M → X such that ι ∗ ([ α ]) + w ( M ) = 0 (8.2) in H ( M, Z ) , here w ( M ) is the second Stiefel-Whitney class of T M . (The con-dition (8.2) is the anomaly cancellation condition for Type I D-branes (Cf. [45]).)(2) If ι ∗ ([ α ]) + w ( M ) = 0 , then the set of equivalence classes of α -twisted Spin structures on M are in one-to-one correspondence with elements in H ( M, Z ) .Let H R be an infinite dimensional, real and separable Hilbert space. The projectiveorthogonal group P O ( H R ) with the norm topology (Cf. [33]) has the homotopy type ofan Eilenberg-MacLane space K ( Z , . The classifying space of P O ( H R ) , as a classifyingspace of principal P O ( H R ) -bundle, is a K ( Z , . Thus, the set of isomorphism classes oflocally trivial principal P O ( H R ) -bundles over X is canonically identified with [ X, K ( Z , ∼ = H ( X, Z ) . Given a KO-twisting α : X → K ( Z , , there is a canonical principal K ( Z , -bundleover X , or equivalently, a locally trivial principal P O ( H R ) -bundle P α over X . Let K R bethe C ∗ -algebra of real compact operators on H R . Denote by C c ( X, P α ( K R )) the C ∗ -algebraof compactly supported sections of the associated bundle P α ( K R ) := P α × P O ( H R ) K R . In [39] (see also [36]), twisted KO-theory is defined for X with a KO-twisting α : X → K ( Z , to be KO i ( X, α ) := KO i (cid:0) C c ( X, P α ( K R )) (cid:1) , Let K R be the 0-th space of the KO-theory spectrum, then there is a base-point preservingaction of K ( Z , on the real K-theory spectrum K ( Z , × K O −→ K O which is represented by the action of real line bundles on ordinary KO-groups. This actiondefines an associated bundle of KO-theory spectrum over X . Denote P α ( K R ) = P α × K ( Z , K R the bundle of based spectra over X with fiber the KO-theory spectra, and { Ω nX P α ( K R ) = P α × K ( Z , Ω n K R } the fiber-wise iterated loop spaces. Then we have an equivalent defi-nition of twisted KO-groups of ( X, α ) (Cf. [39]) KO n ( X, α ) = π (cid:0) C c ( X, Ω nX P α ( K R )) (cid:1) the set of homotopy classes of compactly supported sections of the bundle of K-spectra. Dueto Bott periodicity, we only have eight different twisted K-groups KO i ( X, α ) ( i = 0 , · · · ).Twisted KO-theory is a 8-periodic generalized cohomology theory.One would expect that results in this paper can be extended to twisted KO-theory. Muchof the constructions and arguments in this paper go through in the case of twisted KO-theory. The subtlety is to study the twisted KR -theory for the (co)-tangent bundle with thecanonical involution. This may requires additional arguments.Another interesting generalization is the notion of twisted string structure for a paracom-pact topological space X with a string twisting given by α : X −→ K ( Z , . Definition 8.4.
Let ( X, α ) be a paracompact topological space with a string twisting α : X → K ( Z , . An α -twisted string manifold over X is a quadruple ( M, ν, ι, η ) where(1) M is a smooth compact manifold with a stable spin structure on its normal bundlegiven by ν : M −→ BSpin here
BSpin = lim −→ k BSpin ( k ) is the classifying space of the stable spin structure;(2) ι : M → X is a continuous map;(3) η is an α -twisted string structure on M , that is a homotopy commutative diagram M ι (cid:15) (cid:15) ν / / BSpin η w (cid:127) w w w w ww w w w w p (cid:15) (cid:15) X α / / K ( Z , , where p : BSpin → K ( Z , is the classifying map of the principal K ( Z , -bundle BString → BSpin , representing the generator of H ( BSpin , Z ) , and η is a homotopy between p ◦ ν and α ◦ ι .Two α -twisted String structures η and η ′ on M are called equivalent if there is a homotopybetween η and η ′ . Remark 8.5.
Given a smooth compact spin manifold M and a paracompact space X witha string twisting α : X → K ( Z , . (1) M admits an α -twisted string structure if and only if there is a continuous map ι : M → X such that ι ∗ ([ α ]) + p ( M )2 = 0 (8.3) EOMETRIC CYCLES, INDEX THEORY AND TWISTED K-HOMOLOGY 49 in H ( M, Z ) , here p ( X ) is the first Pontrjagin class of T M .(2) If ι ∗ ([ α ]) + p ( M )2 = 0 , then the set of equivalence classes of α -twisted stringstructures on M are in one-to-one correspondence with elements in H ( M, Z ) .Given a manifold X with a twisting α : X → K ( Z , , one can form a bordism category,called the α -twisted string bordism over ( X, α ) , whose objects are compact smooth spinmanifolds over X with an α -twisted string structure. The corresponding bordism group Ω String ∗ ( X, α ) is called the α -twisted string bordism group of X . We will study these α -twisted string bordism groups and their applications elsewhere. Acknowledgments
The author likes to thank Paul Baum, Alan Carey, Matilde Marcolli, Jouko Mickelsson,Michael Murray, Thomas Schick and Adam Rennie for useful conversations and their com-ments on the manuscript. The author thanks Alan Carey for his continuous support andencouragement. The author also thanks the referee for the suggestions to improve the man-uscript. The work is supported in part by Carey-Marcolli-Murray’s ARC Discovery ProjectDP0769986. R
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