A topological approach to indices of geometric operators on manifolds with fibered boundaries
aa r X i v : . [ m a t h . K T ] A ug A TOPOLOGICAL APPROACH TO INDICES OF GEOMETRICOPERATORS ON MANIFOLDS WITH FIBERED BOUNDARIES
MAYUKO YAMASHITA
Abstract.
In this paper, we investigate topological aspects of indices oftwisted geometric operators on manifolds equipped with fibered boundaries.We define K -groups relative to the pushforward for boundary fibration, andshow that indices of twisted geometric operators, defined by complete Φ oredge metrics, can be regarded as the index pairing over these K -groups. Wealso prove various properties of these indices using groupoid deformation tech-niques. Using these properties, we give an application to the localization prob-lem of signature operators for singular fiber bundles. Contents
1. Introduction 22. Preliminaries 62.1. Representable K -theory 62.2. C l -invertible perturbations 72.3. Groupoids 92.4. b , Φ, e -calculus and corresponding groupoids 193. Indices of geometric operators on manifolds with fibered boundaries :the case without perturbations 223.1. The definition of indices 223.2. Properties 243.3. The cases of twisted spin c and signature operators 304. Indices of geometric operators on manifolds with fibered boundaries :the case with fiberwise invertible perturbations 324.1. The general situation 334.2. The connecting elements of G Φ and G e e -indices 395. The index pairing 445.1. The case of spin c -Dirac operators 445.2. The case of signature operators 556. The local Signature 596.1. Settings 596.2. The universal index class and the pullback of C l -invertibleperturbations 606.3. The local signature 617. Examples 64 Graduate School of Mathematical Sciences, the University of Tokyo, 3-8-1 KomabaMeguro-ku Tokyo 153-8914, Japan
E-mail address : [email protected] . Acknowledgment 67References 671.
Introduction
In this paper, we consider pairs of the form (
M, π : ∂M → Y ), where M is acompact manifold with boundary ∂M which is a closed manifold, and π : ∂M → Y is a smooth submersion, equivalently a fiber bundle structure, to a closed manifold Y . We call such pairs manifolds with fibered boundaries . We investigate topologicalaspects of indices of geometric operators, namely spin c -Dirac operators, signatureoperators and their twisted versions, on such manifolds. There are two purposesof this paper. The first one is to formulate the index pairing on such manifolds.We define K -groups relative to the pushforward for boundary fibration, and showthat indices of twisted geometric operators, defined by complete metrics of the form(1.2), can be regarded as the index pairing over these K -groups. The second one isto prove properties of these indices using groupoid deformation techniques. Usingthese properties, we give an application to the localization problem of signatureoperators for singular fiber bundles.Singular spaces arise in various areas in mathematics. In particular, stratifiedpseudomanifolds include many important examples of singular spaces, such as man-ifolds with corners and algebraic varieties. Manifolds with fibered corners arise asresolutions of stratified pseudomanifolds [ALMP12], and the simplest case, strati-fied manifolds of depth 1, corresponds to manifolds with fibered boundaries. Thereare some classes of metrics which is suited to encode the singularities of such spaces,including (complete) Φ-metrics and edge metrics. To study pseudodifferential op-erators with respect to such metrics, the corresponding pseudodifferential calculi,called Φ-calculus and e -calculus, were introduced by [MM98] and [Maz91]. Sincethen, analysis of elliptic operators in these calculi, in particular Fredholm theoryand spectral theory of geometric operators, has been developed by many authorsand there have been many applications to geometry of singular spaces, for examplesee [ALMP12], [DLR15] and [LMP06].Most of those works are analytic in nature, in the sense that they analyze in-dividual operators under these calculi. On the other hand, it is natural to expectmore topological description of Fredholm indices of these operators, as in the caseof closed manifolds. One of related works in this direction is [MR06], in which theyformulate the index theorem for fully elliptic operators, as an equality of analyticand topological indices defined on abelian groups of stable homotopy classes of fullsymbols K Φ − cu ( φ ), which corresponds to K (Σ ˚ M ( G Φ )) in our paper. We go in thisdirection further, and show that, once we fix a class ⋆ of geometric operators weare concerned with (for example ⋆ can be spin c or sign ), the indices of twistedoperators can be formulated in terms of the pairing on more primary K -groups, K ∗ ( A ⋆π ), “ K -groups relative to the ⋆ pushforward for boundary fibration”. Thispaper is considered as a step to understand elliptic theory on singular spaces froma more topological, or K -theoretical viewpoint.In order to explain our index pairing on manifolds with fibered boundaries, firstwe recall the index pairing on closed manifolds. Let M be a closed even dimensionalsmooth manifold. Suppose we are given a complex vector bundle E over M , and NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 3 a Clifford module bundle W over M , with a Dirac type operator D M . Then wehave the corresponding classes [ E ] ∈ K ( M ) and [ D M ] ∈ K ( M ) in the K -theoryand the K -homology of M , and the index pairing of K ( M ) and K ( M ) sends thispair to the index of the twisted Dirac operator, D EM , acting on the Clifford modulebundle W ˆ ⊗ E , h· , ·i : K ( M ) ⊗ K ( M ) → Z h [ E ] , [ D M ] i = Ind( D EM ) . (1.1)Many examples of such an operator D M arise as “geometric operators” on compactmanifolds, such as spin c -Dirac operators and signature operators. Fixing a class ofgeometric operator ⋆ and a corresponding geometric structure on M that determinesthe operator D M , the above pairing is equivalently described as the pushforward p ⋆ ! : K ( M ) → Z in K -theory.The first main purpose of this paper is to generalize this index pairing to the caseof compact manifolds with fibered boundaries. It is stated in terms of K -theoryfor some C ∗ -algebras. The C ∗ -algebras depend on the “geometric structure” wechoose to deal with, so here we explain the case of spin c -structures.Assume we are given a compact even dimensional manifold with fibered boundary( M, π : ∂M → Y ), and assume that the fibers of π has dimension n . If π is equippedwith a spin c -structure, we associate a C ∗ -algebra A π , whose K -groups fit into thelong exact sequence · · · → K ∗ ( M ) π ! ◦ i ∗ −−−→ K ∗− n ( Y ) → K ∗− n ( A π ) → K ∗ +1 ( M ) π ! ◦ i ∗ −−−→ · · · where π ! is the Gysin map in K -theory π ! : K ∗ ( ∂M ) → K ∗− n ( Y ) defined by thefiberwise spin c -structure, and i is the inclusion i : ∂M → M (Definition 5.4 andProposition 5.5). Thus K -groups of this C ∗ -algebra, K ∗ ( A π ), can be regarded asthe K -groups relative to the spin c -pushforward of the boundary fibration.From now on, we assume n , the dimension of M , is even. A pair of the form( E, ˜ D Eπ ), where E is a complex vector bundle over M satisfying π ! [ E ] = 0 ∈ K ( Y ),and the operator ˜ D Eπ is an invertible perturbation of the fiberwise spin c -Diracoperators by lower order odd self-adjoint operators, gives a class [( E, [ ˜ D Eπ ])] ∈ K ( A π ) (Lemma 5.7; the bracket in [ ˜ D Eπ ] means that we actually only have toconsider the homotopy equivalence class of invertible perturbations). Furthermore,a pair ( P ′ M , P ′ Y ) of (equivalence classes of) spin c -structures on M and Y which iscompatible with the one on π at the boundary, gives a class in K ( A π ). This is theelement [( P ′ M , P ′ Y )] ⊗ Σ ˚ M ( G Φ ) ∂ ˚ M ( G Φ ) ∈ KK ( A π , C ) appearing in Theorem 5.24.On the other hand, from the data ( E, ˜ D Eπ , P ′ M , P ′ Y ) we can construct a Fredholmoperator by using Φ or edge metrics, as we now explain. For a manifold with fiberedboundary ( M, π : ∂M → Y ), natural classes of complete riemannian metrics on theinterior arise as follows. First fix a splitting T ∂M = π ∗ T Y ⊕ T V ∂M and a collarstructure near the boundary. Consider metrics on ˚ M which are on the collar of theform(1.2) g Φ = dx x ⊕ π ∗ g Y x ⊕ g π and g e = dx x ⊕ π ∗ g Y x ⊕ g π . Here g Y and g π are some riemannian metrics on T Y and T V ∂M , respectively, and x is a normal coordinate of the collar. These are called rigid Φ-metrics and rigid edgemetrics in the literature, respectively. In this paper, we adopt pseudodifferential M. YAMASHITA calculus on Lie groupoids, which is due to [NWX99]. As constructed in [Nis00], thegroupoids corresponding to Φ and edge metrics are of the form G Φ = ˚ M × ˚ M ⊔ ∂M × π ∂M × π T Y × R ⇒ MG e = ˚ M × ˚ M ⊔ ∂M × π ∂M × π ( T Y ⋊ R ∗ + ) ⇒ M. Using the given (equivalence classes of) spin c -structures, we can consider the spin c -Dirac operator twisted by E under these metrics, denoted by D E Φ and D Ee . ForΦ-case, the operator D E Φ restricts to the boundary operator of the form D Eπ ˆ ⊗ ⊗ D T Y × R . If we perturb this boundary operator to the invertible operator ˜ D Eπ ˆ ⊗ ⊗ D T Y × R , we get the Fredholmness of the operator on the interior and get theindex, denoted by Ind Φ ( P ′ M , P ′ Y , E, [ ˜ D Eπ ]) (Definition 4.16). The e -case is analogous,and we define Ind e ( P ′ M , P ′ Y , E, [ ˜ D Eπ ]).Our main theorem, Theorem 5.24, proves the equalityInd Φ ( P ′ M , P ′ Y , E, [ ˜ D Eπ ]) = [( E, [ ˜ D Eπ ])] ⊗ A π [( P ′ M , P ′ Y )] ⊗ Σ ˚ M ( G Φ ) ∂ ˚ M ( G Φ ) , which can be regarded as a generalization of the index pairing (1.1).In the case of signature operators, the arguments proceed in parallel. If we aregiven a pair ( M, π : ∂M → Y ) such that T V M are oriented, then we associate a C ∗ -algebra A sign π , whose K -groups can be regarded as the K -groups relative to thesignature pushforward of the boundary fibration. Given an orientation on M , weget the corresponding index pairing formula, Theorem 5.38,Sign Φ ( M, E, [ ˜ D sign ,Eπ ]) = [( E, [ ˜ D sign ,Eπ ])] ⊗ A sign π [ M sign ] ⊗ Σ ˚ M ( G Φ ) ∂ ˚ M ( G Φ ) . These indices, being defined as indices of operators on Lie groupoids, can beanalyzed in terms of groupoids. We call groupoid deformation technique the follow-ing type of arguments. Suppose we are given a compact manifold M and two Liegroupoids G ⇒ M and G ⇒ M , equipped with geometric structures that deter-mine the geometric operators D i ∈ Diff ∗ ( G i ; E i ). If one can define a Lie groupoidstructure on G = G × { } ⊔ G × (0 , ⇒ M × [0 , G that restricts to ones on G and G by evaluation at 0 and 1 respectively,then the associated geometric operator D satisfies D| M ×{ i } = D i . Under some niceassumption on the groupoids, the element [ ev ] ∈ KK ( C ∗ ( G ) , C ∗ ( G )) is a KK -equivalence. Then, for example if the operators are elliptic, their index classes,Ind( D i ) ∈ K ( C ∗ ( G i )), are related asInd( D ) ⊗ C ∗ ( G ) [ ev ] − ⊗ C ∗ ( G ) [ ev ] = Ind( D ) ∈ K ( C ∗ ( G )) . Furthermore, assuming we have a closed saturated subset V ⊂ M such that D i | V are invertible for i = 0 ,
1, we get the index classes Ind M \ V ( D i ) ∈ K ( C ∗ ( G i | M \ V )).If we can give G a geometric structure such that the associated operator D isinvertible when restricted to V × [0 , M \ V ( D ) ⊗ C ∗ ( G | M \ V ) [ ev ] − ⊗ C ∗ ( G| M \ V × [0 , ) [ ev ] = Ind M \ V ( D ) ∈ K ( C ∗ ( G | M \ V )) . This argument, though very simple, turns out to be useful in proving various prop-erties of indices considered here, without any difficult analysis involved.For example, in Proposition 3.8 (also see (4.17)), we prove that the indices definedby Φ-metrics and e -metrics actually coincide for our settings,Ind Φ ( P ′ M , P ′ Y , E, [ ˜ D Eπ ]) = Ind e ( P ′ M , P ′ Y , E, [ ˜ D Eπ ]) . NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 5
The proof is an application of the groupoid deformation technique, by consideringa groupoid of the form G Φ × { } ⊔ G e × (0 , e ) indices of signature operators defined by the fiberwise invertibleperturbations, can be used to solve the localization problem of signature for thesingular fiber bundles. Suppose we are given a smooth map π : M k → X ev between closed oriented manifolds and X is partitioned into compact manifoldswith closed boundaries as X = U ∪ ∪ mi =1 V i . Suppose that each V i are disjoint, andthe restriction of π to U is a fiber bundle structure with structure group containedin some nice subgroup G ⊂ Diff + ( F ) of the orientation-preserving diffeomorphismgroup of the typical fiber F . The submanifold U is regarded as the “regular part”of this singular fiber bundle structure π . The localization problem is to define areal number σ ( M i , V i , π | M i ) ∈ R , which only depends on the data ( M i , V i , π | M i ),and write Sign( M ) = m X i =1 σ ( M i , V i , π | M i ) . This problem originates from algebraic geometry for the case where the typicalfiber is two dimensional, and the local signatures are constructed and calculated invarious areas of mathematics, including topology, algebraic geometry and complexanalysis. For example see [Mat96], [End00] for topological approaches and [Fur99]for a differential geometric approaches. Also see [AK02] and the introduction of[Sat13] for more survey on this problem.In this situation, for each i the pair ( π − ( V i ) , π : π − ( ∂V i ) → ∂V i ) is a compactmanifold with fibered boundary, and π has the structure group G . The idea isto fix an invertible perturbation of universal family of signature operators definedon the classifying space, and pullback the perturbation to define the Φ-indices ofsignature operators for each V i . We verify this idea and construct functions σ withthe desired properties in the main theorem of Section 6, Theorem 6.2. In Section 7,we give a particular example of this localization problem, where the typical fiber isthe two dimensional oriented closed manifold with genus g ≥
2, and the group G isthe hyperelliptic diffeomorphism group. This is similar to the situation consideredin [End00] for the case where M is four dimensional, but we consider a more generalsituation where the dimension of M can be higher.This paper is organized as follows. In Section 2, we give preliminaries on repre-sentable K -theory, Lie groupoids and Φ, e -calculi. In Section 3 and Section 4, wedefine the indices of twisted geometric operators in Φ and edge metrics, and provevarious properties, using the groupoid deformation technique. Section 3 is aboutthe case without the invertible perturbations, and Section 4 is for the case withinvertible perturbations. Although the results in Section 3 are covered by those inSection 4, we separate this primitive case, because the author believes it makes iteasier to understand what is going on. We note that the properties proved in thesesections are not used in Section 5, so the readers who are only interested in theindex pairing need only to check the definitions of indices given in Definition 4.16and Definition 4.24, and proceed to Section 5. In Section 5, we give the formulationof the indices as index pairings over the K -groups relative to the boundary pushfor-ward. In Section 6, we give the application of the those indices to the localization M. YAMASHITA problem of signature for singular fiber bundles, and in Section 7, we apply this tothe case of singular hyperelliptic fiber bundles.2.
Preliminaries
Representable K -theory. In this subsection, we recall the definitions forrepresentable K -theory in [AS]. We only work with complex coefficients.Let H be a separable infinite dimensional Hilbert space. Let ˆ H := H ⊕ H bethe Z -graded separable infinite dimensional Hilbert space. Let B ( H ) and K ( H )denote the spaces of bounded operators and compact operators on H , respectively.For two topological spaces X and Y , let [ X, Y ] denote the set of homotopy classesof continuous maps from X to Y . Definition 2.1. (0) Let Fred (0) ( ˆ H ) denote the space of self-adjoint odd boundedFredholm operators ˜ A on ˆ H such that ˜ A − I ∈ K ( ˆ H ), with the topologycoming from its embeddingFred (0) ( ˆ H ) → B ( ˆ H ) c . o . × K ( ˆ H ) norm , ˜ A ( ˜ A, ˜ A − . Here we denoted by B ( ˆ H ) c . o . the space of bounded operators equipped withcompact open topology, and by K ( ˆ H ) norm the space of compact operatorsequipped with norm topology.(1) Let Fred (1) ( H ) denote the space of self-adjoint bounded Fredholm operators A on H such that A − I ∈ K ( H ), with the topology coming from itsembeddingFred (1) ( H ) → B ( H ) c . o . × K ( H ) norm , A ( A, A − . Fact 2.2 ([AS04, Section3]) . Fred (0) ( ˆ H ) and Fred (1) ( H ) are classifying spaces ofthe functors K and K , respectively, i.e., we have for any space X , K ( X ) = [ X, Fred (0) ( ˆ H )] and K ( X ) = [ X, Fred (1) ( H )] . Fact 2.3 ([AS04, Proposition A2.1]) . The space of unitary operators on H equippedwith compact open topology, denoted by U ( H ) c . o . , is contractible. Define the following spaces as GL (0) ( ˆ H ) := GL ( ˆ H ) ∩ Fred (0) ( ˆ H ) and U (0) ( ˆ H ) := U ( ˆ H ) ∩ Fred (0) ( ˆ H )(2.4) GL (1) ( H ) := GL ( H ) ∩ Fred (1) ( H ) and U (1) ( H ) := U ( H ) ∩ Fred (1) ( H ) , equipped with the topology induced by the ones on Fred (0) ( ˆ H ) and Fred (1) ( H ). Corollary 2.5.
The spaces GL (0) ( ˆ H ) , U (0) ( ˆ H ) , GL (1) ( H ) and U (1) ( H ) are con-tractible.Proof. By Fact 2.3, the spaces U (0) ( ˆ H ) and U (1) ( H ) are contractible. The map( A, t ) → A | A | − t for t ∈ [0 ,
1] gives a retraction from GL (0) ( ˆ H ) to U (0) ( ˆ H ) andfrom GL (1) ( H ) to U (1) ( H ), respectively. So we get the result. (cid:3) The definition of Hilbert bundles, which is suitable for our purposes, is as follows.
Definition 2.6 (Hilbert bundles) . Let X be a space. A separable infinite dimen-sional Hilbert bundle H → X is a fiber bundle whose typical fibers are separableinfinite dimensional Hilbert space H , with structure group U ( H ) c . o . . NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 7 A Z -graded separable infinite dimensional Hilbert bundle ˆ H → X is a fiberbundle whose typical fibers are Z -graded separable infinite dimensional Hilbertspace H , with structure group U ( ˆ H ) c . o . .By [AS04, Proposition 3.1], the action of U ( ˆ H ) c . o . on Fred (0) ( ˆ H ) is continu-ous. Thus given a Z -graded Hilbert bundle ˆ H → X , we also get the associatedFred (0) ( ˆ H )-bundle Fred (0) ( ˆ H ) → X . The analogous construction applies to theungraded case.By Fact 2.3, we have the following. Corollary 2.7.
Any separable infinite dimensional Hilbert bundle is trivial, andany choices of trivialization are homotopic. C l -invertible perturbations. In this subsection, we discuss C l -invertibleperturbations for a family of Z -graded Fredholm operators parametrized by apossibly noncompact space. The symbol K i denotes the representable K -theory.The setting is as follows. • Let X be a topological space. • Let ˆ H = { ˆ H x } x ∈ X → X be a Z -graded separable Hilbert bundle (seeDefinition 2.6). • Let γ be the involution on ˆ H defining the Z -grading. • Let Fred (0) ( ˆ H ) = { Fred (0) ( ˆ H x ) } x ∈ X → X be the Fred (0) ( ˆ H )-fiber bundleassociated to ˆ H . • Assume we are given an element F ∈ Γ( X ; Fred (0) ( ˆ H )).Let pr : X × [0 , → X be the canonical projection, and consider the Hilbertbundle pr ∗ ˆ H → X × [0 , H → X × [0 , C l -invertible perturbation for F is defined to be a homotopy from F to an invertible family, as follows. Definition 2.8 ( C l -invertible perturbations) . Let ( X, ˆ H , F ) as above. An oper-ator ˜ F : Γ( X × [0 , H ) → Γ( X × [0 , H ) is called a C l -invertible perturbationfor F if • ˜ F ∈ Γ( X × [0 , (0) ( ˆ H )) • ˜ F | X ×{ } = F . • ˜ F | X ×{ } is a family of invertible operators.Let us denote the set of C l -invertible perturbations for F by ˜ I ( F ).We introduce a natural homotopy equivalence relation on ˜ I ( F ), Definition 2.9.
Let ˜ F and ˜ F ′ be two elements in ˜ I ( F ). We say ˜ F and ˜ F ′ arehomotopic if there exists an operator ˜ F ′′ : Γ( X × [0 , × [0 , H ) → Γ( X × [0 , × [0 , H ) such that • ˜ F ′′ ∈ Γ( X × [0 , × [0 , (0) ( ˆ H )). • ˜ F ′′ | X × [0 , ×{ } = ˜ F and ˜ F ′′ | X × [0 , ×{ } = ˜ F ′ . • ˜ F ′′ | X × [0 , ×{ u } ∈ ˜ I ( F ) for all u ∈ [0 , I ( F ) the set of homotopy classes of elements in ˜ I ( F ).The following lemma follows directly from Fact 2.2. M. YAMASHITA
Lemma 2.10.
The element F admits a C l -invertible perturbation if and only if [ F ] = 0 ∈ K ( X ) . Lemma 2.11.
Suppose F satisfies [ F ] = 0 ∈ K ( X ) . Then I ( F ) has a naturalstructure of affine space over K − ( X )(:= [ X, ΩFred (0) ( ˆ H )]) .Proof. Assume we are given two elements in I ( F ). By Corollary 2.7, we choose atrivialization of the Hilbert bundle ˆ H ≃ ˆ H × X , which is unique up to homotopy.Take any representative of these elements and denote them by ˜ F , ˜ F ∈ ˜ I ( F ),respectively. We explain the definition of the difference class [˜ F − ˜ F ] ∈ K − ( X ).Define the continuous map F as follows. F : X × [0 , → Fred (0) ( ˆ H ) F| X × [0 , = ( ˜ F | X ×{ − t } for t ∈ [0 , / F | X ×{ t − } for t ∈ [1 / , . The image of F| X ×{ , } is contained in GL (0) ( ˆ H ). Since GL (0) ( ˆ H ) is contractibleby Corollary 2.5, the map F gives the desired element[˜ F − ˜ F ] := [ F ] ∈ [ X, ΩFred (0) ( ˆ H )] = K − ( X ) . The well-definedness is obvious.Conversely, if we are given an element [˜ F ] ∈ I ( F ) and an element [ F ] ∈ K − ( X ),it is easy to construct the unique element [˜ F ] ∈ I ( F ) such that [˜ F − ˜ F ] = [ F ].Also it is easy to see that this defines an affine structure of I ( F ) over K − ( X ). (cid:3) Let us turn to the case where the parameter space X is a smooth compactmanifold (possibly with boundaries or corners), the Hilbert bundle ˆ H and the familyof operators D come from a fiber bundle over X , and the family D is unbounded.More precisely, we consider the following situations. • Let π : M → X be a smooth fiber bundle with closed fibers, equipped witha smooth fiberwise riemannian metric g π . • Let E → M be a smooth hermitian Z -graded vector bundle. • Let D = { D x } x ∈ X , D x : C ∞ ( π − ( x ); E | π − ( x ) ) → C ∞ ( π − ( x ); E | π − ( x ) ) bea smooth family of odd formally self-adjoint elliptic operators of positiveorder. • Let us denote ˆ H = { ˆ H x = L ( π − ( x ); E | π − ( x ) ) } x ∈ X with the naturalHilbert bundle structure over X . The operator D also denotes the closedextension to D : Γ( X ; ˆ H ) → Γ( X ; ˆ H ).For such a family D , the bounded transform ψ ( D ) := D/ √ D is a smoothfamily pseudodifferential operators of order 0, and defines an element [ ψ ( D )] ∈ K ( X ). We call this class the family index class of D , and abuse the notation towrite [ D ] := [ ψ ( D )] ∈ K ( X ). Definition 2.12 ( I sm ( D )) . In the above situations, an operator ˜ D : C ∞ ( M ; E ) → C ∞ ( M ; E ) is called a C l -smooth invertible perturbation of D if • ˜ D = { ˜ D x } x ∈ X is a smooth family of invertible odd formally self-adjointoperators. • ˜ D − D = A = { A x } x ∈ X , where A x is a pseudodifferential operator of order0 for each x ∈ X . NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 9
Let us denote the set of C l -smooth invertible perturbations for D by ˜ I sm ( D ). Wecan introduce the obvious homotopy equivalence relations in ˜ I sm ( D ). We denote I sm ( D ) the set of homotopy classes of elements in ˜ I sm ( D ).There is a canonical map˜ I sm ( D ) → ˜ I ( ψ ( D ))(2.13) ( ˜ D = D + A ) ( t ∈ [0 , ψ ( D + tA )) . (2.14)In fact this map induces an isomorphism I sm ( D ) ≃ I ( ψ ( D )), by the following fact. Fact 2.15 ([MP97]) . The family D admits a C l -smooth invertible perturbation ifand only if [ D ] = 0 ∈ K ( X ) .If [ D ] = 0 ∈ K ( X ) , then I sm ( D ) has a natural structure of an affine space over K − ( X ) , described as follows.Let Q i ∈ I sm ( D ) , i = 0 , . Choose a representative ˜ D i for Q i . Consider thefamily D [0 , of operators parametrized by X × [0 , , defined as D [0 , | X ×{ u } := u ˜ D + (1 − u ) ˜ D . Since the family D [0 , is invertible on X × { , } , it defines a family index class in [ D [0 , ] ∈ K ( X × [0 , X × { , } ) ≃ K ( X × (0 , ≃ K − ( X ) . We have [ Q − Q ] = [ D [0 , ] ∈ K − ( X ) . Every element in K − ( X ) can be written as the index of some operator of the form D [0 , above.Remark . Actually, in [MP97] they define C l -invertible perturbations as per-turbations by fiberwise smoothing operators. Our class of smooth C l -invertibleperturbations in Definition 2.12 is larger because we allow the perturbations tobe zeroth order operators. But divided by the homotopy equivalences, they arecanonically isomorphic.Since the above affine structure corresponds to the affine structure on I ( ψ ( D ))under the canonical map (2.13), we have the following corollary. Corollary 2.17.
In the above situations, we have a canonical isomorphism I sm ( D ) ≃ I ( ψ ( D )) between affine spaces over K − ( X ) , induced by the map (2.13). With an abuse of notation we write I ( D ) := I ( ψ ( D )) for a positive order ellipticfamily D .2.3. Groupoids.
Basic definitions.
We recall basic definitions on groupoids and pseudodiffer-ential calculus on them. The material is taken from [DL10].
Definition 2.18 (Groupoids) . Let G and G (0) be two sets. A groupoid structureon G over G (0) is given by the following maps. • An injective map u : G (0) → G , called the unit map. We often identify G (0) with its image u ( G (0) ) ⊂ G . G (0) is called the space of units. • Two surjective maps r, s : G → G (0) , satisfying r ◦ u = s ◦ u = id G (0) . Theseare called range and source map, respectively. • An involution i : G → G, γ γ − , called the inverse map. It satisfies s ◦ i = r . • A map m : G (2) → G, ( γ , γ ) γ · γ , called product, where G (2) = { ( γ , γ ) ∈ G × G | s ( γ ) = r ( γ ) } . Moreover for ( γ , γ ) ∈ G (2) , we have r ( γ · γ ) = r ( γ ) and s ( γ · γ ) = s ( γ ).The following properties must be satisfied: • The product is associative: for any γ , γ , γ in G such that s ( γ ) = r ( γ )and s ( γ ) = r ( γ ), the following equality holds.( γ · γ ) · γ = γ · ( γ · γ ) . • For any γ in G , we have r ( γ ) · γ = γ · s ( γ ) = γ and γ · γ − = r ( γ ).A groupoid structure on G over G (0) is usually denoted by G ⇒ G (0) , where thearrows stand for the source and range maps.For A, B ⊂ G (0) , we use the following notations. G A := s − ( A ) , G B := r − ( B ) , G BA := G A ∩ G B and G | A := G AA . We say a subset A ⊂ G (0) is saturated if it satisfies G A = G A = G | A .Suppose that G ⇒ G (0) is a locally compact groupoid and φ : X → G (0) is anopen surjective map, where X is a locally compact space. The pull back groupoid is the groupoid ∗ φ ∗ ( G ) ⇒ X, where ∗ φ ∗ ( G ) = { ( x, γ, y ) ∈ X × G × X | φ ( x ) = r ( γ ) and φ ( y ) = s ( γ ) } with s ( x, γ, y ) = y , r ( x, γ, y ) = x , ( x, γ , y ) · ( y, γ , z ) = ( x, γ · γ , z ) and ( x, γ, y ) − =( y, γ − , x ). This endows ∗ φ ∗ ( G ) with a structure of locally compact groupoid. More-over the groupoids G and ∗ φ ∗ ( G ) are Morita equivalent (see [DL10, Section 1.2]). Definition 2.19 (Lie groupoids) . We call G ⇒ G (0) a Lie groupoid when G and G (0) are second-countable smooth manifolds with G (0) Hausdorff, and all the struc-tural homomorphisms are smooth and s is a submersion (for definitions of submer-sions between manifolds with corners, we refer to [LN01, Definition 1]).Note that by requiring s to be a submersion, for each x ∈ G (0) , the s -fiber G x isa smooth manifold without boundary or corners.For a Lie groupoid G , let us denote Ω (ker( ds ) ⊕ ker( dr )) → G the half densitybundle of the vector bundle ker( ds ) ⊕ ker( dr ) → G . We also denote this vectorbundle by Ω → G . Then C ∞ c ( G ; Ω ) has a structure of a ∗ -algebra with • The involution given by f ∗ ( γ ) = f ( γ − ). • The convolution product given by f ∗ g ( γ ) = R G s ( γ ) f ( γη − ) g ( η ).For all x ∈ G (0) there is a ∗ -homomorphism λ x : C ∞ c ( G ; Ω ) → B ( L ( G x ; Ω ( G x )))defined by λ x ( f ) ξ ( γ ) = Z G x f ( γη − ) ξ ( η ) . NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 11
Definition 2.20 (Reduced groupoid C ∗ -algebras) . Let G be a Lie groupoid. Thereduced C ∗ -algebra of G , denoted by C ∗ ( G ), is the completion of C ∞ c ( G ; Ω ) withrespect to the norm k f k r = sup x ∈ G (0) k λ x ( f ) k x , where kk x is the operator norm on B ( L ( G x ; Ω ( G x ))). Remark . In general, there are many possible C ∗ -completion of C ∞ c ( G ; Ω )which are not necessarily isomorphic to C ∗ ( G ). For example the full C ∗ -algebra of G is the completion of C ∞ c ( G ; Ω ) with respect to all continuous representations.All the groupoids we actually use in this paper are amenable, so the full and reduced C ∗ -algebras coincide. We use reduced C ∗ -algebras in this paper, in order to makethe argument in subsection 2.3.3 work. Definition 2.22 (Lie algebroids) . A Lie algebroid A = ( p : A → M, [ · , · ] A ) ona smooth manifold M is a vector bundle equipped with a Lie bracket [ · , · ] A : C ∞ ( M ; A ) × C ∞ ( M ; A ) → C ∞ ( M ; A ) together with a homomorphism of fiberbundle p : A → T M called the anchor map, satisfying the following. • The bracket [ · , · ] A is R -bilinear, antisymmetric and satisfies the Jacobi iden-tity. • [ X, f Y ] A = f [ X, Y ] A + p ( X )( f ) Y for all X, Y ∈ C ∞ ( M ; A ) and f ∈ C ∞ ( M ). • p ([ X, Y ] A ) = [ p ( X ) , p ( Y )] for all X, Y ∈ C ∞ ( M ; A ).Given a Lie groupoid G , we associate a Lie algebroid as follows. The vectorbundle is given by ker( ds ) | G (0) = ∪ x ∈ G (0) T G x → G (0) . This has the structure of aLie algebroid over G (0) with the anchor map dr . We denote this Lie algebroid by A G and call it the Lie algebroid of G .For a Lie groupoid G , a submanifold V ⊂ G (0) is said to be transverse to G iffor each x ∈ V , the composition p x ◦ ♯ x : A x G → ( N MV ) x = T x M/T x V is surjective. Definition 2.23 ( G -operators) . Let G be a Lie groupoid. Let E, F → G (0) be twovector bundles. A linear G -operator D is a continuous linear operator D : C ∞ c ( G ; r ∗ E ⊗ Ω ) → C ∞ ( G ; r ∗ F ⊗ Ω )satisfying the following. • The operator D restricts to a continuous family { D x } x ∈ G (0) of linear oper-ators D x : C ∞ c ( G x ; r ∗ E ⊗ Ω ) → C ∞ ( G x ; r ∗ F ⊗ Ω ) such that Df ( γ ) = D s ( γ ) f s ( γ ) ( γ ) ∀ f ∈ C ∞ c ( G ; r ∗ E ⊗ Ω ) . • The following equivariance property holds: U γ D s ( γ ) = D r ( γ ) U γ , where U γ is the map induced by the right multiplication by γ .A linear G -operator D is called pseudodifferential of order m if it satisfies thefollowing. • Its Schwartz kernel k D is a distribution on G that is smooth outside G (0) . • For every distinguished chart ψ : U ⊂ G → Ω × s ( U ) ⊂ R n − p × R p of G , U s ! ! ❇❇❇❇❇❇❇❇ ψ / / Ω × s ( U ) p z z ttttttttt s ( U )the operator ( ψ − ) ∗ Dψ ∗ : C ∞ c (Ω × s ( U ); ( r ◦ ψ − ) ∗ E ) → C ∞ c (Ω × s ( U ); ( r ◦ ψ − ) ∗ F ) is a smooth family parametrized by s ( U ) of pseudodifferentialoperators of order m on Ω.We say that D is smoothing if k D is smooth and that D is compactly supportedif k D is compactly supported. We denote the space of compactly supported order m G -pseudodifferential operators from E to F by Ψ mc ( G ; E, F ). We also denoteΨ mc ( G ; E ) = Ψ mc ( G ; E, E ) and when E is the trivial bundle we denote Ψ mc ( G ) =Ψ mc ( G ; E ).One can show that the space Ψ ∗ c ( G ; E ) of compactly supported pseudodifferential G -operators on E is an involutive algebra.Let us denote the cosphere bundle of A G → G (0) as S ∗ ( G ) → G (0) . Givena G -pseudodifferential operator D , we can associate its principal symbol σ ( D ) ∈ C ∞ c ( S ∗ ( G ); Hom( E ; F )) as follows. Recall that D is given by a family { D x } x ∈ G (0) of pseudodifferential operators on G x . We define σ ( D, ξ ) := σ pr ( D x )( x, ξ ) , where σ pr ( D x ) denotes the principal symbol of the pseudodifferential operator D x .Now we give important examples of Lie groupoids which are building blocks ofgroupoids appearing in this paper. For more examples including the ones below,see [DL10, Example 6.2 and Example 6.4]. Example . If we are given a smooth vector bundle π : E → X , we get a Lie groupoid E ⇒ X by setting s = r = π and multiplicationinduced from the addition on E x for each x . Choosing any smooth family of fiber-wise riemannian metric on E , the C ∗ -algebra C ∗ ( E ) is the fiberwise convolutionalgebra of E , and we have C ∗ ( E ) ≃ C ( E ∗ ) by the fiberwise Fourier transform.An E -pseudodifferential operator D E is equivalent to a family of pseudodifferentialoperators { D x } x ∈ X parametrized by X , and each D x is an operator on the space E x which is translation invariant. Example . If we are given a smoothfiber bundle π : M → X , we get a Lie groupoid M × π M = { ( m, n ) ∈ M × M | π ( m ) = π ( n ) } ⇒ M . Here s ( m, n ) = n , r ( m, n ) = m and ( m, n ) · ( n, l ) =( m, l ). Choosing any smooth family of fiberwise riemannian metric for π , the C ∗ -algebra C ∗ ( M × π M ) is isomorphic to K ( L X ( M )), where L X ( M ) is the Hilbert C ( X )-module given by the completion of C ∞ c ( M ) by the canonical C ( X )-valuedinner product, and the symbol K denotes the C ∗ -algebra of compact operatorsin the sense of a Hilbert module. We have the canonical Morita equivalence (forthe notion of Morita equivalence, see [DL10, Section 1.2]) between C ∗ ( M × π M )and C ( X ). An M × π M -pseudodifferential operator D π is equivalent to a familyof pseudodifferential operators { D x } x ∈ X parametrized by X , and each D x is anoperator on π − ( x ). NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 13
Geometric operators.
Here we define geometric operators, such as spin Diracoperators and signature operators, on a given Lie groupoid G . For detailed discus-sion and other examples, we refer to [LN01].In this paper, we often deal with Z -graded vector bundles and algebras. If weare given two Z -graded vector bundles V and W , or algebras A and B , we alwaysconsider their graded tensor product V ˆ ⊗ W and A ˆ ⊗ B , following the conventions in[LM89, Section 1.1].In this subsubsection, for an Euclidean space E , we denote Cliff( E ) by the ∗ -algebra over R , generated by the elements of E and relations e = − e ∗ , and e = −|| e || · e ∈ E. This construction applies to Euclidean vector bundles as well.First we define spin Dirac operators. In order to do this, we first define ourconvention on spin and spin c structures on vector bundles. Denote ^ GL + k ( R ) → GL + k ( R ) the unique non-trivial covering of GL + k ( R ) for k ≥
2. For k = 1, denote ^ GL + k ( R ) := GL + k ( R ) × Z → GL + k ( R ) the projection to the first factor. Definition 2.26. (Spin/Pre-spin structures on vector bundles)Let E → X be a real vector bundle on a space X with rank k . • A pre-spin structure on E consists of the following data ( o, P ′ ). – An orientation o on the vector bundle E → X . – A principal ^ GL + k ( R )-bundle P ′ → X equipped with a bundle map P ′ → P GL + ( E ) which is equivariant with respect to the canonicalhomomorphism ^ GL + k ( R ) → GL + k ( R ). Here we denoted P GL + ( E ) theoriented frame bundle of E defined by o . • A spin structure on E consists of the following data ( o, g, P ). – An orientation o and a riemannian metric g on the vector bundle E → X . – A principal
Spin k -bundle P → X equipped with a bundle map P → P SO ( E ) which is equivariant with respect to the canonical homomor-phism Spin k → SO k .Note that a pre-spin structure on E together with any riemannian metric g on E defines a spin structure on E uniquely. A pre-spin structure on E can also beregarded as a homotopy class of spin structures on E . See [LM89, pp.131–132]. Definition 2.27 ( Spin c /Pre- spin c structures on vector bundles) . Let E → X bea real vector bundle on a space X . • A pre- spin c structure on E consists of the following data ( o, P ′ ). – An orientation o on the vector bundle E → X . – A principal ^ GL + k ( R ) × Z C ∗ -bundle P ′ → X equipped with a bundlemap P ′ → P GL + ( E ) which is equivariant with respect to the canonicalhomomorphism ^ GL + k ( R ) × Z C ∗ → GL + k ( R ). • A spin c structure on E consists of the following data ( o, g, P ). – An orientation o and a riemannian metric g on the vector bundle E → X . – A principal
Spin ck -bundle P → X equipped with a bundle map P → P SO ( E ) which is equivariant with respect to the canonical homomor-phism Spin ck → SO k . If E has a spin c -structure, by the group homomorphism p : Spin ck = Spin k × Z U (1) → U (1) , [ g, z ] z we get a hermitian line bundle L := P × p C → X. We call L the determinant line bundle associated to the spin c -structure. • A differential spin c structure on E consists of the following data ( o, g, P, ∇ L ). – A spin c structure ( o, g, P ) on E . – A unitary connection ∇ L on the determinant line bundle L . Definition 2.28.
A spin (pre-spin, spin c , pre- spin c , differential spin c ) structure ona Lie groupoid G is a spin (pre-spin, spin c , pre- spin c , differential spin c ) structureon its Lie algebroid A G → G (0) (regarded as a vector bundle).Suppose we are given a metric g on A G . For each x ∈ G (0) , since we have T G x ≃ ( r ∗ A G ) | G x canonically, g induces a riemannian metric on G x . Levi-Civitaconnection on each G x , denoted by ∇ x : C ∞ ( G x ; T G x ) → C ∞ ( G x ; T G x ⊗ T ∗ G x ),combines to give a linear map(2.29) ∇ LC : C ∞ ( G ; r ∗ A G ) → C ∞ ( G ; r ∗ A G ⊗ r ∗ A ∗ G ) . For each X ∈ C ∞ ( G (0) ; A G ), r ∗ X ∈ C ∞ ( G ; r ∗ A G ) gives a first order differentialoperator ∇ LCr ∗ X : C ∞ ( G ; r ∗ A G ) → C ∞ ( G ; r ∗ A G ) , and it is right invariant, i.e., ∇ r ∗ X ∈ Diff ( G ; A G ).Suppose that we are given a spin structure on G . Let S → G (0) be the associatedcomplex spinor bundle. The Levi-Civita connection on r ∗ A G lifts uniquely to aconnection ∇ S : C ∞ ( G ; r ∗ S ) → C ∞ ( G ; r ∗ S ⊗ r ∗ A ∗ G ) and it has a right invarianceproperty as above. Let us denote c : Cliff( A G ) → End( S ) the Clifford action onthe spinor bundle. Definition 2.30 (Spin Dirac operators on Lie groupoids) . Let G be a Lie groupoidequipped with a spin structure. Let { e α } α be a local orthonormal frame of A G → G (0) . The differential operator D S on C ∞ ( G ; r ∗ S ), locally defined as D S := X α c ( e α ) ∇ Sr ∗ e α , gives an element D S ∈ Diff ( G ; S ). We call it the spin Dirac operator on G . Ifthe rank of A G is even, the spinor bundle is naturally Z -graded and the Diracoperator is odd with respect to this grading.Equivalently, the definition of D S can also be described as follows. Given a spinstructure on G , for each x ∈ G (0) the spin structure on G x is associated. If wedenote D Sx the spin Dirac operator for each G x , the family D S = { D Sx } x ∈ G (0) formsa right invariant family, and coincides with the definition given above.This construction generalizes to Clifford modules and Dirac operators on a Liegroupoid G , defined as follows. Definition 2.31 (Clifford modules, connections and Dirac operators on Lie groupoids) . Let G be a Lie groupoid equipped with an orientation and a metric on A G . LetCliff( A G ) → G (0) denote the Clifford bundle of A G . Let W → G (0) be a Cliff( A G )-module bundle. Let us denote c : Cliff( A G ) → End( W ) the Clifford action. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 15 • A continuous linear map ∇ W : C ∞ ( G ; r ∗ W ) → C ∞ ( G ; r ∗ W ⊗ r ∗ A ∗ G ) iscalled an admissible connection if ∇ WX ( c ( Y ) ξ ) = c ( ∇ LCX Y ) ξ + c ( Y ) ∇ WX ( ξ ) , for all ξ ∈ C ∞ ( G ; r ∗ W ) and X, Y ∈ C ∞ ( G ; r ∗ A G ). • A right invariant admissible connection ∇ W is called a Clifford connectionon W . • For a Cliff( A G )-module bundle W equipped with a Clifford connection ∇ W ,the Dirac operator D W ∈ Diff ( G ; W ) is defined by D W := X α c ( e α ) ∇ Wr ∗ e α , using a local orthonormal frame { e α } α for A G .In other words, a Clifford connection is given by a smooth family of Clifford con-nections on r ∗ W → G x for each x ∈ G (0) , satisfying the right invariance. Theassociated Dirac operators D Wx form a right invariant family, and define the ele-ment D W ∈ Diff ( G ; W ) which coincides with the above definition. Example
Spin c -Dirac operators) . Let G be a Lie groupoid equipped with adifferential spin c -structure. The spin c -structure on A G gives the spinor bundle S ( A G ) → G (0) with a Cliff( A G )-module structure. Moreover, as in the classicalcase, the unitary connection ∇ L on the determinant line bundle, together withthe fiberwise Levi-Civita connection ∇ LC for G as in (2.29), determines a Cliffordconnection ∇ S on the complex spinor bundle of A G , denoted by S ( A G ). We callthe associated Dirac operator D S ∈ Diff ( G ; S ( A G )) the spin c -Dirac operator . Example spin c Dirac operators) . Let G be a Lie groupoid equippedwith a differential spin c structure. Let E → G (0) be a Z -graded hermitian vectorbundle with unitary connection ∇ E which preserves the grading. If we denoteby c : Cliff( A G ) → End( S ( A G )) the Clifford action on the spinor bundle, c ˆ ⊗ A G ) → End( S ( A G ) ˆ ⊗ E ) gives a Clifford module structure on S ( A G ) ˆ ⊗ E .For each x ∈ G (0) , denote r x := r | G x : G x → G (0) . We consider the pullbackconnection r ∗ x ∇ E on r ∗ x E → G x for each x ∈ X . These combine to give a rightinvariant continuous linear map r ∗ ∇ E : C ∞ ( G ; r ∗ E ) → C ∞ ( G ; r ∗ E ⊗ r ∗ A G ) . The map ∇ S ˆ ⊗ E : C ∞ ( G ; r ∗ ( S ( A G ) ˆ ⊗ E )) → C ∞ ( G ; r ∗ (( A G ) ˆ ⊗ E ) ⊗ r ∗ A ∗ G ) ∇ S ˆ ⊗ E := ∇ S ˆ ⊗ ⊗ r ∗ ∇ E gives a Clifford connection on S ˆ ⊗ E . We call the associated Dirac operator D S ˆ ⊗ E ∈ Diff ( G ; S ( A G ) ˆ ⊗ E ) the spin Dirac operator twisted by ( E, ∇ E ). Example . Let G be a Lie groupoid equipped with ametric on A G . As in the classical case, the complexified exterior algebra bundle ∧ C A ∗ G → G (0) has the Cliff( A G )-module structure. The fiberwise Levi-Civitaconnection as in (2.29) induces a Clifford connection on ∧ C A ∗ G → G (0) . We callthe associated Dirac operator D sign ∈ Diff ( G ; ∧ C A ∗ G ) the signature operator on G . Of course this is the family consisting of the signature operator on G x for each x ∈ G (0) . If the rank of A G is even (let us denote it by n ), the exterior algebra bundle ∧ C A ∗ G is Z -graded by the Hodge star operator. We only consider thisgrading on complexified exterior algebra bundles of even-rank real vector bundlesin this paper. Under this grading, the signature operators are odd.2.3.3. Ellipticity and index classes.
From now on we assume that G (0) is compact.A G -pseudodifferential operator D is called elliptic if σ ( D ) is invertible. If D ∈ Ψ mc ( G ; E, F ) is elliptic, as in the classical situations, it has a parametrix Q ∈ Ψ − mc ( G ; F, E ) such that DQ − Id ∈ Ψ −∞ c ( G ; F ) and QD − Id ∈ Ψ −∞ c ( G ; E ).For an elliptic operator F ∈ Ψ c ( G ; E, F ), we can define the index class
Ind( F ) ∈ K ( C ∗ ( G )) as follows. For simplicity we work in the case where coefficient bundlesare trivial; for the general case we use the nontrivial-coefficient version of alge-bras (such as C ∗ ( G ; E )) which are Morita equivalent to trivial coefficient versions( C ∗ ( G )), and proceed exactly in the same way. We have Ψ c ( G ) ⊂ M ( C ∗ ( G )),where M ( C ∗ ( G )) denotes the multiplier algebra of C ∗ ( G ). We denote Ψ c ( G ) thecompletion of Ψ c ( G ) by the norm induced from M ( C ∗ ( G )). The ∗ -homomorphism σ : Ψ c ( G ) → C ∞ ( S ∗ ( G )) extends to the ∗ -homomorphism σ : Ψ c ( G ) → C ( S ∗ ( G ))and fits into the exact sequence0 → C ∗ ( G ) → Ψ c ( G ) σ → C ( S ∗ ( G )) → . We denote the connecting element associated to the above exact sequence byind G (0) ( G ) ∈ KK ( C ( S ∗ ( G )) , C ∗ ( G )).For F ∈ Ψ c ( G ), we say it is elliptic if σ ( F ) is invertible. When F is elliptic, itdefines a class [ σ ( F )] ∈ K ( C ( S ∗ ( G ))). We define the index class of the ellipticoperator F as Ind( F ) := [ σ ( F )] ⊗ ind G (0) ( G ) ∈ K ( C ∗ ( G )) . Suppose there is a compact space Y and a continuous map f : G (0) → Y suchthat f − ( y ) is saturated for G for all y ∈ Y . Then C ( Y ) acts canonically on C ∗ -algebras associated to G above, namely Ψ c ( G ), C ∗ ( G ) and C ( S ∗ ( G )). Theseactions commute with elements in these algebras, so for an elliptic operator F ∈ Ψ c ( G ), we get a finer index class,(2.35) Ind Y ( F ) ∈ KK ( C ( Y ) , C ∗ ( G )) , which maps to Ind( F ) by the ∗ -homomorphism C → C ( Y ).Next, we consider the case where an elliptic operator F ∈ Ψ c ( G ) is invertiblein some closed saturated subset of G (0) . Recall that we call a subset A ⊂ G (0) saturated if G A = G AA = G A . Assume X ⊂ G (0) is closed and saturated, and G X is NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 17 amenable. We have the following diagram, whose rows and columns are all exact.(2.36) 0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / C ∗ ( G | G (0) \ X ) / / (cid:15) (cid:15) C ∗ ( G ) / / (cid:15) (cid:15) C ∗ ( G | X ) / / (cid:15) (cid:15) / / Ψ c ( G | G (0) \ X ) / / σ (cid:15) (cid:15) Ψ c ( G ) / / σ (cid:15) (cid:15) Ψ c ( G | X ) / / σ (cid:15) (cid:15) / / C ( S ∗ G | G (0) \ X ) / / (cid:15) (cid:15) C ( S ∗ G ) / / (cid:15) (cid:15) C ( S ∗ G | X ) / / (cid:15) (cid:15)
00 0 0Throughout this article, we denote the connecting element of the top row of theabove exact sequence by ∂ G (0) \ X ( G ) ∈ KK ( C ∗ ( G | X ) , C ∗ ( G | G (0) \ X )). We de-note the pullback C ∗ -algebra of the downright corner of the above diagram byΣ G (0) \ X ( G ) := Ψ c ( G | X ) ⊕ X C ( S ∗ G ). We have the exact sequence(2.37) 0 → C ∗ ( G | G (0) \ X ) → Ψ c ( G ) σ f,X → Σ G (0) \ X ( G ) → . We call σ f,X the full symbol map with respect to X . Since we are assumingthat G | X is amenable, this exact sequence is semisplit. Denote ind G (0) \ X ( G ) ∈ KK (Σ G (0) \ X ( G ) , C ∗ ( G | G (0) \ X )) the connecting element of the exact sequence (2.37).Suppose F ∈ Ψ c ( G ) is elliptic and its restriction to G | X , F | X ∈ Ψ c ( G | X ),is invertible. We call such an operator as fully elliptic with respect to X . Thismeans that σ f,X ( F ) ∈ Σ G (0) \ X ( G ) is invertible, so it defines a class [ σ f,X ( F )] ∈ K (Σ G (0) \ X ( G )). The classInd G (0) \ X ( F ) := [ σ f,X ( F )] ⊗ ind G (0) \ X ( G ) ∈ K ( C ∗ ( G | G (0) \ X ))is called the full index class of the operator F with respect to G (0) \ X .In the case of an elliptic positive order pseudodifferential operator D ∈ Ψ mc ( G ),we also define the index class as follows. Let us denote ψ ( x ) = x √ x and considerthe operator ψ ( D ). This operator satisfies ψ ( D ) ∈ Ψ c ( G ), as shown in [V] (notethat it is not in Ψ c ( G ) in general). Then we define its index class asInd( D ) := Ind( ψ ( D )) ∈ K ( C ∗ ( G )) . In the case D ∈ Ψ mc ( G ) is invertible on a closed saturated subset X ⊂ G (0) , thebounded transform ψ ( D ) is fully elliptic with respect to X. In this case we also saythat D is fully elliptic with respect to X, and define its full index class asInd G (0) \ X ( D ) := Ind G (0) \ X ( ψ ( D )) ∈ K ( C ∗ ( G | G (0) \ X )) . Deformation goupoids and blowup groupoids.
Here we recall the two con-structions of groupoids; deformation to the normal cone and blowup. For detailswe refer to [DS17].
First we explain these constructions for manifolds. Let Y be a manifold and X a locally closed submanifold. First we explain in the case X does not intersect withthe boundary of Y . Denote by N YX the normal bundle of X in Y . The deformationto the normal cone , denoted by DN C ( Y, X ), is a smooth manifold which is obtainedby gluing N YX × { } with Y × R ∗ . Choose an exponential map θ : U ′ → U , where U ′ is an open neighborhood of the 0-section in N YX and U is an open neighborhoodof X in Y . The smooth structure is defined in the way that the following maps arediffeormorphisms onto open subsets of DN C ( Y, X ). • the inclusion Y × R ∗ → DN C ( Y, X ). • the map Θ : Ω ′ := { (( x, ξ ) , λ ) ∈ N YX × R | ( x, λξ ) ∈ U ′ } → DN C ( Y, X )defined by Θ(( x, ξ ) ,
0) = (( x, ξ ) ,
0) and Θ(( x, ξ ) , λ ) = ( θ ( x, λξ ) , λ ) ∈ Y × R ∗ if λ = 0.This condition defines the smooth structure on DN C ( Y, X ) uniquely and it does notdepend on the choice of θ . We also denote by DN C + ( Y, X ) := Y × ( R ∗ + ) ⊔ N YX ×{ } ∈ DN C ( Y, X ).There exists a canonical action of the group R ∗ on the manifold DN C ( Y, X ),called the gauge action . This is defined by, for an element λ ∈ R ∗ , λ. ( w, t ) = ( w, λt )and λ. (( x, ξ ) ,
0) = (( x, λ − ξ ) ,
0) (with t ∈ R ∗ , w ∈ Y , x ∈ X and ξ ∈ (( N YX ) x )).This action is free and locally proper on the open subset DN C ( Y, X ) \ X × R .The DN C -construction has the functoriality as follows. Let f : ( Y, X ) → ( Y ′ , X ′ ) be a smooth map between the pair as above. We can show that f in-duces a smooth map DN C ( f ) : DN C ( Y, X ) → DN C ( Y ′ , X ′ ) . This map is equivariant with respect to the gauge action by R ∗ .The blowup Blup ( Y, X ) is a smooth manifold which is a union of Y \ X with P ( N YX ), the projective space of the normal bundle N YX . We also define the sphericalblowup SBlup ( Y, X ), which is a manifold with boundary obtained by gluing Y \ X with the sphere bundle S ( N YX ). The definition is as follows. Blup ( Y, X ) := (
DN C ( Y, X ) \ X × R ) / R ∗ SBlup ( Y, X ) := (
DN C + ( Y, X ) \ X × R + ) / R ∗ + . Here we take quotient by the gauge action.The functoriality of
Blup is described as follows. Let f : ( Y, X ) → ( Y ′ , X ′ ) be asmooth map between the pair as above. Let U f := DN C ( Y, X ) \ DN C ( f ) − ( X ′ × R ). Denote Blup f ( Y, X ) := U f / R ∗ . Then we obtain a smooth map Blup ( f ) : Blup f ( Y, X ) → Blup ( Y ′ , X ′ ). Similarly we obtain a smooth map SBlup ( f ) : SBlup f ( Y, X ) → SBlup ( Y ′ , X ′ ).Let us explain the case where Y is a manifold with corners and X meets ∂Y . X is called an interior p -submanifold of Y if it is a smooth submanifold which meetsall the boundary faces of Y transversally, and covered by coordinate neighborhoods { U, ( v, w ) } in Y such that v is a tuple of boundary defining functions on Y and U ∩ X = { w i = 0 | i = 1 , · · · , codim X } . If X is an interior p -submanifold of Y ,we consider the inward normal bundle ( N YX ) + and we can define DN C + ( Y, X ) =( N YX ) + × { } ⊔ Y × R ∗ + . This manifold admits the gauge action by R ∗ + . We define SBlup ( Y, X ) by the same formula as above.Now we apply these constructions to inclusions of Lie groupoids. Let Γ be aclosed Lie subgroupoid of a Lie groupoid G . First we assume that Γ does not meet NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 19 the boundary of G . Using the functoriality of the DN C construction, we get a Liegroupoid
DN C ( G, Γ) ⇒ DN C ( G (0) , Γ (0) ) , where the source and range maps are DN C ( s ) and DN C ( r ), and the multiplicationis DN C ( m ). Denoting the subset ^ DN C ( G, Γ) := U r ∩ U s ⊂ DN C ( G, Γ), we define
Blup r,s ( G, Γ) := ^ DN C ( G, Γ) / R ∗ . We get a Lie groupoid Blup r,s ( G, Γ) ⇒ Blup ( G (0) , Γ (0) ) , where the source and range maps are Blup ( s ) and Blup ( r ), and the multiplica-tion is Blup ( m ). Using ^ DN C + ( G, Γ) := ^ DN C ( G, Γ) ∩ DN C + ( G, Γ) instead of ^ DN C ( G, Γ) in the above construction and taking the quotient by the gauge actionof R ∗ + , we get a Lie groupoid SBlup r,s ( G, Γ) ⇒ SBlup r,s ( G (0) , Γ (0) ) . In the case Γ meets the boundary of G , if Γ is a p -submanifold of G , we can definethe Lie groupoids DN C + ( G, Γ) and
SBlup r,s ( G, Γ) in the same way as above.2.4. b , Φ , e -calculus and corresponding groupoids. In this subsection, werecall the basics of b , Φ, and e -calculus, in terms of the groupoid approach. Thesettings are as follows. • Let (
M, ∂M ) be a compact manifold with closed boundaries. Here closedmeans that ∂M is a compact manifold without boundary. • Let ∂M = ⊔ mi =1 H i be the decomposition into connected components. • Let π i : H i → Y i be a smooth oriented fiber bundle structure with closedfibers. The typical fibers are allowed to vary from one component to an-other. We also denote Y = ⊔ i Y i and π : ⊔ i π i . • Let x ∈ C ∞ ( M ) be a boundary defining function. Here a boundary definingfunction is a smooth function x on M such that x − (0) = ∂M , x > M and dx ( p ) = 0 for all p ∈ ∂M .Define V ( M ) := C ∞ ( M ; T M ). Consider the subspaces V b ( M ), V Φ ( M ) and V e ( M ) of V ( M ), defined as follows. V b ( M ) := { ξ ∈ V ( M ) | ξ | ∂M is tangent to ∂M }V Φ ( M ) := { ξ ∈ V ( M ) | ξ | ∂M is tangent to the fibers of π and ξ ( x ) ∈ x C ∞ ( M ) }V e ( M ) := { ξ ∈ V ( M ) | ξ | ∂M is tangent to the fibers of π } . These are Lie subalgebras of V ( M ). Using the Serre-Swan theorem, we see thatthere exist smooth vector bundles T b M , T Φ M , and T e M over M such that V b ( M ) = C ∞ ( M ; T b M ), V Φ ( M ) = C ∞ ( M ; T Φ M ), and V e ( M ) = C ∞ ( M ; T e M ). Note that,restricted to ˚ M , these vector bundles are canonically isomorphic to T ˚ M .A b , Φ, e -metric on M is a smooth riemannian metric on the vector bundles T b M , T Φ M , and T e M , respectively. We also call a riemannian metric on ˚ M a b ,Φ, e -metric, if it extends to a smooth metric on these vector bundles. Examplesof such metrics are described as follows. Let T ∂M ≃ π ∗ T Y ⊕ T V ∂M be a fixedsplitting for the boundary fibration. We consider three classes of metrics on M , which have the following forms near the boundary. g b = dx x ⊕ g ∂M g Φ = dx x ⊕ π ∗ g Y x ⊕ g π (2.38) g e = dx x ⊕ π ∗ g Y x ⊕ g π . (2.39)Here g ∂M and g Y are some riemannian metrics on ∂M and Y respectively, and g π is a fiberwise riemannian metric for π . These are examples of b , Φ, e -metricsrespectively, and a metric of the form above is called rigid .Denote by Ω b , Ω Φ and Ω e the bundle of smooth densities on the vector bundle T b M , T Φ M and T e M , respectively, and we call them b , Φ, e -density bundles,respectively.We define the space of b , Φ, and e -pseudodifferential operators. Let Diff ∗ b ( M )denote the filtered algebra generated by V b ( M ) and C ∞ ( M ). An element in thisalgebra is called a b -differential operator. The space of b -pseudodifferential oper-ators contains this algebra. We define the algebra Diff ∗ Φ ( M ) and Diff ∗ e ( M ) in theanalogous way, and the analogous result holds. This space of pseudodifferential op-erators can be described in two ways, microlocal approach and groupoid approach .The microlocal approach originates from Melrose [Mel93] for the b -case, and the Φ-case was given by Mazzeo and Melrose [MM98] and the e -case was given by Mazzeo[Maz91]. In this paper, we use the groupoid approach, which is more suited with K -thoretic approach using C ∗ -algebras, as explained below. For relations betweenthese two approaches, see [PZ19, Section 6.6].2.4.1. The groupoid approach.
Here we recall the groupoid approach. We can con-struct groupoids G b , G Φ , G e associated to a manifold with fibered boundary, anddefine b , Φ, e -pseudodifferential operators as operators in Ψ ∗ c ( G b ), Ψ ∗ c ( G Φ ) andΨ ∗ c ( G e ), respectively. The groupoid corresponding to b -calculus is introduced by[Mon99], and a general construction by [Nis00] includes the Φ and e cases. Herewe use the description using the blowup construction of groupoids. We use theblowup construction for groupoids explained in the subsubsection 2.3.4. For thisdescription, also see [PZ19, Section 13]. • The b -groupoids.We start with the pair groupoid M × M ⇒ M . Note that this does notsatisfy the definition of Lie groupoid given in Definition 2.19, since s is not asubmersion; however it is easy to see that the spherical blowup constructionis also valid in this case. Consider the subgroupoid ⊔ i ( H i × H i ) ⇒ ∂M of M × M . Then b -groupoid of M is defined by G b := SBlup r,s ( M × M, ⊔ i ( H i × H i )) ⇒ M. ˚ M and H i , 1 ≤ i ≤ m are saturated subsets of G b , and we have G b = ˚ M × ˚ M ⊔ ⊔ i ( H i × H i × R ) ⇒ M. • The Φ-groupoids.Consider the subgroupoid ∂M × π ∂M = ⊔ i ( H i × π i H i ) ⇒ ∂M of G b .Then Φ-groupoid of M is defined by G Φ := SBlup r,s ( G b , ∂M × π ∂M ) ⇒ M. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 21
Let us look at the singular part. The inward normal bundle groupoid of H i × π i H i in G b is H i × π i T Y i × π i H i × R × R + ⇒ H i × R + s ( x, v, y, a, b ) = ( y, b ) r ( x, v, y, a, b ) = ( x, b ) m (( x, v, y, a, b ) , ( y, w, z, a ′ , b )) = ( x, v + w, z, a + a ′ , b )And the gauge action by λ ∈ R ∗ + is given by ( x, v, y, a, b ) ( x, λv, y, a, λb ).Thus dividing by this gauge action, we get an isomorphism G Φ | H i ≃ H i × π i H i × π i T Y i × R (this can be seen by restricting to b = 1). In other words we have G Φ = ˚ M × ˚ M ⊔ ∂M × π ∂M × π T Y × R ⇒ M. • The e -groupoids.Consider the groupoid M × M ⇒ M and its subgroupoid ∂M × π ∂M = ⊔ i ( H i × π i H i ) ⇒ ∂M . Then e -groupoid of M is defined by G e = SBlup r,s ( M × M, ∂M × π ∂M ) ⇒ M. Let us look at the singular part. The inward normal bundle groupoid of H i × π i H i in M × M is H i × π i T Y i × π i H i × ( R + ) ⇒ H i × R + s ( x, v, y, a, b ) = ( y, b ) r ( x, v, y, a, b ) = ( x, a ) m (( x, v, y, a, b ) , ( y, w, z, b, c )) = ( x, v + w, z, a, c )And the gauge action by λ ∈ R ∗ + is given by ( x, v, y, a, b ) ( x, λv, y, λa, λb ).So dividing by this action, we get G e | H i ≃ H i × π i H i × π i ( T Y i ⋊ R ∗ + )where R ∗ + acts on T Y i by multiplication.We apply the general construction of the subsubsection 2.3.3 to these groupoids.Recall that a G ✷ operator P ∈ Ψ c ( G ) is called elliptic if its symbol σ ( P ) ∈ C ( S ∗ G ✷ ) is invertible. Note that ∂M ⊂ M = G (0) ✷ is a closed saturated sub-manifold. Applying the construction in (2.37) to the case ∂M = X , we get theexact sequence 0 → C ∗ ( G ✷ | ˚ M ) → Ψ c ( G ✷ ) σ f,∂M → Σ ˚ M ( G ✷ ) → . We say P ∈ Ψ c ( G ) is fully elliptic if σ f,∂M ( P ) ∈ Σ ˚ M ( G ✷ ) is invertible. Recallthat if P is fully elliptic it defines the index class Ind ˚ M ( P ) ∈ K ( C ∗ ( G ✷ | ˚ M )) = K ( K ( L ( ˚ M ))) ≃ Z . By the exact sequence above, the restriction of a fully ellipticoperator P to ˚ M is Fredholm, and its Fredholm index corresponds to Ind ˚ M ( P ) ∈ Z . Indices of geometric operators on manifolds with fiberedboundaries : the case without perturbations
The definition of indices.
In subsections 3.1 and 3.2, for simplicity we onlyconsider spin Dirac operators, without any twists or perturbations. For our conven-tions on spin structures and pre-spin structures on vector bundles, see Definition2.26.For a given even dimensional compact manifold with fibered boundary ( M ev , π : ∂M → Y ) equipped with pre-spin structures on T M and
T Y as well as a riemannianmetric on the vertical tangent bundle of the boundary fibration, T V ∂M , for whichthe fiberwise spin Dirac operator forms an invertible family, we associate its indexin Z . This index can be realized using either Φ-metrics or e -metrics. In the nextsection, we show that they actually coincide. Also we show some properties of thisindex, using groupoid deformation techniques. For simplicity, we only work in thecase where Y is odd dimensional. The case where Y is even dimensional can betreated similarly. Remark . For a manifold with fibered boundary (
M, π : ∂M → Y ), assume thatwe are given pre-spin structures on T M and
T Y . The pre-spin structure on
T M induces a pre-spin structure on
T ∂M . Choose any splitting
T ∂M = T V ∂M ⊕ π ∗ T Y .We introduce the pullback pre-spin structure on π ∗ T Y . Then a pre-spin structureon T V ∂M is induced, and it does not depend on the choice of the splitting of T ∂M .We always consider this choice of pre-spin structure on T V ∂M . In particular, whenwe are given pre-spin structures on T M and
T Y as well as a riemannian metric on T V ∂M , a spin structure on T V ∂M is canonically induced and the fiberwise spinDirac operator D π is defined.First, we show that for a fixed spin structure on T Φ M or T e M which has a prod-uct decomposition at the boundary, we get the Fredholmness from the invertibilityof the fiberwise Dirac operators.Let ( M ev , π : ∂M → Y odd ) be a compact manifold with fibered boundary,equipped with pre-spin structures on T M and
T Y , as well as a riemannian metric g π on T V ∂M .Fix some riemannian metric g Y on Y . Choose a smooth riemannian metric g Φ ( g e ) for A G Φ ( A G e ), whose restriction to A G Φ | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R x ( A G e | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R x ) can be written as g Φ | ∂M = g π ⊕ π ∗ g Y ⊕ dx g e | ∂M = g π ⊕ π ∗ g Y ⊕ dx . For example rigid metrics as in (2.38) and (2.39) on the interior ˚ M extends tometrics on A G Φ and A G e satisfying this condition. Let D Φ ∈ Diff ( G Φ ; S ( A G Φ )), D e ∈ Diff ( G e ; S ( A G e )) be the spin Dirac operators associated to the metrics g Φ and g e , respectively. Denote D π the fiberwise spin Dirac operators for the boundaryfibration structure π ( D π is a family of operators parametrized by Y ). Proposition 3.2.
In the above settings, assume that the family D π is invertible.Then both D Φ and D e are Fredholm, as operators on ˚ M with metric induced from g Φ and g e , respectively.Proof. First we prove in the Φ-case. We have the decomposition(3.3) G Φ = ˚ M × ˚ M ⊔ ∂M × π ∂M × π T Y × R ⇒ M. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 23
The restriction of D Φ to the singular part ∂M × π ∂M × π T Y × R is a family ofoperators D Φ | ∂M = { D Φ ,y } y ∈ Y parametrized by Y , and each D Φ ,y is given by D Φ ,y : C ∞ c ( π − ( y ) × T y Y × R ; S ( π − ( y )) ˆ ⊗ S ( T y Y × R )) → C ∞ c ( π − ( y ) × T y Y × R ; S ( π − ( y )) ˆ ⊗ S ( T y Y × R )) D Φ ,y = D π ˆ ⊗ ⊗ D T y Y × R (3.4)Here S ( T y Y × R ) and D T y Y × R is the translation invariant spinor bundle and theDirac operator over the Euclidean space T y Y × R with respect to the metric g Y ⊕ dx .The operators D π ˆ ⊗ ⊗ D T y Y × R anticommute, and since D π is invertible, wesee that D Φ ,y is invertible for all y ∈ Y . So D Φ | ∂M is invertible. Thus D Φ is fullyelliptic and we get the Fredholm indexInd ˚ M ( D Φ ) ∈ K ( C ∗ ( G Φ | ˚ M )) ≃ Z . Next, we prove the Proposition in the e -case. The restriction of D e to theboundary component G e | ∂M = ∂M × π ∂M × π ( T Y ⋊ R ∗ + ) is described as follows. D e | ∂M is a family of operators parametrized by Y , D e | ∂M = { D e,y } y ∈ Y , and foreach y ∈ Y , we have D e,y : C ∞ ( π − ( y ) × ( T y Y ⋊ R ∗ + ); S ( π − i ( y )) ˆ ⊗ S ( T y Y ⋊ R ∗ + )) → C ∞ ( π − ( y ) × ( T y Y ⋊ R ∗ + ); S ( π − i ( y )) ˆ ⊗ S ( T y Y ⋊ R ∗ + )) D e,y = D π − ( y ) ˆ ⊗ ⊗ D T y Y ⋊R ∗ + . (3.5)Here, S ( T y Y ⋊ R ∗ + ) and D T y Y ⋊R ∗ + denotes the spinor bundle and its Dirac operatoron the Lie group T y Y ⋊ R ∗ + with the translation invariant spin structure and metric g Y ⊕ dx . From the same argument as in the Φ-case above, we get the full ellipticityof D e . (cid:3) Next we show that the index only depends on the choice of the fiberwise metric g π for the boundary fibration, and does not depend on the choice of base metricsas well as interior metrics. We consider the following situations.(1) Pre-spin structures P ′ M and P ′ Y on T M and
T Y , respectively, are fixed.(2) A riemannian metric g π on T V ∂M is fixed. Assume that the associatedfiberwise spin Dirac operator D π is invertible.(3) A smooth riemannian metric g Φ for A G Φ ≃ T Φ M → M , whose restrictionto A G Φ | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R can be written as g Φ | ∂M = g π ⊕ π ∗ g T Y ⊕ R , where g T Y ⊕ R is some riemannian metric on the vector bundle T Y ⊕ R → Y .(4) A smooth riemannian metric g e for A G e ≃ T e M → M , whose restrictionto A G e | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R can be written as g e | ∂M = g π ⊕ π ∗ g T Y ⊕ R , where g T Y ⊕ R is some riemannian metric on the vector bundle T Y ⊕ R → Y .(5) Let us denote the spin Dirac operators associated to g Φ and g e by D Φ and D e , respectively. Proposition 3.6 (Stability) . Under the above situations,
Ind ˚ M ( D Φ ) and Ind ˚ M ( D e ) only depend on the data (1) and (2) above. It does not depend on the choice of g Φ and g e which satisfy the conditions (3) and (4) above.Proof. This can be proved by a simple homotopy argument. We prove in the Φ-case. The e -case is similar. Let g and g be two choices of smooth metrics on T Φ M which satisfies the condition (3) (for the same fiberwise metric g π ). Let usdenote D and D the spin Dirac operator with respect to these metrics. Letting g t Φ = tg + (1 − t ) g for t ∈ [0 , T Φ M connecting g and g . Note that for all t ∈ [0 , g t Φ satisfies the condition(3).Let us consider the groupoid G Φ × [0 , ⇒ M × [0 , { g t Φ } t ∈ [0 , givea smooth metric on A G Φ × [0 , M , we get the spin Dirac operator D [0 , . Since D [0 , | ∂M ×{ t } is invertible forall t , we get the indexInd ˚ M × [0 , ( D [0 , ) ∈ K ( C ∗ ( G Φ | ˚ M ) × [0 , ≃ Z . and we have, denoting the ∗ -homomorphisms ev t : C ∗ ( G Φ | ˚ M ) × [0 , → C ∗ ( G| ˚ M ×{ t } ) for t ∈ [0 , ev ) ∗ Ind ˚ M × [0 , ( D [0 , ) = Ind ˚ M ( D ) ∈ K ( C ∗ ( G Φ | ˚ M )) ≃ Z , and( ev ) ∗ Ind ˚ M × [0 , ( D [0 , ) = Ind ˚ M ( D ) ∈ K ( C ∗ ( G Φ | ˚ M )) ≃ Z . Since ( ev t ) ∗ : K ( C ∗ ( ˚ M × ˚ M × [0 , ≃ Z → K ( C ∗ ( ˚ M × ˚ M × { t } )) ≃ Z is theidentity map on Z for all t ∈ [0 , (cid:3) By Proposition 3.6, in order to define the indices of spin Dirac operator D Φ and D e , we only have to specify the data (1) and (2) listed before the Proposition 3.6.So we define the index of the triple ( P ′ M , P ′ Y , g π ) by the above number. Definition 3.7.
Let ( M ev , π : ∂M → Y odd ) be a compact manifold with fiberedboundary. For a triple ( P ′ M , P ′ Y , g π ), where P ′ M and P ′ Y are pre-spin structureson T M and
T Y , respectively, and g π is a riemannian metric on T V ∂M such thatthe associated fiberwise spin Dirac operator is an invertible family, we define itsΦ-index and e -index as Ind Φ ( P ′ M , P ′ Y , g π ) := Ind ˚ M ( D Φ )Ind e ( P ′ M , P ′ Y , g π ) := Ind ˚ M ( D e ) . Here D Φ ∈ Diff ( G Φ ; S ( A G Φ )) and D e ∈ Diff ( G e ; S ( A G e )) are spin Dirac opera-tors which are defined by arbitrary choices of data (3) and (4).3.2. Properties.
First, we show that two indices Ind Φ ( P ′ M , P ′ Y , g π ) and Ind e ( P ′ M , P ′ Y , g π )actually coincide. Proposition 3.8 (Equality of Φ and e -indices) . For a compact manifold with fiberedboundary ( M ev , π : ∂M → Y odd ) , assume that we are given pre-spin structures P ′ M and P ′ Y on T M and
T Y , respectively, and a riemannian metric g π on T V M , forwhich the fiberwise spin Dirac operator D π is an invertible family. Then we have Ind Φ ( P ′ M , P ′ Y , g π ) = Ind e ( P ′ M , P ′ Y , g π ) . NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 25
Proof.
Let us fix a splitting
T ∂M ≃ π ∗ T Y ⊕ T V ∂M , a boundary defining function x , and a metric g Y on Y . Fix a collar neighborhood U of ∂M and also fix anidentification ∂M × [0 , a ) ≃ U which is compatible with x . Then consider themetric g Φ and g e defined as (2.38) and (2.39): these satisfy the conditions above.The idea is to consider the family of metrics(3.9) g e ( t ) = dx x ( x + t ) ⊕ π ∗ g Y x ⊕ g π . and justify the limit t →
0. This can be realized as follows.Consider the Lie algebra
T M × [0 , → M × [0 ,
1] with the canonical Lie bracket(not to be confused with T ( M × [0 , C ∞ ( M × [0 , C ∞ ( M × [0 , T M × [0 , V := (cid:26) V ∈ C ∞ ( M × [0 , T M × [0 , V | ∂M × [0 , ∈ C ∞ ( ∂M × [0 , T V ∂M × [0 , , and V ( x ) ∈ x ( x + t ) C ∞ ( M × [0 , t ) (cid:27) . This is a Lie subalgebra of C ∞ ( M × [0 , T M × [0 , A → M × [0 , C ∞ ( M × [0 , A ) ≃ V as a C ∞ ( M × [0 , p : C ∞ ( M × [0 , A ) → C ∞ ( M × [0 , T ( M × [0 , V ֒ → C ∞ ( M × [0 , T M × [0 , ֒ → C ∞ ( M × [0 , T ( M × [0 , A → M × [0 ,
1] with anchor p . We have the following. • A | M ×{ t } = A G e for all t ∈ (0 , • A | M ×{ } = A G Φ . • The metric g A on A , defined as (see (3.9)) g A := ( g e ( t ) on M × (0 , t g Φ on M × { } , gives a smooth metric on A .Since p | ˚ M × [0 , is injective, ( A , p ) is an almost injective Lie algebroid. By [Deb01],there exists a smooth Lie groupoid G → M × [0 ,
1] such that its Lie algebroid A G is isomorphic to ( A , p ).We give the explicit definition of G . As a set, G = G Φ × { } ⊔ G e × (0 , M × ˚ M × [0 , ⊔ ∂M × π ∂M × π ( T Y ⋊ R ∗ + ) × (0 , ⊔ ∂M × π ∂M × π T Y × R × { } , We describe the smooth structure as follows. Recall we have fixed a tubularneighborhood U of ∂M . Outside the collar neighborhood, the smooth structure G \ G U × [0 , U × [0 , is given in the canonical way. On U , recall we have fixed the iso-morphism U ≃ ∂M × [0 , a ) and T U ≃ ( T V ∂M ⊕ π ∗ T Y ⊕ R ) × [0 , a ). Also fixan exponential map for T ∂M . On G U × [0 , U × [0 , , we consider the following exponentialmap. ( T V ∂M ⊕ π ∗ T Y ⊕ R ) × ∂M × [0 , a ) × [0 , → G U × [0 , U × [0 , ( ∂ z , ∂ y , ξ, p, x, t ) ((exp p ( ∂ z + x∂ y ) , x + e x ( x + t ) ξ ) , ( p, x ) , t ) ∈ U × U × [0 ,
1] for x > (exp p ( ∂ z ) , p, ∂ y , e tξ , t ) ∈ ∂M × π ∂M × π ( T Y ⋊ R ∗ + ) × (0 ,
1] for x = 0 , t ∈ (0 , (exp p ( ∂ z ) , p, ∂ y , ξ, ∈ ∂M × π ∂M × π T Y × R × { } for x = t = 0 . We define the smooth structure on G U × [0 , U × [0 , so that the above map is a diffeomor-phism. This smooth structure does not depend on any of the choices. Note alsothat, restricted to ∂M × [0 , ⊂ M × [0 , G ∂M × [0 , ≃ DN C ( ∂M × π ∂M × π ( T Y ⋊R ∗ + ) , ∂M × π ∂M × π T Y ) | ∂M × [0 , ⇒ ∂M × [0 , . We consider the spin structure on A induced from the one on T M and the metric g A , and denote the associated spin Dirac operator by D ∈
Diff ( G ; S ( A )). Wecan show that D| ∂M × [0 , ∈ Diff ( G ∂M × [0 , ; S ( A ∂M × [0 , )) is invertible, as follows.By the invertibility of D π , there exists c > D π ≥ c . The operator D| ∂M × [0 , is given by a family of operators { D y,t } ( y,t ) ∈ Y × [0 , parametrized by Y × [0 , D y,t has the form (3.4) for t = 0 and (3.5) for t ∈ (0 , D y,t ≥ D π ˆ ⊗ ≥ c . Thus D| ∂M × [0 , is invertible.So we get the index classInd ˚ M × [0 , ( D ) ∈ K ( C ∗ ( G| ˚ M × [0 , )) ≃ Z . and we have, denoting the ∗ -homomorphisms ev t : C ∗ ( G| ˚ M × [0 , ) → C ∗ ( G| ˚ M ×{ t } )for t ∈ [0 , ev ) ∗ Ind ˚ M × [0 , ( D ) = Ind ˚ M ( D Φ ) ∈ K ( C ∗ ( G Φ | ˚ M )) ≃ Z , and( ev ) ∗ Ind ˚ M × [0 , ( D ) = Ind ˚ M ( D e ) ∈ K ( C ∗ ( G e | ˚ M )) ≃ Z . Since ( ev t ) ∗ : K ( C ∗ ( ˚ M × ˚ M × [0 , ≃ Z → K ( C ∗ ( ˚ M × ˚ M × { t } )) ≃ Z is theidentity map on Z for all t ∈ [0 , (cid:3) Next we show the gluing formula.
Proposition 3.10 (The gluing formula) . Consider the following situation. • M and M are manifolds with fibered boundaries as above, equipped withpre-spin structures P ′ M i and P ′ Y i on T M i and T Y i , respectively, and ariemannian metric g π i on T V ∂M i , for i = 0 , . • Assume that on some components of ∂M and − ∂M , we are given iso-morphisms of the data ( π i , P ′ M i , P ′ Y i , g π i ) restricted there. • ( M, ∂M, π ′ , Y ′ ) : the manifold with fibered boundary obtained by the aboveisomorphism of some boundary components. This manifold is equipped withthe pre-spin structures P ′ M and P ′ Y ′ on T M and
T Y ′ , respectively, and ariemannian metric g π ′ on T V ∂M induced by the ones on M i . • Assume that on each boundary components of M and M , the fiberwisespin Dirac operators are invertible.Then, we have Ind e ( P ′ M , P ′ Y ′ , g π ′ ) = Ind e ( P ′ M , P ′ Y , g π ) + Ind e ( P ′ M , P ′ Y , g π ) . Proof.
We use a similar argument to the one in Proposition 3.8. For simplicity weconsider the case where the boundary of each M and M consists of one compo-nent, and the isomormorphism is given between ∂M and − ∂M . In particular,the resulting manifold M is a closed manifold in this case. The general case canbe shown in an analogous way. We denote the image of ∂M ≃ − ∂M in M by H ⊂ M , which is a closed hypersurface. Also we denote π : H → Y the fiber bundlestructure induced from the ones on ∂M ≃ − ∂M and the given fiberwise metricas g π . NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 27
Consider the Lie algebra
T M × [0 , → M × [0 ,
1] with the canonical Lie bracket.Consider the following C ∞ ( M × [0 , C ∞ ( M × [0 , T M × [0 , V := { V ∈ C ∞ ( M × [0 , T M × [0 , | V | H ×{ } ∈ C ∞ ( H × { } ; T V H × { } ) } . This is a Lie subalgebra of C ∞ ( M × [0 , T M × [0 , A → M × [0 , C ∞ ( M × [0 , A ) ≃ V as a C ∞ ( M × [0 , p : C ∞ ( M × [0 , A ) → C ∞ ( M × [0 , T ( M × [0 , V ֒ → C ∞ ( M × [0 , T M × [0 , ֒ → C ∞ ( M × [0 , T ( M × [0 , A → M × [0 ,
1] with anchor p . We have the following. • A | M ×{ t } = A ( M × M ) = T M for all t ∈ (0 , • A | M ×{ } = A G e ∪ H A G e . Here we denoted the e -groupoid of M i by G ie for i = 0 , • The metric g A on A , defined as g A := dx x + t ⊕ π ∗ g Y x + t ⊕ g π gives a smooth metric on A . Here x is a defining function for H ⊂ M and t is the [0 , M × [0 , g Y can be any metric on Y .Since p | M × [0 , \ ( Y ×{ } ) is injective, ( A , p ) is an almost injective Lie algebroid andby [Deb01] we can integrate this to get a Lie groupoid G ⇒ M × [0 , G = ( G e ∪ H G e ) × { } ⊔ M × M × (0 , ⇒ M × [0 , . The description is similar to the one in the proof of the Proposition 3.8. Weconsider the spin Dirac operator
D ∈ Ψ ( G ; S ( A )) with respect to the given spinstructure and metric g A . The submanifold H ×{ } ⊂ M × [0 ,
1] is a closed saturatedsubmanifold for G . The restriction D| H ×{ } is of the form (3.5), and since we areassuming that D π is invertible, we see that D| H ×{ } is invertible. Thus we get theindex class Ind M × [0 , \ ( H ×{ } ) ( D ) ∈ K ( C ∗ ( G| M × [0 , \ ( H ×{ } ) )) . Note that we have( ev ) ∗ Ind M × [0 , \ ( H ×{ } ) ( D ) = (Ind( D e ) , Ind( D e )) ∈ K ( C ∗ ( G e | ˚ M )) ⊕ K ( C ∗ ( G e | ˚ M )) ≃ Z ⊕ Z ( ev ) ∗ Ind M × [0 , \ ( H ×{ } ) ( D ) = Ind( D M ) ∈ K ( C ∗ ( M × M )) ≃ Z . Here we denoted D M the spin Dirac operator on M . This coincides with D e in thestatement of this proposition. Note that [ ev ] ∈ KK ( C ∗ ( G| M × [0 , \ ( H ×{ } ) ) , C ∗ ( G e | ˚ M ) ⊕ C ∗ ( G e | ˚ M )) is a KK -equivalence. Thus, it is enough to show that [ ev ] − ⊗ [ ev ] : Z ⊕ Z → Z is given by addition.The groupoid G| M × [0 , \ ( H ×{ } ) is an open subgroupoid of M × M × [0 , ⇒ M × [0 , / / C ∗ ( M × M × (0 , / / (cid:15) (cid:15) C ∗ ( G| M × [0 , \ ( H ×{ } ) ) / / (cid:15) (cid:15) C ∗ ( G e | ˚ M ) ⊕ C ∗ ( G e | ˚ M ) / / (cid:15) (cid:15) / / C ∗ ( M × M × (0 , / / C ∗ ( M × M × [0 , / / C ∗ ( M × M ) / / , where the rows are exact. The element [ ev ] − ⊗ [ ev ] ∈ KK ( C ∗ ( G e | ˚ M ) ⊕ C ∗ ( G e | ˚ M ) , C ∗ ( M × M )) coincides with the connecting element of the top row. By the functorial-ity of connecting maps, we see that [ ev ] − ⊗ [ ev ] = [ j ⊕ j ], where j i denotesthe inclusion j i : C ∗ ( G ie | ˚ M i ) ֒ → C ∗ ( M × M ) for i = 0 ,
1. Since the inclusion G ie | ˚ M i = ˚ M i × ˚ M i ֒ → M × M is a Morita equivalence for i = 0 ,
1, we see that[ ev ] − ⊗ [ ev ] = [ j ] ⊕ [ j ] : Z ⊕ Z → Z induced between the K -groups is given byaddition. (cid:3) Next we show that the Φ-index can be written as a limit of the Atiyah-Patodi-Singer (APS) indices. For a manifold with fibered boundary (
M, π : ∂M → Y )as above, we fix riemannian metrics g Y and g π for Y and T V ∂M . For µ >
0, weconsider a b -metric of the form(3.11) g b,µ = 1 µ ( dx x ⊕ π ∗ ( g Y )) ⊕ g π on a collar neighborhood of the boundary. Denote the Dirac operator associatedto this metric by D µ . As always we assume that D π is an invertible family. Theboundary operator of D µ is the Dirac operator on ∂M with respect to the metric µ − π ∗ g Y ⊕ g π . It has the form D µ,∂ = D π ˆ ⊗ µD Y + µ R where D Y is a first order differential operator whose principal symbol is equal tothe Clifford multiplication by T Y , and R is an operator of order 0, coming fromthe curvature of the fibration π . For the precise formula, we refer to [BC89, Section4]. As explained in the proof of Proposition 4.41 in [BC89], the anticommutator[ D π ˆ ⊗ , D Y + µR ] is a fiberwise operator, so using fiberwise elliptic estimate andinvertibility of D π , we see that for 0 < µ << D µ,∂ is invertible. When theboundary operator D µ,∂ is invertible, the APS index Ind APS ( D µ ) of the b -operator D µ is, by definition, the Fredholm index of D µ as an operator on the L -space withrespect to the metric g b,µ . Since D µ,∂ stays invertible for µ > µ → +0 Ind
APS ( D µ ) ∈ Z . (The existence of the limit can also be seen as a consequence of the proof of Propo-sition 3.12 below. ) Proposition 3.12 (The limit of the APS index is the Φ-index) . We have lim µ → +0 Ind
APS ( D µ ) = Ind Φ ( P ′ M , P ′ Y , g π ) . Proof.
Again we use a similar argument as in Proposition 3.8. Consider the Liealgebra
T M × [0 , → M × [0 ,
1] with the canonical Lie bracket (not to be confusedwith T ( M × [0 , C ∞ ( M × [0 , C ∞ ( M × [0 , T M × [0 , V := (cid:26) V ∈ C ∞ ( M × [0 , T M × [0 , V | ∂M ×{ } ∈ C ∞ ( ∂M × { } ; T V ∂M × { } ) , and V ( x ) ∈ x ( x + t ) C ∞ ( M × [0 , t ) (cid:27) . This is a Lie subalgebra of C ∞ ( M × [0 , T M × [0 , A → M × [0 , M × (0 ,
1] and a Φ-metric g Φ on M ×{ } , which has the form g Φ | ∂M = g π ⊕ π ∗ g T Y ⊕ R , NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 29 gives a smooth metric g A on A → M × [0 , µ . We can construct a groupoid G ⇒ M × [0 ,
1] which integrates A and can be written as G = G Φ × { } ⊔ G b × (0 , ⇒ M × [0 , . We denote the spin Dirac operator on G with respect to the metric g A by D . Asexplained above, there exists a positive number 0 < ǫ ≤ D| ∂M × [0 ,ǫ ] isinvertible. Thus we get the index classInd ˚ M × [0 ,ǫ ] ( D| M × [0 ,ǫ ] ) ∈ K ( C ∗ ( G ˚ M × [0 ,ǫ ] )) ≃ Z . We have Ind
APS ( D µ ) = ( ev µ ) ∗ Ind ˚ M × [0 ,ǫ ] ( D| M × [0 ,ǫ ] ) ∈ K ( C ∗ ( G ˚ M ×{ µ } )) ≃ Z forall 0 < µ ≤ ǫ . Moreover, the restriction of D to M × { } is exactly the sameas the operator D Φ defining the index Ind Φ ( P ′ M , P ′ Y , g π ) = Ind ˚ M ( D Φ ). We haveInd ˚ M ( D Φ ) = ( ev ) ∗ Ind ˚ M × [0 ,ǫ ] ( D| M × [0 ,ǫ ] ) ∈ K ( C ∗ ( G ˚ M ×{ } )) ≃ Z . Since ev µ in-duces the identity map on Z for all µ ∈ [0 , ǫ ], we get the result. (cid:3) Next we show the vanishing formula for the case where the spin fiber bundlestructure (preserving the boundary) extends to the whole manifold, and the fiber-wise operators are invertible for the whole family.
Proposition 3.13 (The vanishing formula) . We consider the following situation. • Let ( M ev , ∂M, π : ∂M → Y odd ) be a compact manifold with fibered bound-ary, equipped with pre-spin structures P ′ M and P ′ Y on T M and
T Y , respec-tively, and a riemannian metric g π on T V ∂M , for which the fiberwise spinDirac operator D π is an invertible family. • There exist data ( π ′ , X, P ′ X , g π ′ ) such that – X is a compact manifold with boundary ∂X , with a fixed diffeormor-phism ∂X ≃ Y . We identify ∂X with Y . – π ′ : ( M, ∂M ) → ( X, ∂X ) is a fiber bundle structure which preservesthe boundary, and π ′ | ∂M = π . Note that the typical fibers of π and π ′ are the same. – P ′ X is a pre-spin structure on T X which satisfies P ′ X | Y = P ′ Y . – g π ′ is a riemannian metric on T V M (the fiberwise tangent bundle ofthe fiber bundle π ′ ) satisfying g π ′ | Y = g π . We denote D π ′ the familyof fiberwise spin Dirac operators for π ′ .Assume that D π ′ is invertible. Then we have Ind Φ ( P ′ M , P ′ Y , g π ) = Ind e ( P ′ M , P ′ Y , g π ) = 0 . Proof.
The first equality follows from Proposition 3.8. Consider the subgroupoid M × π ′ M ⊂ G e and define G := DN C ( G e , M × π ′ M ) | M × [0 , . Denote the closedsaturated subset M := M × { } ∪ ∂M × [0 , ⊂ M × [0 ,
1] for this groupoid.Note that we have G| ∂M × [0 , = ∂M × π ∂M × π DN C ( T Y ⋊ R ∗ + , Y ) | Y × [0 , . Wecan also see that the restriction G| M ×{ } is of the form M × π ′ M × π ′ E X , where E X → X is a vector bundle over X . In particular, there exists canonical directsum decomposition of A G| M such that one component is T V M . Choose anyriemannian metric g A on A G such that, on M , the two direct sum components areorthogonal, and T V M -component is equal to g π ′ ∪ g π × [0 , A G defined by the given data and metric g A chosen above, andconsider the spin Dirac operator D ∈
Diff ( G ; S ( A G )). Then, the restriction of D to M has the product form as in (3.4) and (3.5).Since we are assuming that the fiberwise operator D π ′ is invertible, we see that D| M is invertible. So we get the index classInd ˚ M × (0 , ( D ) ∈ K ( C ∗ ( G| ˚ M × (0 , )) , and we see that Ind e ( P ′ M , P ′ Y , g π ) = ( ev ) ∗ Ind ˚ M × (0 , ( D ). However, since C ∗ ( G| ˚ M × (0 , ) = C ∗ ( ˚ M × ˚ M ) ⊗ C ((0 , K -group is trivial and we get the re-sult. (cid:3) The cases of twisted spin c and signature operators. The above argu-ment easily generalizes to the cases of twisted spin c -Dirac operators and twistedsignature operators, as follows. Let ( M ev , π : ∂M → Y odd ) be a compact manifoldwith fibered boundary, and E → M be a Z -graded complex vector bundle.3.3.1. Twisted spin c Dirac operators.
For our conventions on spin c /pre- spin c /differential spin c structures, see Definition 2.27. In order to define the Φ and e -indices of the spin c -Dirac operator on M twisted by E , we need the following data.(D1) Pre- spin c structures P ′ M and P ′ Y on T M and
T Y , respectively.(D2) A differential spin c structure P π on T V ∂M , which is compatible with thepre- spin c -structure induced from P ′ M and P ′ Y (see Remark 3.1).(D3) A hermitian structure on E | ∂M as well as a smooth family of fiberwiseunitary connection for the boundary fibration, i.e., a continuous map ∇ Eπ : C ∞ ( ∂M ; E | ∂M ) → C ∞ ( ∂M ; E | ∂M ⊗ ( T V ∂M ) ∗ )given by a family of unitary connections {∇ Ey } y ∈ Y on the vector bundle E | π − ( y ) → π − ( y ) for each y ∈ Y .(D4) Denote the fiberwise twisted spin c -Dirac operators for π by D Eπ = { D Eπ − ( y ) } y ∈ Y .Here D Eπ − ( y ) acts on C ∞ ( π − ( y ); S ( π − ( y )) ˆ ⊗ E ). We assume that D Eπ forms an invertible family.Additional data which are needed to define an operator are as follows.(d1) A differential spin c structure on A G Φ ( A G e ) such that • it is compatible with the pre- spin c structures in (D1). • it has a product structure with respect to the decomposition A G Φ | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R ( A G e | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R ) at the boundary. • the T V ∂M -component coincides with the one in (D2).(d2) A hermitian structure on E which restricts to the one given in (D3), and aunitary connection ∇ E which restricts to ∇ Eπ in (D3).From these data, we get the twisted spin c -Dirac operators D S ˆ ⊗ E Φ ∈ Diff ( G Φ ; S ( A G Φ ) ˆ ⊗ E )and D S ˆ ⊗ Ee ∈ Diff ( G e ; S ( A G e ) ˆ ⊗ E ). By the assumption on the invertibility of D Eπ in (D4), we get the fredholmness of these operators as in Proposition 3.2,as follows. We only explain it in the Φ-case. It is enough to see that the re-striction to the boundary, D S ˆ ⊗ E Φ | ∂M ∈ Diff ( G Φ | ∂M ; S ( A G Φ ) | ∂M ˆ ⊗ E | ∂M ), is in-vertible. This operator is given by a family { D Ey } y ∈ Y parametrized by Y , andeach D Ey is the spin c -Dirac operator twisted by E on the groupoid G Φ | π − ( y ) = π − ( y ) × π − ( y ) × T y Y × R , in the sense of Example 2.33. We have the isomor-phism S ( A G Φ | π − ( y ) ) ≃ S ( π − ( y )) ˆ ⊗ S ( T y Y × R ) by the assumption (d1). Define r : π − ( y ) × T y Y × R → π − ( y ) , ( x, ξ, t ) x. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 31
By the construction of twisted spin c -Dirac operators on groupoids explained in Ex-ample 2.33, we introduce the connection on the hermitian vector bundle r ∗ ( E | π − ( y ) ) → π − ( y ) × T y Y × R as the pullback r ∗ ∇ E of the connection ∇ E on E . By the as-sumption in (d2), it coincides with the pullback r ∗ ∇ Eπ . Thus the operator D Ey iswritten as D Ey : C ∞ c ( π − ( y ) × T y Y × R ; ( S ( π − ( y )) ˆ ⊗ E | π − ( y ) ) ˆ ⊗ S ( T y Y × R )) → C ∞ c ( π − ( y ) × T y Y × R ; ( S ( π − ( y )) ˆ ⊗ E | π − ( y ) ) ˆ ⊗ S ( T y Y × R )) D Ey = D Eπ − ( y ) ˆ ⊗ ⊗ D T y Y × R . (3.14)Here the operator D T y Y × R is the spin c -Dirac operator on the Euclidean space T y Y × R defined by (d1). The operators D Eπ ˆ ⊗ ⊗ D T y Y × R anticommute, and bythe assumption (D4), we get the invertibility of the family D S ˆ ⊗ E Φ | ∂M = { D Ey } y ∈ Y .So we get their indicesInd ˚ M ( D S ˆ ⊗ E Φ ) , Ind ˚ M ( D S ˆ ⊗ Ee ) ∈ Z . These indices depend only on the data (D1) ∼ (D4) and do not depend on theadditional data (d1) or (d2), as in Proposition 3.6. So we can define the Φ and e -indices of the data ( P ′ M , P ′ Y , P π , E, ∇ Eπ ) as follows. Definition 3.15.
Given data ( P ′ M , P ′ Y , P π , E, ∇ Eπ ) as in (D1) ∼ (D4) above, wechoose additional data (d1) and (d2) arbitrarily and defineInd Φ ( P ′ M , P ′ Y , P π , E, ∇ Eπ ) := Ind ˚ M ( D S ˆ ⊗ E Φ ) , Ind e ( P ′ M , P ′ Y , P π , E, ∇ Eπ ) := Ind ˚ M ( D S ˆ ⊗ Ee ) . These do not depend on the choice in (d1) or (d2).We can show the equality Ind Φ ( P ′ M , P ′ Y , P π , E, ∇ Eπ ) = Ind e ( P ′ M , P ′ Y , P π , E, ∇ Eπ )as in Proposition 3.8. The gluing formula as in Proposition 3.10 and the vanishingproperty as in Proposition 3.13 hold analogously.3.3.2. Twisted signature operators.
For twisted signature operators, we need thefollowing data. Let ( M ev , π : ∂M → Y odd ) be a compact manifold with fiberedboundaries, where both M and Y are oriented. We call such ( M, π : ∂M → Y ) oriented ; note that this includes the orientation on Y . These orientations inducean orientation on T V ∂M . The data needed to define the Φ and e -signature are asfollows.(D1) A riemanian metric g π on T V ∂M .(D2) A hermitian structure on E | ∂M as well as a smooth family of fiberwiseunitary connection for the boundary fibration, i.e., a continuous map ∇ Eπ : C ∞ ( ∂M ; E | ∂M ) → C ∞ ( ∂M ; E | ∂M ⊗ ( T V ∂M ) ∗ )given by a smooth family of unitary connections {∇ Ey } y ∈ Y on E | π − ( y ) → π − ( y ) for each y ∈ Y .(D3) Denote the fiberwise twisted signature operators for π by D sign ,Eπ = { D sign ,Eπ − ( y ) } y ∈ Y .Here D sign ,Eπ − ( y ) acts on C ∞ ( π − ( y ); ∧ C T ∗ ( π − ( y )) ˆ ⊗ E ). We assume that D sign ,Eπ forms an invertible family.Additional data which are needed to define an operator are as follows. (d1) A smooth riemannian metric g Φ for A G Φ ≃ T Φ M → M , whose restrictionto A G Φ | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R can be written as g Φ | ∂M = g π ⊕ π ∗ g T Y ⊕ R , where g T Y ⊕ R is some riemannian metric on T Y ⊕ R → Y .(d1) ′ A smooth riemannian metric g e for A G e ≃ T e M → M , whose restrictionto A G e | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R can be written as g e | ∂M = g π ⊕ π ∗ g T Y ⊕ R , where g T Y ⊕ R is some riemannian metric on T Y ⊕ R → Y .(d2) A hermitian structure on E which restricts to the one given in (D3), and aunitary connection ∇ E which restricts to ∇ Eπ in (D3).From these data, we get the twisted signature operators D sign ,E Φ ∈ Diff ( G Φ ; ∧ C A ∗ G Φ ˆ ⊗ E )and D sign ,Ee ∈ Diff ( G e ; ∧ C A ∗ G e ˆ ⊗ E ). By the assumption on the invertibility of D sign ,Eπ in (D4), we get the fredholmness of these operators as in the subsubsection3.3.1. Note that in this case, the boundary operator D sign ,E Φ | ∂M = { D sign ,Ey } y ∈ Y isgiven by D sign ,Ey : C ∞ c ( π − ( y ) × T y Y × R ; ( ∧ C T ∗ ( π − ( y )) ˆ ⊗ E | π − ( y ) ) ˆ ⊗ ∧ C ( T y Y × R ) ∗ ) → C ∞ c ( π − ( y ) × T y Y × R ; ( ∧ C T ∗ ( π − ( y )) ˆ ⊗ E | π − ( y ) ) ˆ ⊗ ∧ C ( T y Y × R ) ∗ ) D y = D sign ,Eπ − ( y ) ˆ ⊗ ⊗ D sign T y Y × R . Here D sign T y Y × R is the Euclidean signature operator defined by the metric g T Y ⊕ R in(d1).So we get their indicesInd ˚ M ( D sign ,E Φ ) , Ind ˚ M ( D sign ,Ee ) ∈ Z . These indices depend only on the data (D1) ∼ (D3) and do not depend on theadditional data (d1), (d1) ′ or (d2), as in Proposition 3.6. So we can define the Φand e -indices of the data ( g π , E, ∇ Eπ ) as follows. Definition 3.16.
Given a compact oriented manifold with boundary ( M ev , π : ∂M → Y odd ) with data ( g π , E, ∇ Eπ ) as in (D1) ∼ (D3) above, we choose additionaldata (d1), (d1) ′ and (d2) arbitrarily and defineSign Φ ( M, g π , E, ∇ Eπ ) := Ind ˚ M ( D sign ,E Φ ) , Sign e ( M, g π , E, ∇ Eπ ) := Ind ˚ M ( D sign ,Ee ) . This does not depend on the choice in (d1), (d1) ′ or (d2).We can show the equality Sign Φ ( M, g π , E, ∇ Eπ ) = Sign e ( M, g π , E, ∇ Eπ ) as inProposition 3.8. The gluing formula as in Proposition 3.10 and the vanishing prop-erty as in Proposition 3.13 holds analogously.4. Indices of geometric operators on manifolds with fiberedboundaries : the case with fiberwise invertible perturbations
Next we consider operators with fiberwise invertible perturbations on the bound-ary family. The idea is that, if we are given a pair ( P ′ M , P ′ Y , g π ) as in Definition3.7, even if we do not have the invertibility of fiberwise Dirac operator D π for theboundary fibration, if we are given an invertible perturbation ˜ D π by a lower orderfamily, then we can construct fully elliptic Φ/ e -operators ˜ D such that NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 33 • on the interior ˚ M , ˜ D differs from D Φ ( D e ) by an operator of order 0. • the boundary operator of ˜ D is given by ˜ D π ˆ ⊗ ⊗ D T Y × R ( ˜ D π ˆ ⊗ ⊗ D T Y ⋊R ).We would like to define the index of this operator as the index of the pair( P ′ M , P ′ Y , g π ) defined by the fiberwise invertible perturbation ˜ D π . This index has asimpler description, as below.4.1. The general situation.
In this subsection, we recall the well-known generalconstruction of indices, defined using invertible perturbations of an operator ona closed saturated subset for a Lie groupoid. We start with a general setting asfollows. • Let M be a compact manifold possibly with boundaries and corners. • Let G ⇒ M be a Lie groupoid. • Let V ⊂ M be a closed saturated subset for G . • Let ( σ M , ˜ F V ) ∈ C ( S ∗ G M ) ⊕ V Ψ c ( G V ) be an invertible element.Denote the full symbol algebra Σ M \ V ( G ) := C ( S ∗ G M ) ⊕ V Ψ c ( G V ) as in subsub-section 2.3.3. We consider the following exact sequence.0 → C ∗ ( G M \ V ) → Ψ c ( G ) σ f,V → Σ M \ V ( G ) → . We denote the connecting element for this short exact sequence as ind M \ V ( G ) ∈ KK (Σ M \ V ( G ) , C ∗ ( G M \ V )). The element ( σ M , ˜ F V ) gives a class in K (Σ M \ V ( G )),so defines the index class asInd M \ V (( σ M , ˜ F V )) := [( σ M , ˜ F V )] ⊗ ind M \ V ( G ) ∈ K ( C ∗ ( G M \ V )) . This index can be generalized to the case where we are given a path from theoperator F V to an invertible operator. The settings are as follows. • Let ( σ M , F V ) ∈ Σ M \ V ( G ) be an element such that σ M ∈ C ( S ∗ G M ) isinvertible. • Let F V × [0 , = { F V ×{ t } } t ∈ [0 , be a continuous path of operators F V ×{ t } ∈ Ψ c ( G V ) parametrized by t ∈ [0 ,
1] such that – F V ×{ } = F V . – F V ×{ t } is elliptic for all t ∈ [0 , – F V ×{ } is invertible.We call such a path “an invertible perturbation for F V ”. Remark . In the following, we often work in the situation where we are given • An element F V ∈ Ψ c ( G V ) for which σ ( F V ) ∈ C ( S ∗ G V ) is invertible, and • An invertible element ˜ F V ∈ Ψ c ( G V ) which satisfies ˜ F V − F V ∈ C ∗ ( G V ).In this case, we have a canonical choice, up to homotopy, of path F V × [0 , = { F V ×{ t } } t ∈ [0 , such that F V ×{ } = F V and F V ×{ } = ˜ F V . Namely, we chooseany such continuous path which satisfies F V ×{ t } − F V ∈ C ∗ ( G V ) for all t ∈ [0 , F V “an invertible perturbation for F V ”and actually consider such path of operators.From the data above, we define Ind M \ V ( σ M , F V × [0 , ) ∈ K ( C ∗ ( G M \ V )) as fol-lows. Denote(4.2) M := M ∪ V ×{ } V × [0 ,
1] and ˚ M := M ∪ V ×{ } V × [0 , G := G ∪ V ×{ } G V × [0 , ⇒ M and ˚ G := G | ˚ M . Although M is not a manifold, G is a longitudinally smooth groupoid, so weabuse the notations such as C ∗ ( G ) := C ∗ ( G ) ⊕ V ×{ } C ∗ ( G V × [0 , → C ∗ (˚ G ) → Ψ c ( G ) σ f,V ×{ } −−−−−−→ Σ ˚ M ( G ) → . We denote the associated connecting element as ind ˚ M ( G ) ∈ KK (Σ ˚ M ( G ) , C ∗ (˚ G )).Consider the canonical ∗ -homomorphism σ ′ f,V ×{ } : Σ M \ V × [0 , ( G ) → Σ ˚ M ( G ) , defined by applying the symbol map on G V × [0 , M \ V × [0 , ( G ) = C ( S ∗ G ) ⊕ V × [0 , Ψ c ( G V × [0 , ≃ C ( S ∗ G ) ⊕ V ×{ } Ψ c ( G V × [0 , σ M , F V × [0 , ) ∈ Σ M \ V × [0 , ( G ) as above, by the conditions, theelement σ ′ f,V ×{ } ( σ M , F V × [0 , ) ∈ Σ ˚ M ( G ) is invertible. So we get a class[ σ ′ f,V ×{ } ( σ M , F V × [0 , )] ∈ K (Σ ˚ M ( G )) . Furthermore the inclusion i : C ∗ ( G M \ V ) → C ∗ (˚ G ) gives a KK -equivalence [ i ] ∈ KK ( C ∗ ( G M \ V ) , C ∗ (˚ G )). So we define the index as follows. Definition 4.3.
Ind M \ V ( σ M , F V × [0 , ) = [ σ ′ f,V ×{ } ( σ M , F V × [0 , )] ⊗ ind ˚ M ( G ) ⊗ [ i ] − ∈ K ( C ∗ ( G M \ V )) . Next we prove the following relative formula for this index. Recall that, if we aregiven two invertible perturbations F iV × [0 , , i = 0 , F V , they definethe difference class in K ( C ∗ ( G V )) as follows. Let F ′ V × [0 , = { F ′ V ×{ t } } t ∈ [0 , be acontinuous path of operators F ′ V ×{ t } ∈ Ψ c ( G V ) defined by(4.4) F ′ V ×{ t } = ( F V ×{ − t } if t ∈ [0 , . F V ×{ t − } if t ∈ [0 . , . i.e., first follow the path F V × [0 . in the reversed direction and next follow F V × [0 , .This operator satisfies F ′ V × [0 , ∈ Ψ c ( G V × [0 , → C ∗ ( G V × (0 , → Ψ c ( G V × [0 , σ f,V ×{ , } −−−−−−−→ Σ V × (0 , ( G V × [0 , → . By assumption σ f,V ×{ , } ( F ′ V × [0 , ) is invertible. Thus we get the index classInd V × (0 , ( F ′ V × [0 , ) ∈ K ( C ∗ ( G V × (0 , ≃ K ( C ∗ ( G V )) . We define this class as the difference class of the invertible perturbations F V × [0 , and F V × [0 , :[ F V × [0 , − F V × [0 , ] := Ind V × (0 , ( F ′ V × [0 , ) ∈ K ( C ∗ ( G V )) . Remark . As in subsection 2.2, we denote by ˜ I ( F V ) the set of invertible pertur-bations for the operator F V . This set has the obvious homotopy relation, and wedenote I ( F V ) the set of homotopy classes of elements in ˜ I ( F V ). We can show that I ( F V ) is nonempty if and only if Ind( F V ) = 0 ∈ K ( C ∗ ( G V )). The above defini-tion of the difference class induces the affine space structure on I ( F V ) modeled on K ( C ∗ ( G V )). NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 35
Remark . In Remark 4.1, we explained that an operator ˜ F V such that ˜ F V − F V ∈ C ∗ ( G V ) can be regarded as an invertible perturbation of F V . Assume we have twoinvertible perturbations ˜ F V and ˜ F V of F V in this sense. Then the difference classdefined above between these perturbations, which we denote by [ ˜ F V − ˜ F V ], canbe described as follows. We take any path F ′ V × [0 , ∈ Ψ c ( G V × [0 , F ′ V ×{ i } = ˜ F iV for i = 0 , F ′ V ×{ t } − ˜ F V ∈ C ∗ ( G V ) for all t ∈ [0 , F V − ˜ F V ] = Ind V × (0 , ( F ′ V × [0 , ) ∈ K ( C ∗ ( G V )) . Two different choices of such path are homotopic, and the one which is obtainedby the construction in (4.4) is one of such choices.
Proposition 4.7 (The general relative formula) . Let G ⇒ M be a longitudinallysmooth Lie groupoid over a compact manifold M , and V ⊂ M be a closed saturatedsubset. Let ( σ M , F V ) ∈ Σ M \ V be an element such that σ M ∈ C ( S ∗ G M ) is invert-ible. Suppose we are given two invertible perturbations F iV × [0 , , i = 0 , for F | V .Then we have Ind M \ V ( σ M , F V × [0 , ) − Ind M \ V ( σ M , F V × [0 , ) = [ F V × [0 , − F V × [0 , ] ⊗ ∂ M \ V ( G ) ∈ K ( C ∗ ( G M \ V )) . Here, the element ∂ M \ V ( G ) ∈ KK ( C ∗ ( G V ) , C ∗ ( G M \ V )) is the connecting elementof the short exact sequence, → C ∗ ( G M \ V ) → C ∗ ( G ) → C ∗ ( G V ) → . as defined in subsection 2.3.3. In particular, the element Ind M \ V ( σ M , F V × [0 , ) ∈ K ( C ∗ ( G M \ V )) only depends on the class of F V × [0 , in I ( F V ) .Proof. We use the notations • M t := M ∪ V ×{ } V × [0 , t ] and ˚ M t := M ∪ V ×{ } V × [0 , t ), • G t := G ∪ V ×{ } G V × [0 , t ] ⇒ M t and ˚ G t := G t | ˚ M t • The inclusion which gives a KK -equivalence i t : C ∗ ( G M \ V ) → C ∗ (˚ G t ).for t >
0. Consider the path of operators F ′ V × [0 , defined in (4.4). We changethe parameters t ∈ [0 , t + 1 ∈ [1 ,
2] and consider it as an operator F ′ V × [1 , ∈ Ψ c ( G V × [1 , F V × [0 , ∪ V ×{ } F ′ V × [1 , is a continuouspath of elliptic operators and defines an element in Ψ c ( G V × [0 , σ M = σ M ∪ σ V × [0 , ( F V × [0 , ∪ F ′ V × [1 , ) ∈ C ( S ∗ ( G )). The pair ( σ M , F V ×{ } ⊔ F ′ V ×{ } )gives an element in Σ M \ V ×{ , } ( G ). By construction, this is invertible. Thus weget the index classInd M \ V ×{ , } ( σ M , F V ×{ } ⊔ F ′ V ×{ } ) ∈ K ( C ∗ ( G | M \ V ×{ , } ))= K ( C ∗ (˚ G )) ⊕ K ( C ∗ ( G V × (1 , . (4.8)We denote by p ˚ M and p V × (1 , the projections to the first and second factor on thegroup appearing in the right hand side of the above equation (4.8). By construction,we have p ˚ M (Ind M \ V ×{ , } ( σ M , F V ×{ } ⊔ F ′ V ×{ } )) = Ind M \ V ( σ M , F V × [0 , ) ⊗ [ i ] ∈ K ( C ∗ (˚ G )) .p V × (1 , (Ind M \ V ×{ , } ( σ M , F V ×{ } ⊔ F ′ V ×{ } )) = [ F V × [0 , − F V × [0 , ] ∈ K ( C ∗ ( G V × (1 , . Moreover, we see that under the inclusion j : C ∗ (˚ G ) ⊕ C ∗ ( G V × (1 , → C ∗ (˚ G ) , we haveInd M \ V ( σ M , F V × [0 , ) ⊗ [ i ] = Ind ˚ M ( σ M , F ′ V ×{ } )= Ind M \ V ×{ , } ( σ M , F V ×{ } ⊔ F ′ V ×{ } ) ⊗ [ j ] . So we haveInd M \ V ( σ M , F V × [0 , ) = Ind M \ V ×{ , } ( σ M , F V ×{ } ⊔ F ′ V ×{ } ) ⊗ [ j ] ⊗ [ i ] − = Ind M \ V ( σ M , F V × [0 , ) + [ F V × [0 , − F V × [0 , ] ⊗ [ j ] ⊗ [ i ] − . Thus it is enough to show that ∂ M \ V ( G ) = [ j ] ⊗ [ i ] − ∈ KK ( C ∗ ( G V × (1 , , C ∗ ( G M \ V )).But this is well-known, since in general the element [ ∂ φ ] ∈ KK ( B, J ) associatedto an extention of C ∗ -algebra 0 → J → A φ −→ B → , where B is nuclear, is given by [ ∂ φ ] = [ j ] ⊗ [ i ] − , where j : B ⊗ C ((0 , → A ⊕ φ ( B ⊗ C ([0 , i ] ∈ KK ( J, A ⊕ φ ( B ⊗ C ([0 , KK -equivalence (see [Bla98]).If we have two invertible perturbations F iV × [0 , ( i = 0 ,
1) which define thesame class in I ( F V ), the difference class [ F V × [0 , − F V × [0 , ] vanishes, so we haveInd M \ V ( σ M , F V × [0 , ) = Ind M \ V ( σ M , F V × [0 , ) (cid:3) Remark . If we deal with a positive order elliptic operator D ∈ Ψ ∗ c ( G ), we con-sider the bounded transform ψ ( D ) := D/ (1 + D ∗ D ) − / ∈ Ψ c ( G ) and do the samearguments. More generally we can deal with an elliptic operator F ′ ∈ Ψ c ( G ; E , E )acting between two vector bundles in an essentially the same way. Namely, we con-sider the vector bundle E ⊕ E with the Z grading so that E is the even partand E is the odd part. Consider the odd self-adjoint operator on E ⊕ E definedas F := (cid:18) F ′ F ′∗ (cid:19) . We construct the C ∗ -algebras with coefficients in E ⊕ E , such as C ∗ ( G ; E ⊕ E )and Σ M \ V ( G ; E ⊕ E ), with the Z -grading associated to the grading on E ⊕ E .These C ∗ -algebras are Morita equivalent to the corresponding algebras with trivialcoefficients. Associated to an elliptic symbol and an invertible perturbation as be-fore, we get an odd self-adjoint invertible element ( σ M , F V × [0 , ) ∈ Σ M \ V ( G ; E ⊕ E ) (c.f. [CS84, Definition 1.3]). Thus we get the class [( σ M , F V × [0 , )] ∈ K (Σ M \ V × [0 , ( G ; E ⊕ E )) = K (Σ M \ V × [0 , ( G )) and the same argument applies.4.2. The connecting elements of G Φ and G e . In this preparatory subsection,we show that the connecting elements of the exact sequences0 → C ∗ ( G Φ | ˚ M ) → C ∗ ( G Φ ) → C ∗ ( G Φ | ∂M ) → → C ∗ ( G e | ˚ M ) → C ∗ ( G e ) → C ∗ ( G e | ∂M ) → . Given a unital graded C ∗ -algebra A and an odd self-adjoint unitary operator u ∈ A , we canconstruct a unital graded ∗ -homomorphism C l → A by sending the generator ǫ to u . The K class of this element, [ u ] ∈ K ( A ) ≃ KK ( C l , A ) is defined to be the class given by this graded ∗ -homomorphism. The space of odd self-adjoint invertible elements on A retracts to the space ofodd self-adjoint unitary elements, so an odd self-adjoint invertible element also defines the classin K ( A ) this way. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 37 correspond to the Poincar´e dual to the element [ C Y ] ∈ KK ( C , C ( Y )). This resultis used in the proof of relative formulas for Φ and e -indices in Proposition 4.20 andProposition 4.26. Lemma 4.10 (The connecting elements of G Φ and G e ) . Consider the exact se-quences → C ∗ ( G Φ | ˚ M ) → C ∗ ( G Φ ) → C ∗ ( G Φ | ∂M ) → → C ∗ ( G e | ˚ M ) → C ∗ ( G e ) → C ∗ ( G e | ∂M ) → . (4.12) Denote by ∂ ˚ M ( G Φ ) ∈ KK ( C ∗ ( G Φ | ∂M ) , C ∗ ( G Φ | ˚ M )) and ∂ ˚ M ( G e ) ∈ KK ( C ∗ ( G e | ∂M ) , C ∗ ( G e | ˚ M )) the connecting elements associated to the above exact sequences. Denote by C Y : C → C ( Y ) the canonical ∗ -homomorphism. Denote by [ σ Y ] ∈ KK ( C ∗ ( T Y ) , C ) theelement which is Poincar´e dual to [ C Y ] ∈ KK ( C , C ( Y )) . (Φ) Under the Morita equivalence between G Φ | ∂M and T Y × R , the element ∂ ˚ M ( G Φ ) ∈ KK ( C ∗ ( G Φ | ∂M ) , C ∗ ( G Φ | ˚ M )) ≃ KK ( C ∗ ( T Y ) , C ) identifieswith the element [ σ Y ] . ( e ) Under the Morita equivalence between G Φ | ∂M and T Y ⋊ R ∗ + and the KK -equivalence between C ∗ ( T Y ⋊ R ∗ + ) and C ∗ ( T Y ) given by the Connes-Thomisomorphism, the element ∂ ˚ M ( G e ) ∈ KK ( C ∗ ( G e | ∂M ) , C ∗ ( G e | ˚ M )) ≃ KK ( C ∗ ( T Y ) , C ) identifies with the element [ σ Y ] .Proof. First we show that it is enough to consider the case M = Y × R + and π : ∂M = Y × { } → Y is the identity map. Indeed, fixing a tubular neighborhood U ≃ ∂M × R + of ∂M in M , U ⊂ M is a transverse submanifold of both G e and G Φ . Thus the connecting element of (4.11) is equal to the connecting element ofthe exact sequence(4.13) 0 → C ∗ ( G Φ | ˚ U ) → C ∗ ( G Φ | U ) → C ∗ ( G Φ | ∂M ) → , and analogously for (4.12). We consider the manifold with fibered boundary ( Y × R + , id Y : Y × { } → Y ) and denote its Φ and e -groupoids as ˜ G Φ and ˜ G e . Denote˜ π := π × id R + : U ≃ ∂M × R + → Y × R + . We easily see that G Φ | U ≃ ∗ ˜ π ∗ ˜ G Φ and G e | U ≃ ∗ ˜ π ∗ ˜ G e . Under this Morita equivalence, the connecting element of theexact sequence (4.13) is equal to the connecting element of the corresponding exactsequence of ˜ G Φ , and analogously for the e -case. Thus it is enough to consider thecase of manifold with fibered boundary ( Y × R + , id Y : Y × { } → Y ), as stated.From now on, in this proof we denote the b , Φ and e groupoids of ( Y × R + , id Y : Y × { } → Y ) by G b , G Φ and G e , respectively.From now on in this proof, we use symbols such as R ∗ + or ˆ R + in order to distin-guish various R -factors which have different roles. First we show in the e -case. Re-call the definition of G e given in subsubsection 2.4.1; G e is defined by the sphericalblowup construction of the pair groupoid Y × Y × R + × R + ⇒ Y × R + by the sub-groupoid Y ×{ (0 , } ⇒ Y ×{ } , i.e., G e = SBlup r,s ( Y × Y × R + × R + , Y ×{ (0 , } ).Recall the Connes tangent groupoid ([Con94]) of Y , T Y = T Y ×{ }⊔ Y × Y × R ∗ + ⇒ Y × R + . Its Lie groupoid structure is described as T Y = DN C + ( Y × Y, Y ) (cf.[DS17, section 5.3.2]). We easily see that G e ≃ T Y ⋊ R ∗ + , where R ∗ + ∋ λ acts on T Y as multiplication by λ and on R ∗ + as multiplication by 1 /λ (cf. [DS17, section5.3.3]. Apply the construction there for G = Y × Y ). Thus we have a commutative diagram in KK -theory,0 / / C ∗ ( G e | Y × R ∗ + ) / / ≃ C ∗ ( G e ) / / ≃ C ∗ ( G e | Y ×{ } ) / / ≃ / / C ∗ ( Y × Y × R ∗ + ⋊ R ∗ + ) / / KK C ∗ ( T Y ⋊ R ∗ + ) / / KK C ∗ ( T Y ⋊ R ∗ + ) / / KK / / C ∗ ( Y × Y × R ∗ + ) / / C ∗ ( T Y ) / / C ∗ ( T Y ) / / , where the rows are exact and the vertical maps between the middle and the bottomrows are KK -equivalences given by the Connes-Thom isomorphism. The connect-ing element of the bottom row is equal to [ σ Y ] ⊗ [ Bott ] ∈ KK ( C ∗ ( T Y ) , C ( R ∗ + ))(see [Con94, Lemma 6 in Chapter 2, Section 5]), so we get the result.Next we prove the Φ-case. Recall that G Φ is defined as G Φ = SBlup r,s ( G b , Y ),where we regard Y ⇒ Y as a subgroupoid Y × { } × { } ⇒ Y × { } of thegroupoid G b = Y × Y × R × { } ⊔ Y × Y × R ∗ + × R ∗ + ⇒ Y × R + . We define ∂G b := G b | Y ×{ } = Y × Y × R and ˚ G b = G b | Y × R ∗ + = Y × Y × R ∗ + × R ∗ + . Notingthat Y × { } is a closed saturated submanifold for G b , we have a commutativediagram 0 (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) (cid:15) / / C ∗ (˚ G b × R ∗ + ) / / (cid:15) (cid:15) C ∗ ( ^ DN C + ( G b , Y )) / / (cid:15) (cid:15) C ∗ ( ˚ N G b Y ) ≃ ( C ∗ ( T Y × R ) ⊗ C ( ˆ R ∗ + )) / / (cid:15) (cid:15) / / C ∗ ( G b × R ∗ + ) / / (cid:15) (cid:15) C ∗ ( DN C + ( G b , Y )) / / (cid:15) (cid:15) C ∗ ( N G b Y ) ≃ ( C ∗ ( T Y × R ) ⊗ C ( ˆ R + )) / / (cid:15) (cid:15) / / C ∗ ( ∂G b × R ∗ + ) / / (cid:15) (cid:15) C ∗ ( DN C + ( ∂G b , Y )) / / (cid:15) (cid:15) C ∗ ( N ∂G b Y ) ≃ C ∗ ( T Y × R ) / / (cid:15) (cid:15)
00 0 0where the rows and columns are exact. We easily see that
DN C + ( ∂G b , Y ) ≃ T Y × R ⇒ Y × R + , where the factor R does not act on the base. Thus the connect-ing element of the bottom row is equal to [ σ Y ] ⊗ C id R ⊗ C [ Bott ] ∈ KK ( C ∗ ( T Y × R ) , C ∗ ( Y × Y × R × R ∗ + )). The connecting element of the right column is equalto id C ∗ ( T Y × R ) ⊗ C [ [ Bott ] ∈ KK ( C ∗ ( T Y × R ) , C ∗ ( T Y × R ) ⊗ C ( ˆ R ∗ + )). The con-necting element of the left column is equal to [ Bott ] − ⊗ C id R ∗ + ∈ KK ( C ∗ ( ∂G b × R ∗ + ) , C ∗ (˚ G b × R ∗ + )) = KK ( C ∗ ( R ) ⊗ C ( R ∗ + ) , C ( R ∗ + )) (this well-known fact is aspecial case of the e -case above).On the other hand, recalling that SBlup r,s is defined as the quotient by the R ∗ + -action on ^ DN C + (see subsubsection 2.3.4), we have a commutative diagram in NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 39 KK -theory,0 / / C ∗ ( G Φ | Y × R ∗ + ) / / KK C ∗ ( G Φ ) / / KK C ∗ ( G Φ | Y ×{ } ) / / KK / / C ∗ (˚ G b × R ∗ + ) / / C ∗ ( ^ DN C + ( G b , Y )) / / C ∗ ( ˚ N G b Y ) / / , where the rows are exact and vertical arrows are KK -equivalences by the compo-sition of the Connes-Thom isomorphism and the Morita equivalence between thecrossed product and the quotient. Combining these, we get the result. (cid:3) The definitions and relative formulas for the Φ and e -indices. Weapply this general construction to our settings.4.3.1.
Twisted spin c -Dirac operators. Here we explain the case for twisted spin c -Dirac operator. First we give a fundamental remark on the space of C l -invertibleperturbations of geometric operators. Remark . Let X be a closed manifold equipped with a pre- spin c structure, and E → X be a Z -graded complex vector bundle. In order to define the twisted spin c -Dirac operator D E , we have to specify a differential spin c -structure, a hermitianmetric on E and a unitary connection on E . However, since the space of thesechoices is contractible, the sets of homotopy classes of C l -invertible perturbations, I ( D E ), for two different choices are canonically isomorphic.An analogous remark applies when we consider a family of twisted spin c -Diracoperators. Suppose we are given a fiber bundle π : N → Y whose typical fiberis a closed manifold, a pre- spin c -structure P ′ π for π , and a complex vector bundle E → N . Choosing the additional data to define a twisted spin c -Dirac operator D Eπ , we define I ( P ′ π , E ) := I ( D Eπ ) . These sets for two different choices of additional data are canonically isomorphic.For a family of signature operators analogous remark applies. Suppose a fiberbundle π : N → Y whose typical fiber is a closed manifold, is oriented, and let E → N be a Z -graded hermitian vector bundle. We define I sign ( π, E ) := I ( D sign ,Eπ )where D sign ,Eπ is the twisted signature operator defined by any fiberwise metric,hermitian metric on E and unitary connection on E .Let ( M, π : ∂M ev → Y odd , E → M ) be a compact manifold with fibered bound-aries, equipped with a complex vector bundle. The data needed to define the indexare the following.(D1) Pre- spin c structures P ′ M and P ′ Y on T M and
T Y , respectively. Theseinduce a pre- spin c structure on T V ∂M , denoted by P ′ π .(D2) A homotopy class of C l -invertible perturbation Q π ∈ I ( P ′ π , E ).The additional data needed to construct operators are as follows.(d1) A differential spin c structure on A G Φ ( A G e ) such that • it is compatible with the pre- spin c structures in (D1). • it has a product structure with respect to the decomposition A G Φ | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R ( A G e | ∂M = T V ∂M ⊕ π ∗ T Y ⊕ R ) at the boundary. (d2) A hermitian structure on E and a unitary connection ∇ E . Denote thefiberwise twisted spin c -Dirac opeartor D Eπ .(d3) A family of operators ˜ D Eπ ∈ ˜ I sm ( D Eπ ) which is a representative of the class Q π ∈ I ( P ′ π , E ) in (D2).Let us denote by D S ˆ ⊗ E Φ and D S ˆ ⊗ Ee the twisted spin Dirac operators constructedfrom the above data, respectively. Recall that, under the assumption (d1) above,the restriction of D S ˆ ⊗ E Φ to G Φ | ∂M is given by a family { D Ey } y ∈ Y parametrized by Y , of the form D Ey : C ∞ c ( π − ( y ) × T y Y × R ; ( S ( π − ( y )) ˆ ⊗ E | π − ( y ) ) ˆ ⊗ S ( T y Y × R )) → C ∞ c ( π − ( y ) × T y Y × R ; ( S ( π − ( y )) ˆ ⊗ E | π − ( y ) ) ˆ ⊗ S ( T y Y × R )) D Ey = D Eπ − ( y ) ˆ ⊗ ⊗ D T y Y × R . as in (3.14).Using the C l -invertible perturbation ˜ D Eπ = { ˜ D Eπ − ( y ) } y ∈ Y in the data (d3)above, we define an operator ˜ D S ˆ ⊗ E Φ ,∂M ∈ Ψ c ( G Φ | ∂M ; S | ∂M × E | ∂M ) as a family { ˜ D Ey } y ∈ Y , given by ˜ D Ey := ˜ D Eπ − ( y ) ˆ ⊗ ⊗ D T y Y × R . This gives an invertible operator on G Φ | ∂M , which satisfies D S ˆ ⊗ E Φ | ∂M − ˜ D S ˆ ⊗ E Φ ,∂M ∈ Ψ c ( G Φ | ∂M ; S | ∂M ). It is easy to see that the class [ ˜ D S ˆ ⊗ E Φ ,∂M ] ∈ I ( D S ˆ ⊗ E Φ | ∂M ) doesnot depend on the choice of the explicit operator ˜ D Eπ representing the class Q π ∈I ( P π , E ). Applying the bounded transform, it defines a class(4.15) [( σ M ( D S ˆ ⊗ E Φ ) , ψ ( ˜ D S ˆ ⊗ E Φ ,∂M ))] ∈ K (Σ ˚ M ( G Φ )) . This class only depends on the data (D1) and (D2), and does not depend on theadditional data (d1), (d2), or (d3).In the e -case, D e | ∂M also has the product form as in (3.5), so we define aninvertible operator ˜ D S ˆ ⊗ Ee,∂M in an analogous way.
Definition 4.16.
Given the data (D1) and (D2) as above, choose any additionaldata (d1), (d2) and (d3). We define the Φ and e -indices, defined by the boundaryfiberwise invertible perturbations asInd Φ ( P ′ M , P ′ Y , E, Q π ) := Ind ˚ M ( σ M ( D S ˆ ⊗ E Φ ) , ψ ( ˜ D S ˆ ⊗ E Φ ,∂M )) ∈ K ( C ∗ ( G Φ | ˚ M )) ≃ Z , Ind e ( P ′ M , P ′ Y , E, Q π ) := Ind ˚ M ( σ M ( D S ˆ ⊗ E Φ ) , ψ ( ˜ D S ˆ ⊗ E Φ ,∂M )) ∈ K ( C ∗ ( G e | ˚ M )) ≃ Z . This number only depends on the data (D1) and (D2), and does not depend on theadditional data (d1), (d2), or (d3).For this index we also have the equality(4.17) Ind Φ ( P ′ M , P ′ Y , E, Q π ) = Ind e ( P ′ M , P ′ Y , E, Q π )as in Proposition 3.8. Also, similar results to Proposition 3.10 and 3.13 hold inthis case. For the vanishing formula, the assumption becomes that “the fibrationextends to the whole manifold and the fiberwise invertible perturbation extends tothe whole family”. We give the precise formulation of these properties, as follows. Proposition 4.18 (The gluing formula) . We consider the following situations.
NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 41 • Let ( M , π : ∂M → Y , E → M ) and ( M , π : ∂M → Y , E → M ) be manifolds with fibered boundaries equipped with complex vector bundles. • Assume we are given data ( P ′ M i , P ′ Y i , Q π i ) satisfying the conditions (D1)and (D2) above for each i = 0 , . • Assume that on some components of ∂M and − ∂M , we are given iso-morphisms of the data ( π i , P ′ M i , P ′ Y i , E i , Q π i ) restricted there. • Let us denote ( M, π ′ : ∂M → Y ′ ) the manifold with fibered boundary ob-tained by identifying isomorphic boundary components. This manifold isequipped with data ( P ′ M , P ′ Y ′ , E, Q π ′ ) induced from those on M and M .Then, we have Ind Φ ( P ′ M , P ′ Y ′ , E, Q π ′ ) = Ind e ( P ′ M , P ′ Y ′ , E, Q π ′ )= Ind e ( P ′ M , P ′ Y , E , Q π ) + Ind e ( P ′ M , P ′ Y , E , Q π ) Proposition 4.19 (The vanishing formula) . We consider the following situations. • Let ( M ev , ∂M, π : ∂M → Y odd ) be a compact manifold with fibered bound-ary, equipped with a complex vector bundle E → M . • Let ( P ′ M , P ′ Y , Q π ) be data satisfying the conditions in (D1) and (D2). • Assume that there exists data ( π ′ , X, P ′ X , Q π ′ ) such that – A compact manifold X with boundary ∂X , with a fixed diffeormor-phism ∂X ≃ Y . We identify ∂X with Y . – A fiber bundle structure π ′ : ( M, ∂M ) → ( X, ∂X ) which preserves theboundary, and π ′ | ∂M = π . Note that the typical fibers of π and π ′ arethe same. – A pre- spin c structure P ′ X on T X which restricts to P ′ Y . – Assume that the induced pre- spin c -structure induced on T V M restrictsto P ′ π at the boundary. – An element Q π ′ in I ( P ′ π ′ , E ) which satisfies Q π ′ | ∂M = Q π .Then we have Ind Φ ( P ′ M , P ′ Y , E, Q π ) = Ind e ( P ′ M , P ′ Y , E, Q π ) = 0 . Next we show the relative formula for such indices. Recall that, for a family D π of Z -graded self-adjoint operators parametrized by Y , if we are given two elements Q π and Q π in I ( D π ), their difference class [ Q π − Q π ] is defined in K − ( Y ). Proposition 4.20 (The relative formula) . Let ( M, ∂M, π ) as before, and Q π and Q π be two elements in I ( P ′ π , E ) . Then we have Ind Φ ( P ′ M , P ′ Y , E, Q π ) − Ind Φ ( P ′ M , P ′ Y , E, Q π )= Ind e ( P ′ M , P ′ Y , E, Q π ) − Ind e ( P ′ M , P ′ Y , E, Q π ) = h [ Q π − Q π ] , [ D Y ] i . Here [ D Y ] ∈ K ( Y ) is the class of spin c -Dirac operator on Y defined by the data(D1) and (D2), and h· , ·i : K ( Y ) ⊗ K ( Y ) → Z denotes the index pairing.Proof. The first equality follows from (4.17). Choose any additional data (d1),(d2) and (d3) to define the operator D Ee . For each i = 0 ,
1, choose any repre-sentative ˜ D E,iπ ∈ ˜ I sm ( P ′ π , E ) for the class Q iπ ∈ I ( P ′ π , E ). By the general rela-tive formula, Proposition 4.7, it is enough to show that the difference class of theinvertible perturbations D E,i∂M := ˜ D E,iπ ˆ ⊗ ⊗ D T Y ⋊R ∗ + for i = 0 ,
1, defined in K ( C ∗ ( G e | ∂M ))( ≃ K ( C ∗ ( T Y ⋊ R ∗ + )) ≃ K ( Y )), maps to h [ Q π − Q π ] , [ D Y ] i underthe boundary map ∂ ˚ M ( G e ) : K ( C ∗ ( G e | ∂M )) → K ( C ∗ ( G e | ˚ M )). Consider the operator D on the groupoid G e | ∂M × [0 , t ⇒ ∂M × [0 , t definedby the family D| ∂M ×{ t } := ( t ˜ D E, π + (1 − t ) ˜ D E, π ) ˆ ⊗ ⊗ D T Y ⋊R ∗ + . The restriction to ∂M × { , } is invertible. Thus we get the index class of D in K ( C ∗ ( G e | ∂M × (0 , D E,i∂M coincides with this class:[ D E, ∂M − D E, ∂M ] = Ind ∂M × (0 , ( D ) ∈ K ( C ∗ ( G e | ∂M × (0 , . Denote the Connes-Thom element [ th ] ∈ KK ( C ∗ ( G e | ∂M ) , C ∗ ( ∂M × π ∂M × π T Y )). Consider the following self-adjoint ungraded operator on the groupoid ∂M × π ∂M × π T Y × (0 , ⇒ ∂M × (0 , D ′ | ∂M ×{ t } := ( t ˜ D E, π + (1 − t ) ˜ D E, π ) ˆ ⊗ ⊗ D T Y , for each t ∈ [0 , ∂M × (0 , ( D ′ ) ∈ K ( C ∗ ( ∂M × π ∂M × π T Y × (0 , ∂M × (0 , ( D ) ⊗ [ th ] = Ind ∂M × (0 , ( D ′ ) . We consider the following elements. • [ ˜ D E, π − ˜ D E, π ] ∈ K ( C ∗ (( ∂M × π ∂M ) × (0 , ≃ K ( Y ). • [ D Y ] ∈ K ( Y ). • Ind Y ( D T Y ) ∈ KK ( C ( Y ) , C ∗ ( T Y )) represented by the ungraded Kasparov C ( Y )- C ∗ ( T Y ) bimodule ( C ∗ ( T Y ; S ( T Y )) , multi , ψ ( D T Y )), where multi isthe multiplication by C ( Y ). This is an ungraded version of (2.35). • m ∈ KK ( C ( Y ) ⊗ C ∗ ( T Y ) , C ∗ ( T Y )) represented by the Kasparov C ( Y ) ⊗ C ∗ ( T Y )- C ∗ ( T Y ) bimodule ( C ∗ ( T Y ) , multi ⊗ id C ∗ ( T Y ) , • [ σ Y ] ∈ KK ( C ∗ ( T Y ) , C ).The element m ⊗ C ∗ ( T Y ) σ Y ∈ KK ( C ( Y ) ⊗ C ∗ ( T Y ) , C ) is the element which gives thePoincar´e duality between C ∗ ( T Y ) and C ( Y ). Also we have Ind( D T Y ) ⊗ C ∗ ( T Y ) m =Ind Y ( D T Y ), since the element Ind( D T Y ) ∈ KK ( C , C ∗ ( T Y )) is represented by theKasparov module ( C ∗ ( T Y ; S ( T Y )) , , D T Y ). Since Ind( D T Y ) is the Poincar´e dualto [ D Y ], we have[ D Y ] = Ind( D T Y ) ⊗ C ∗ ( T Y ) m ⊗ C ∗ ( T Y ) σ Y = Ind Y ( D T Y ) ⊗ C ∗ ( T Y ) σ Y . Next, we show the following equality.(4.23) Ind ∂M × (0 , ( D ′ ) = [ ˜ D E, π − ˜ D E, π ] ⊗ C ( Y ) Ind Y ( D T Y ) ∈ KK ( C , C ∗ ( T Y )) . Let c > D E,iπ ) ∩ [ − c, c ] is empty for i = 0 , ψ ′ ∈ C ([ −∞ , ∞ ]) such that ψ ′ ≡ c, ∞ ]and ψ ′ ≡ − −∞ , − c ]. We see that ψ ′ ( ˜ D E,iπ ) and ψ ′ ( ˜ D E,iπ ˆ ⊗ ⊗ D T Y ) areself-adjoint unitaries for i = 0 ,
1. Using this, the classes appearing in (4.23) arerepresented by the Kasparov modules,[ ˜ D E, π − ˜ D E, π ] = [( C ∗ ( ∂M × π ∂M ; S ( T V ∂M )) ⊗ C (0 , , , { ψ ′ ( t ˜ D E, π + (1 − t ) ˜ D E, π ) } t )]Ind ∂M × (0 , ( D ′ ) = [( C ∗ ( ∂M × π ∂M × π T Y ; S ( T V M ) ˆ ⊗ C ( Y ) S ( T Y )) ⊗ C (0 , , , ψ ′ ( D ′ ))] , where the first one is graded and the second one is ungraded. We have C ∗ ( ∂M × π ∂M ; S ( T V ∂M )) ⊗ C ( Y ) C ∗ ( T Y ; S ( T Y )) = C ∗ ( ∂M × π ∂M × π T Y ; S ( T V M ) ⊗ C ( Y ) S ( T Y )). Since the above operators commute with the multiplication by elementsin C ( Y ), the computation of this Kasparov product is the family version of the NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 43 product over C (more precisely, it is the product in R KK ( Y ; · , · ); see the paragraphpreceeding Proposition 5.18 below). Here operators satisfy the relation (4.21), bythe same argument as in [HR00, Section 10.7 and 10.8], we get the equality (4.23).By Lemma 4.10, we know that the connecting element ∂ ˚ M ( G e ) ∈ KK ( G e | ∂M , G e | ˚ M )satisfies [ σ Y ] = [ th ] − ⊗ ∂ ˚ M ( G e ) ∈ KK ( C ∗ ( T Y ) , C ) , Thus we have[ D E, ∂M − D E, ∂M ] ⊗ C ∗ ( T Y ) ∂ ˚ M ( G e ) = Ind ∂M × (0 , ( D ) ⊗ C ∗ ( T Y ) ∂ ˚ M ( G e )= Ind ∂M × (0 , ( D ′ ) ⊗ C ∗ ( T Y ) [ th ] − ⊗ [ th ] ⊗ [ σ Y ]= [ ˜ D E, π − ˜ D E, π ] ⊗ C ( Y ) Ind Y ( D T Y ) ⊗ C ∗ ( T Y ) [ σ Y ]= [ ˜ D E, π − ˜ D E, π ] ⊗ C ( Y ) [ D Y ]= h [ ˜ D E, π − ˜ D E, π ] , [ D Y ] i . So we get the result. (cid:3)
Twisted signature operators.
Here we explain in the case of twisted signatureoperators. The argument is parallel to that in the case for twisted spin c -Dirac op-erators. Let ( M ev , π : ∂M → Y odd ) be a compact oriented manifold with fiberedboundaries equipped with a Z -graded complex vector bundle E → M . Assume weare given an element Q π ∈ I sign ( π, E ). We choose additional data as in subsubsec-tion 3.3.2, and define the twisted signature with respect to the fiberwise invertibleperturbation, analogously as in the twisted spin c Dirac operator case.
Definition 4.24.
Given an element Q π ∈ I sign ( π, E ), we defineSign Φ ( M, E, Q π ) and Sign e ( M, E, Q π ) ∈ Z , in an analogous way to that in Definition 3.15.We also have the equality of Φ and e -signatures as(4.25) Sign Φ ( M, E, Q π ) = Sign e ( M, E, Q π ) . The gluing formula analogous to Proposition 4.18, as well as the vanishing propo-sition analogous to Proposition 4.19 also holds for this case.The relative formula for the twisted signature case is as follows.
Proposition 4.26.
Let ( M ev , π : ∂M → Y odd ) as before, and Q π and Q π be twoelements in I sign ( π, E ) . Then we have Sign Φ ( M, E, Q π ) − Sign Φ ( M, E, Q π )= Sign e ( M, E, Q π ) − Sign e ( M, E, Q π ) = 2 h [ Q π − Q π ] , [ D sign Y ] i . Here [ D sign Y ] ∈ K ( Y ) is the class of odd signature operator on Y , and h· , ·i : K ( Y ) ⊗ K ( Y ) → Z denotes the index pairing.Proof. The proof is analogous to that for Proposition 4.20. The factor 2 in theabove formula is due to the following observation.First of all, recall the definition of odd signature operators acting on odd di-mensional manifolds ([RW06, Definition and Notation 1]). On an odd dimensionalriemannian manifold Y , the essentially self-adjoint operator d + d ∗ acting on ∧ C T ∗ Y commutes with the Hodge star τ , so we define the odd signature operator D sign Y to be the operator d + d ∗ restricted to the +1-eigenbundle of τ . So the total signatureoperator is isomorphic to the direct sum of two copies of D sign Y . We define odd sig-nature operators for Lie groupoids whose dimensions of s -fibers are odd dimensionalanalogously.The signature operator D sign T Y ⋊R ∗ + on the groupoid T Y ⋊ R ∗ + ⇒ Y defines a classInd( D sign T Y ⋊R ∗ + ) ∈ K ( C ∗ ( T Y ⋊ R ∗ + )). The signature operator D sign T Y on the groupoid
T Y ⇒ Y defines a class Ind( D sign T Y ) ∈ K ( C ∗ ( T Y )). We denote the Connes-Thomelement [ th ] ∈ KK ( C ∗ ( T Y ⋊ R ∗ + ) , C ∗ ( T Y )). Then these elements are related by(4.27) Ind( D sign T Y ⋊R ∗ + ) ⊗ [ th ] = 2 · Ind( D sign T Y ) ∈ K ( C ∗ ( T Y )) . Indeed, under Connes-Thom isomorphism K ( C ∗ ( T Y ⋊ R ∗ + )) ≃ K ( C ∗ ( T Y × R )),the element Ind( D sign T Y ⋊R ∗ + ) maps to Ind( D sign T Y × R ). By the same argument as in theproof of [RW06, Lemma 6], we see that 2 · Ind( D sign T Y ) ⊗ Ind( D sign R ) = Ind( D sign T Y × R ) ∈ K ( C ∗ ( T Y × R )). Since by definition Ind( D sign R ) ∈ K ( C ∗ ( R )) is equal to the Bottelement, (4.27) follows.So the factor 2 appears in the equation corresponding to (4.22). (cid:3) The index pairing
In this section, we give a description of the indices defined above, as the indexpairing on the K -theory “relative to the boundary pushforward”. In the following,we use the following notations. • For a C ∗ -algebra A , the symbol M ( A ) denotes its multiplier algebra. • For a C ∗ -algebra A and a Hilbert A -module H A , the symbols B ( H A ) and K ( H A ) denote the C ∗ -algebras of adjointable operators and compact oper-ators on H A , respectively. • For a Euclidean space E , let us denote by C l ( E ) the ∗ -algebra over C ,generated by the elements of E and relations e = e ∗ and e = || e || · e ∈ E. This construction applies to Euclidean vector bundles as well. • Let us denote by ǫ ∈ C l = C l ( R ) the element corresponding to the unitvector in R . In other words, this element is a generator of C l , which isodd, self-adjoint and unitary.5.1. The case of spin c -Dirac operators. In this subsection, we consider the caseof spin c -Dirac operators. First we consider the following setting. • The pair (
M, π : ∂M → Y ) is a compact manifold with fibered boundary. • The fiber bundle π is equipped with a pre- spin c structure P ′ π .In order to formulate the index pairing in this setting, we proceed in the followingfour steps. In the following, let n be the dimension of the fiber of π .(1) We define a C ∗ -algebra A π whose K -groups fit in the exact sequence · · · → K ∗ ( M ) π ! ◦ i ∗ −−−→ K ∗− n ( Y ) → K ∗− n ( A π ) → K ∗ +1 ( M ) π ! ◦ i ∗ −−−→ · · · . (Definition 5.4 and Proposition 5.5). The groups K ∗ ( A π ) are regarded as K -groups relative to the boundary pushforward. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 45 (2) For a pair (
E, Q π ) where E is a Z -graded complex vector bundle over M and Q π ∈ I ( P π , E ), we show that it naturally defines a class [( E, Q π )] ∈ K n − ( A π ) (Lemma 5.7).(3) Assume n is even. For a pair ( P ′ M , P ′ Y ) of pre- spin c structures on T M and
T Y which satisfies P ′ M | ∂M = π ∗ P ′ Y ⊕ P ′ π , we show that it naturally definesa class [( P ′ M , P ′ Y )] ∈ KK ( A π , Σ ˚ M ( G Φ )) (Definition 5.23).(4) We show the equalityInd Φ ( P ′ M , P ′ Y , E, Q π ) = [( E, Q π )] ⊗ A π [( P ′ M , P ′ Y )] ⊗ Σ ˚ M ( G Φ ) ind ˚ M ( G Φ ) ∈ Z . (Theorem 5.24). This is the desired index pairing formula.The most difficult point of the proof of Theorem 5.24 is to relate the invertible op-erators ψ ( ˜ D Eπ ) and ψ ( ˜ D Eπ ˆ ⊗ ⊗ D T Y × R ), since they are not “directly related”, forexample by a ∗ -homomorphism. In order to overcome this difficulty, we construct a C ∗ -algebra D π which “connects A π and Σ ˚ M ( G Φ )” using an asymptotic morphismgiving the KK -equivalence between C ( T Y ⊕ R ; C l ( T Y ⊕ R )) and C ( Y ), and con-struct an invertible element in D π which, under suitable ∗ -homomorphisms, mapsto [( E, [ ˜ D Eπ ])] ∈ K ( A π ) and [( σ ( D Φ ) , ψ ( ˜ D Eπ ˆ ⊗ ⊗ D T Y × R )] ∈ K (Σ ˚ M ( G Φ )).Let N be a compact space. Let π : N → Y be a fiber bundle whose fibers haveclosed manifold structure, and π is equipped with a pre- spin c -structure. Chooseany differential spin c -structure representing the given pre- spin c structure (choos-ing any other choice, we get canonically KK -equivalent C ∗ -algebras below). Let S ( T V N ) → N denote the spinor bundle of vertical tangent bundle and D π denotethe fiberwise spin c -Dirac operators acting on S ( T V N ). Let L Y ( N ; S ( T V N )) denotethe Hilbert C ( Y )-module which is obtained by the completion of C ∞ c ( N ; S ( T V N ))with the natural C ( Y )-valued inner product. Note that L Y ( N ; S ( T V N )) is nat-urally Z -graded if the typical fiber of π is even dimensional. In this setting wedefine a C ∗ -algebra Ψ( D π ). We separate the definition in two cases, depending onthe parity of the dimension of the fiber of π .(1) Assume that the typical fiber of π is odd dimensional. Define χ ∈ C ([ −∞ , ∞ ])as χ ( x ) := (1 + x/ √ x ). Let Ψ( D π ) denote the C ∗ -subalgebra of B ( L Y ( N ; S ( T V N ))) generated by { χ ( D π ) } , C ( N ) and K ( L Y ( N ; S ( T V N ))).(2) Assume that the typical fiber of π is even dimensional. Define the oddfunction ψ ∈ C ([ −∞ , ∞ ]) by ψ ( x ) := x/ √ x . Let Ψ( D π ) denote the Z -graded C ∗ -subalgebra of B ( L Y ( N ; S ( T V N ))) generated by { ψ ( D π ) } , C ( N ) and K ( L Y ( N ; S ( T V N ))). Lemma 5.1. (1)
When the typical fiber of π is odd dimensional, the algebra Ψ( D π ) fits into the exact sequence → K ( L Y ( N ; S ( T V N ))) → Ψ( D π ) → C ( N ) → . The connecting element of this extension coincides with the class π ! ∈ KK ( C ( N ) , C ( Y )) . (2) When the typical fiber of π is even dimensional, the algebra Ψ( D π ) fits intothe exact sequence of graded C ∗ -algebras (5.2) 0 → K ( L Y ( N ; S ( T V N ))) → Ψ( D π ) → C ( N ) ⊗ C l → . The connecting element of this extension coincides with the class π ! ∈ KK ( C ( N ) , C ( Y )) . Proof.
We prove the case (2). The case (1) can be proved analogously. Denote byΓ the groupoid N × π N ⇒ N . Recall that we have a Z -graded exact sequence0 → C ∗ (Γ; S ( A Γ)) → Ψ c (Γ; S ( A Γ)) σ −→ C ( S ∗ Γ; End( S ( A Γ))) → . Of course we have C ∗ (Γ; S ( A Γ)) = K ( L Y ( N ; S ( T V N ))). Consider the restrictionof the symbol map σ to the C ∗ -subalgebra Ψ( D π ) ⊂ Ψ c (Γ; S ( A Γ)). Its image isthe C ∗ -subalgebra of C ( S ∗ Γ; End( S ( A Γ))), generated by { σ ( ψ ( D π )) } and C ( N ).Since σ ( ψ ( D π )) is an odd self-adjoint unitary element commuting with elements in C ( N ), we get the canonical isomorphism between this C ∗ -algebra and C ( N ) ⊗ C l ,by mapping σ ( ψ ( D π )) to the odd self-adjoint unitary generator ǫ ∈ C l . Thus weget the desired graded exact sequence (5.2).Next we describe the connecting element of (5.2). We have the following com-mutative diagram,0 / / K ( H Y ) / / Ψ( D π ) / / (cid:127) _ (cid:15) (cid:15) C ( N ) ⊗ C l / / (cid:127) _ φ (cid:15) (cid:15) / / K ( H Y ) / / Ψ c (Γ; S ( A Γ)) / / C ( S ∗ Γ; End( S ( A Γ))) / / , where the rows are exact and the inclusion φ is explained above. The bottomrow is Morita equivalent to the pseudodifferential extension for the groupoid Γ,so the connecting element is the element ind N (Γ) ∈ KK ( C ( S ∗ Γ) , C ∗ (Γ)). Bythe naturality of connecting elements, the connecting element of (5.2) is equal to[ φ ] ⊗ C ( S ∗ Γ) ind N (Γ).Let us consider the following KK -elements. • The element [ σ ( D π )] ∈ K ( C ( S ∗ Γ)). This element coincides with the ele-ment in KK ( C l , C ( S ∗ Γ; End( S ( A Γ))) given by the unital ∗ -homomorphismwhich maps ǫ ∈ C l to σ ( ψ ( D π )) (see Remark 4.9). • The element [ m ] ∈ KK ( C ( N ) ⊗ C ( S ∗ Γ) , C ( S ∗ Γ))) given by the ∗ -homomorphism f ⊗ ξ f · ξ .We see the equality [ φ ] = [ σ ( D π )] ⊗ C ( S ∗ Γ) [ m ]. On the other hand, the elementind N (Γ) ∈ KK ( C ( S ∗ Γ) , K ( H Y )) ≃ KK ( C ( S ∗ Γ × R ∗ + ) , C ( Y )) is the element giv-ing the family index map. If we denote by [ p.d. ] ∈ KK ( C ( N ) ⊗ C (( T V N ) ∗ ) , C ( Y ))the element which gives the fiberwise Poincar´e duality and by q : C ( S ∗ Γ × R ∗ + ) → C (( T V N ) ∗ ) the inclusion, we have the equation[ m ] ⊗ C ( S ∗ Γ) ind N (Γ) = [ q ] ⊗ C (( T V N ) ∗ ) [ p.d. ] ∈ KK ( C ( N ) ⊗ C ( S ∗ Γ) , C ( Y )) . (see [CS84, pp.1159–1162]). Thus we see that the product [ φ ] ⊗ C ( S ∗ Γ) ind N (Γ) =[ σ ( D π )] ⊗ C ( S ∗ Γ) [ q ] ⊗ C (( T V N ) ∗ ) [ p.d. ] is the element π ! ∈ KK ( C ( N ) , C ( Y )), namelythe element given by the Kasparov module ( L Y ( N ; S ( T V N )) , multi , ψ ( D π )), wheremulti denotes the multiplication by C ( N ). (cid:3) Remark . As we work in KK -theory in this section, we only need C ∗ -algebrasto be defined up to KK -equivalence. As in Lemma 5.1, in order to define C ∗ -algebras in terms of operators, we need to fix rigid structures, such as differential spin c -structures. However, the KK -equivalence class is determined by homotopyequivalence class of those structures, such as pre- spin c -structure (c.f. Remark 4.14).In order to simplify the arguments, we often omit this procedure of “choosing a rigid NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 47 structure, defining algebras and forgetting the structure to get a KK -equivalenceclass”, but the reader should note that we always need such steps. Definition 5.4 ( A π ) . Let (
M, π : ∂M → Y ) be a compact manifold with fiberedboundary. Assume that π is equipped with a pre- spin c -structure. Denote i : ∂M → M the inclusion.(1) Assume that the typical fiber of π is odd dimensional. We define A π to bethe C ∗ -algebra defined by the pullback (c.f. Remark 5.3) A π / / (cid:15) (cid:15) ❴✤ Ψ( D π ) (cid:15) (cid:15) C ( M ) i ∗ / / C ( ∂M )(2) Assume that the typical fiber of π is even dimensional. We define A π to bethe Z -graded C ∗ -algebra defined by the pullback (c.f. Remark 5.3) A π / / (cid:15) (cid:15) ❴✤ Ψ( D π ) (cid:15) (cid:15) C ( M ) ⊗ C l i ∗ / / C ( ∂M ) ⊗ C l Proposition 5.5.
Let ( M, π : ∂M → Y ) be a compact manifold with fibered bound-ary. Assume that π is equipped with a pre- spin c -structure. The K -groups of the C ∗ -algebra A π naturally fits in the exact sequence · · · → K ∗ ( M ) π ! ◦ i ∗ −−−→ K ∗− n ( Y ) → K ∗− n ( A π ) → K ∗ +1 ( M ) π ! ◦ i ∗ −−−→ · · · , where n is the dimension of the fiber of π .Proof. We only prove the Proposition in the case n is even. The odd case is similar.By Lemma 5.1 and the surjectivity of the restriction i ∗ : C ( M ) ⊗ C l → C ( ∂M ) ⊗ C l , we get the graded exact sequence(5.6) 0 → K ( L Y ( N ; S ( T V N ))) → A π → C ( M ) ⊗ C l → . By Lemma 5.1, the connecting element associated to the above exact sequence isequal to [ i ] ⊗ π ! ∈ KK ( C ( M ) , C ( Y )) ≃ KK ( C ( M ) ⊗ C l , K ( L Y ( N ; S ( T V N )))).Thus the long exact sequence of K -groups associated to the above short exactsequence gives the desired sequence. (cid:3) Lemma 5.7.
Let ( M, π : ∂M → Y ) be a compact manifold with fibered boundary.Denote Γ the groupoid ∂M × π ∂M ⇒ ∂M . Assume that π is equipped with apre- spin c -structure P ′ π . Let E be a Z -graded complex vector bundle over M . Let Q π ∈ I ( P ′ π , E ) (see Remark 4.14). Then the pair ( E, Q π ) naturally defines a class [( E, Q π )] ∈ K n − ( A π ) , where n is the dimension of the fiber of π .Proof. We only prove the Lemma in the case n is even. Let us denote D Eπ the fiber-wise spin c -Dirac operators twisted by E (defined using any additional choice; seeRemark 4.14). Let Ψ( D Eπ ) denote the Z -graded C ∗ -subalgebra of B ( L Y ( N ; S ( T V N ) ˆ ⊗ E ))generated by { ψ ( D Eπ ) } , C ( N ; End( E )) and K ( L Y ( N ; S ( T V N ) ˆ ⊗ E )). Consider the graded C ∗ -algebra A π ( E ) defined by the pullback(5.8) A π ( E ) / / (cid:15) (cid:15) ❴✤ Ψ( D Eπ ) (cid:15) (cid:15) C ( M ; End( E )) ˆ ⊗ C l i ∗ / / C ( ∂M ; End( E )) ˆ ⊗ C l as in Definition 5.4. There is a canonical Morita equivalence between A π and A π ( E ).Given a pair ( E, Q π ) where Q π ∈ I ( P ′ π , E ), we define an element in K ( A π ( E ))as follows. Let us choose a representative ˜ D Eπ ∈ ˜ I sm ( D Eπ ). Since ˜ D Eπ is an in-vertible family which differs from D Eπ by a lower order family, ψ ( ˜ D Eπ ) is an invert-ible operator in Ψ( D Eπ ). Thus the element (1 M ˆ ⊗ ǫ, ψ ( ˜ D Eπ )) ∈ A π ( E ) is invertible.The K class defined by this element does not depend on the choice of ˜ D Eπ . Bycomposing with the KK -equivalence between A π and A π ( E ), we get the element[( E, Q π )] := [(1 M ˆ ⊗ ǫ, ψ ( ˜ D Eπ ))] ∈ K ( A π ). (cid:3) Now we assume that M is even dimensional and Y is odd dimensional. Weconstruct an element [( P ′ M , P ′ Y )] ∈ KK n ( A π , Σ ˚ M ( G Φ )) for given P ′ M , P ′ Y pre- spin c -structures on T M and
T Y which are compatible with the given fiberwisepre- spin c structure P ′ π at the boundary, i.e., P ′ M | ∂M = π ∗ P ′ Y ⊕ P ′ π .The next lemma can be proved in the same way as Lemma 5.1. Lemma 5.9.
Let ( M ev , π : ∂M → Y odd ) be a compact manifold with fibered bound-ary, equipped with a pre- spin c -structure P ′ π on T V ∂M . Let P ′ M , P ′ Y be pre- spin c -structures on T M and
T Y respectively. We assume that the pre- spin c -structuresare compatible at the boundary.Choose differential spin c structures on T Y and T V M representing P ′ Y and P ′ π ,and denote the associated spin c -Dirac operator on G Φ | ∂M = ∂M × π ∂M × π T Y × R ⇒ ∂M as, D Φ ,∂ := D π ˆ ⊗ ⊗ D T Y × R . Let Ψ( D Φ ,∂ ) denote the Z -graded C ∗ -subalgebra of Ψ c ( G Φ | ∂M ; S ( A G Φ | ∂M )) gen-erated by { ψ ( D Φ ,∂ ) } , C ( ∂M ) and C ∗ ( G Φ | ∂M ; S ( A G Φ | ∂M )) . If we choose any otherdifferential spin c -structures on T Y and T V M representing P ′ Y and P ′ π , the resulting C ∗ -algebras are canonically KK -equivalent.This C ∗ -algebra fits into the graded exact sequence → C ∗ ( G Φ | ∂M ; S ( A G Φ | ∂M )) → Ψ( D Φ ,∂ ) → C ( ∂M ) ⊗ C l → . The connecting element of this extension coincides with the class π ! ⊗ C ( Y ) Ind Y ( D T Y × R ) ∈ KK ( C ( ∂M ) , C ∗ ( G Φ | ∂M )) ( Ind Y is defined in (2.35)). Definition 5.10 ( B π ) . In the situations in Lemma 5.9, we define a graded C ∗ -algebra by the pullback (c.f. Remark 5.3) B π / / (cid:15) (cid:15) ❴✤ Ψ( D Φ ,∂ ) (cid:15) (cid:15) C ( M ) ⊗ C l i ∗ / / C ( ∂M ) ⊗ C l Choosing a differential spin c structure of G Φ representing P ′ M , we get a canonicalinjective ∗ -homomorphism ι : B π → Σ ˚ M ( G Φ ; S ( A G Φ )) as follows. Denote the NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 49 spin c -Dirac operator on G Φ by D Φ ∈ Diff ( G Φ ; S ( A G Φ )) and the principal symbolof the bounded transform of this operator by σ ( ψ ( D Φ )) ∈ C ( S ∗ G Φ ; End( S ( A G Φ ))),which is an odd self-adjoint unitary element. Thus the ∗ -homomorphism C ( M ) ⊗ C l → C ( S ∗ G Φ ; End( S ( A G Φ ))) f ⊗ f ⊗ ǫ σ ( ψ ( D Φ ))is well-defined. Recall we have Σ ˚ M ( G Φ ; S ( A G Φ )) = C ( S ∗ G Φ ; End( S ( A G Φ ))) ⊕ ∂M Ψ c ( G Φ | ∂M ; S ( A G Φ | ∂M )). It is easy to see that the inclusion Ψ( D Φ ,∂ ) → Ψ c ( G Φ | ∂M ; S ( A G Φ | ∂M ))is compatible with the above ∗ -homomorphism at ∂M , so they combine to in-duce the desired ∗ -homomorphism ι : B π → Σ ˚ M ( G Φ ; S ( A G Φ )). The KK -element[ ι ] ∈ KK ( B π , Σ ˚ M ( G Φ )) is independent of the differential spin c -structure represent-ing P ′ M .Next, in the settings in Lemma 5.9, we construct an element µ ∈ KK ( A π , B π )which fits into a commutative diagram in KK -theory,0 / / C ∗ (Γ; S ( A Γ)) / / (cid:15) (cid:15) A π / / µ (cid:15) (cid:15) C ( M ) ⊗ C l / / / / C ∗ ( G Φ | ∂M ; S ( A G Φ | ∂M )) / / B π / / C ( M ) ⊗ C l / / . Here we denote by Γ the groupoid N × π N ⇒ N .First we consider a general setting. Suppose we are given a compact manifold Y , and an oriented real vector bundle V over Y . For simplicity we only considerthe case where the rank m of this vector bundle V is odd. We consider the Cliffordalgebra bundle C l ( V ) over V (trivial on each fiber of V → Y ) and the Z -graded C ∗ -algebra C ( V ; C l ( V )), where the algebra structure comes from the pointwiseoperations and grading comes from the grading on C l ( V ). We consider a variantof the construction of an asymptotic morphism in [GH04, Section 1], which givesa KK -equivalence between C ( V ; C l ( V )) and C ( Y ). We are going to apply thefollowing general construction to the real vector bundle V = T ∗ Y in our Φ- spin c -setting. The construction below is also used in the next subsection for signatureoperators.We fix a riemannian metric on V . On the Hilbert C ( Y )-module L Y ( V ; C l ( V )),we consider unbounded, odd essentially self-adjoint operators C V , D V , B V definedas follows. They are families of operators parametrized by Y , and for each y ∈ Y ,the operator acts on L ( V y ; C l ( V y )) as D V,y := X i ˆ e i ∂∂x i C V,y := X i x i e i B V := D V + C V , by choosing an oriented orthonormal basis { e i } i of V y and denoting the correspond-ing orthonormal coodinates by x i on V y . Here we denote the actions of elements in V y on C l ( V ) by e ( x ) = e · x ˆ f ( x ) = ( − deg ( x ) x · f. Consider the operator ω y := ( − ( m +1) / ˆ e ˆ e · · · ˆ e m ∈ End( C l ( V y )). This elementdoes not depend on the choice of an oriented orthonormal basis, and gives an oddelement ω ∈ End( Y ; C l ( V )). Recall that we have assumed that m is odd. Thisoperator satisfies ω = 1, ω = ω ∗ , ω y e i = − e i ω y and ω y ˆ e i = ˆ e i ω y . So we see that D V ω is an odd essentially self-adjoint operator, D V ω = ωD V and C V ω = − ωC V .It is well-known that, the operator B V , called the harmonic oscillator, is Fred-holm and has rank 1-kernel in the even part and zero kernel in the odd part. More-over, this kernel bundle has a canonical trivialization, given by the global section { e −k v k / ∈ L ( V y ; C l ( V y )) } y . We let P ∈ K ( L Y ( V ; C l ( V ))) to be the projectionto the kernel of B V . We denote the asymptotic algebra of K ( L Y ( V ; C l ( V )) by A µ ( K ( L Y ( V ; C l ( V ))) := C b ([ µ, ∞ ); K ( L Y ( V ; C l ( V ))) /C ([ µ, ∞ ); K ( L Y ( V ; C l ( V )))for µ ∈ R . Of course for any µ ∈ R , the above algebra A µ ( K ( L Y ( V ; C l ( V ))) arecanonically isomorphic.We define a Z -graded ∗ -homomorphism φ by φ : C ( R ; C l ) ˆ ⊗ C ( V ; C l ( V )) → A ( K ( L Y ( V ; C l ( V ))) f ˆ ⊗ h [[1 , ∞ ) ∋ t f ( t − D V ω ) · M h t ] f ǫ ˆ ⊗ h [[1 , ∞ ) ∋ t f ( t − D V ω ) ω · M h t ] , for f ∈ C ( R ) and h ∈ C ( V ; C l ( V )). Here we denote by h t ∈ C ( V ; C l ( V )) thefunction h t ( v ) := h ( t − v ) and by M h t the pointwise Clifford multiplication operatorby h t . The following lemma is an analogue of [GH04, Proposition 1.5]. Lemma 5.11.
The map φ defines a ∗ -homomorphism.Proof. This can be proved in an analogous way to the proof of [GH04, Proposition1.5]. Instead of repeating the proof, we only point out that the essential point isthat we have the following relations, e i D V = − D V e i , e i ω y = − ω y e i , and D V ω = ωD V . (cid:3) Remark . In [GH04], they use the C ∗ -algebra S := ( C ( R ) with even-odd grading).For an Euclidean space V , they define a Z -graded ∗ -homomorphism˜ φ : S ˆ ⊗ C ( V ; C l ( V )) → A ( K ( L ( V ; C l ( V ))) f ˆ ⊗ h [ t f ( t − D V ) · M h t ] . This ∗ -homomorphism defines an element in the E -theory group [ ˜ φ ] ∈ E ( C ( V ; C l ( V )) , C ),since their definition of E -theory groups is E ( A, B ) := [[ S ˆ ⊗ A ˆ ⊗K ( ˆ H ) , B ˆ ⊗K ( ˆ H )]]([GH04, Sectiton 2.1]). Here we denoted the abelian group of equivalence classes ofasymptotic morphisms from A to B by [[ A, B ]]. On the other hand, an asymptoticmorphism from A to B which admits a completely positive lifting naturally definesan element in KK ( A, B ). Thus their asymptotic morphism ˜ φ gives an element in KK ( S ˆ ⊗ C ( V ; C l ( V )) , C ), which is not the desired element. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 51
We would like to treat KK -elements arising from asymptotic morphisms di-rectly in our construction, so we do not use the C ∗ -algebra S and consider theabove asymptotic morphism φ , which indeed gives the KK -equivalence between C ( R ; C l ) ˆ ⊗ C ( V ; C l ( V )) and C ( Y ) (see Proposition 5.15 below).Define the C ∗ -algebra T P := { f ∈ C b ([0 , ∞ ); K ( L Y ( V ; C l ( V )))) | f (0) = P f (0) = f (0) P } . Recall that P is the orthogonal projection to ker B V , which is a rank one bundle on Y with a canonical trivialization. So we can identify C ( Y ) with P K ( L Y ( V ; C l ( V ))) P ,to get ∗ -homomorphism ev : T P → C ( Y ). We define a C ∗ -algebra C by the pull-back(5.13) C / / ev C∞ (cid:15) (cid:15) ❴✤ T P (cid:15) (cid:15) C ( R ; C l ) ˆ ⊗ C ( V ; C l ( V )) φ / / A ( K ( L Y ( V ; C l ( V ))) ≃ / / A ( K ( L Y ( V ; C l ( V )))We define the ∗ -homomorphism ev C : C → C ( Y ) by the composition of the top rowof (5.13) and ev . Lemma 5.14.
Let g be a function in C ( R ) . For t ∈ (0 , ∞ ) , consider the func-tional calculus g ( t − B V ) ∈ K ( L Y ( V ; C l ( V ))) . This family canonically extends toan element in T P by setting g ( t − B V ) | t =0 = g (0) P . We abuse the notation andstill denote this element by g ( t − B V ) ∈ T P .We define the function X : R → R , x x . Note that Xǫ and C V are oddunbounded multipliers on C ( R ; C l ) and C ( V ; C l ( V )) , respectively. Then we have φ ( g ( Xǫ ˆ ⊗ ⊗ C V )) = [ g ( t − B V )] ∈ A ( K ( L Y ( V ; C l ( V ))) . In other words, for a function g ∈ C ( R ) we have an element [ g ( Xǫ ˆ ⊗ ⊗ C V ) , g ( t − B V )] ∈C . Moreover, the operator B V := [ Xǫ ˆ ⊗ ⊗ C V , t − B V ] (defined to be at ∈ [0 , ∞ ) ) is an unbounded multiplier (see [GH04, pp. 168–169] ) of C .Proof. The essential point is that B V has discrete spectrum with finite multiplicity.The lemma is proved by checking on the generators e − x and xe − x of C ( R ), andthe computations are essentially the same as in [GH04, Section 1.13]. (cid:3) Proposition 5.15.
The ∗ -homomorphisms ev C : C → C ( Y ) , and ev C∞ : C → C ( R ; C l ) ˆ ⊗ C ( V ; C l ( V )) ≃ C ( V ⊕ R ; C l ( V ⊕ R )) induce KK -equivalences among C ( Y ) , C and C ( V ⊕ R ; C l ( V ⊕ R )) . Moreover, wehave (5.16) [ ev C ] − ⊗ [ ev C∞ ] = [ ψ ( Xǫ ˆ ⊗ ⊗ C V )] ∈ KK ( C ( Y ) , C ( V ⊕ R ; C l ( V ⊕ R ))) . Proof.
The fact that ev C∞ is a KK -equivalence is seen by checking that the kernelof this ∗ -homomorphism, ker( ev C∞ ) = { f ∈ C ([0 , ∞ ); K ( L Y ( V ; C l ( V )))) | f (0) = P f (0) = f (0) P } , is KK -contractible. Let us use the notation H Y := L Y ( V ; C l ( V )) in this proof. We have the following commutative diagram,0 / / C ((0 , ∞ ); K ( H Y )) / / ker( ev C∞ ) (cid:127) _ (cid:15) (cid:15) ev / / C ( Y ) / / (cid:127) _ (cid:15) (cid:15) / / C ((0 , ∞ ); K ( H Y )) / / C ([0 , ∞ ); K ( H Y )) ev / / K ( H Y ) / / , where the rows are exact. The right vertical inclusion is given by C ( Y ) ≃ P K ( H Y ) P ֒ →K ( H Y ) so this is a KK -equivalence. By the five lemma in KK -theory, we see thatthe middle vertical arrow is a KK -equivalence. Since C ([0 , ∞ ); K ( H Y )) is KK -contractible, so is ker( ev C∞ ).On the other hand, by Lemma 5.14, we see that ψ ( B V ) is a self-adjoint mul-tiplier of C which satisfies ψ ( B V ) − Id C ∈ C , and by construction ψ ( B V ) com-mutes with the action of C ( Y ) on C by multiplication. So we get the element[ ψ ( B V )] ∈ KK ( C ( Y ) , C ). It satisfies [ ψ ( B V )] ⊗ [ ev C∞ ] = [ ψ ( Xǫ ˆ ⊗ ⊗ C V )] ∈ KK ( C ( Y ) , C ( R ⊕ V ; C l ( R ⊕ V ))) and [ ψ ( B V )] ⊗ [ ev C ] = id C ( Y ) ∈ KK ( C ( Y ) , C ( Y ))(this is because ψ ( t − B V ) | t =0 = ψ (0) P = 0). Thus we have [ ψ ( Xǫ ˆ ⊗ ⊗ C V )] ⊗ [ ev C∞ ] − ⊗ [ ev C ] = id C ( Y ) ∈ KK ( C ( Y ) , C ( Y )). Since the element [ ψ ( Xǫ ˆ ⊗ ⊗ C V )]is the Thom element which is a KK -equivalence, we see that [ ev C ] ∈ KK ( C , C ( Y ))is a KK -equivalence, which proves (5.16). (cid:3) Now, we return to settings of manifolds with fibered boundaries (
M, π : ∂M → Y ) equipped with pre- spin c -structures. We assume that M is even dimensional and Y is odd dimensional. We apply the above constructions for V := T ∗ Y which is anoriented vector bundle over Y . Let us denote by Γ the groupoid ∂M × π ∂M ⇒ ∂M .Choosing differential spin c -structures on T V ∂M and T Y representing the givenpre- spin c structure, define a C ∗ -algebra Ψ( B V ˆ ⊗ ⊗ D π ) to be the C ∗ -subalgebraof M ( C ˆ ⊗ C ( Y ) C ∗ (Γ; S ( A Γ))) generated by { ψ ( B V ˆ ⊗ ⊗ D π ) } , C ˆ ⊗ C ( Y ) C ∗ (Γ; S ( A Γ))and C ( ∂M × [0 , ∞ ]). Note that we have an exact sequence0 → C ˆ ⊗ C ( Y ) C ∗ (Γ; S ( A Γ)) → Ψ( B V ˆ ⊗ ⊗ D π ) → C ( ∂M × [0 , ∞ ]) ⊗ C l → Definition 5.17 ( D π ) . In the above settings, we define the C ∗ -algebra D π by thepullback (c.f. Remark 5.3) D π / / (cid:15) (cid:15) ❴✤ Ψ( B V ˆ ⊗ ⊗ D π ) (cid:15) (cid:15) C ( M × [0 , ∞ ]) ⊗ C l i ∗ / / C ( ∂M × [0 , ∞ ]) ⊗ C l We have a ∗ -homomorphism ev := ev C ⊗ C ( Y ) id C ∗ (Γ) : C ˆ ⊗ C ∗ (Γ; S ( A Γ)) → C ∗ (Γ; S ( A Γ)). This ∗ -homomorphism extends to a ∗ -homomorphism D π → A π , bysending ψ ( B V ˆ ⊗ ⊗ D π ) to ψ ( D π ) and by evaluating at 0 ∈ [0 , ∞ ] on C ( M × [0 , ∞ ]) ⊗ C l , since ψ (0) = 0 and P is the projection to the kernel of B V . We alsodenote this ∗ -homomorphism by ev .On the other hand, using the isomorphism C ( R ⊕ T ∗ Y ; C l ( R ⊕ T ∗ Y )) ≃ C ∗ ( T Y × R ; S ( T Y × R )), we have a ∗ -homomorphism ev ∞ := ev C∞ ⊗ C ( Y ) id C ∗ (Γ) : C ˆ ⊗ C ( Y ) C ∗ (Γ; S ( A Γ)) → C ∗ ( T Y × R ; S ( T Y × R )) ˆ ⊗ C ( Y ) C ∗ (Γ , S ( A Γ)) ≃ C ∗ ( G Φ | ∂M ; S ( A G Φ | ∂M )). This ∗ -homomorphism extends to a ∗ -homomorphism D π → B π by sending ψ ( B V ˆ ⊗ NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 53 ⊗ D π ) to ψ ( D T Y × R ˆ ⊗ ⊗ D π ) = ψ ( D Φ ,∂ ) and evaluation at ∞ ∈ [0 , ∞ ] on C ( M × [0 , ∞ ]) ⊗ C l , since Xǫ ˆ ⊗ ⊗ C V corresponds to D T Y × R under the Fouriertransform. We also denote this ∗ -homomorphism by ev ∞ .In the next proposition, we need to use the R KK -theory for C ( Y )-algebras([Kas88, Section 2.19]). Given a compact space Y and two C ( Y )-algebras A and B , we get an abelian group R KK ( Y ; A, B ), roughly by requiring Kasparov modulesto be compatible with the action by C ( Y ). We also have the Kasparov product inthis theory, R KK ( Y ; A , B ˆ ⊗ C ( Y ) D ) ⊗ R KK ( Y ; D ˆ ⊗ C ( Y ) A , B ) → R KK ( Y ; A ˆ ⊗ C ( Y ) A , B ˆ ⊗ C ( Y ) B ) . With an abuse of notation, we also denote this product by ⊗ D . Note that Ind Y ( D T Y × R ) ∈ KK ( C ( Y ) , C ∗ ( T Y × R )) lifts canonically to an element in R KK ( C ( Y ) , C ∗ ( T Y × R )), which is also denoted by Ind Y ( D T Y × R ). Proposition 5.18.
Assume that M is even dimensional and Y is odd dimensional.Consider the following commutative diagram, (5.19)0 / / C ∗ (Γ; S ( A Γ)) / / A π / / C ( M ) ⊗ C l / / / / C ˆ ⊗ C ( Y ) C ∗ (Γ; S ( A Γ)) ev O O ev ∞ (cid:15) (cid:15) / / D π / / ev O O ev ∞ (cid:15) (cid:15) C ( M × [0 , ∞ ]) ⊗ C l / / ev O O ev ∞ (cid:15) (cid:15) / / C ∗ ( G Φ | ∂M ; S ( A G Φ | ∂M )) / / B π / / C ( M ) ⊗ C l / / , where the rows are exact. The vertical arrows are KK -equivalences, and we define (5.20) µ := [ ev ] − ⊗ D π [ ev ∞ ] ∈ KK ( A π , B π ) . Then this element fits into the commutative diagram in KK -theory, (5.21) 0 / / C ∗ (Γ; S ( A Γ)) / / Ind Y ( D TY × R ) ⊗ id C ∗ (Γ) (cid:15) (cid:15) A π / / µ (cid:15) (cid:15) C ( M ) ⊗ C l / / / / C ∗ ( G Φ | ∂M ; S ( A G Φ | ∂M )) / / B π / / C ( M ) ⊗ C l / / . Here the left vertical arrow is defined by taking Kasparov product of elements
Ind Y ( D T Y × R ) ∈ R KK ( Y ; C ( Y ) , C ∗ ( T Y × R )) and id C ∗ (Γ) ∈ R KK ( Y ; C ∗ (Γ) , C ∗ (Γ)) ,and then forgetting the C ( Y ) -algebra structure.Proof. For ease of notations, we drop the coefficient bundle in this proof and write,for example, C ∗ (Γ) for C ∗ (Γ; S ( A Γ)). The commutativity of the diagram (5.19)directly follows from the definition.That the arrows in the left column of (5.19) are KK -equivalences is the directconsequence of Proposition 5.15, by noting that ev = ev C ⊗ C ( Y ) id C ∗ (Γ) and ev ∞ = ev C∞ ⊗ C ( Y ) id C ∗ (Γ) . By (5.16), we also have the equality(5.22) [ ev ] − ⊗ [ ev ∞ ] = Ind Y ( D T Y × R ) ⊗ id C ∗ (Γ) ∈ R KK ( Y ; C ∗ (Γ) , C ∗ ( G Φ | ∂M )) , by noting that the operator D T Y × R corresponds to Xǫ ˆ ⊗ ⊗ C T ∗ Y = C T ∗ Y ⊕ R under the Fourier transform. Next let us look at the middle column of the diagram (5.19). By the com-mutativity of the diagram and the five lemma in KK -theory, the above KK -equivalence result on the left column implies that the middle vertical arrows arealso KK -equivalences. Finally, the commutativity of the diagram (5.21) followsfrom (5.22). (cid:3) Definition 5.23.
Let ( M ev , π : ∂M → Y odd ) be a compact manifold with fiberedboundary, equipped with a pre- spin c -structures P ′ π , P ′ M and P ′ Y on T V ∂M , T M and
T Y , respectively. We assume that these pre- spin c structures are compatibleat the boundary. Then we define[( P ′ M , P ′ Y )] := µ ⊗ [ ι ] ∈ KK ( A π , Σ ˚ M ( G Φ )) . Here the element [ ι ] ∈ KK ( B π , Σ ˚ M ( G Φ )) is defined in Definition 5.10 and µ ∈ KK ( A π , B π ) is defined in Proposition 5.18. Theorem 5.24 (The index pairing formula for spin c -Dirac operators) . Let ( M ev , π : ∂M → Y odd ) be a compact manifold with fibered boundary. Let P ′ π , P ′ M and P ′ Y be pre- spin c -structures on T V ∂M , T M and
T Y respectively. We assume that thepre- spin c -structures are compatible at the boundary. Let E be a complex vectorbundle over M . Let Q π ∈ I ( P ′ π , E ) . Then we have Ind Φ ( P ′ M , P ′ Y , E, Q π ) = [( E, Q π )] ⊗ A π [( P ′ M , P ′ Y )] ⊗ Σ ˚ M ( G Φ ) ind ˚ M ( G Φ ) ∈ Z . Here the element ind ˚ M ( G Φ ) ∈ KK (Σ ˚ M ( G Φ ) , C ) is defined in subsection 2.3.3.Proof. We only prove the Theorem in the case where the dimension of the fiber of π is even. As in Lemma 5.7, a pair ( E, Q Φ ,∂ ), where E → M is a Z -graded complexvector bundle and Q Φ ,∂ ∈ I ( D E Φ ,∂ ), naturally defines a class [( E, Q Φ ,∂ )] ∈ K ( B π ).Here we denoted by D E Φ ,∂ the spin c Dirac operator twisted by E and I ( D E Φ ,∂ ) isthe set of homotopy classes of C l -invertible perturbations of the operator D E Φ ,∂ on G Φ | ∂ .First we remark that, when we are given a twisting bundle E , it is convenient touse C ∗ -algebras A π ( E ), B π ( E ) and D π ( E ) which are Morita equivalent to A π , B π and D π , respectively; the first one appears in (5.8) and the definitions of the otheralgebras are self-explanatory. The corresponding element µ ∈ KK ( A π ( E ) , B π ( E ))is realized as [ ev ] − ⊗ D π ( E ) [ ev ∞ ] as in Proposition 5.18, and a ∗ -homomorphism ι E : B π ( E ) → Σ ˚ M ( G Φ ; S ( A G Φ ) ˆ ⊗ E ) is constructed analogously to Definition 5.10.It is enough to prove the following. Given an element Q π ∈ I ( P π , E ), take somerepresentative ˜ D Eπ for Q π and consider the class Q Φ ,∂ := [ ˜ D Eπ ˆ ⊗ ⊗ D T Y × R ] ∈I ( D Eπ ˆ ⊗ ⊗ D T Y × R ) (this class does not depend on the choice). Then we have(5.25) [( E, Q π )] ⊗ A π µ = [( E, Q Φ ,∂ )] ∈ KK ( C l , B π ) . For, in view of the definition of ι E , we have the equality[( E, Q Φ ,∂ )] ⊗ ι ⊗ ind ˚ M ( G Φ ) = [( σ ( D E Φ ) , ψ ( ˜ D Eπ ˆ ⊗ ⊗ D T Y × R ))] ⊗ ind ˚ M ( G Φ )= Ind Φ ( P ′ M , P ′ Y , E, Q π ) . So let us prove (5.25). For simplicity we assume E is the trivial bundle. For a givenrepresentative ˜ D π for Q π ∈ I ( P π ), we can construct an invertible element in D π NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 55 defined as (1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D π )) ∈ D π . By definition we have ev ((1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D π ))) = (1 ˆ ⊗ ǫ, ψ ( ˜ D π )) ∈ A π ev ∞ ((1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D π ))) = (1 ˆ ⊗ ǫ, ψ ( ˜ D Φ ,∂ )) ∈ B π . Thus we see that [(1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D π ))] ⊗ [ ev ] = [( C , Q π )] ∈ K ( A π ) and[(1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D π ))] ⊗ [ ev ∞ ] = [( C , Q Φ ,∂ )] ∈ K ( B π ). By Proposition 5.18, wesee that [( C , Q π )] ⊗ A π µ = [( C , Q π )] ⊗ A π [ ev ] − ⊗ D π [ ev ∞ ] = [( C , Q Φ ,∂ )] ∈ K ( B π ),so we get (5.25). (cid:3) The case of signature operators.
In this subsection, we consider the caseof signature operators. The arguments are parallel to those in subsection 5.1. Let(
M, π : ∂M → Y ) be a compact manifold with fibered boundaries, with fixedorientation on T M and
T Y . For simplicity we only consider the case where thefibers of π are even dimensional.We have the element Ind( D sign π ) ∈ KK ( C ( ∂M ) , C ( Y )) given by the fiberwisefamily of signature operators. This element gives the “signature pushforward”homomorphism, π sign! := ⊗ C ( ∂M ) Ind( D sign π ) : K ∗ ( ∂M ) → K ∗ ( Y ) . First, exactly in the analogous way to that in the last subsection, we define a C ∗ -algebra A sign π whose K -group fits in the exact sequence · · · → K ∗ ( M ) π sign! ◦ i ∗ −−−−−→ K ∗ ( Y ) → K ∗ ( A sign π ) → K ∗ +1 ( M ) π sign! ◦ i ∗ −−−−−→ · · · . Let N be a compact space. Let π : N → Y be a fiber bundle whose fibersare equipped with even dimensional closed manifold structure, and an orientationof π is fixed. Choose any fiberwise riemannian metric, and denote the fiberwisesignature operator by D sign π . Let L Y ( N ; ∧ C ( T V N ) ∗ ) denote the Z -graded Hilbert C ( Y )-module which is obtained by the completion of C ∞ c ( N ; ∧ C ( T V N ) ∗ ) with thenatural C ( Y )-valued inner product.Denote the odd function ψ ( x ) := x/ √ x . Let Ψ( D sign π ) denote the Z -graded C ∗ -subalgebra of B ( L Y ( N ; ∧ C ( T V N ) ∗ )) generated by { ψ ( D sign π ) } , C ( N )and K ( L Y ( N ; ∧ C ( T V N ) ∗ )). Lemma 5.26.
The algebra Ψ( D sign π ) fits into the exact sequence of graded C ∗ -algebras (5.27) 0 → K ( L Y ( N ; ∧ C ( T V N ) ∗ )) → Ψ( D sign π ) → C ( N ) ⊗ C l → . The connecting element of this extension coincides with the class π sign! ∈ KK ( C ( N ) , C ( Y )) . Definition 5.28 ( A sign π ) . Let (
M, π : ∂M → Y ) be a compact manifold withfibered boundary. Assume that π is oriented and the fibers are even dimensional.Denote i : ∂M → M the inclusion.We define A sign π to be the Z -graded C ∗ -algebra defined by the pullback (c.f.Remark 5.3) A sign π / / (cid:15) (cid:15) ❴✤ Ψ( D sign π ) (cid:15) (cid:15) C ( M ) ⊗ C l i ∗ / / C ( ∂M ) ⊗ C l
16 M. YAMASHITA
We can prove that this C ∗ -algebra induces the desired long exact sequence,analogously to Proposition 5.5. Proposition 5.29.
Let ( M, π : ∂M → Y ) be a compact manifold with fiberedboundary. Assume that π is oriented and the dimension of fibers is even. The K -groups of the C ∗ -algebra A sign π naturally fits in the exact sequence · · · → K ∗ ( M ) π sign! ◦ i ∗ −−−−−→ K ∗ ( Y ) → K ∗ ( A sign π ) → K ∗ +1 ( M ) π sign! ◦ i ∗ −−−−−→ · · · . Then analogously to Lemma 5.7, we have the following.
Lemma 5.30.
Let ( M, π : ∂M → Y ) be a compact manifold with fibered boundary.Let us denote by Γ the groupoid ∂M × π ∂M ⇒ ∂M . Assume that π is orientedand the fibers are even dimensional. Let E be a Z -graded complex vector bundleover M . Assume we are given an element Q π ∈ I sign ( π, E ) . Then the pair ( E, Q π ) naturally defines a class [( E, Q π )] ∈ K ( A sign π ) . Now we assume that M is even dimensional and oriented. We construct anelement [ M sign ] ∈ KK ( A sign π , Σ ˚ M ( G Φ )). Lemma 5.31.
Let ( M ev , π : ∂M → Y odd ) be a compact manifold with fiberedboundary, equipped with orientations on T M and T V ∂M .Choose any metric on A G Φ which has a direct sum decomposition at the bound-ary, and denote the associated signature operator on G Φ | ∂M = ∂M × π ∂M × π T Y × R ⇒ ∂M as, D signΦ ,∂ := D sign π ˆ ⊗ ⊗ D sign T Y × R . Let Ψ( D signΦ ,∂ ) denote the Z -graded C ∗ -subalgebra of Ψ c ( G Φ | ∂M ; ∧ C ( A G Φ | ∂M ) ∗ ) gen-erated by { ψ ( D signΦ ,∂ ) } , C ( ∂M ) and C ∗ ( G Φ | ∂M ; ∧ C ( A G Φ | ∂M ) ∗ ) . This C ∗ -algebra fitsinto the graded exact sequence → C ∗ ( G Φ | ∂M ; ∧ C ( A G Φ | ∂M ) ∗ ) → Ψ( D signΦ ,∂ ) → C ( ∂M ) ⊗ C l → . The connecting element of this extension coincides with the class π sign! ⊗ C ( Y ) Ind Y ( D sign T Y × R ) ∈ KK ( C ( ∂M ) , C ∗ ( G Φ | ∂M )) . Definition 5.32 ( B sign π ) . We define a graded C ∗ -algebra by the pullback (c.f. Re-mark 5.3) B sign π / / (cid:15) (cid:15) ❴✤ Ψ( D signΦ ,∂ ) (cid:15) (cid:15) C ( M ) ⊗ C l i ∗ / / C ( ∂M ) ⊗ C l Choosing a metric of G Φ , we get a canonical injective ∗ -homomorphism ι : B sign π → Σ ˚ M ( G Φ ; ∧ C ( A G Φ ) ∗ ). The KK -element [ ι ] ∈ KK ( B sign π , Σ ˚ M ( G Φ )) is independentof the choice of a metric.We are going to construct a KK -element µ sign ∈ KK ( A sign π , B sign π ) analogousto µ in Proposition 5.18. Note that in the signature case, this element is not a KK -equivalence.We construct a C ∗ -algebra D sign π , using the C ∗ -algebra C constructed in the lastsubsection. Recall that the construction of C does not need any spin c -structure NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 57 on vector bundle V → Y . We apply the constructions in the last subsection for V = T ∗ Y . Denote Γ the groupoid ∂M × π ∂M ⇒ ∂M .Choosing any riemannian metrics on T V ∂M and T Y , define the C ∗ -algebraΨ( B V ˆ ⊗ ⊗ D sign π ) to be the C ∗ -subalgebra of M ( C ˆ ⊗ C ∗ (Γ; ∧ C ( A Γ) ∗ )) generatedby { ψ ( B V ˆ ⊗ ⊗ D sign π ) } , C ˆ ⊗ C ∗ (Γ; ∧ C ( A Γ) ∗ ) and C ( ∂M × [0 , ∞ ]). Note that wehave an exact sequence0 → C ˆ ⊗ C ∗ (Γ; ∧ C ( A Γ) ∗ ) → Ψ( B V ˆ ⊗ ⊗ D sign π ) → C ( ∂M × [0 , ∞ ]) ⊗ C l → . Definition 5.33 ( D sign π ) . In the above settings, we define the C ∗ -algebra D sign π bythe pullback (c.f. Remark 5.3) D sign π / / (cid:15) (cid:15) ❴✤ Ψ( B V ˆ ⊗ ⊗ D sign π ) (cid:15) (cid:15) C ( M × [0 , ∞ ]) ⊗ C l i ∗ / / C ( ∂M × [0 , ∞ ]) ⊗ C l Analogously to the last subsection, the ∗ -homomorphism ev C : C → C ( Y ) in-duces a ∗ -homomorphism ev : C ˆ ⊗ C ∗ (Γ; ∧ C ( A Γ) ∗ ) → C ∗ (Γ; ∧ C ( A Γ) ∗ ). This ∗ -homomorphism extends to a ∗ -homomorphism D sign π → A sign π , by sending ψ ( B V ˆ ⊗ ⊗ D sign π ) to ψ ( D sign π ) and evaluation at 0 on C ( M × [0 , ∞ ]) ⊗ C l . We denote this ∗ -homomorphism by ev .On the other hand, contrary to the last subsection, the ∗ -homomorphism ev C∞ : C → C ( R ; C l ) ˆ ⊗ C ( V ; C l ( V )) induces a ∗ -homomorphism ev ∞ : C ˆ ⊗ C ( Y ) C ∗ (Γ; ∧ C ( A Γ) ∗ ) → C ( V ⊕ R ; C l ( V ⊕ R )) ˆ ⊗ C ( Y ) C ∗ (Γ , ∧ C ( A Γ) ∗ ), and the range of this homomorphismis not isomorphic to C ∗ ( G Φ | ∂M ; ∧ C ( A G Φ | ∂M ) ∗ ). To overcome this difference, weneed an intermediate C ∗ -algebra ˜ B sign π . Definition 5.34 ( ˜ B sign π ) . Let us denote V = T ∗ Y . Consider the C ∗ -algebra C ( V ⊕ R ; C l ( V ⊕ R )) ˆ ⊗ C ( Y ) C ∗ (Γ , ∧ C ( A Γ) ∗ ) and the unbounded multiplier C V ⊕ R ˆ ⊗ ⊗ D sign π of this C ∗ -algebra. Let Ψ( C V ⊕ R ˆ ⊗ ⊗ D sign π ) denote the Z -graded C ∗ -subalgebraof the multiplier algebra M ( C ( V ⊕ R ; C l ( V ⊕ R )) ˆ ⊗ C ( Y ) C ∗ (Γ , ∧ C ( A Γ) ∗ )), generatedby { ψ ( C V ⊕ R ˆ ⊗ ⊗ D sign π ) } , C ( ∂M ) and C ( V ⊕ R ; C l ( V ⊕ R )) ˆ ⊗ C ( Y ) C ∗ (Γ , ∧ C ( A Γ) ∗ ).This C ∗ -algebra fits into the graded exact sequence0 → C ( V ⊕ R ; C l ( V ⊕ R )) ˆ ⊗ C ( Y ) C ∗ (Γ , ∧ C ( A Γ) ∗ ) → Ψ( C V ⊕ R ˆ ⊗ ⊗ D sign π ) → C ( ∂M ) ⊗ C l → . We define the C ∗ -algebra ˜ B sign π by the pullback (c.f. Remark 5.3)˜ B sign π / / (cid:15) (cid:15) ❴✤ Ψ( C V ⊕ R ˆ ⊗ ⊗ D sign π ) (cid:15) (cid:15) C ( M ) ⊗ C l i ∗ / / C ( ∂M ) ⊗ C l Next, we construct a ∗ -homomorphism b : ˜ B sign π → B sign π . Since the vector bundle ∧ C ( T Y ⊕ R ) ∗ → Y is a C l ( T Y ⊕ R )-module bundle, if we define a spin structureon T Y ⊕ R locally, we can write ∧ C ( T Y ⊕ R ) ∗ = S ( T Y ⊕ R ) ˆ ⊗ W, with some Z -graded hermitian vector bundle W , and the Clifford multiplicationcan be written as c ˆ ⊗
1. Note that the vector bundle End( W ) is canonically definedindependently on the chosen local spin structure, and extends to a vector bundle over the whole Y , still denoted by End( W ). Under the Fourier transform, theoperator D sign T Y ⊕ R corresponds to the unbounded multiplier C V ⊕ R ˆ ⊗ C ( V ⊕ R ; End( ∧ C ( T Y ⊕ R ) ∗ )) ≃ C ( V ⊕ R ; C l ( V ⊕ R ) ˆ ⊗ End( W )).Thus the ∗ -homomorphism id C l ( V ⊕ R ) ˆ ⊗ W : C ( V ⊕ R ; C l ( V ⊕ R )) → C ( V ⊕ R ; C l ( V ⊕ R ) ˆ ⊗ End( W )) , induces the ∗ -homomorphism b ′ : Ψ( C V ⊕ R ˆ ⊗ ⊗ D sign π ) → Ψ( D signΦ ,∂ )by sending ψ ( C V ⊕ R ˆ ⊗ ⊗ D sign π ) to ψ ( D signΦ ,∂ ), and induces the desired ∗ -homomorphism b : ˜ B sign π → B sign π . Using this intermediate algebra, we easily see the following proposition, which isthe signature version of Proposition 5.18.
Proposition 5.35.
Consider the following commutative diagram, / / C ∗ (Γ; ∧ C ( A Γ) ∗ ) / / A sign π / / C ( M ) ⊗ C l / / / / C ˆ ⊗ C ( Y ) C ∗ (Γ; ∧ C ( A Γ) ∗ ) ev O O ev ∞ (cid:15) (cid:15) / / D sign π / / ev O O ev ∞ (cid:15) (cid:15) C ( M × [0 , ∞ ]) ⊗ C l / / ev O O ev ∞ (cid:15) (cid:15) / / C ( V ⊕ R ; C l ( V ⊕ R )) ˆ ⊗ C ( Y ) C ∗ (Γ , ∧ C ( A Γ) ∗ ) / / ˆ ⊗ W ˆ ⊗ id C ∗ (Γ) (cid:15) (cid:15) ˜ B sign π / / b (cid:15) (cid:15) C ( M ) ⊗ C l / / / / C ∗ ( G Φ | ∂M ; ∧ C ( A G Φ | ∂M ) ∗ ) / / B sign π / / C ( M ) ⊗ C l / / where the rows are exact. The arrows connecting the first, second and third rows,denoted by ev and ev ∞ , are KK -equivalences, and we define (5.36) µ sign := [ ev ] − ⊗ D sign π [ ev ∞ ] ⊗ ˜ B sign π [ b ] ∈ KK ( A sign π , B sign π ) . Then this element fits into the commutative diagram in KK -theory, / / C ∗ (Γ; ∧ C ( A Γ) ∗ ) / / Ind Y ( D sign TY × R ) ⊗ id C ∗ (Γ) (cid:15) (cid:15) A sign π / / µ sign (cid:15) (cid:15) C ( M ) ⊗ C l / / / / C ∗ ( G Φ | ∂M ; ∧ C ( A G Φ | ∂M ) ∗ ) / / B sign π / / C ( M ) ⊗ C l / / . Here the left vertical arrow is defined by taking Kasparov product of elements
Ind Y ( D sign T Y × R ) ∈ R KK ( Y ; C ( Y ) , C ∗ ( T Y × R )) and id C ∗ (Γ) ∈ R KK ( Y ; C ∗ (Γ) , C ∗ (Γ)) ,and then forgetting the C ( Y ) -algebra structure. Finally we define the element [ M sign ] ∈ KK ( A sign π , Σ ˚ M ( G Φ )) as follows. Definition 5.37.
Let ( M ev , π : ∂M → Y odd ) be a compact manifold with fiberedboundary, equipped with orientations on T M and T V ∂M . Then we define[ M sign ] := µ sign ⊗ B sign π [ ι ] ∈ KK ( A sign π , Σ ˚ M ( G Φ )) . Here the element [ ι ] ∈ KK ( B sign π , Σ ˚ M ( G Φ )) is defined in Definition 5.32 and µ sign is defined in Proposition 5.35. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 59
Then, we can describe the Φ-signature as follows.
Theorem 5.38 (The index pairing formula for signature operators) . Let ( M, π : ∂M → Y ) be a compact even dimensional manifold with fibered boundary, equippedwith orientations on T M and T V ∂M . Let E be a complex vector bundle over M .Let Q π ∈ I sign ( π, E ) . Then we have Sign Φ ( M, E, Q π ) = [( E, Q π )] ⊗ A sign π [ M sign ] ⊗ Σ ˚ M ( G Φ ) ind ˚ M ( G Φ ) ∈ Z . Proof.
As in the proof of Theorem 5.24, it is enough to prove the following. Givenan element Q π ∈ I sign ( π, E ), take some representative ˜ D sign ,Eπ for Q π and considerthe class Q Φ ,∂ := [ ˜ D sign ,Eπ ˆ ⊗ ⊗ D sign T Y × R ] ∈ I ( D sign ,Eπ ˆ ⊗ ⊗ D sign T Y × R ) (this classdoes not depend on the choice). Then we have[( E, Q π )] ⊗ A sign π µ sign = [( E, Q Φ ,∂ )] ∈ KK ( C l , B sign π ) . For simplicity we assume E is the trivial bundle. For a given representative˜ D sign π for Q π ∈ I sign ( π ), we can construct an invertible element in D π defined as(1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D sign π )) ∈ D π . By definition we have ev ((1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D sign π ))) = (1 ˆ ⊗ ǫ, ψ ( ˜ D sign π )) ∈ A sign π b ◦ ev ∞ ((1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D sign π ))) = b ((1 ˆ ⊗ ǫ, ψ ( C V ⊕ R ˆ ⊗ ⊗ ˜ D sign π )))= (1 ˆ ⊗ ǫ, ψ ( D sign T Y ⊕ R ˆ ⊗ ⊗ ˜ D sign π )) ∈ B sign π . Thus we see that [(1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D sign π ))] ⊗ [ ev ] = [( C , Q π )] ∈ K ( A sign π ) and[(1 ˆ ⊗ ǫ, ψ ( B V ˆ ⊗ ⊗ ˜ D sign π ))] ⊗ [ ev ∞ ] ⊗ [ b ] = [( C , Q Φ ,∂ )] ∈ K ( B sign π ). By Proposition5.35, we see that [( C , Q π )] ⊗ A sign π µ sign = [( C , Q π )] ⊗ A sign π [ ev ] − ⊗ D sign π [ ev ∞ ] ⊗ ˜ B π [ b ] =[( C , Q Φ ,∂ )] ∈ K ( B sign π ), so we get the result. (cid:3) The local Signature
Settings.
The settings for the localization problem for signature are the fol-lowing. • Let F be an oriented closed even dimensional smooth manifold. • Let G be a subgroup of the orientation-preserving diffeomorphism groupDiff + ( F ) of F . • Let Z ⊂ BG be a subspace of the classifying space of G . A particular caseof interest is when Z is the k -skeleton of a CW -complex model of BG forsome integer k .We can define the universal family signature class Sign( F univ ) ∈ K ( BG ), where K ( BG ) denotes the representable K -theory of BG (see subsection 2.1). Thisclass is constructed by considering the fiberwise signature operators on the uni-versal F -fiber bundle π univ over BG . This construction is explained in detail insubsection 6.2. Moreover, if there exists a positive integer n such that the re-striction of Sign( F univ ) to Z is n -torsion in K ( Z ), the set of homotopy equiva-lence classes of C l -invertible perturbations of n -direct sum of signature operators, I sign ( π univ | Z , C n ), is nonempty and has a canonical affine space structure modeledon K − ( Z ). Definition 6.1.
Let S F,G,Z denote the set of isomorphism classes of pairs (
M, π : ∂M → Y ) satisfying the following conditions: • The pair (
M, π : ∂M → Y ) is a compact oriented manifold with fiberedboundaries, and assume that M is even dimensional. • Assume that π is an F -fiber bundle structure with structure group G . • Assume that i ∗ : [ Y, Z ] → [ Y, BG ], induced by the inclusion i : Z → BG , isan isomorphism.Our main theorem of this section is the following. Theorem 6.2.
Assume that a positive integer n satisfies n · i ∗ Sign( F univ ) = 0 ∈ K ( Z ) . For each element Q Z ∈ I sign ( π univ | Z , C n ) , we have a natural map σ Q Z : S F,G,Z → Z n such that the following holds. • (vanishing)For ( M, π ) ∈ S F,G,Z , if there exist a compact oriented manifold withboundary ( X, ∂X ) with a fixed diffeomorphism ∂X ≃ Y , and an F -fiberbundle structure π ′ : M → X with structure group G which satisfies π ′ | ∂M = π such that i ∗ : [ X, Z ] → [ X, BG ] is surjective, then we have σ Q Z ( M, π ) = 0 . • (additivity)For ( M , π ) and ( M , π ) in S F,G,Z , assume that there exists a de-composition ∂M i = ∂M + i ⊔ − ∂M − i for i = 0 , , and there exists an iso-morphism of the fiber bundle φ : π | ∂M +0 ≃ π | − ∂M − . We can form ( M, π ) = ( M , π ) ∪ φ ( M , π ) ∈ S F,G,Z . Then we have σ Q Z ( M, π ) = σ Q Z ( M , π ) + σ Q Z ( M , π ) . • (compatibility with signature)An oriented even dimensional closed manifold M can be regarded as anelement in S F,G,Z . For this element, we have σ Q Z ( M ) = Sign( M ) . Moreover, if we have two elements Q Z , Q Z ∈ I sign ( π univ | Z , C n ) , the difference be-tween σ Q Z and σ Q Z is described as follows. Let ( M, π : ∂M → Y ) be an element in S F,G,Z , and denote the classifying map for π by [ f ] ∈ [ Y, Z ] ≃ [ Y, BG ] . Recall thatwe have the difference class [ Q Z − Q Z ] ∈ K − ( Z ) . Denote the K -homology class ofsignature operator on Y by [ D sign Y ] ∈ K ( Y ) . We have, for each ( M, π ) ∈ S F,G,Z , (6.3) σ Q Z ( M, π ) − σ Q Z ( M, π ) = 2 n h f ∗ ([ Q Z − Q Z ]) , [ D sign Y ] i . Here we denoted the index pairing by h· , ·i : K ( Y ) ⊗ K ( Y ) → Z . The universal index class and the pullback of C l -invertible per-turbations. Let F be an oriented closed even dimensional smooth manifold and G ⊂ Diff + ( F ) be a subgroup. We define the universal signature class Sign( F univ ) ∈ K ( BG ) as follows. We have the universal F -fiber bundle over BG , π univ : EG × G F → BG.
Fix any continuous family of fiberwise smooth metrics g univ over this fiber bundle.Then this defines a Z -graded Hilbert bundle (see Definition 2.6)ˆ H univ := { L ( π − ( x ); ∧ C T ∗ F g univ,x ) } x ∈ BG → BG.
NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 61
Also the metric g univ defines the fiberwise signature operator D signuniv acting on ˆ H univ ,and the bounded transform of this operator, ψ ( D signuniv ), gives an element ψ ( D signuniv ) ∈ Γ( BG ; Fred (0) ( ˆ H univ )) (see subsection 2.2). So this operator defines a classSign univ ( F ) := [ ψ ( D signuniv )] ∈ K ( BG ) , where the symbol K denotes the representable K -theory. Since the space of choicesof fiberwise metric g univ is contractible, this class does not depend on the choice of g univ .Given a subset Z ⊂ BG , we can restrict this class and get the universal signatureclass over Z , Sign univ ( F ) := i ∗ Sign univ ( F ) ∈ K ( Z ) , where i : Z → BG denotes the inclusion. We abuse the notation and use the samesymbol for the universal signature class over Z and BG without any confusion.Next we give fundamental remarks on pullbacks of C l -invertible perturbationsfor signature operators. Suppose that we are given a continuous map f : X → X between topological spaces, and a continuous oriented fiber bundle structure π : M → X with fiber F . The fiberwise signature class, Sign( π ) ∈ K ( X ), isdefined as above. Consider the pullback bundle f ∗ π : f ∗ ( M ) → X . The fiberwisesignature class of this bundle satisfies Sign( f ∗ π ) = f ∗ Sign( π ) ∈ K ( X ).Suppose that we have Sign( π ) = 0 ∈ K ( X ). Then by Lemma 2.10, we see thatthe set I sign ( π ) is nonempty (see Remark 4.14; it is easy to see the remark appliesto the case where we do not assume smooth structure on the base X ). By thecontractibility of the space of fiberwise metrics, the pullback map, f ∗ : I sign ( π ) → I sign ( f ∗ π ) , is well-defined. Moreover we can easily see that this map is actually a homo-morphism of affine spaces, with respect to the homomorphism f ∗ : K − ( X ) → K − ( X ).We can also generalize this construction to the signature operators twisted bythe trivial rank n bundle C n , i.e., the n -direct sum of signature operators. Supposethat n · Sign( f ∗ π ) = f ∗ Sign( π ) ∈ K ( X ). Then the set I sign ( π, C n ) is nonempty,and we have a well-defined affine space homomorphism(6.4) f ∗ : I sign ( π, C n ) → I sign ( f ∗ π, C n ) . The local signature.
In this subsection, we return to the settings of sub-section 6.1. First we explain the pullback of C l -invertible perturbations by theclassifying maps. We cannot apply the procedure explained in the last sectiondirectly, because the classifying map is defined up to homotopy . Proposition 6.5.
Suppose we are given a F -fiber bundle π : M → X over atopological space X with structure group G . (1) Suppose that the universal signature class satisfies n · Sign( F univ ) = 0 ∈ K ( BG ) . Then the classifying map [ f ] ∈ [ X, BG ] induces a well-definedaffine space homomorphism [ f ] ∗ : I sign ( π univ , C n ) → I sign ( π, C n ) . (2) Let Z ⊂ BG be a subspace and assume that i ∗ : [ X, Z ] → [ X, BG ] inducedby the inclusion i : Z → BG is an isomorphism. Also assume that n · i ∗ Sign( F univ ) = 0 ∈ K ( Z ) . Then the classifying map [ f ] ∈ [ X, Z ] inducesa well-defined affine space homomorphism [ f ] ∗ : I sign ( π univ | Z , C n ) → I sign ( π, C n ) . Proof.
First we prove the case (1). We recall the construction of the classifyingmap for the fiber bundle π . Let ˜ π : P → X be the principal G -bundle such that π = ( P × G F → X ). We have a fiber bundle(6.6) P × G EG → X. Since the fiber of the bundle (6.6) is contractible, we can take a section s : X → P × G EG , and any choice of section is homotopic to each other. If we fix a section s , we get the associated maps f s : X → BG ; x ¯ π ( s ( x ))(6.7) φ ′ s : P → EG ; s (˜ π ( p )) = [ p, φ ′ s ( p )] for p ∈ P. (6.8)Here we denoted by ¯ π the canonical map P × G EG → BG . This defines a bundlemap ( φ ′ s , f s ) : ( P, X ) → ( EG, BG ). This induces a bundle map between the associ-ated bundles π : M = P × G F → X and π univ : M univ = EG × G F → BG , denotedby ( φ s , f s ) : ( M, X ) → ( M univ , BG ). Note that fixing a bundle map as above isequivalent to fixing an identification f ∗ s M univ ≃ M as a fiber bundle over X . As in(6.4), this induces an affine space homomorphism( φ s , f s ) ∗ : I sign ( π univ , C n ) → I sign ( π, C n ) . Since any two choices of the section s are homotopic, we can easily see that thishomomorphism does not depend on the choice of s .Next we prove the case (2). Denote π − ( Z ) = ˜ Z ⊂ M univ . In this case, bythe next Lemma 6.9, we see that we can take a section s : X → P × G ˜ Z , and anychoice of section is homotopic to each other. Thus we can apply exactly the sameargument as in the case (1) and get the result. (cid:3) Lemma 6.9. If i ∗ : [ X, Z ] ≃ [ X, BG ] , the space Γ( X ; P × G ˜ Z ) is nonempty andpath-connected.Proof. First we prove the nonemptiness. Choose a section s : X → P × G EG .As in (6.7), we get the associated maps f : X → BG and φ : P → EG . Since i ∗ : [ X, Z ] ≃ [ X, BG ], we can find a continuous map F ′ : X × [0 , → BG such that F ′ | X ×{ } = f and Im( F ′ | X ×{ } ) ⊂ Z . We denote F = id X × F ′ : X × [0 , → X × BG . Note that s is a lift of F | X ×{ } for the fiber bundle Π : P × G EG → X × BG . By the homotopy lifting property applied to the diagram P × G EG Π (cid:15) (cid:15) X × [0 , F / / X × BG NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 63 we can lift F to ˜ F : X × [0 , → P × G EG . ˜ F | X ×{ } : X → P × G ˜ Z gives anelement in Γ( X ; P × G ˜ Z ).Next we prove the path-connectedness. Let us denote the canonical projectionby Π = (Π X , Π BG ) : P × G EG → X × BG . Suppose we are given two sections s , s ∈ Γ( X ; P × G ˜ Z ). Since the fiber of the fiber bundle Π X : P × G EG → X iscontractible, we can choose a path { s t } t ∈ [0 , in Γ( X ; P × G EG ) connecting s and s . We denote S : X × [0 , → P × G EG : ( x, t ) s t ( x ) . Π X ◦ S : X × [0 , → BG satisfies Im(Π X ◦ S | X ×{ , } ) ⊂ Z . Since i ∗ : [ X, Z ] ≃ [ X, BG ], we can take a continuous map F : X × [0 , t × [0 , u → BG satisfying F| X × [0 , ×{ } = Π X ◦ S , F| X ×{ i }×{ u } = F| X ×{ i }×{ } for all u ∈ [0 , i = 0 ,
1, and Im( F| X × [0 , ×{ } ) ⊂ Z . In the diagram P × G EG Π (cid:15) (cid:15) ( X × [0 , t ) × [0 , u id X ×F / / X × BG we see that S is a lift of ( id X ×F ) | X × [0 , ×{ } . By the homotopy lifting property, weget a lift of id X × F . Its restriction to X × [0 , × { } is a map X × [0 , → P × G ˜ Z ,and by the construction, this gives a path in Γ( X ; P × G ˜ Z ) connecting s and s . (cid:3) We proceed to give a proof of Theorem 6.2. Under the assumption n · i ∗ Sign( F univ ) =0 ∈ K ( Z ), the set I sign ( π univ | Z , C ) is nonempty. For each Q Z ∈ I sign ( π univ | Z , C ),we are going to construct a map σ Q Z : S F, Γ ,Z → Z n satisfying the conditions in the Theorem 6.2. Definition 6.10.
Assume that n · i ∗ Sign( F univ ) = 0 ∈ K ( Z ). For a given element Q Z ∈ I sign ( π univ | Z , C n ), we define the map σ Q Z : S F,G,Z → Z n as, for ( M, π : ∂M → Y ) ∈ S F,G,Z , σ Q Z ( M, π : ∂M → Y ) = 1 n Sign Φ ( M, C n , [ f ] ∗ ( Q Z )) , where [ f ] ∈ [ Y, Z ] is the classifying map for the fiber bundle π .We check that this map satisfies the conditions in Theorem 6.2. Proof. (of Theorem 6.2) First we prove the vanishing condition. Suppose we aregiven an element (
M, π ) ∈ S F,G,Z such that π extends to an F -fiber bundle structure π ′ : M → X with structure group G and i ∗ : [ X, Z ] → [ X, BG ] is surjective. Denotethe classifying map of π ′ by [ f ′ ] ∈ [ X, BG ]. Take any lift of [ f ′ ] to an element of[ ˜ f ′ ] ∈ [ X, Z ], and realize this map as a bundle map ( ˜ φ ′ , ˜ f ′ ) : ( M, X ) → ( ˜ Z, Z ).Then as in (6.4), we can pullback the element Q Z ∈ I sign ( π univ | Z , C n ) by thebundle map ( ˜ φ ′ , ˜ f ′ ) to get an element ( ˜ φ ′ , ˜ f ′ ) ∗ Q Z ∈ I sign ( π ′ , C n ). This element restricts to [ f ] ∗ ( Q Z ) ∈ I sign ( π, C n ) at the boundary. Thus applying the vanishingproposition, the signature version of Proposition 4.19, we get the result.The additivity follows from the gluing formula, the twisted signature version ofProposition 4.18.The compatibility with signature is obvious by definition.The equation (6.3) follows from the relative formula for Φ-signature, Proposition4.26. (cid:3) Examples
In this section, as an application of Theorem 6.2, we consider the followinglocalization problem for singular surface bundles.Fix a positive integer k and an integer g ≥
0. Let
M, X be 4 k, (4 k − π : M → X bea smooth map and X = U ∪ ∪ mi =1 V i be a partition into compact manifolds withclosed boundaries, i.e., U and V i are compact manifolds with closed boundaries,and each two of them intersect only on their boundaries. Assume { V i } i are dis-joint. Denote M U := π − ( U ) and M i := π − ( V i ). Assume that π | M U : M U → U defines a smooth fiber bundle with fiber Σ g (closed oriented surface with genus g ).Then the localization problem is stated as follows: Problem 7.1 (Localization problem for signature of singular surface bundles) . Can we define a real number σ ( M i , V i , π | M i ) ∈ R , which only depends on the data ( M i , V i , π | M i ) , and write Sign( M ) = m X i =1 σ ( M i , V i , π | M i ) ?We call the real number σ ( M i , V i , π | M i ) the local signature . The answer to thisproblem is positive in the case g = 0 , ,
2. However the answer is negative for g ≥ g -fiber bundle M → X over a closed surface X withSign( M ) = 0. However, if we assume some structure on the fiber bundle π | M U ,the answer can be positive. There are some examples of “structures” for which thelocalization problem has a positive answer, and the local signatures are constructedand calculated in various areas of mathematics, including topology, algebraic ge-ometry and complex analysis. See [AK02] and the introduction of [Sat13] for moredetailed survey on this problem. In this paper, we consider the case g ≥ hyperellipticity (Definition 7.5) as the “structure” imposed on the regular part ofthe fibration. This is the analogue to the setting in [End00], where the case k = 1 isconsidered (note that in [End00] the case g = 0 , Problem 7.2.
Let g ≥ and k be fixed as above. Assume that π | M U : M U → U defines a hyperelliptic Σ g -fiber bundle structure (see Definition 7.5). Can we definea real number σ ( M i , V i , π | M i ) ∈ R , which only depends on the data ( M i , V i , π | M i ) ,and write Sign( M ) = m X i =1 σ ( M i , V i , π | M i ) ?We construct such a function σ using Theorem 6.2. We apply the notations inthe last section for the following. NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 65 • Let F = Σ g , a closed oriented closed 2-dimensional manifold of genus g . • Denote by MCG + ( F ) the orientation-preserving mapping class group of F . Let G = Diff Hg := p − ( H g ) ⊂ Diff + (Σ g ), where H g ⊂ MCG + ( G ) isthe hyperelliptic mapping class group (Definition 7.3) and p : Diff + ( F ) → MCG + ( F ) is the quotient map. Definition 7.3.
Let c ∈ MCG + (Σ g ) denote the class of hyperelliptic involution([End00, p.240]) on Σ g . The hyperelliptic mapping class group, denoted by H g , isdefined as follows. H g := { γ ∈ MCG + (Σ g ) | γc = cγ } . For detailed descriptions of this group, see [End00].
Remark . If g = 0 , ,
2, MCG + (Σ g ) = H g . But in the case g ≥ H g is asubgroup of infinite index in MCG + (Σ g ). Definition 7.5.
Let X be a topological space. A Σ g -fiber bundle π : M → X withstructure group Diff Hg is called a hyperelliptic fiber bundle .We have the following facts about the groups H g and Diff Hg . Fact 7.6. (1)
The rational group cohomology of H g satisfies H i ( H g ; Q ) = 0 for all i ≥ ( [Kaw97] ) . (2) For g ≥ , the unit component of Diff Hg is contractible. In particular wehave a homotopy equivalence between BH g and B Diff Hg ( [EE67] ). (3) For all g ≥ , H g is of type F P ∞ . That is, BH g has a realization as aCW-complex whose m -skeleton are finite for all m ≥ .It well-known that the mapping class group of an oriented compact sur-face of genus g with s punctures and n boundary components is of type F P ∞ (For example see [Luc05] ). This case can be seen by noting that an exten-sion of a type F P ∞ group by a type F P ∞ group is also of type F P ∞ , andthat we have an extension by Birman-Hilden theorem (see [Kaw97, equation(2.1)] ) → Z → H g → π Diff + ( S , { (2 g + 2) − punctures } ) → . Remark . The reason for assuming g ≥ g = 0 the inclusion SO (3) → Diff H (= Diff + ( S )) is a homotopy equivalence,and for g = 1 the unit component of Diff H = Diff + ( T ) is homotopy equivalent to S × S ([EE67]). These groups have torsion in group cohomology, so the argumentbelow does not work in these cases.From now on, we fix an integer g ≥
2. From Fact 7.6 (2) and (3), we see that B Diff Hg has a realization as a CW-complex whose m -skeleton are finite for all m ≥ m -skeleton by Z g,m . Lemma 7.8.
For g ≥ , the universal fiberwise signature class for hyperellipticfiber bundle, Sign ( F univ ) ∈ K ( B Diff Hg ) , maps to ∈ K ( B Diff Hg ; Q ) under thecanonical homomorphism K ( B Diff Hg ) → K ( B Diff Hg ; Q ) . Here the symbol K denotes the representable K -theory. Proof.
We have the rational Chern character isomorphism Ch : K ( BH g ; Q ) ≃ H ev ( H g ; Q ). Using the Fact 7.6 (1) and (2), we have K ( B Diff Hg ; Q ) ≃ K ( BH g ; Q ) ≃ Q and this isomorphism is given by a map ∗ → B Diff Hg . Since the manifold Σ g istwo dimensional, the virtual rank of the class Sign( F univ ) ∈ K ( B Diff Hg ) is 0. Thuswe have Sign( F univ ) = 0 ∈ K ( B Diff Hg ; Q ) and the lemma follows. (cid:3) For each m , we also denote the restriction of the class Sign( F univ ) to Z g,m bySign( F univ ) ∈ K ( Z g,m ). By Lemma 7.8, we have Sign( F univ ) = 0 ∈ K ( Z g,m ; Q ).Since Z g,m is compact, we have K ( Z g,m ; Q ) ≃ K ( Z g,m ) ⊗ Q , so the class Sign( F univ )is of finite order in K ( Z g,m ). Definition 7.9.
For each positive integer m , let n g,m denote the order of theclass Sign( F univ ) ∈ K ( Z g,m ). i.e., n g,m is the smallest positive integer satisfying n g,m · Sign( F univ ) = 0 ∈ K ( Z g,m ).We are in the situation where Theorem 6.2 applies. Definition 7.10.
Let k be a positive integer and g ≥
2. Let S g,k be the set ofisomorphism classes of pairs ( M, π : ∂M → Y ) such that • The pair (
M, π : ∂M → Y ) is a compact oriented manifold with fiberedboundaries, and M is 4 k -dimensional. • The fiber bundle π : ∂M → Y is a hyperelliptic fiber bundle with fiber Σ g .For ( M, π : ∂M → Y ) ∈ S g,k , Y is a (4 k − Y, Z g, k − ] ≃ [ Y, B
Diff Hg ]. We see that S g,k ⊂ S Σ g , Diff Hg ,Z g, k − . We applyTheorem 6.2 to the case F = Σ g , G = Diff Hg , Z = Z g, k − , and n = n g, k − . Notethat we have K − ( Z g, k − ) ⊗ Q = 0 because of Fact 7.6, (1) and (2). Thus, choosingany element Q Z k − ∈ I sign ( π univ | Z k − , C n ), we get the same map σ Q Z k − by (6.3)in Theorem 6.2. So we set σ := σ Q Z k − . Corollary 7.11.
Let k be a positive integer and g ≥ . We have a canonical map σ : S g,k → Z n g, k − satisfying the following. • (vanishing)For ( M, π : ∂M → Y ) ∈ S g,k , assume that π extends to an orientedhyperelliptic Σ g -fiber bundle structure π ′ : ( M, ∂M ) → ( X, ∂X ) preserv-ing boundaries. Here X is an oriented smooth compact oriented (4 k − -dimensional manifold and an orientation preserving diffeomorphism ∂X ≃ Y is fixed. Then we have σ ( M, π ) = 0 . • (additivity)For ( M , π ) and ( M , π ) in S g,k , assume that there exists a decom-position ∂M i = ∂M + i ⊔ − ∂M − i for i = 0 , , and there exists an isomor-phism of the fiber bundle φ : π | ∂M +0 ≃ π | − ∂M − . We form the union ( M, π ) = ( M , π ) ∪ φ ( M , π ) ∈ S g,k . Then we have σ ( M, π ) = σ ( M , π ) + σ ( M , π ) . NDICES ON MANIFOLDS WITH FIBERED BOUNDARIES 67 • (compatibility with signature)An oriented k -dimensional closed manifold M can be regarded as anelement in S g,k . For this element we have σ ( M ) = Sign( M ) . This solves Problem 7.2 as follows. For each i , we have ( M i , π | ∂M i : ∂M i → ∂V i ) ∈ S g,k . We define σ ( M i , V i , π | M i ) := σ ( M i , π | ∂M i : ∂M i → ∂V i ), where theright hand side is defined by Corollary 7.11. We have, by the additivity propertyand the compatibility with signature proved in Corollary 7.11,Sign( M ) = σ ( M U , π | ∂M U : ∂M U → ∂U ) + m X i =1 σ ( M i , V i , π | M i ) . On the other hand, by the vanishing property, we have σ ( M U , π | ∂M U : ∂M U → ∂U ) = 0. Thus we get the equalitySign( M ) = m X i =1 σ ( M i , V i , π | M i ) . Remark . We remark that Corollary 7.11 “solves” the localization problem,Problem 7.2, in the sense that we have shown the existence of local signature func-tion. However, this construction is abstract and does not give an explicit formula forthe local signature. In contrast, in [End00] the author provides an explicit formulafor the local signature in the case k = 1. In order to find applications of the aboveresults, we would definitely need to find an explicit formula. To proceed further, weneed more geometric insight to signature class and their invertible perturbationson mapping class groups. In future works, the author hopes to investigate more onthis aspect. Acknowledgment
This paper is written for master’s thesis of the author. The author would liketo thank her supervisor Yasuyuki Kawahigashi for his support and encouragement.She also would like to thank Georges Skandalis, Mikio Furuta and Yosuke Kubotafor fruitful advice and discussions. This work is supported by Leading GraduateCourse for Frontiers of Mathematical Sciences and Physics, MEXT, Japan.
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