Factorization of Dirac operators on almost-regular fibrations of spin c manifolds
FFACTORIZATION OF DIRAC OPERATORS ONALMOST-REGULAR FIBRATIONS OF SPIN c MANIFOLDS
JENS KAAD AND WALTER D. VAN SUIJLEKOM
Abstract.
We establish the factorization of the Dirac operatoron an almost-regular fibration of spin c manifolds in unboundedKK-theory. As a first intermediate result we establish that anyvertically elliptic and symmetric first-order differential operatoron a proper submersion defines an unbounded Kasparov module,and thus represents a class in KK-theory. Then, we generalizeour previous results on factorizations of Dirac operators to properRiemannian submersions of spin c manifolds. This allows us to showthat the Dirac operator on the total space of an almost-regularfibration can be written as the tensor sum of a vertically ellipticfamily of Dirac operators with the horizontal Dirac operator, up toan explicit ‘obstructing’ curvature term. We conclude by showingthat the tensor sum factorization represents the interior Kasparovproduct in bivariant K-theory. Introduction
In this paper we study factorizations of Dirac operators on singularfibrations of spin c manifolds into vertical and horizontal components.We restrict ourselves to almost-regular fibrations of spin c manifolds;these are defined to be Riemannian spin c manifolds M for which aproper Riemannian submersion to a spin c manifold B is defined on the(open, dense) complement M of a finite collection of compact embed-ded submanifolds which all have codimension strictly greater than 1.We will show that the Dirac operator D M on the total space can befactorized as a tensor sum of a (vertically elliptic) family of Dirac op-erators D V with the horizontal Dirac operator D B , up to an explicitcurvature term Ω. Thus, we obtain that (up to unitary equivalenceand on a core of D M ) we can write D M = D V ⊗ ⊗ ∇ D B + Ω , where ∇ is a suitable metric connection on vertical spinors. Date : November 7, 2018.2010
Mathematics Subject Classification.
Key words and phrases.
Unbounded Kasparov modules, Half-closed chains,Dirac operators, Spin c -manifolds, Proper Riemannian submersions, KK-theory, Un-bounded KK-theory, Kasparov product, Unbounded Kasparov product. a r X i v : . [ m a t h . F A ] O c t JENS KAAD AND WALTER D. VAN SUIJLEKOM
We show that our factorization results fit well into the context ofnoncommutative geometry [10] by proving (Theorem 28) that the ten-sor sum represents the interior Kasparov product in bivariant K-theory(or KK-theory) [37]. That is to say, we show that D V defines an un-bounded Kasparov module (aka unbounded KK-cycle [3]) that repre-sents a class in KK ( C ( M ) , C ( B )) whereas D B defines a half-closedchain (in the sense of [25]), representing the fundamental class of B in KK ( C ( B ) , C ). Similarly, since we do not assume that the Riemannianmanifold M is complete, the Dirac operator D M defines a half-closedchain (and not necessarily a spectral triple) representing the funda-mental class of M in KK ( C ( M ) , C ). The tensor sum factorization isthen an unbounded representative of the interior Kasparov product (cid:98) ⊗ C ( B ) : KK ( C ( M ) , C ( B )) × KK ( C ( B ) , C ) → KK ( C ( M ) , C )in the sense that ı ∗ [ D M ] = [ D V ] (cid:98) ⊗ C ( B ) [ D B ] , where ı ∗ is the pullback homomorphism in bivariant K-theory of the ∗ -homomorphism ı : C ( M ) → C ( M ) given by extension by zero. Westress that the curvature term Ω is not visible at the level of boundedKK-theory. Indeed, it is the great advantage of working at the levelof unbounded KK-cycles and half-closed chains that geometric infor-mation remains intact. In the course of the proof of the above result,we will establish (Proposition 30) that ı ∗ [ D M ] = [ D M ], in terms of theDirac operator D M on M . Then, the above factorization result followsfrom the result (Theorem 22) that the Dirac operator D M representsthe same class in KK ( C ( M ) , C ) as the above tensor sum does. Thisis a generalization of our previous result [35, Theorem 24] to properRiemannian submersions (not necessarily between compact manifolds),for which we make heavy use of a generalization to half-closed chains ofa theorem by Kucerovsky [38, Theorem 13], which we proved recentlyin [34].Let us spend a few words on the main motivation for this paper. Thiscomes from a class of almost-regular fibrations derived from actions oftori on Riemannian manifolds (see Examples 25 and 27 below for moredetails). Consider a torus G acting on a Riemannian spin c manifold N (thus respecting the metric and spin c structure on M ). Let N ⊆ N denote the principal stratum for the action and suppose that all otherorbits are singular ( i.e. there are no exceptional orbits). Then ourmain factorization result (Theorem 28) implies that the Dirac operator D N on N can be written as D N = D V ⊗ ⊗ ∇ D N /G + Ω , in terms of a vertically elliptic family of Dirac operators D V , the Diracoperator D N /G on the principal orbit space, and the curvature Ω of ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 3 the proper Riemannian submersion N → N /G . Moreover, this ten-sor sum factorization is an unbounded representative of the interiorKasparov product: ı ∗ [ D N ] = [ D V ] ⊗ C ( N /G ) [ D N /G ] , where ı ∗ is the pullback homomorphism in KK-theory of the embeddingmap ı : N (cid:44) → N .In particular, we are motivated by the implications the above factor-ization results will have for Dirac operators on toric noncommutativemanifolds [12, 11, 47].Interestingly, this very same class of examples also arises in the studyof holonomy groupoids for singular foliations. Indeed, the orbits of anisometric Lie group action constitute an example of singular Riemann-ian foliations [45, 1] and, more generally, orbits of any Lie group actionare an example of the almost-regular foliations that are analyzed usingholonomy groupoids in [2, 15]. We expect that —at least in the con-text of these examples— our factorization results of Dirac operators inunbounded KK-theory and the appearance of curvature can be appliedto index theory on singular foliations as studied in [9, 13, 26, 17, 18].Another potential application of our results is to equivariant indextheory. For instance, in [27] a decomposition appears of a G -equivariantDirac operator D M on a spin c manifold M carrying a proper G -action.More precisely, after identifying M ∼ = G × K N as a bundle over G/K for a suitable subgroup K ⊆ G and submanifold N ⊆ M , they write D M = D G/K + D N with D G/K a so-called G -differential operator and D N differentiating only in the vertical direction. It is an interestingquestion to see whether this formula can be cast as a tensor sum andrepresents the interior Kasparov product of the corresponding classesin bivariant K-theory.On a more general level our paper could indicate how to proceed inthe search for an unbounded, geometric version of bivariant K-theory.This is intimately related to the construction of a category of spec-tral triples, initiated in [41] and ongoing in [32, 19, 22, 30, 43]. It isclear that having examples (or, even better, classes of examples) ofunbounded representatives of the interior Kasparov product is of vitalimportance for finding the correct geometrical enrichment of the KK-category [14, 23, 42]. A list of such examples includes [6, 7, 8, 21, 33, 35],and to which we now add the present general construction. Acknowledgements.
We would like to thank Peter Hochs for a veryuseful suggestion on how to prove the compactness of the resolvent inthe context of fiber bundles.We gratefully acknowledge the Syddansk Universitet Odense and theRadboud University Nijmegen for their financial support in facilitatingthis collaboration.
JENS KAAD AND WALTER D. VAN SUIJLEKOM
During the initial stages of this research project the first author wassupported by the Radboud excellence fellowship.The first author was partially supported by the DFF-Research Project2 “Automorphisms and Invariants of Operator Algebras”, no. 7014-00145B and by the Villum Foundation (grant 7423).The second author was partially supported by NWO under VIDI-grant 016.133.326.2.
Unbounded Kasparov modules and fiber bundles
Let π : M → B be a smooth fiber bundle with a compact modelfiber F . We do not put any compactness restrictions on the mani-folds M and B (but they are not allowed to have a boundary). ByEhresmann’s fibration Theorem ( cf. [44, Lemma 17.2] for a proof) theabove is equivalent to demanding that π : M → B is a proper andsurjective submersion. We will assume that M comes equipped witha Riemannian metric in the fiber direction. In particular, we have afixed hermitian form (cid:104)· , ·(cid:105) V : X V ( M ) × X V ( M ) → C ∞ ( M )on the C ∞ ( M )-module of (complex) vertical vector fields X V ( M ) ⊆ X ( M ). We will moreover assume that we have a smooth hermitiancomplex vector bundle E → M . We denote the C ∞ ( M )-module ofsmooth sections of E by E := Γ ∞ ( M, E ) and the C ∞ c ( M )-module ofcompactly supported sections of E by E c := Γ ∞ c ( M, E ). We denote thehermitian form by (cid:104)· , ·(cid:105) E : E × E → C ∞ ( M ) . We remark that E can be considered as a C ∞ ( M )- C ∞ ( B )-bimodulewhere the right action is given by ( s · f )( x ) := s ( x ) · f ( π ( x )) for all s ∈ E , f ∈ C ∞ ( B ), x ∈ M . We finally fix a first-order differentialoperator D : E → E which only differentiates in the fiber direction, or in other words: D is C ∞ ( B )-linear.We let σ D : Hom C ∞ ( M ) (cid:0) X ( M ) , C ∞ ( M ) (cid:1) → End C ∞ ( M ) ( E ) denotethe principal symbol of D such that σ D ( df ) = [ D , f ] f ∈ C ∞ ( M ) , where df ∈ Hom C ∞ ( M ) (cid:0) X ( M ) , C ∞ ( M ) (cid:1) denotes the exterior deriva-tive of f ∈ C ∞ ( M ). We let d V f ∈ Hom C ∞ ( M ) (cid:0) X V ( M ) , C ∞ ( M ) (cid:1) de-note the restriction of df to the C ∞ ( M )-submodule X V ( M ) ⊆ X ( M ). Definition 1.
We say that D is vertically elliptic when it holds forall x ∈ M that the symbol σ D ( df )( x ) : E x → E x is invertible whenever f ∈ C ∞ ( M ) satisfies that ( d V f )( x ) : T V ( M ) x → C is non-trivial, where T V ( M ) → M denotes the complex vertical tangent bundle. ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 5
The fiber-wise Riemannian metric (cid:104)· , ·(cid:105) V : X V ( M ) × X V ( M ) → C ∞ ( M ) gives rise to a Riemannian metric (cid:104)· , ·(cid:105) b : X ( M b ) × X ( M b ) → C ∞ ( M b ) on each of the fibers M b := π − ( { b } ) ⊆ M , b ∈ B . Indeed,the inclusion i b : M b → M induces an isomorphism di b : X ( M b ) → X V ( M ) ⊗ C ∞ ( M ) C ∞ ( M b ) of C ∞ ( M b )-modules for all b ∈ B . In partic-ular, we have an associated measure µ b on M b for all b ∈ B . We maythus define the C ∞ c ( B )-linear map ρ : C ∞ c ( M ) → C ∞ c ( B ) ρ ( f )( b ) := (cid:90) M b f | M b dµ b b ∈ B by integration over the fiber. This operation provides us with a C ∗ -correspondence X from C ( M ) to C ( B ) defined as the completion of E c with respect to the inner product(2.1) (cid:104) s, t (cid:105) X := ρ ( (cid:104) s, t (cid:105) E ) s, t ∈ E c . Notice that the bimodule structure on X is induced by the C ∞ c ( M )- C ∞ c ( B )-bimodule structure on E c . We let L ( X ) denote the C ∗ -algebraof bounded adjointable operators on X and denote the ∗ -homomorphismproviding the left action on X by m : C ( M ) → L ( X ).We may promote our first-order differential operator D : E → E toan unbounded operator D : E c → X D ( s ) := D ( s ) . We immediately record that the commutator[ D , m ( f )] : E c → X extends to a bounded operator δ ( f ) : X → X for all f ∈ C ∞ c ( M ).We emphasize that this is true even though the first-order differentialoperator D : E → E is not assumed to have finite propagation speed. Definition 2.
We say that D : E → E is symmetric when D : E c → X is symmetric in the sense that (cid:104) D ( s ) , t (cid:105) X = (cid:104) s, D ( t ) (cid:105) X for all s, t ∈ E c . When D : E → E is symmetric we denote the closure of D : E c → X by D := D : Dom( D ) → X .Our first main result can now be formulated. The full proof willoccupy the remainder of this section. Theorem 3.
Suppose that the first-order differential operator D : E → E is vertically elliptic and symmetric. Then the triple (cid:0) C ∞ c ( M ) , X, D (cid:1) is an odd unbounded Kasparov module from C ( M ) to C ( B ) . More-over, (cid:0) C ∞ c ( M ) , X, D (cid:1) is even when E comes equipped with a Z / Z -grading operator γ ∈ End C ∞ ( M ) ( E ) which anti-commutes with D (inthis case γ induces the grading operator on X ).Proof. We already know that m ( f ) : X → X preserves the domain of D and that the commutator [ D, m ( f )] : Dom( D ) → X extends to abounded operator on X whenever f ∈ C ∞ c ( M ). JENS KAAD AND WALTER D. VAN SUIJLEKOM
We need to show that D : Dom( D ) → X is selfadjoint and regularand that m ( f ) · ( i + D ) − : X → X is a compact operator (in thesense of Hilbert C ∗ -modules) for all f ∈ C ∞ c ( M ). The selfadjointnessand regularity is proved in Proposition 5 and the compactness resultfollows from Proposition 7 and Proposition 11. (cid:3) Symmetry and regularity.
For each b ∈ B we define the local-ization of X at the point b ∈ B as the interior tensor product X b := X (cid:98) ⊗ C ( B ) C , where the left action of C ( B ) on C is defined via the ∗ -homomorphismev b : C ( B ) → C given by evaluation at the point b ∈ B . We remarkthat X b is a Hilbert space.For each b ∈ B , we denote the smooth sections of the smooth her-mitian complex vector bundle E | M b → M b by E b := Γ ∞ ( M b , E | M b ) ∼ = E ⊗ C ∞ ( M ) C ∞ ( M b ) . We use the notation (cid:104)· , ·(cid:105) E b : E b × E b → C ∞ ( M b )for the hermitian form (inherited from the hermitian form on E ) andwe let L ( E b ) denote the Hilbert space obtained as the completion of E b with respect to the inner product (cid:104) s, t (cid:105) := (cid:90) M b (cid:104) s, t (cid:105) E b dµ b . We remark that the map E c → E b defined by s (cid:55)→ s | M b , or alter-natively by s (cid:55)→ s ⊗
1, is a surjection. The next lemma can now beverified by computing the inner products involved:
Lemma 4.
Let b ∈ B . The linear map E c ⊗ C ∞ c ( B ) C → E b s ⊗ λ (cid:55)→ s | M b · λ induces a unitary isomorphism of Hilbert spaces: X b ∼ = L ( E b ) . We recall that the model fiber F of our smooth fiber bundle π : M → B is assumed to be compact. This plays an important role inthe following proposition. Proposition 5.
Suppose that D : E c → X is symmetric. Then theclosure D := D : Dom( D ) → X is selfadjoint and regular.Proof. By [46, Theorem 1.18] and [31, Theorem 5.8] it suffices to verifythat the induced unbounded operator (the localization) D (cid:98) ⊗ D (cid:98) ⊗ → X (cid:98) ⊗ C ( B ) C ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 7 is selfadjoint for all points b ∈ B (recall that any pure state on C ( B )is of the form ev b : C ( B ) → C for some b ∈ B ). Moreover, we recallthat the image of the map E c → X b s (cid:55)→ s (cid:98) ⊗ D (cid:98) ⊗ b ∈ B . We denote this core by E c ⊗ C ∞ c ( B ) C ⊆ X b .Let thus b ∈ B be fixed. Under the identification X b ∼ = L ( E b ) (ofLemma 4) we have that the core E c ⊗ C ∞ c ( B ) C corresponds to E b ⊆ L ( E b ) and that D ⊗ E c ⊗ C ∞ c ( B ) C → X b corresponds to a first-order symmetric differential operator ( D ) b : E b → L ( E b ). Since M b is compact we have that the closure D b := ( D ) b is selfadjoint, see[24, Corollary 10.2.6]. Moreover, we obtain that D (cid:98) ⊗ D b under the unitary isomorphism of Lemma 4. But this means that thelocalization D (cid:98) ⊗ b ∈ B and the proposition istherefore proved. (cid:3) Restriction to an open subset.
We continue in this subsectionunder the assumption that D : E → E is symmetric. We thus knowfrom Proposition 5 that the closure of D , D := D : Dom( D ) → X is a selfadjoint and regular unbounded operator. We are now going toprovide a local criteria that will later on allow us to verify the localcompactness of the resolvent of D (under an extra ellipticity condition). Remark 6.
One might of course believe that the local compactnessof the resolvent of D : Dom( D ) → X would follow immediately fromthe compactness of the resolvents of all the localized operators D b :Dom( D b ) → L ( E b ) , b ∈ B . This kind of argumentation is howevererroneous. Indeed, consider the simple case where both the fiber bundle π : M → B and the vector bundle E → M are trivial. In this casewe may identify X with the standard module X ∼ = C ( B ) (cid:98) ⊗ L ( F ) ⊕ k and the compactness of the resolvents of all the localized operators D b amounts to the pointwise compactness of ( i + D ) − . However, thecompact operators on X are given by the operator norm continuous maps B → K ( L ( F ) ⊕ k ) which vanish at infinity, see [36, Lemma 4] .The selfadjointness and regularity of D only implies that the resolvent ( i + D ) − : B → L ( L ( F ) ⊕ k ) is continuous with respect to the ∗ -strongtopology .In general, the task of proving the local compactness of the resolventof D is also made more complicated by the fact that our vector bundle E → M need not be trivial over open subsets of the form π − ( V ) ⊆ M . For any open subset U ⊆ M we let E U := Γ ∞ ( U, E | U ) denote thesmooth sections of the restriction of E → M to U . The notation E cU := Γ ∞ c ( U, E | U ) refers to the smooth compactly supported sections.We define the C ∗ -correspondence X U from C ( U ) to C ( π ( U )) as the JENS KAAD AND WALTER D. VAN SUIJLEKOM completion of E cU with respect to (the norm coming from) the innerproduct (cid:104) s, t (cid:105) X U := (cid:104) i U ( s ) , i U ( t ) (cid:105) X ∈ C (cid:0) π ( U ) (cid:1) s, t ∈ E cU , where i U : E cU → E c denotes the inclusion given by extension byzero. We remark that the bimodule structure on X U is induced bythe C ∞ c ( U )- C ∞ c ( π ( U ))-bimodule structure on E cU . Our first-order dif-ferential operator D : E → E then restricts to a first-order differentialoperator D U : E U → E U which we may promote to a symmetric un-bounded operator ( D U ) : E cU → X U . We emphasize that the closure D U := ( D U ) : Dom( D U ) → X U need not be selfadjoint.We equip Dom( D U ) with the structure of a Hilbert C ∗ -module over C ( π ( U )) by defining the inner product (cid:104) s, t (cid:105) D U := (cid:104) s, t (cid:105) X U + (cid:104) D U ( s ) , D U ( t ) (cid:105) X U , and the right action induced by the right action of C ( π ( U )) on X U .The inclusion i D U : Dom( D U ) → X U is then a bounded operator be-tween the two Hilbert C ∗ -modules. Remark that the existence of anadjoint to this inclusion is equivalent to the regularity of D U so for themoment we only know that the inclusion is bounded.We denote the left actions by m : C ( M ) → L ( X ) and m U : C ( U ) → L ( X U ) . Our localization result for resolvents can now be stated and proved:
Proposition 7.
Suppose that, for each p ∈ M we may find an opensubset U ⊆ M with p ∈ U such that the bounded operator Dom( D U ) i DU −−−→ X U m U ( χ ) −−−−→ X U is compact for all χ ∈ C ∞ c ( U ) . Then we have that m ( f ) · ( i + D ) − : X → X is compact for all f ∈ C ∞ c ( M ) .Proof. Let us fix an f ∈ C ∞ c ( M ). Without loss of generality we mayassume that supp( f ) ⊆ U for some open subset U ⊆ M satisfying theassumptions to this proposition. Moreover, we may assume that f ≥ f has a smooth square root.The proof runs in four steps.(1) Consider the interior tensor product X U (cid:98) ⊗ C ( π ( U )) C ( B ) where C ( π ( U )) acts on C ( B ) via the ∗ -homomorphism C ( π ( U )) → C ( B ) given by extension by zero. We start by showing thatthe compositionDom( D U (cid:98) ⊗ i DU (cid:98) ⊗ −−−−→ X U (cid:98) ⊗ C ( π ( U )) C ( B ) m U ( √ f ) (cid:98) ⊗ −−−−−−→ X U (cid:98) ⊗ C ( π ( U )) C ( B ) ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 9 is a compact operator. To this end, we remark that Dom( D U (cid:98) ⊗ D U ) (cid:98) ⊗ C ( π ( U )) C ( B ) and that themap i D U (cid:98) ⊗ : Dom( D U (cid:98) ⊗ → X U (cid:98) ⊗ C ( π ( U )) C ( B ) corresponds to i D U (cid:98) ⊗ D U ) (cid:98) ⊗ C ( π ( U )) C ( B ) → X U (cid:98) ⊗ C ( π ( U )) C ( B ) underthis unitary isomorphism. It thus suffices to show that thebounded operator (cid:0) m U ( (cid:112) f ) · i D U (cid:1) (cid:98) ⊗ D U ) (cid:98) ⊗ C ( π ( U )) C ( B ) → X U (cid:98) ⊗ C ( π ( U )) C ( B )is compact. But this follows from [39, Proposition 4.7] since m U ( √ f ) · i D U : Dom( D U ) → X U is compact by assumption andsince C ( π ( U )) acts on C ( B ) by compact operators.(2) The map E cU ⊗ C ∞ c ( π ( U )) C ∞ c ( B ) → E c defined by s ⊗ g (cid:55)→ i U ( s ) · ( g ◦ π ) induces a C ( B )-linear isometry j : X U (cid:98) ⊗ C ( π ( U )) C ( B ) → X and we may thus conclude (using (1)) that the compositionDom( D U (cid:98) ⊗ ( m U ( √ f ) (cid:98) ⊗ · i DU (cid:98) ⊗ −−−−−−−−−−−→ X U (cid:98) ⊗ C ( π ( U )) C ( B ) j −−−→ X is a compact operator. Remark that we do not need j : X U (cid:98) ⊗ C ( π ( U )) C ( B ) → X to be adjointable since the compact operators are compatiblewith left multiplication by bounded operators that are linearover the base (the corresponding statement for right multiplica-tion need not be true).(3) The multiplication operator r U ( √ f ) : E c → E cU ⊗ C ∞ c ( π ( U )) C ∞ c ( B )induces a bounded adjointable operator r U ( (cid:112) f ) : Dom( D ) → Dom( D U (cid:98) ⊗ . To prove this claim, we recall that δ ( √ f ) : X → X de-notes the bounded adjointable extension of the commutator[ D , m ( √ f )] : E c → X .The fact that r U ( √ f ) is bounded then follows since (cid:107) r U ( (cid:112) f )( s ) (cid:107) Dom( D U (cid:98) ⊗ = (cid:107) ( i + D )( (cid:112) f · s ) (cid:107) X ≤ (cid:107) m ( (cid:112) f ) (cid:107) ∞ · (cid:107) s (cid:107) Dom( D ) + (cid:107) δ ( (cid:112) f ) (cid:107) ∞ · (cid:107) s (cid:107) X for all s ∈ E c .It can then be verified by a direct computation that the ad-joint of r U ( √ f ) is given by the expression: r U ( (cid:112) f ) ∗ = m ( (cid:112) f ) · j − ( i + D ) − · δ ( (cid:112) f ) · j − (1 + D ) − · δ ( (cid:112) f ) · ( i + D ) · j on the dense subspace of Dom( D U (cid:98) ⊗
1) provided by the tensorproduct E cU ⊗ C ∞ c ( π ( U )) C ∞ c ( B ).This proves the claim. (4) We end the proof of the proposition by concluding that thecomposition Dom( D ) i D −−−→ X m ( f ) −−−→ X is a compact operator. Indeed, this composition can be rewrit-ten as the compositionDom( D ) r U ( √ f ) −−−−→ Dom( D U (cid:98) ⊗ i DU (cid:98) ⊗ −−−−→ X U (cid:98) ⊗ C ( π ( U )) C ( B ) m U ( √ f ) (cid:98) ⊗ −−−−−−→ X U (cid:98) ⊗ C ( π ( U )) C ( B ) j −−−→ X but this latter composition is compact by a combination of (2)and (3). (cid:3) Compactness of local resolvents.
Throughout this subsectionwe will suppose that the first-order differential operator D : E → E isvertically elliptic and symmetric.Let us fix an open subset U ⊆ M such that the following holds: Assumption 1. (1)
There exists a diffeomorphism ψ : U → π ( U ) × B δ (0) such that p ◦ ψ = π where p : π ( U ) × B δ (0) → π ( U ) de-notes the projection onto the first component and where B δ (0) ⊆ R dim( F ) denotes the open ball with center and radius δ > . (2) There exists a unitary smooth trivialization φ : E | U → U × C k of the restriction of the vector bundle E → M to U . We define the vertical coordinates associated to ψ by y i := r i ◦ p ◦ ψ : U → R , i = 1 , . . . , dim( F ), where r i : B δ (0) → R are restrictions of thestandard coordinates on R dim( F ) and p : π ( U ) × B δ (0) → B δ (0) is theprojection onto the second component. We then have the associatedsmooth map g V : U → GL + ( R dim( F ) ) g Vij := (cid:104) ∂/∂y i , ∂/∂y j (cid:105) V . The next lemma follows by a straightforward computation of innerproducts.
Lemma 8.
The C ∞ c ( π ( U )) -module isomorphism α : E cU → C ∞ c (cid:0) π ( U ) × B δ (0) (cid:1) ⊕ k defined by α ( s ) := (cid:0) ( φ ◦ s ) · det( g V ) / (cid:1) ◦ ψ − induces a unitary isomor-phism of Hilbert C ∗ -modules over C ( π ( U )) : α : X U → C (cid:0) π ( U ) (cid:1) (cid:98) ⊗ L (cid:0) B δ (0) (cid:1) ⊕ k . We are going to study the closed and symmetric unbounded operator D α := α ◦ D U ◦ α − : Dom( D α ) = α (cid:0) Dom( D U ) (cid:1) → C (cid:0) π ( U ) (cid:1) (cid:98) ⊗ L (cid:0) B δ (0) (cid:1) ⊕ k . ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 11
We immediately remark that C ∞ c ( π ( U ) × B δ (0)) ⊕ k is a core for D α .Moreover, we can find smooth maps A j , B : π ( U ) × B δ (0) → M k ( C ), j = 1 , . . . , dim( F ), such that D α = dim( F ) (cid:88) j =1 A j · ∂∂r j + B : C ∞ c ( π ( U ) × B δ (0)) ⊕ k → C (cid:0) π ( U ) (cid:1) (cid:98) ⊗ L (cid:0) B δ (0) (cid:1) ⊕ k , where D α denotes the restriction of D α to the core C ∞ c ( π ( U ) × B δ (0)) ⊕ k .For each x ∈ π ( U ) we define the first-order differential operator D αx := dim( F ) (cid:88) j =1 A j ( x, · ) · ∂∂r j + B ( x, · ) : C ∞ ( B δ (0)) ⊕ k → C ∞ ( B δ (0)) ⊕ k . The vertical ellipticity of D : E → E then implies that D αx is ellipticfor all x ∈ π ( U ). For each x ∈ π ( U ), we let D αx : Dom( D αx ) → L ( B δ (0)) ⊕ k denote the closure of the unbounded operator ( D αx ) : C ∞ c ( B δ (0)) ⊕ k → L ( B δ (0)) ⊕ k induced by the first-order differential operator D αx .By passing to a smaller open subset U (cid:48) ⊆ U if necessary we mayassume that the following holds: Assumption 2. (1)
For each x ∈ π ( U ) , the smooth maps A j ( x, · ) , B ( x, · ) : B δ (0) → M k ( C ) , j = 1 , . . . , dim( F ) , are bounded. (2) The maps A j , B : π ( U ) → C b ( B δ (0) , M k ( C )) , j = 1 , . . . , dim( F ) ,are continuous, where C b ( B δ (0) , M k ( C )) is equipped with thesupremum norm. (3) D α satisfies the following uniform G˚arding’s inequality: Thereexists a constant C > such that (cid:107) ξ (cid:107) + (cid:107) D αx ( ξ ) (cid:107) ≥ C · (cid:0) (cid:107) ξ (cid:107) + dim( F ) (cid:88) j =1 (cid:107) ∂ξ/∂r j (cid:107) (cid:1) , for all ξ ∈ C ∞ c ( B δ (0)) ⊕ k and all x ∈ π ( U ) , where (cid:107) · (cid:107) denotesthe L -norm on L (cid:0) B δ (0) (cid:1) ⊕ k . Indeed, this is a consequence ofthe proof of the usual G˚arding’s inequality, see for example [24,Theorem 10.4.4] . Let us apply the notation ∇ : Dom( ∇ ) → L (cid:0) B δ (0) (cid:1) ⊕ dim( F ) · k for the closure of the gradient ∇ : C ∞ c ( B δ (0)) ⊕ k → L (cid:0) B δ (0) (cid:1) ⊕ dim( F ) · k ∇ ( ξ ) = ∂ξ/∂r ... ∂ξ/∂r dim( F ) . The above assumptions imply the following:
Lemma 9.
We have that
Dom( ∇ ) = Dom( D αx ) and there exist con-stants C , C > such that C · (cid:107) ξ (cid:107) ∇ ≥ (cid:107) ξ (cid:107) D αx ≥ C · (cid:107) ξ (cid:107) ∇ , for all x ∈ π ( U ) . Moreover, we have that D αx ( ξ ) = A ( x, · ) · ∇ ( ξ ) + B ( x, · )( ξ ) , for all ξ ∈ Dom( ∇ ) and all x ∈ π ( U ) , where A ( x, · ) := ( A ( x, · ) , . . . , A dim( F ) ( x, · )): L (cid:0) B δ (0) (cid:1) ⊕ dim( F ) · k → L (cid:0) B δ (0) (cid:1) ⊕ k . Define the map R : π ( U ) → L (cid:0) L ( B δ (0)) ⊕ k (cid:1) R ( x ) := (cid:0) D αx ) ∗ D αx (cid:1) − as well as its square root R / : x (cid:55)→ R ( x ) / , x ∈ π ( U ). We remark thatit follows by Lemma 9 that Im( R / ( x )) = Dom( ∇ ) for all x ∈ π ( U )and that the map ∇ · R / : π ( U ) → L (cid:0) L ( B δ (0)) ⊕ k , L ( B δ (0)) ⊕ dim( F ) · k (cid:1) ∇ · R / ( x ) := ∇ · (cid:0) D αx ) ∗ D αx (cid:1) − / is well-defined. Moreover, Lemma 9 implies thatsup x ∈ π ( U ) (cid:107)∇ · R / ( x ) (cid:107) ∞ < ∞ , where (cid:107) · (cid:107) ∞ refers to the operator norm. Lemma 10.
The maps R : π ( U ) → L (cid:0) L ( B δ (0)) ⊕ k (cid:1) and ∇ · R : π ( U ) → L (cid:0) L ( B δ (0)) ⊕ k , L ( B δ (0)) ⊕ dim( F ) · k (cid:1) are continuous in opera-tor norm.Proof. We will only prove the statement for the map ∇ · R since theargument in the case of R is similar but easier.Thus, let ξ ∈ L ( B δ (0)) ⊕ k and η ∈ L ( B δ (0)) ⊕ dim( F ) · k as well as x, y ∈ π ( U ) be given. To ease the notation, we define the boundedoperator T ( y ) := R / ( y ) (cid:0) ∇ · R / ( y ) (cid:1) ∗ : L ( B δ (0)) ⊕ dim( F ) · k → L ( B δ (0)) ⊕ k and remark that T ( y ) ∗ = ∇ · R ( y ) : L ( B δ (0)) ⊕ k → L ( B δ (0)) ⊕ dim( F ) · k . Using the resolvent identity we have that ∇ · ( R ( x ) − R ( y )) = ∇ · R / ( y )( D αy R / ( y )) ∗ D αy R ( x )+ ∇ R ( y ) R ( x ) − ∇ R ( y )= ( D αy T ( y )) ∗ D αy R ( x ) + T ( y ) ∗ R ( x ) − T ( y ) ∗ = ( D αy T ( y )) ∗ D αy R ( x ) − T ( y ) ∗ ( D αx ) ∗ D αx R ( x ) ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 13 and thus that (cid:104)∇ · ( R ( x ) − R ( y )) ξ, η (cid:105) = (cid:10) D αy R ( x ) ξ, D αy T ( y ) η (cid:11) − (cid:10) ( D αx ) ∗ D αx R ( x ) ξ, T ( y ) η (cid:11) = (cid:10)(cid:0) A ( y, · ) − A ( x, · ) (cid:1) ∇ · R ( x ) ξ, D αy T ( y ) η (cid:11) + (cid:10)(cid:0) B ( y, · ) − B ( x, · ) (cid:1) R ( x ) ξ, D αy T ( y ) η (cid:11) + (cid:10) D αx R ( x ) ξ, (cid:0) A ( y, · ) − A ( x, · ) (cid:1) ∇ · T ( y ) η (cid:11) + (cid:10) D αx R ( x ) ξ, (cid:0) B ( y, · ) − B ( x, · ) (cid:1) T ( y ) η (cid:11) . But this implies that (cid:107)∇ · ( R ( x ) − R ( y )) (cid:107) ∞ ≤ · (cid:107) A ( y, · ) − A ( x, · ) (cid:107) ∞ · K + 2 · (cid:107) B ( y, · ) − B ( x, · ) (cid:107) ∞ · K , where K := sup z ∈ π ( U ) (cid:107)∇ · R / ( z ) (cid:107) ∞ . This estimate proves the lemma. (cid:3) Proposition 11.
Suppose that D : E → E is vertically elliptic andsymmetric. For each p ∈ M , there exists an open subset U ⊆ M with p ∈ U such that (1) The closed and symmetric unbounded operator D U : Dom( D U ) → X U is regular. (2) The bounded adjointable operator m U ( χ ) · (1 + D ∗ U D U ) − : X U → X U is compact for all χ ∈ C ∞ c ( U ) .In particular, it holds that the composition Dom( D U ) i DU −−−→ X U m U ( χ ) −−−−→ X U is a compact operator for all χ ∈ C ∞ c ( U ) .Proof. We choose the open subset U ⊆ M with p ∈ U such that As-sumption 1 and Assumption 2 are satisfied.Let us first show that D α is regular. Since D U and D α are unitarilyequivalent this is equivalent to the regularity of D U . We show thatthe operator norm continuous map R : π ( U ) → L (cid:0) L ( B δ (0)) ⊕ k (cid:1) satis-fies that Im( R ) ⊆ Dom (cid:0) ( D α ) ∗ D α (cid:1) and that (cid:0) D α ) ∗ D α (cid:1) R ( ξ ) = ξ for all ξ ∈ C (cid:0) π ( U ) (cid:1) (cid:98) ⊗ L ( B δ (0)) ⊕ k . Remark here that R is identi-fied with a bounded adjointable operator on C (cid:0) π ( U ) (cid:1) (cid:98) ⊗ L ( B δ (0)) ⊕ k in the obvious way. Now, by Lemma 9 and Lemma 10 we have thatIm( R ) ⊆ Dom( D α ). For each ξ ∈ C ( π ( U )) (cid:98) ⊗ L ( B δ (0)) ⊕ k and each η ∈ Dom( D α ) we then have that (cid:104) D α R ( ξ ) , D α ( η ) (cid:105) ( x ) = (cid:104) D αx R ( x )( ξ ( x )) , D αx ( η ( x )) (cid:105) = (cid:104) (1 − R )( ξ ) , η (cid:105) ( x )for all x ∈ π ( U ). But this shows that Im( D α R ) ⊆ Dom (cid:0) ( D α ) ∗ (cid:1) andhence that Im( R ) ⊆ Dom (cid:0) ( D α ) ∗ D α (cid:1) . Moreover, we may conclude that ( D α ) ∗ D α R = 1 − R . We have thus proved that D α is regular withresolvent (1 + ( D α ) ∗ D α ) − = R .To show that m U ( χ ) · (1 + D ∗ U D U ) − : X U → X U is a compactoperator for all χ ∈ C ∞ c ( U ) it now suffices to show that m ( f ) · R ( x ) : L (cid:0) B δ (0) (cid:1) ⊕ k → L (cid:0) B δ (0) (cid:1) ⊕ k is a compact operator for all x ∈ π ( U )and all f ∈ C ∞ c ( B δ (0)). But this follows from Rellich’s Lemma since D αx : C ∞ (cid:0) B δ (0) (cid:1) ⊕ k → C ∞ (cid:0) B δ (0) (cid:1) ⊕ k is an elliptic first-order differentialoperator, see [24, Lemma 10.4.3]. (cid:3) Factorization in unbounded KK-theory
We remain in the setting described in Section 2, thus we consider asmooth fiber bundle π : M → B with a compact model fiber F . We willassume that both M and B are Riemannian and that the submersion π : M → B is Riemannian thus that the derivative dπ ( x ) : ( T V M ) ⊥ x → ( T B ) π ( x ) is an isometry for all x ∈ M ( T V M → M denotes the smoothhermitian vector bundle of vertical tangent vectors). We will finallyassume that M and B are both spin c manifolds .The second main result of this paper is the factorization in un-bounded KK-theory of the Dirac operator D M on the total manifold M in terms of a vertical operator D V and the Dirac operator D B onthe base manifold B . This factorization result holds up to an explicitcurvature term, which is invisible at the level of bounded KK-theory.We work in the case where M and B are both even dimensional, butnotice that our results can be readily translated to the remaining threecases (counting parity of dimensions).The Riemannian metrics on M and B will be denoted by (cid:104)· , ·(cid:105) M : X ( M ) × X ( M ) → C ∞ ( M ) and (cid:104)· , ·(cid:105) B : X ( B ) × X ( B ) → C ∞ ( B ),respectively.3.1. The vertical unbounded Kasparov module.
In order to linkthe Dirac operators D B and D M via a tensor-sum factorization, westart by constructing a vertical Dirac operator D V : E V → E V . Thisvertical Dirac operator will be an odd symmetric, vertically elliptic,first-order differential operator acting on the smooth sections E V of a Z / Z -graded smooth hermitian vector bundle E V → M (see Definition2 and Definition 1). In particular, using Theorem 3, we will obtain aneven unbounded Kasparov module from C ( M ) to C ( B ).We follow the approach and notation of [35], where we studied —actually, as a preparation for the present paper— Riemannian submer-sions of compact Riemannian spin c manifolds.Working with the unital function algebras C ∞ ( M ) and C ∞ ( B ), weconsider the smooth sections E M := Γ ∞ ( M, E M ) and E B := Γ ∞ ( B, E B ) ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 15 of the Z / Z -graded spinor bundles E M → M and E B → B , respec-tively. The spin c -structures provide us with even isomorphisms c M : Cl( M ) → End C ∞ ( M ) ( E M ) , c B : Cl( B ) → End C ∞ ( B ) ( E B ) . We fix even hermitian Clifford connections ∇ E M : Γ ∞ ( M, E M ) → Γ ∞ ( M, E M ⊗ T ∗ M ) and ∇ E B : Γ ∞ ( B, E B ) → Γ ∞ ( B, E B ⊗ T ∗ B ).We letCl V ( M ) := Γ ∞ (cid:0) M, Cl( T V M ) (cid:1) , Cl H ( M ) := Γ ∞ (cid:0) M, Cl( T H M ) (cid:1) denote the Clifford algebras of vertical and horizontal vector fields, re-spectively. Remark that the horizontal vector fields X H ( M ) are definedas the smooth sections of the smooth vector bundle T H M → M withfiber ( T V M ) ⊥ x ⊆ ( T M ) x at each x ∈ M .Then, as in [35], we arrive at • a Z / Z -graded horizontal spinor bundle E H := π ∗ E B , togetherwith a hermitian Clifford connection ∇ E H for Clifford multipli-cation c H by horizontal vector fields on M . • a Z / Z -graded vertical spinor bundle E V := E ∗ H ⊗ Cl( T H M ) E M ,together with a hermitian Clifford connection ∇ E V for Cliffordmultiplication c V by vertical vector fields on M .The explicit formulae for these operations can be found in [35, Section3]. We let E H := Γ ∞ ( M, E H ) and E V := Γ ∞ ( M, E V ) denote the smoothsections of the horizontal and vertical spinor bundle, respectively.The vertical Dirac operator D V : E V → E V is defined by the localexpression D V ( ξ ) = i dim( F ) (cid:88) j =1 c V ( e j ) ∇ E V e j ( ξ ) ξ ∈ E V , where { e j } is a local orthonormal frame of real vertical vector fields.Clearly, D V is an odd first-order differential operator, which only dif-ferentiates in the fiber direction.As in Section 2.1 we let X denote the Z / Z -graded C ∗ -correspondencefrom C ( M ) to C ( B ) obtained as the C ∗ -completion of the compactlysupported sections in E V (the inner product is defined in Equation (2.1)and the grading is induced by the grading on E V ). We promote D V toan odd unbounded operator( D V ) : E cV → X .
As in [35, Lemma 15] we obtain the following symmetry result:
Lemma 12.
The odd unbounded operator ( D V ) : E cV → X is symmet-ric. The closure of ( D V ) will be denoted by D V : Dom( D V ) → X . Proposition 13.
The triple ( C ∞ c ( M ) , X, D V ) is an even unboundedKasparov module from C ( M ) to C ( B ) . Proof.
Let f ∈ C ∞ ( M ) and let x ∈ M . By Lemma 12 and Theorem 3it is enough to verify that the principal symbol σ D V ( df )( x ) : ( E V ) x → ( E V ) x is invertible whenever ( d V f )( x ) : ( T V M ) x → C is non-trivial.We compute this principal symbol to be given by the local expression σ D V ( df ) = [ D V , f ] = i dim( F ) (cid:88) j =1 c V ( e j ) e j ( f ) . Hence σ D V ( df )( x ) = c V (( d V f ) (cid:93) )( x ), where (cid:93) : Hom C ∞ ( M ) ( X V ( M ) , C ∞ ( M )) → X V ( M ) (cid:93) : (cid:104) e j | (cid:55)→ e j denotes the musical isomorphism. This proves the proposition. (cid:3) We apply the notation L ( E cM ) and L ( E cB ) for the Z / Z -gradedHilbert space completions of the smooth compactly supported sectionsof the spinor bundles E M and E B , respectively. The inner productscome from the Riemannian metrics and the hermitian forms in theusual way.For later use, we record the following result. The proof is the sameas the proof of [35, Proposition 14]. Notice, when reading the state-ment, that we are tacitly applying the identifications End C ∞ ( M ) ( E H ) ∼ =Cl H ( M ) ⊆ Cl( M ) ∼ = End C ∞ ( M ) ( E M ). Proposition 14.
The even left C ∞ c ( M ) -module isomorphism W : E cV ⊗ C ∞ c ( B ) E cB → E cM W : ( (cid:104) ξ | ⊗ s ) ⊗ r (cid:55)→ (cid:0) | r ◦ π (cid:105)(cid:104) ξ | (cid:1) ( s ) defined for ξ ∈ E cH , s ∈ E cM and r ∈ E cB , induces an even unitaryisomorphism W : X (cid:98) ⊗ C ( B ) L ( E cB ) → L ( E cM ) of Z / Z -graded Hilbert spaces. Lift of the Dirac operator on the base.
Our next ingredientis the Dirac operator on the base defined by the local expression, D B = i dim( B ) (cid:88) α =1 c B ( f α ) ∇ E B f α : E B → E B , for any local orthonormal frame { f α } for X ( B ) consisting of real vectorfields. This Dirac operator is an odd, symmetric and elliptic, first-orderdifferential operator. We are interested in the associated symmetricunbounded operator ( D B ) : E cB → L ( E cB )and we denote its closure by D B : Dom( D B ) → L ( E B ). Since theRiemannian manifold B is not assumed to be complete, it can happenthat D B is not selfadjoint and the triple ( C ∞ c ( B ) , L ( E cB ) , D B ) is there-fore in general not a spectral triple over C ( B ). Instead, as in [4] and ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 17 [25], the above triple forms an even half-closed chain , representing thefundamental class [ B ] ∈ KK ( C ( B ) , C ).In order to form the unbounded Kasparov product of the vertical andthe horizontal components we need to lift the Dirac operator D B to asymmetric unbounded operator on the Hilbert space X (cid:98) ⊗ C ( B ) L ( E cB ).It turns out that the hermitian Clifford connection ∇ E V on E V does notdefine a metric connection on E cV ⊆ X , due to correction terms thatcome from the measure on the fibers M b , b ∈ B .To obtain a metric connection for the C ∞ c ( B )-valued inner prod-uct (cid:104)· , ·(cid:105) X on E cV , we recall that the second fundamental form S ∈ Γ ∞ ( M, T ∗ V M ⊗ T ∗ V M ⊗ T ∗ H M ) can be defined by(3.1) S ( X, Y, Z ) := 12 (cid:0) Z ( (cid:104) X, Y (cid:105) M ) − (cid:104) [ Z, X ] , Y (cid:105) M − (cid:104) [ Z, Y ] , X (cid:105) M (cid:1) for real vertical vector fields X, Y and real horizontal vector fields Z on M . Moreover, the mean curvature k ∈ Hom C ∞ ( M ) (cid:0) X H ( M ) , C ∞ ( M ) (cid:1) is given as the trace k = (Tr ⊗ S ) . As in [35, Definition 18], we define a metric connection on E cV ⊆ X by ∇ XZ ( ξ ) = ∇ E V Z H ( ξ ) + 12 k ( Z H ) · ξ ∈ E cV ⊆ X for any real vector field Z on B , with horizontal lift Z H ∈ X H ( M ) ∼ =Γ ∞ ( M, π ∗ T B ), and any ξ ∈ E cV ⊆ X . Lemma 15.
The local expression (1 ⊗ ∇ D B ) ( ξ ⊗ r ):= ξ ⊗ D B ( r ) + i (cid:88) α ∇ Xf α ( ξ ) ⊗ c B ( f α )( r ) ξ ∈ E cV , r ∈ E cB defines a symmetric unbounded operator (1 ⊗ ∇ D B ) : E cV ⊗ C ∞ c ( B ) E cB → X (cid:98) ⊗ C ( B ) L ( E cB ) . We denote the closure of (1 ⊗ ∇ D B ) by1 ⊗ ∇ D B : Dom(1 ⊗ ∇ D B ) → X (cid:98) ⊗ C ( B ) L ( E cB ) . The selfadjoint and regular unbounded operator D V : Dom( D V ) → X induces a selfadjoint unbounded operator( D V ⊗ : Dom( D V ) ⊗ C ( B ) L ( E cB ) → X (cid:98) ⊗ C ( B ) L ( E cB )and we let D V ⊗ D V ⊗ → X (cid:98) ⊗ C ( B ) L ( E cB ) denote its closure.We now compute the commutator of D V ⊗ ⊗ ∇ D B (whenrestricted to E cV ⊗ C ∞ c ( B ) E cB ⊆ X (cid:98) ⊗ C ( B ) L ( E cB )). This computation willbe crucial in the proof of our main Theorem 22 below. Lemma 16.
Suppose that ξ ∈ E cV ⊆ X and r ∈ E cB ⊆ L ( E cB ) . Thenwe have the local expression [ D V ⊗ , ⊗ ∇ D B ]( ξ ⊗ r )= − (cid:88) j,k,α S ( e k , e j , ( f α ) H ) (cid:0) c V ( e j ) ∇ E V e k (cid:1) ( ξ ) ⊗ c B ( f α )( r ) − (cid:88) j,α c V ( e j ) (cid:16) Ω E V ( e j , ( f α ) H ) + 12 e j (cid:0) k (( f α ) H ) (cid:1)(cid:17) ( ξ ) ⊗ c B ( f α )( r ) , where Ω E V : Γ ∞ ( M, E V ) → Γ ∞ ( M, E V ⊗ T ∗ M ∧ T ∗ M ) is the curvatureform of the hermitian connection ∇ E V .Proof. We let ∇ V = ( P ⊗ ∇ M P : Γ ∞ ( M, T V M ) → Γ ∞ ( M, T V M ⊗ T ∗ M ) denote the compression of the Levi–Civita connection on M tovertical vector fields (thus, P : X ( M ) → X ( M ) denotes the orthogo-nal projection with image X V ( M ) ⊆ X ( M )).We insert the definition of D V and 1 ⊗ ∇ D B in the above expressionand compute[ D V ⊗ , ⊗ ∇ D B ] = − (cid:88) j,α (cid:104) c V ( e j ) ∇ E V e j , ∇ Xf α (cid:105) ⊗ c B ( f α )= (cid:88) j,α (cid:16) c V ( ∇ V ( f α ) H ( e j )) ∇ E V e j − c V ( e j ) (cid:104) ∇ E V e j , ∇ E V ( f α ) H (cid:105) − c V ( e j ) e j (cid:0) k (( f α ) H ) (cid:1)(cid:17) ⊗ c B ( f α ) . We consider the second term after the last equality sign, for which (cid:104) ∇ E V e j , ∇ E V ( f α ) H (cid:105) = ∇ E V [ e j , ( f α ) H ] + Ω E V ( e j , ( f α ) H )in terms of the curvature form Ω E V of the connection ∇ E V .For each α ∈ { , . . . , dim( B ) } , we proceed by computing the first-order differential operator (cid:88) j (cid:0) c V ( ∇ V ( f α ) H ( e j )) ∇ E V e j + c V ( e j ) ∇ E V [( f α ) H ,e j ] (cid:1) = (cid:88) j,k c V ( e j ) (cid:0)(cid:10) ∇ M ( f α ) H ( e k ) , e j (cid:11) M + (cid:10) [( f α ) H , e j ] , e k (cid:11) M (cid:1) ∇ E V e k = (cid:88) j,k c V ( e j ) (cid:18) (cid:10) [( f α ) H , e k ] , e j (cid:11) M + 12 (cid:10) [( f α ) H , e j ] , e k (cid:11) M (cid:19) ∇ E V e k , where we used Koszul’s formula for the Levi–Civita connection ∇ M on M , together with the fact that the Lie-bracket [( f α ) H , e j ] is a vertical vector field for all α, j , see [35, Lemma 1]. When expressed in terms ofthe second fundamental form of Equation (3.1) this leads to the desiredformula. (cid:3) ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 19
As a consequence of the above lemma we obtain that the commutator[ D V ⊗ , ⊗ ∇ D B ] is relatively bounded by D V ⊗ on compact subsetsof M . Lemma 17.
Suppose that K ⊆ M is a compact subset. Then thereexists a constant C > such that (cid:107) [ D V ⊗ , ⊗ ∇ D B ]( η ) (cid:107) ≤ C · (cid:0) (cid:107) η (cid:107) + (cid:107) ( D V ⊗ η ) (cid:107) (cid:1) , for all η ∈ E cV ⊗ C ∞ c ( B ) E cB with support contained in K .Proof. Throughout this proof, we will suppress the left C ∞ ( M )-moduleisomorphism W : E cV ⊗ C ∞ c ( B ) E cB → E cM from Proposition 14.Without loss of generality, suppose that K ⊆ U , where U ⊆ M is anopen subset supporting an orthonormal frame { e j } for ( T V M ) | U andwhere π ( U ) ⊆ B supports an orthonormal frame { f α } for ( T B ) | π ( U ) .Recall from Proposition 13 that D V : E V → E V is a vertically ellipticfirst-order differential operator. In particular, we have that D V ⊗ E M . It therefore follows from G˚arding’s inequality that there exists aconstant C > dim( F ) (cid:88) j =1 (cid:107) ( ∇ E V e j ⊗ η ) (cid:107) ≤ C · (cid:0) (cid:107) η (cid:107) + (cid:107) ( D V ⊗ η ) (cid:107) (cid:1) , for all η ∈ E cV ⊗ C ∞ c ( B ) E cB with support contained in K ⊆ U .By Lemma 16 there exist A , . . . , A dim( F ) , B ∈ Γ ∞ (cid:0) U, End( E M ) | U (cid:1) such that[ D V ⊗ , ⊗ ∇ D B ]( η ) = dim( F ) (cid:88) j =1 A j ( ∇ E V e j ⊗ η ) + B ( η ) , for all η ∈ E cV ⊗ C ∞ c ( B ) E cB with support contained in K ⊆ U .This proves the present lemma. (cid:3) Factorization of the Dirac operator.
The tensor sum we areafter is given by the symmetric unbounded operator( D V × ∇ D B ) := ( D V ⊗ + ( γ X ⊗ ⊗ ∇ D B ) : Dom( D V × ∇ D B ) → X (cid:98) ⊗ C ( B ) L ( E cB ) , where the domain is the image of E cV ⊗ C ∞ c ( B ) E cB in X (cid:98) ⊗ C ( B ) L ( E cB ) andwhere γ X : X → X denotes the Z / Z -grading operator on X . Theclosure of the symmetric unbounded operator ( D V × ∇ D B ) will bedenoted by D V × ∇ D B .We are going to compare this tensor sum with the Dirac operatoron the spin c manifold M . As mentioned earlier, these two unboundedoperators agree up to an explicit error term given by the curvature form of the proper Riemannian submersion π : M → B . We recall thatthis curvature form Ω ∈ Γ ∞ ( M, T ∗ H M ∧ T ∗ H M ⊗ T ∗ V M ) is defined byΩ( X, Y, Z ) := (cid:104) [ X, Y ] , Z (cid:105) M for any real horizontal vector fields X, Y and any real vertical vec-tor field Z . We represent this curvature form as an endomorphism of E M via the Clifford multiplication c M : X ( M ) → End C ∞ ( M ) ( E M ) asfollows: c : Γ ∞ ( M, T ∗ H M ∧ T ∗ H M ⊗ T ∗ V M ) → End C ∞ ( M ) ( E M ) c ( ω ∧ ω ⊗ ω ) := (cid:2) c M ( ω (cid:93) ) , c M ( ω (cid:93) ) (cid:3) · c M ( ω (cid:93) ) . Notice that the sharps refer to the musical isomorphisms (cid:93) : Ω H ( M ) → X H ( M ) and (cid:93) : Ω V ( M ) → X V ( M ). We emphasize that the corre-sponding operator c (Ω) : E cM → L ( E cM ) can be unbounded .We recall that the Dirac operator on M is defined by the local ex-pression D M = dim( M ) (cid:88) k =1 c M (cid:0) ( dx k ) (cid:93) (cid:1) ∇ E M ∂/∂x k : E M → E M . As usual, we let D M : Dom( D M ) → L ( E cM ) denote the closure of thesymmetric unbounded operator ( D M ) : E cM → L ( E cM ) induced by D M .Let γ B : L ( E cB ) → L ( E cB ) denote the grading operator on L ( E cB ).The grading operator on X (cid:98) ⊗ C ( B ) L ( E cB ) is then given by γ := γ X ⊗ γ B .We define the even selfadjoint unitary isomorphismΓ := ( γ X ⊗
1) 1 + γ − γ X (cid:98) ⊗ C ( B ) L ( E cB ) → X (cid:98) ⊗ C ( B ) L ( E cB ) . Proposition 18.
Under the even unitary isomorphism given by W Γ : X (cid:98) ⊗ C ( B ) L ( E cB ) → L ( E cM ) we have the identity (3.2) W Γ( D V × ∇ D B )Γ W ∗ = ( D M ) − i c (Ω) . Proof.
We first notice thatΓ( D V × ∇ D B )Γ( η ) = ( D V ⊗ γ B )( η ) + (1 ⊗ ∇ D B )( η ) , for all η ∈ E cV ⊗ C ∞ c ( B ) E cB . As in the proof of [35, Theorem 23] we thenestablish that W Γ( D V × ∇ D B )Γ W ∗ ( ξ ) = (cid:0) D M − i c (Ω) (cid:1) ( ξ )for all ξ ∈ E cM . The result of the proposition now follows since E cM is a core for both of the unbounded operators appearing in Equation(3.2). (cid:3) Lemma 19.
The triple (cid:0) C ∞ c ( M ) , X (cid:98) ⊗ C ( B ) L ( E cB ) , D V × ∇ D B (cid:1) is aneven half-closed chain over C ( M ) . Moreover, this even half-closedchain represents the fundamental class [ M ] in KK ( C ( M ) , C ) . ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 21
Proof.
By Proposition 18, ( D V × ∇ D B ) is unitarily equivalent to theodd, symmetric and elliptic, first-order differential operator ( D M ) − i c (Ω) : E cM → L ( E cM ). This establishes that the triple (cid:0) C ∞ c ( M ) , X (cid:98) ⊗ C ( B ) L ( E cB ) , D V × ∇ D B (cid:1) is an even half-closed chain, see [4, 25].To end the proof, we need to show that the even half-closed chains (cid:0) C ∞ c ( M ) , L ( E cM ) , D M (cid:1) and (cid:0) C ∞ c ( M ) , L ( E cM ) , ( D M ) − i c (Ω) (cid:1) (3.3)represent the same class in KK ( C ( M ) , C ).For a function x ∈ C ( M ), we let (cid:104) x, C ∞ c ( M ) (cid:105) ⊆ C ( M ) denote thesmallest ∗ -subalgebra of C ( M ) containing both C ∞ c ( M ) and x . Wethen choose a positive function x ∈ C ( M ) such that x · C ( M ) ⊆ C ( M ) is norm-dense and such that (cid:0) (cid:104) x, C ∞ c ( M ) (cid:105) , L ( E cM ) , D M (cid:1) and (cid:0) (cid:104) x, C ∞ c ( M ) (cid:105) , L ( E cM ) , ( D M ) − i c (Ω) (cid:1) are still even half-closed chains from C ( M ) to C . Moreover, we mayarrange that x (( D M ) − i c (Ω)) x ( ξ ) = xD M x ( ξ ) − i xc (Ω) x ( ξ )for all ξ ∈ E cM and that c (Ω) x is a bounded operator on L ( E cM ).Clearly, the passage from C ∞ c ( M ) to (cid:104) x, C ∞ c ( M ) (cid:105) does not change thecorresponding classes in KK-theory.We now localize our symmetric unbounded operators with respectto the positive function x , obtaining the essentially selfadjoint un-bounded operators xD M x : E cM → L ( E cM ) and xD M x − i xc (Ω) x : E cM → L ( E cM ), see [34, Proposition 11]. Moreover, we have that (cid:0) (cid:104) x, C ∞ c ( M ) (cid:105) , L ( E cM ) , xD M x (cid:1) and (cid:0) (cid:104) x, C ∞ c ( M ) (cid:105) , L ( E cM ) , xD M x − i xc (Ω) x (cid:1) (3.4)are even spectral triples over C ( M ) and that these spectral triplesrepresent the same classes in KK-theory as our original even half-closedchains in Equation (3.3), see [34, Theorem 13 and Theorem 19].But the two spectral triples in Equation (3.4) clearly represent thesame class in KK-theory since the unbounded selfadjoint operators xD M x and xD M x − i xc (Ω) x are bounded perturbations of each other.This ends the proof of the lemma. (cid:3) Remark 20.
The result of the above lemma can not be proved di-rectly using the recent work of van den Dungen, since we are lacking an “adequate approximate identity” for the unbounded symmetric op-erator D M : Dom( D M ) → L ( E cM ) , see [20] . Instead we rely on thegeneral localization techniques initiated in [28] and developped furtherin [29, 34] . For any compact subset K ⊆ M , we introduce the subspace L ( E KM ) := (cid:8) ξ ∈ L ( E cM ) | ψ · ξ = ξ , ∀ ψ ∈ C ∞ c ( M ) with ψ | K = 1 (cid:9) . Lemma 21.
Suppose that K ⊆ M is a compact subset. Then Dom( D V × ∇ D B ) ∩ L ( E KM ) ⊆ Dom( D V ⊗ . Moreover, there exists a constant C K > such that (cid:107) ( D V ⊗ η ) (cid:107) ≤ C K · (cid:0) (cid:107) ( D V × ∇ D B )( η ) (cid:107) + (cid:107) η (cid:107) (cid:1) for all η ∈ Dom( D V × ∇ D B ) ∩ L ( E KM ) .Proof. Throughout this proof we will suppress the left C ∞ c ( M )-moduleisomorphism W : E cV ⊗ C ∞ c ( B ) E cB → E cM from Proposition 14.Let L ⊆ M be a compact subset. Since ( D V × ∇ D B ) : E cM → L ( E cM ) is induced by an elliptic first-order differential operator andsince ( D V ⊗ : E cM → L ( E cM ) is induced by a first-order differentialoperator we may apply G˚arding’s inequality to find a constant C L > (cid:107) ( D V ⊗ η ) (cid:107) ≤ C L · (cid:0) (cid:107) ( D V × ∇ D B )( η ) (cid:107) + (cid:107) η (cid:107) (cid:1) for all η ∈ E cM with support contained in L .Let now η ∈ Dom( D V × ∇ D B ) ∩ L ( E KM ). Since E cM is a core for D V × ∇ D B we may find a sequence { η n } in E cM such that η n → η and( D V × ∇ D B )( η n ) → ( D V × ∇ D B )( η ) in the norm on L ( E cM ). Moreover,since η ∈ L ( E KM ) we may suppose, without loss of generality, thatthere exists a compact subset L ⊆ M such that the support of η n iscontained in L for all n ∈ N .The result of the lemma now follows from Equation (3.5). (cid:3) Finally, we establish that the tensor sum half-closed chain( C ∞ c ( M ) , X (cid:98) ⊗ C ( B ) L ( E cB ) , D V × ∇ D B )is indeed an unbounded representative of the Kasparov product of thecorresponding classes in bounded KK-theory. Note that the relevantKasparov product and KK-groups are the following: (cid:98) ⊗ C ( B ) : KK ( C ( M ) , C ( B )) × KK ( C ( B ) , C ) → KK ( C ( M ) , C ) . We are thus going to prove the identity[
X, F D V ] (cid:98) ⊗ C ( B ) [ L ( E cB ) , F D B ] = [ X (cid:98) ⊗ C ( B ) L ( E cB ) , F D V × ∇ D B ]in KK ( C ( M ) , C ), where F D := D (1 + D ∗ D ) − / : E → E denotesthe bounded transform of a symmetric and regular unbounded operator D : Dom( D ) → E . ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 23
The proof will be based on a generalization to half-closed chains ofa theorem by Kucerovsky [38, Theorem 13], which we proved recentlyin [34] (see Appendix A Theorem 34 for the main result).
Theorem 22.
Suppose that π : M → B is a proper Riemanniansubmersion of even dimensional spin c manifolds. Then the even half-closed chain ( C ∞ c ( M ) , L ( E cM ) , D M ) is the unbounded Kasparov prod-uct of the even unbounded Kasparov module ( C ∞ c ( M ) , X, D V ) with theeven half-closed chain ( C ∞ c ( B ) , L ( E cB ) , D B ) up to the curvature term − i c (Ω) : E cM → L ( E cM ) .Proof. From Proposition 18 we know that the tensor sum half-closedchain is unitarily equivalent to the half-closed chain (cid:0) C ∞ c ( M ) , L ( E cM ) , ( D M ) − i c (Ω) (cid:1) . Moreover, Lemma 19 says that the tensor sum half-closed chain rep-resents the fundamental class [ M ] in KK ( C ( M ) , C ). We thereforeonly need to verify the connection condition (Definition 31) and thelocal positivity condition (Definition 32) for the tensor sum half-closedchain, the vertical unbounded Kasparov module and the horizontalhalf-closed chain.For the connection condition we work with the core E cV ⊗ C ∞ c ( B ) E cB for D V × ∇ D B and the core E cB for D B . We then compute locally forhomogeneous ξ ∈ E cV ⊆ X and r ∈ E cB ⊆ L ( E cB ) that( D V × ∇ D B )( ξ ⊗ r ) − ( − ∂ξ · ( ξ ⊗ D B ( r ))= D V ξ ⊗ r + ( − ∂ξ · i (cid:88) α ∇ Xf α ( ξ ) ⊗ c B ( f α ) r which clearly extends to a bounded operator from L ( E cB ) to the interiortensor product X (cid:98) ⊗ C ( B ) L ( E cB ).For the localizing subset Λ ⊆ C ∞ c ( M ) we start by choosing a count-able open cover { U m } of the base manifold B and a smooth partitionof unity { χ m } subordinate to that cover and with supp( χ m ) compactfor each m ∈ N . Clearly, { π − ( U m ) } is a countable open cover of M ,and { χ m ◦ π } is a partition of unity. We will then take as a localizingsubset Λ = { χ m ◦ π } for which one readily checks the two first condi-tions of Definition 32. Note that K m := supp( χ m ◦ π ) = π − (supp( χ m ))is compact because π : M → B is assumed to be proper. The finalcondition of Definition 32 follows from Lemma 21.Let m ∈ N and choose a compact subset L m ⊆ M such that K m iscontained in the interior of L m . To verify the local positivity conditionit suffices to show that there exists a κ m > (cid:104) ( D V (cid:98) ⊗ η, ( D V × ∇ D B ) η (cid:105) + (cid:104) ( D V × ∇ D B ) η, ( D V (cid:98) ⊗ η (cid:105) ≥ − κ m (cid:104) η, η (cid:105) for all η ∈ Im (( χ m ◦ π ) ) ∩ Dom( D V × ∇ D B ). By Lemma 17 we may find a constant C m > (cid:107) [ D V ⊗ , ⊗ ∇ D B ]( η ) (cid:107) ≤ C m · (cid:0) (cid:107) η (cid:107) + (cid:107) ( D V ⊗ η ) (cid:107) (cid:1) for all η ∈ E cV ⊗ C ∞ c E cB with support contained in L m .We claim that the inequality in Equation (3.6) is satisfied for κ m := (1 + C m ).Suppose first that η ∈ E cV ⊗ C ∞ c ( B ) E cB with supp( η ) ⊆ K m . Arguingjust as in the proof of [31, Lemma 7.5] and using Equation (3.7) weobtain that ± (cid:104) η, ( γ X ⊗ D V ⊗ , ⊗ ∇ D B ] η (cid:105) = ± (cid:0)(cid:10) C / m η, ( γ X ⊗ D V ⊗ , ⊗ ∇ D B ] C − / m η (cid:11) + (cid:10) ( γ X ⊗ D V ⊗ , ⊗ ∇ D B ] C − / m η, C / m η (cid:11)(cid:1) ≤ C m · (cid:107) [ D V ⊗ , ⊗ ∇ D B ] η (cid:107) + C m · (cid:107) η (cid:107) ≤ · (cid:107) ( D V ⊗ η (cid:107) + κ m · (cid:107) η (cid:107) . This implies that (cid:104) ( D V ⊗ η, ( D V × ∇ D B ) η (cid:105) + (cid:104) ( D V × ∇ D B ) η, ( D V ⊗ η (cid:105) = (cid:104) η, ( γ X ⊗ ⊗ ∇ D B , D V ⊗ η (cid:105) + 2 · (cid:107) ( D V ⊗ η (cid:107) ≥ − κ m (cid:107) η (cid:107) (3.8)and hence that Equation (3.6) holds for all η ∈ E cV ⊗ C ∞ c ( B ) E cB withsupp( η ) ⊆ L m .Suppose now that η ∈ Im (( χ m ◦ π ) ) ∩ Dom( D V × ∇ D B ). We thenchoose a sequence { η n } in E cV ⊗ C ∞ c ( B ) E cB which converges to η in thegraph norm of D V × ∇ D B . We may assume, without loss of generality,that η n has support in L m ⊆ M for all n ∈ N . By Lemma 21 thisimplies that { η n } also converges to η in the graph norm of D V ⊗ (cid:3) Almost regular fibrations
In this section we come to the third main result of this paper, whichis the factorization in unbounded KK-theory of the Dirac operator D M on the total manifold M of a so-called almost-regular fibration of spin c manifolds. This factorization takes place on a dense open submanifold M of M and is given in terms of a vertical Dirac operator D V and theDirac operator D B on a base manifold B . The point is here that allinformation about the total Dirac operator D M can be deduced fromthe behaviour of its restriction D M to M . But let us begin with theprecise definitions. ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 25
Definition 23.
Let M be a Riemannian manifold (not necessarily com-pact but without boundary), together with a finite union P = ∪ mj =1 P j of compact embedded submanifolds P j ⊆ M each without boundary andof codimension strictly greater than . If there exist a Riemannianmanifold without boundary B and a proper Riemannian submersion π : M → B with total space M = M \ P , we call the data ( M , P, B, π ) an almost-regular fibration . Note that M = M \ P is a dense open subset of M (and P hasRiemannian measure zero). Example 24.
A proper Riemannian submersion π : M → B is anexample of an almost-regular fibration when we take P = ∅ . Example 25.
Let G be a torus acting isometrically, but not necessar-ily freely, on a compact Riemannian manifold N such that the orbitspace N/G is connected. Let H prin , H , . . . , H m ⊆ G denote the finitelymany isotropy groups, where H prin is the principal stabilizer. We sup-pose that all the orbits G/H j , j = 1 , . . . , m , are singular and moreover,that each subspace of H j -fixed points N H j is connected. Then, letting N ⊆ N denote the principal stratum, we have that N \ N = ∪ mj =1 N H j and that each N H j ⊆ N is a compact embedded submanifolds of codi-mension strictly greater than , see [16, Theorem 5.11, Theorem 5.14and Proposition 5.15] and [40, Proposition 1.24] . The projection map π : N → N /G is a proper Riemannian submersion hence the data ( N, ∪ mj =1 N H j , N /G, π ) is an almost-regular fibration.In fact, in the above we may restrict our attention to a subset { H j i } of the isotropy groups such that N \ N = ∪ i N H ji . It does moreoversuffice to assume that each quotient space N H ji /G is connected insteadof assuming that each N H ji is connected. In this case we use the con-nected components of the H j i -fixed points as our compact embeddedsubmanifolds instead. Definition 26. An almost-regular fibration of spin c manifolds is analmost-regular fibration ( M , P, B, π ) such that M and B are equippedwith spin c structures. Example 27.
Continuing with Example 25, if we assume that N car-ries a G -equivariant spin c structure, it follows that the open submani-fold N is a G -equivariant spin c manifold. If in addition the action of G on N is effective, the orbit space N /G is a spin c manifold. This formsa key class of examples of almost-regular fibrations of spin c manifolds. Let us consider an almost-regular fibration of even dimensional spin c manifolds ( M , P, B, π ). We let E M → M and E B → B denote the Z / Z -graded spinor bundles. Thus there exist Dirac operators on the spin c manifolds M and B , given as odd unbounded operators( D M ) : Γ ∞ c ( M , E M ) → L ( M , E M ) and( D B ) : Γ ∞ c ( B, E B ) → L ( B, E B ) . Moreover, since the spin c structure on M restricts to a spin c structureon the open submanifold M ⊆ M we also obtain the Dirac operator( D M ) : Γ ∞ c ( M, E M ) → L ( M, E M ) , where the spinor bundle E M → M agrees with the restriction of E M to M ⊆ M . Since P ⊆ M has Riemannian measure 0 we may identifythe Z / Z -graded Hilbert spaces L ( M, E M ) and L ( M , E M ) using theinclusion ι : Γ ∞ c ( M, E M ) → Γ ∞ c ( M , E M ) given by extension by zero.We may then arrange that( D M ) ( ι ( ξ )) = ( D M ) ( ξ ) for all ξ ∈ Γ ∞ c ( M, E M ) . Each of these unbounded operators determine even half-closed chains( C ∞ c ( M ) , L ( M , E M ) , D M )( C ∞ c ( B ) , L ( B, E B ) , D B ) and( C ∞ c ( M ) , L ( M, E M ) , D M ) , representing the fundamental classes [ M ] ∈ KK ( C ( M ) , C ), [ B ] ∈ KK ( C ( B ) , C ) and [ M ] ∈ KK ( C ( M ) , C ), respectively.Using the ∗ -homomorphism ι : C ∞ c ( M ) → C ∞ c ( M ) given by exten-sion by zero, we may pullback the even half-closed chain( C ∞ c ( M ) , L ( M , E M ) , D M )to an even half-closed chain ι ∗ ( C ∞ c ( M ) , L ( M , E M ) , D M ) = ( C ∞ c ( M ) , L ( M , E M ) , D M ) . At the level of bounded KK-theory this pullback operation correspondsto the usual pullback homomorphism ι ∗ : KK ( C ( M ) , C ) → KK ( C ( M ) , C )coming from the ∗ -homomorphism ι : C ( M ) → C ( M ) and the con-travariant functoriality in the first variable.The following theorem is the third main result of this paper. Theproof relies for the main part on Theorem 22 and we apply the notationof that theorem here as well. Theorem 28.
Suppose that ( M , P, B, π ) is an almost regular-fibrationof even dimensional spin c manifolds. Then it holds (up to unitaryequivalence) that (4.1) ι ∗ ( C ∞ c ( M ) , L ( M , E M ) , D M ) = ( C ∞ c ( M ) , L ( M, E M ) , D M ) . Moreover, the even half-closed chain ( C ∞ c ( M ) , L ( M, E M ) , D M ) is theunbounded Kasparov product of the even unbounded Kasparov module ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 27 ( C ∞ c ( M ) , X, D V ) and the even half-closed chain ( C ∞ c ( B ) , L ( B, E B ) , D B ) up to the curvature term − i c (Ω) : Γ ∞ c ( M, E M ) → L ( M, E M ) . In par-ticular it holds that ι ∗ [ M ] = [ M ] = [ X, D V (1 + D V ) − / ] (cid:98) ⊗ C ( B ) [ B ] at the level of bounded KK-theory.Proof. Since π : M → B is a proper Riemannian submersion by as-sumption, we may apply Theorem 22 to establish the second part ofthe present theorem. It therefore suffices to prove the identity in Equa-tion (4.1). But this identity follows immediately from Proposition 30here below. (cid:3) The restriction we impose on the codimension of the compact em-bedded submanifolds P j ⊆ M in the definition of an almost-regularfibration ( M , P, B, π ) guarantees that given a closed extension of afirst-order differential operator —such as the Dirac operator— on M ,its restriction to sections that have compact support contained in M gives the same closure. The following lemma will be useful in thiscontext. Lemma 29.
Let P ⊆ M be a subset of a Riemannian manifold M suchthat P = (cid:83) mj =1 P j is a finite union of compact embedded submanifolds P j ⊆ M , each of codimension strictly greater than . Then there existsan increasing sequence { ψ n } of positive smooth functions on M suchthat (1) supp( ψ n ) ⊆ M \ P for all n ∈ N ; (2) sup n ( ψ n | M \ P ) = 1 M \ P ; (3) The exterior derivative dψ n has compact support for all n ∈ N and the sequence { dψ n } is bounded in L ( M , T ∗ M ) .Proof. We may assume, without loss of generality, that P consists of asingle closed embedded submanifold P ⊆ M . Indeed, suppose that anincreasing sequence { ψ jn } of positive smooth functions satisfying (1), (2)and (3) has been constructed for each compact embedded submanifold P j ⊆ M . Then the sequence { ψ n } := { ψ n · . . . · ψ mn } satisfies (1), (2)and (3) for P = ∪ mj =1 P j .Let k > P ⊆ M . Choose an increasingsequence { (cid:102) ψ n } of positive smooth functions on R k such that • supp( (cid:102) ψ n ) ⊆ R k \ { } ; • supp( d (cid:102) ψ n ) ⊆ B /n (0), where B /n (0) ⊆ R k denotes the ball ofradius 1 /n and center 0; • sup n ( (cid:102) ψ n | R k \{ } ) = 1 | R k \{ } ; • There exists a constant
C > (cid:107) d (cid:102) ψ n (cid:107) ∞ ≤ C · n for all n ∈ N , where (cid:107) · (cid:107) ∞ denotes the supremum norm onΓ ∞ c ( R k , T ∗ R k ) ∼ = C ∞ c ( R k ) ⊕ k . Choose a finite open cover of P ⊆ M by submanifold charts( V , ϕ ) , . . . , ( V N , ϕ N ) , such that V i ⊆ M is compact for all i ∈ { , . . . , N } . Thus, for each i ∈ { , . . . , N } , we have that V i ∩ P = (cid:8) x ∈ V i | ( π ◦ ϕ i )( x ) = 0 (cid:9) , where π : R k × R dim( M ) − k → R k denotes the projection onto the first k coordinates.Put V := M \ P and choose a smooth partition of unity χ , χ , . . . , χ N for M with supp( χ i ) ⊆ V i for all i ∈ { , , . . . , N } . Remark thatsupp( χ i ) is compact for all i ∈ { , . . . , N } but that supp( χ ) need notbe compact. It does however hold that the support of the exteriorderivative dχ is compact.We define ψ n := χ + N (cid:88) i =1 χ i · ( (cid:102) ψ n ◦ π ◦ ϕ i ) for all n ∈ N . We leave it to the reader to verify that the increasing sequence { ψ n } of positive smooth functions satisfies (1), (2) and (3). When verifying(3), notice that (cid:8) (cid:107) d (cid:102) ψ n (cid:107) ∞ · Vol( B /n (0) ⊆ R k ) (cid:9) ∞ n =1 is a bounded sequence since k > (cid:3) The following proposition generalizes the results in [5, Prop. 4.12]and [22, Sect. 2.2.1].
Proposition 30.
Let M be a Riemannian manifold and P = ∪ mj =1 P j bea finite union of compact embedded submanifolds, each of codimensionstrictly greater than ; write the complement as M := M \ P . Let E → M be a smooth hermitian vector bundle and ( D M ) : Γ ∞ c ( M , E ) → L ( M , E ) be a first-order differential operator. Then, if we let ( D M ) :Γ ∞ c ( M, E | M ) → L ( M, E | M ) denote the restriction of ( D M ) to thesmooth compactly supported sections Γ ∞ c ( M, E | M ) , the closure of ( D M ) coincides with the closure of ( D M ) , both as operators in L ( M , E ) .Proof. First of all, we use that both M and M are manifolds withoutboundary so that by [24, Lemma 10.2.1] both ( D M ) and ( D M ) areclosable. We denote the closures by D M and D M , respectively. Since P ⊆ M is a null-set, we can identify L ( M, E | M ) with L ( M , E ) andconsider the operators D M and D M both as being (densely defined) on L ( M , E ). Indeed, the inclusion Γ ∞ c ( M, E | M ) → Γ ∞ c ( M , E ) induces aunitary isomorphism of Hilbert spaces L ( M, E | M ) ∼ = L ( M , E ).Clearly, D M ⊆ D M , so it suffices to show that the core Γ ∞ c ( M , E )for D M is included in the domain of D M . ACTORIZING DIRAC OPERATORS ON ALMOST-REGULAR FIBRATIONS 29
Let thus s ∈ Γ ∞ c ( M , E ) be given. For the increasing sequence ofpositive smooth functions { ψ n } constructed in Lemma 29 we have that ψ n s → s in the norm of L ( M , E ) (recall that P has Riemannianmeasure 0). By construction ψ n s ∈ Γ ∞ c ( M, E | M ) for all n ∈ N and,by [24, Lemma 1.8.1], we may thus conclude that s ∈ Dom( D M ), if wecan establish that { D M ( ψ n s ) } is a bounded sequence in L ( M , E ).Let σ M : Γ ∞ ( M , T ∗ M ) → Γ ∞ ( M ,
End( E )) denote the principal sym-bol of the first-order differential operator D M : Γ ∞ ( M , E ) → Γ ∞ ( M , E ).For each x ∈ M we let (cid:107) σ M ( x ) (cid:107) ∞ denote the norm of the fiber-wise op-erator σ M ( x ) : T ∗ x M → End( E x ) and we let (cid:107) s (cid:107) ∞ denote the supremumnorm of the compactly supported section s : M → E .We then have the estimates (cid:107) D M ( ψ n s ) (cid:107) L ( M,E ) = (cid:107) σ M ( dψ n ) · s + ψ n · D M ( s ) (cid:107) L ( M,E ) ≤ (cid:107) σ M ( dψ n ) · s (cid:107) L ( M,E ) + (cid:107) D M ( s ) (cid:107) L ( M,E ) ≤ sup x ∈ supp( s ) (cid:107) σ M ( x ) (cid:107) ∞ · (cid:107) s (cid:107) ∞ · (cid:107) dψ n (cid:107) L ( M,T ∗ M ) + (cid:107) D M ( s ) (cid:107) L ( M,E ) . Using property (3) of the sequence { ψ n } from Lemma 29, these es-timates show that {(cid:107) D M ( ψ n s ) (cid:107) L ( M,E ) } is a bounded sequence. Thisproves the proposition. (cid:3) Appendix A. On a theorem of Kucerovsky forhalf-closed chains
We summarize the main result of [34], which generalizes a theo-rem by Kucerovsky [38, Theorem 13] to half-closed chains. Let us fixthree C ∗ -algebras A, B and C with A separable and B , C σ -unital.Throughout this section we will assume that ( A , E , D ), ( B , E , D )and ( A , E, D ) are even half-closed chains from A to B , from B to C and from A to C , respectively. We denote the ∗ -homomorphisms as-sociated to the C ∗ -correspondences E , E and E by φ : A → L ( E ), φ : B → L ( E ) and φ : A → L ( E ), respectively. We will more-over assume that E = E (cid:98) ⊗ B E agrees with the interior tensor productof the C ∗ -correspondences E and E . In particular, we assume that φ ( a ) = φ ( a ) ⊗ a ∈ A . We let γ : E → E , γ : E → E and γ := γ (cid:98) ⊗ γ denote the Z / Z -grading operators on E , E and E .We will denote the bounded transforms of our half-closed chainsby ( E , F D ), ( E , F D ) and ( E, F D ) and the corresponding classes inKK-theory by [ E , F D ] ∈ KK ( A, B ), [ E , F D ] ∈ KK ( B, C ) and[
E, F D ] ∈ KK ( A, C ). We may then form the interior Kasparov prod-uct [ E , F D ] (cid:98) ⊗ B [ E , F D ] ∈ KK ( A, C )and it becomes a relevant question to find an explicit formula for thisclass in KK ( A, C ). For each ξ ∈ E , we let T ξ : E → E denote the bounded adjointableoperator given by T ξ ( η ) := ξ ⊗ η for all η ∈ E . Definition 31.
The connection condition demands that there exist adense B -submodule E ⊆ E and cores E and E for D : Dom( D ) → E and D : Dom( D ) → E , respectively, such that (a) For each ξ ∈ E : T ξ ( E ) ⊆ Dom( D ) , T ∗ ξ ( E ) ⊆ Dom( D ) , γ ( ξ ) ∈ E (b) For each homogeneous ξ ∈ E : DT ξ − ( − ∂ξ T ξ D : E → E extends to a bounded operator L ξ : E → E . Definition 32. A localizing subset is a countable subset Λ ⊆ A with Λ = Λ ∗ such that (a) The span span C (cid:8) x · a | a ∈ A , x ∈ Λ (cid:9) ⊆ A is norm-dense in A . (b) The commutator [ D (cid:98) ⊗ , φ ( x )] : Dom( D (cid:98) ⊗ → E is trivial for all x ∈ Λ . (c) We have the domain inclusion
Dom( D ) ∩ Im( φ ( x ∗ x )) ⊆ Dom( D (cid:98) ⊗ for all x ∈ Λ . Definition 33.
Given a localizing subset Λ ⊆ A , the local positivitycondition requires that for each x ∈ Λ , there exists a constant κ x > such that (cid:10) ( D (cid:98) ⊗ φ ( x ∗ ) ξ, Dφ ( x ∗ ) ξ (cid:11) + (cid:104) Dφ ( x ∗ ) ξ, ( D (cid:98) ⊗ φ ( x ∗ ) ξ (cid:105)≥ − κ x · (cid:104) ξ, ξ (cid:105) for all ξ ∈ Im( φ ( x )) ∩ Dom( Dφ ( x ∗ )) . In practice, it is useful to record that the local positivity conditionwould follow if for each x ∈ Λ, there exists a constant κ x > (cid:104) ( D (cid:98) ⊗ η, Dη (cid:105) + (cid:104) Dη, ( D (cid:98) ⊗ η (cid:105) ≥ − κ x (cid:104) η, η (cid:105) , for all η ∈ Im( φ ( x ∗ x )) ∩ Dom( D ). Theorem 34.
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Department of Mathematics and Computer Science, Syddansk Uni-versitet, Campusvej 55, 5230, Odense M, Denmark
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