A geometric approach to K-homology for Lie manifolds
aa r X i v : . [ m a t h . K T ] A p r A GEOMETRIC APPROACH TO K -HOMOLOGY FOR LIEMANIFOLDS KARSTEN BOHLEN, JEAN-MARIE LESCURE
Abstract.
We prove that the computation of the Fredholm index for fully ellipticpseudodifferential operators on Lie manifolds can be reduced to the computation ofthe index of Dirac operators perturbed by smoothing operators. To this end weadapt to our framework ideas coming from Baum-Douglas geometric K -homologyand in particular we introduce a notion of geometric cycles that can be classifiedinto a variant of the famous geometric K -homology groups, for the specific situationhere. We also define comparison maps between this geometric K -homology theoryand relative K -theory. Introduction
The Atiyah-Singer index theorem is a celebrated and fundamental result with numerousapplications in Geometry and Analysis. A particular approach to index theory is due toP. Baum and R. Douglas, cf. [5] and [6]. They constructed a geometric K -homology anda suitable comparison homomorphism between the geometric and analytic K -homologygroups. A complete proof that the comparison map is in fact an isomorphism waspublished recently [7]. In the Baum-Douglas approach to the index theorem, the com-putation of the Fredholm index of an elliptic pseudodifferential operator on a compactclosed manifold can be reduced to the computation of the index of a suitable geometricDirac operator, naturally associated to a geometric cycle. The origin of this work is toaddress the corresponding question for singular manifolds, at least the ones for whicha suitable Lie groupoid permits to well pose the index problem. More precisely, weconsider Lie manifolds ( M, G ) , that is Lie groupoids G over compact manifolds withcorners such that M = M \ ∂M is saturated and G M = M × M . This occurs inmany cases, for example manifolds with corners or fibered corners and manifolds withfoliated boundary. In such a case, there is a well defined notion of full ellipticity foroperators in the corresponding calculus, that ensures the Fredholmness of the associatedoperators on M . The question can now be made more precise. Given a fully ellipticoperator P on ( M, G ) , can we construct a Dirac operator D in the same calculus, whichis Fredholm and with the same index than P ? Contrary to the case of C ∞ compactmanifolds without boundary, we are not able to give an affirmative answer to this ques-tion. Nevertheless, we are able to solve positively the question by allowing tamed Diracoperators, that is, Dirac operators perturbed by smoothing elements in the calculus.Along the way, we prove that if there is no obstruction at the level of K -theory for thefull ellipticity (and thus Fredholmness) of Dirac operators, then a perturbation into aFredholm operator using smoothing operators and elementary Dirac operators alwaysexists. This echoes previous works by Bunke [10] and Carrillo Rouse-Lescure [12]. Also,these considerations bring a notion of geometric cycles that mimetizes the original oneof Baum-Douglas with the following main variations into the choice of ingredients:(1) We only accept submersions of manifolds with corners ϕ : Σ → M instead ofgeneral continuous maps from Spin c manifolds to M ; (2) We replace Dirac operators by tame Dirac operators.The point (1) is apparently rather restrictive but actually sufficient for our purpose, sincethe geometric cycles constructed by the clutching process are of this kind. Furthermore,the classical operations on geometric cycles: isomorphisms, direct sums, cobordisms andvector bundle modifications have a natural analog here. The resulting abelian group,that we call geometric K -homology of ( M, G ) can then be compared with the suit-able relative K -theory group (also known as stable homotopy group of fully ellipticoperators). Our approach is definitely tied to the clutching construction which, in asense, dictated to us the most convenient notion of geometric cycles for our purpose,the latter being, as explained above, the possibility of representing the index of abstractpseudodifferential operators by the one of Dirac operators, perturbed by smoothing op-erators. We mention that other and more general approaches of geometric K -homologyfor groupoids exist, in particular in [13, 15, 24]. Overview.
The article is organized as follows. • In section 2 we study fully elliptic operators contained in the pseudodifferentialcalculus on the tuple ( M, G ) , where G is a Lie groupoid over M . We introducethe group of stable homotopy classes of fully elliptic operators V F Ell( M ) whichis defined to equal the relative K -theory group K ( µ ) , where µ is the homo-morphism of the continuous functions M into the full symbol algebra, givenby the action as multiplication operators. We prove a Poincaré duality typeresult which states that the Fredholm index can be expressed in a precise wayin terms of an index map defined in terms of purely geometric data, given bysuitable deformation groupoids. More precisely, we recall the geometric modelfor the full symbol space of a pseudodifferential operator on a Lie groupoid interms of the deformation groupoid T , referred to as the noncommutative tangentbundle . The K -theory group K ( C ∗ ( T )) turns out to be the natural receptablefor the stable homotopy class of the full symbol of a fully elliptic pseudodif-ferential operator. The Poincaré duality result is the isomorphism of groups V F Ell( M ) ∼ = K ( C ∗ ( T )) via the non-commutative principal symbol homomor-phism σ nc . • In section 3 we investigate so-called tamings of geometric Dirac operators on Liegroupoids. The main result is a
Diracification theorem which states that anyfully elliptic pseudodifferential operator P on a Lie groupoid, whose principalsymbol has the same class in K -theory as a given Dirac operator D , has aso-called taming B = ( D ⊕ D ′ ) + R , consisting of a Dirac operator D ′ and asmoothing operator R , such that the classes of the operators B and P agree inthe relative K -theory group K ( µ ) . We introduce a pushforward operation onthe level of deformation groupoids and show that this operation commutes withthe Fredholm index. • In section 4 we introduce geometric K -homology groups V K geo ( M ) (of evenand odd degree) on a given Lie manifold ( M, G ) . Using the pushforward op-eration, introduced in the previous section, we define the comparison map λ : V K geo ( M ) → K ( µ ) . The main result of this section is a proof that thecomparison map λ is a well-defined group homomorphism. This result is cor-rectly viewed as a generalization of the cobordism invariance of the analyticindex in the standard setting. • In section 5 we define the so-called clutching construction in the setting of Liemanifolds. The resulting quotient map c : K ( C ∗ ( T )) → V K geo ( M ) is induced -HOMOLOGY 3 by the map associating to a full symbol of a (fully elliptic) pseudodifferentialoperator a geometric cycle. We show that c is a well-defined homomorphism ofgroups. Then we finish the proof of the main result of the paper, which statesthat the computation of the Fredholm index of any fully elliptic pseudodiffer-ential operator on a Lie manifold can be reduced to the computation of theindex of a geometric Callias type operator, associated to the geometric data,generating the geometric K -homology group.The view of index theory on Lie manifolds considered in this work can be summarizedin terms of the following commutative diagram: K ( C ∗ ( T )) Z V K geo ( M ) K ( µ ) = V F Ell( M ) ind F c λ pd ≃ ind (1)In conjunction with the recent work of Bohlen-Schrohe [9] the reduction of the indexproblem to first order geometric operators furnishes a corresponding index formula forfully elliptic pseudodifferential operators on Lie manifolds.2. Poincaré duality
Full ellipticity, groupoids and K -theory. Let G ⇒ M be a Lie groupoid withLie algebroid denoted by A . The manifold M will always be compact, and may havecorners. In that case, we assume that the boundary hypersurfaces of M and G areembedded [33], and the source and range maps are submersions between manifolds withcorners [29], also called tame submersions in [39]. We recall that it means, for a C ∞ map f : M → N between manifolds with corners, that at any point x ∈ M we have: df x ( T x M ) = T f ( x ) N and ( df x ) − ( T + f ( x ) N ) = T + x M. (2)Here T M denotes the ordinary tangent vector bundle and T + M its subspace of inwardpointing vectors. Under such assumptions, f preserves the codimension of points andits fibers have no boundary.If E j → M are vector bundles, we denote by Ψ ∗G ( E , E ) the space of compactly G -pseudodifferential operators [14, 36, 34, 41]. The principal symbol map is denoted by: σ pr : Ψ m G ( E , E ) → C ∞ ( S ( A ∗ ) , π ∗ (Hom( E , E ))) . Here S ( A ∗ ) is the sphere bundle of the dual Lie algebroid A ∗ of G and π denotes anyof the projections maps of A ∗ and S ( A ∗ ) onto M .Let Ψ m G ( E , E ) denote the closure of Ψ m G ( E , E )) into Mor( H s ( G , E ) , H s − m ( G , E )) [45] (and we choose once for all, the reduced or maximal completion indifferently). Forsimplicity we denote Ψ G ( E ) = Ψ G ( E ) . We get a short exact sequence of C ∗ -algebras: C ∗ ( G , End( E )) / / / / Ψ G ( E ) σ pr / / / / C ( S ( A ∗ ) , π ∗ End( E )) . (3)Now let F ⊂ M be a closed subspace which is saturated, that is s − ( F ) = r − ( F ) .Then G F is a continuous family groupoid [42, 28]. By restricting over F , we get the F -indicial symbol map: I F : Ψ ∗G ( E , E ) → Ψ ∗G F ( E | F , E | F ) KARSTEN BOHLEN, JEAN-MARIE LESCURE
Gathering both symbol maps, we get the F -joint symbol map : σ F,m = ( σ pr , I F ) : Ψ m G ( E , E ) −→ C ( S ( A ∗ ) , π ∗ Hom( E , E )) × Ψ m G F ( E | F , E | F ) . (4)The range Σ mF ( E , E ) of σ F,m is called the F -joint symbols space and its closure isdenoted by Σ mF ( E , E ) . We will write Σ F for Σ F . This gives the short exact sequenceof C ∗ -algebras [28]: C ∗ ( G O , End( E )) / / / / Ψ G ( E ) σ F / / / / Σ F ( E ) . (5)where O = M \ F and C ∗ ( G O , End( E )) is the closure of C ∞ c ( G O , r ∗ End( E )) into Ψ G ( E ) .We recall that for any elliptic A ∈ Ψ G ( E, E ′ ) , the unbounded operator Λ E = (1 + A ∗ A ) / defined by functional calculus belongs to Ψ G ( E ) , realizes isomorphisms H s ( G, E ) →H s − ( G, E ) and satisfies Λ − E ∈ C ∗ ( G , End( E )) [45]. Definition 2.1.
We say that P ∈ Ψ m G ( E ) is F -fully elliptic if σ F (Λ − mE P ) ∈ Σ F ( E ) × .We now turn our attention to the natural groups constructed out of F -full ellipticoperators [27, 44, 43, 3]. A convenient way to handle them is to use the so-called K -group K ( f ) of a homomorphism f : A → B ([27, II.2.13]). We recall the definition:the set of cycles Γ( f ) is given by triples ( E , E , α ) where the E i are finitely generatedprojective A -modules and α : E ⊗ f B → E ⊗ f B is an isomorphism. The notions ofisomorphisms ( ≃ ) and direct sums ( ⊕ ) in Γ( f ) are the obvious ones. A cycle ( E , E , α ) is elementary if E = E and α is homotopic to the identity within the automorphismsof E ⊗ f B . Finally, the equivalence relation giving K ( f ) = Γ( f ) / ∼ is defined by: σ ∼ σ ′ if there exists elementary τ, τ ′ such that σ ⊕ τ ≃ σ ′ ⊕ τ ′ . We always have K ( f ) ≃ K ( C f ) where C f is the mapping cone of f and K ( f ) ≃ K (ker f ) when f is onto [3]. In [43] these groups are introduced in a slightly differ-ent way, namely as stable homotopy groups of elliptic operators. The comparison isstraightforward and we keep a notation inspired by A. Savin’s one. Definition 2.2.
The group F Ell F ( G ) will be defined as the K -group K ( µ F ) of thenatural homomorphism µ F : C ( M ) −→ Ψ G /C ∗ ( G O ) = Σ F .We also consider another natural homomorphism µ pr : C ( M ) → Ψ G /C ∗ ( G ) ≃ C ( S ( A ∗ )) .If P ∈ Ψ m G ( E , E ) is F -fully elliptic, m ≥ , then it canonically defines classes: [ P ] F = (cid:2) E , E , σ F (Λ − mE P ) (cid:3) ∈ K ( µ F ) = F Ell F ( G ) , [ P ] pr = (cid:2) E , E , σ pr (Λ − mE P ) (cid:3) ∈ K ( µ pr ) ≃ K c ( A ∗ ) , with E j = C ( X, E j ) . The resulting classes do not depend on the choice of Λ E . Indeed,consider Λ A = (1 + A ∗ A ) / and Λ B = (1 + B ∗ B ) / . Since C ( t ) = (1 − t ) A ∗ A + tB ∗ B is elliptic self-adjoint and non negative we get a family Λ t = (1 + C ( t )) / ∈ Ψ G ( E ) ofinvertible elements connecting Λ A to Λ B , and σ F (Λ − mt P ) provides the desired homotopy.The same is true for the principal symbol classes.Note that there is a natural isomorphism C µ pr ≃ C ( A ∗ ) and therefore: K ( µ pr ) ≃ K c ( A ∗ ) . We will denote by [ P ] pr , ev the image of [ P ] pr through this isomorphism. Proposition 2.3.
Let P + ∈ Ψ G ( E + , E − ) be F -fully elliptic. Then P ∗ + is F -fully ellipticand [ P + ] F = − [ P ∗ + ] F ∈ K ( µ F ) . (6) -HOMOLOGY 5 Proof. If P + is F -fully elliptic, then P − := P ∗ + is also F -fully elliptic. Set P = (cid:18) P − P + (cid:19) ∈ Ψ G ( E ) , E = E + + E − . We get by polar decomposition of the invertibleelement p = σ F ( P ) in the C ∗ -algebra Σ F ( E ) the existence of U, T ∈ Ψ G ( E ) such that u = σ F ( U ) is unitary, | p | = σ F ( T ) and p = u | p | . Since | p | is of degree , since p ofdegree with respect to the Z grading and since | p | is invertible, the unitary u isnecessarily of degree with respect to the Z -grading. Therefore, we can assume that U = (cid:18) U − U + (cid:19) and T = (cid:18) T + T − (cid:19) . Since p is selfajoint, we can also assume that theunitary u is self-adjoint, thus u − = u ∗ + = u − . Since p is positive, we get a homotopy p = u | p | ∼ u , therefore [ P + ] F + [ P ∗ + ] F = [ E , E , p ] = [ E , E , u ] = [ E + , E − , u + ] + [ E − , E + , u − ] = 0 ∈ K ( µ F ) . (cid:3) Index and Poincaré duality results.
There is an index map coming with F Ell F ( G ) .Indeed, the commutative diagram C ( M ) µ (cid:15) (cid:15) µ F / / Σ F Id (cid:15) (cid:15) Ψ G σ F / / Σ F gives rise to a homomorphism K ( µ F ) → K ( σ F ) and since σ F is onto, we also havea natural isomorphism K ( σ F ) ≃ K (ker σ F ) = K ( C ∗ ( G O )) [3]. Their composition iscalled the ( F -)index map: ind F : F Ell F ( G ) → K ( C ∗ ( G O )) . (7)It is possible to get a slightly more geometrical description of (7) by establishing akind of Poincaré duality with the help of deformation groupoids. Let us introduce thenecessary objects.The adiabatic groupoid G ad ⇒ M ad := M × [0 , is the natural Lie groupoid in-tegrating the Lie algebroid ( A ad , ̺ ad ) given by: A ad = A × [0 , and ̺ ad : A ad → T M × T [0 , , A ad ∋ ( x, v, t ) ( x, tv, t, ∈ T M × T [0 ,
1] =
T M ad . More precisely, G ad = A × { } ∪ G × (0 , and A ( G ad ) ∼ = A ad . Out of the adiabatic groupoid we construct the so-called F -Fredholm groupoid : G F F := G ad \ ( G F × { } ) ⇒ M F F = ( M × [0 , \ ( F × { } ) (8)This is again a Lie groupoid (as an open subset of G ad ). The non-commutative tangentbundle is defined by: T F M := G F F \ ( G O × (0 , ⇒ M F = M F F \ ( O × (0 , . (9)It is a C ∞ , groupoid [42]. The exact sequence: C ∗ ( G O × (0 , / / / / C ∗ ( G F ) e / / / / C ∗ ( T F M ) (10)possessing a contractible kernel, we get an isomorphism e : K ( C ∗ ( G F F )) → K ( C ∗ ( T F M )) ,and if (10) has a completely positive section, then e ∈ KK ( C ∗ ( G F F ) , C ∗ ( T F M )) pro-vides a KK -equivalence (that is, is invertible). Considering the restriction e : C ∗ ( G F F ) → KARSTEN BOHLEN, JEAN-MARIE LESCURE C ∗ ( G O ) , we get another index map: ind F F := ( e ) ∗ ◦ ( e ) − ∗ : K ( C ∗ ( T F M )) → K ( C ∗ ( G O )) . We think to Σ F as a noncommutative cosphere bundle , relative to F , and to T F M asthe noncommutative tangent bundle , relative to F , associated with G ⇒ M . We end upwith a Poincaré duality like theorem. Theorem 2.4.
There is a group isomorphism: σ nc F : F Ell F ( G ) → K ( C ∗ ( T F M )) (11) such that e ( σ nc F [ P ] F ) = [ P ] pr , ev ∈ K c ( A ∗ ) and ind F F ( σ nc F [ P ] F ) = ind F ([ P ] F ) . Here e is the restriction map C ∗ ( T F M ) → C ∗ ( A ) .Proof. By [27, 44, 43] we have a natural isomorphism: F Ell F ( G ) = K ( µ F ) ≃ K ( C µ F ) (12)where C f denotes the mapping cone of f . The isomorphism K ( C µ F ) ≃ K ( C ∗ ( T F M )) can be proved following verbatim the proof of Theorem 10.6 in [21, Theorem 10.6].Alternatively, we use the following short argument . Denote by G ad the restriction of G ad to M × [0 , . Since Ψ G ad /C ∗ ( G ad ) ≃ C ( S ( A ∗ ) × [0 , is contractible, the inclusion C ∗ ( G ad ) → Ψ G ad is an isomorphism in K -theory, as well as the inclusion C ∗ ( T F M ) = C ∗ ( G ad ) /C ∗ ( G O × (0 , ⊂ Ψ G ad /C ∗ ( G O × (0 , as well. If µ : C ( M ) → Ψ G denotes the natural homomorphism, then we have ahomomorphism κ : C µ → Ψ G ad given by ( f, P ) P . Using on the one hand thecommutative diagram C ((0 , , Ψ G ) / / / / C µ κ (cid:15) (cid:15) / / / / C ( M ) κ (cid:15) (cid:15) C ((0 , , Ψ G ) / / / / Ψ G ad / / / / Ψ A . and on the other hand that κ : C ( M ) −→ Ψ A is a KK -equivalence, we get by the fivelemma that κ is a KK -equivalence too. Quotienting both algebras C µ and Ψ G ad bythe ideal C ∗ ( G O × (0 , , we get that the homomorphism C µ F −→ Ψ G ad /C ∗ ( G O × (0 , is a KK -equivalence. The remaining assertions in Theorem 2.4 are then easy. (cid:3) From abstract operators to tame Dirac operators
Dirac bundles over groupoids.
Let G ⇒ M be a Lie groupoid with Lie algebroid A .We equip A with a euclidean structure and denote by Cl( A ) the corresponding bundle ofClifford algebras over M . We define Dirac bundles as in [10, Definition 3.1] by replacing T M by A in the definition of c , in other words we deal with a Cl( A ) -module complexvector bundle E over M with Clifford multiplication induced by c , compatible hermitianmetric h , admissible connection ∇ and a Z -grading z when A has even rank. Now by[29], there is an associated Dirac operator D ∈ Diff G ( E ) . For simplicity, we will denotesuch a Dirac bundle by ( E, D ) , the necessary data being understood. communicated to us by G. Skandalis -HOMOLOGY 7 K -tamings. One can check that, for any F -fully elliptic P ∈ Ψ G ( E , E ) , we have: [ P ] F, ev := σ ncF ([ P ] F ) = h C ∗ ( T F M, E ⊕ E ) , (cid:18) Q| T F M P| T F M (cid:19) i ∈ KK ( C , C ∗ ( T F M )) . Here P is a F × { } -fully elliptic lift of P , that is P ∈ Ψ G ad , P| t =1 = P and P| t =0 = σ pr ( P ) ; while Q is a similar lift of a full parametrix Q of P . We refer to this situationas the even case.Now, if P ∈ Ψ G ( E ) is selfadjoint and F -fully elliptic, which we refer to as the odd case,we will consider instead of [ P ] F and [ P ] pr , the classes: [ P ] F, odd = h C ∗ ( T F M, E ) , P| T F M i ∈ KK ( C , C ∗ ( T F M )) , [ P ] pr , odd = h C ∗ ( A , E ) , σ pr ( P ) i ∈ KK ( C , C ∗ ( A )) , The convention is the same for positive order operators after using suitable order re-duction operator Λ . Definition 3.1.
We say that an elliptic operator P ∈ Ψ ∗G is F -tameable if [ P ] pr , ∗ is inthe range of τ = (ev t =0 ) ∗ : K ∗ ( C ∗ ( T F M )) −→ K ∗ ( C ∗ ( A )) . (13)Using the K -equivalences G F F ∼ K T F M and G ad ∼ K A , we see that the above conditionis equivalent to be in the range of i ∗ : K ∗ ( C ∗ ( G F F )) −→ K ∗ ( C ∗ ( G ad )) . (14)where i : C ∗ ( G F F ) ֒ → C ∗ ( G ad ) is the inclusion.Also, in the even case, the natural homomorphism q F : Σ F → C ( S A ∗ ) induces a map: q F : K ( µ F ) −→ K ( µ pr ) (15)which coincides with τ under the suitable isomorphisms. In particular q F ([ P ] F ) = [ P ] pr and F -tameability of an elliptic operator P ∈ Ψ ∗G ( E , E ) is equivalent to [ P ] pr ∈ im q F . Reduction to Dirac.
We call ( S, D ) an even Dirac bundle for G if S = S + ⊕ S − is a Z -graded Cliff( A ) -module vector bundle over X and D = Antidiag( D + , D ∗ + ) ∈ Ψ G ( S ) is a Dirac operator. Odd Dirac bundles are defined in the same way without gradings. Definition 3.2.
Any F -fully elliptic operator of the form: B = ( D ⊕ D ′ ) + R where(1) ( S ′ , D ′ ) is a Dirac bundle of the same parity as ( S, D ) ,(2) R ∈ Ψ −∞G ( S ⊕ S ′ ) ,(3) [ B ] pr , ∗ = [ D ] pr , ∗ ,is called a F -taming of D . Theorem 3.3.
Let ( S, D ) be a F -tameable Dirac bundle. Then for any F -fully ellipticoperator P ∈ Ψ G such that [ P ] pr , ∗ = [ D ] pr , ∗ ∈ K ∗ ( A ) , there exists a F -taming B of ( S, D ) such that [ B ] F, ∗ = [ P ] F, ∗ ∈ K ∗ ( C ∗ ( T F M )) ≃ K ( µ F ) . (16) KARSTEN BOHLEN, JEAN-MARIE LESCURE
Proof of the Theorem.
We begin with the even case and use for that the relative K -theory description. Far any integer N , we can give Cliff( A ) -modules structure to thetrivial bundles X × C k (by embedding isometrically the bundle A into some trivial bundleover X ) where k ≥ N is chosen in order to get an even Dirac bundle ( M × C k , D k ) .Then [ D k, + ⊕ ( − D k, + )] pr = 0 ∈ K ( µ pr ) and we set D ′ = D k + ( − D k ) .Let P + ∈ Ψ G ( E + , E − ) be a F -fully elliptic element such that [ P + ] pr = [ D + ] pr . Bydefinition of K ( µ pr ) , there exist elementary elements ( ξ, ξ, α ) , ( ξ ′ , ξ ′ , α ′ ) ∈ Γ( µ pr ) such that ( E + , E − , σ pr ( P + )) + ( ξ, ξ, α ) ≃ ( S + , S − , σ pr ( D + )) + ( ξ ′ , ξ ′ , α ′ ) . Adding to both sides another elementary element, and renaming, we can assume that ξ ′ = C ( S ( A ∗ ) , C k ) . We rename S ± ⊕ C ( S ( A ∗ ) , C k ) into S ± and identify E ± ⊕ ξ ≃ S ± .Replacing α by and α ′ by σ pr ( D ′ + ) after homotopies, and renaming P + ⊕ into P + ,we get: (cid:0) S + , S − , σ pr ( P + ) (cid:1) ∼ (cid:0) S + , S − , σ pr (( D ⊕ D ′ ) + ) (cid:1) , and in particular σ pr ( P + ) − σ pr (( D ⊕ D ′ ) + ) ∼ Id within Aut( S + ) . (17)Now we use the homotopies lifting argument of [23, Proposition 4.3]. Since S ± = p ± C ( S ( A ∗ )) N for some projectors p ± ∈ M N ( C ( M )) , the surjectivity of q : Σ F −→ C ( S ( A ∗ )) implies the surjectivity of e q : L ( p + (Σ NF )) −→ L ( S + ) . By the open mapping theorem, e q is then open. Thus, e q (Aut( p ± (Σ F )) (0) ) is an open,and therefore closed, subgroup of Aut( S + ) , therefore: e q (Aut( p ± Σ NF ) (0) ) = Aut( S ± ) (0) . By (17), we can choose y ∈ Aut( p + Σ NF ) (0) such that σ pr ( P + ) e q ( y ) = σ pr ( P + ) y = σ pr (( D ⊕ D ′ ) + ) . (18)Let Y ∈ Ψ G ( S + ) such that σ F ( Y ) = y . Let us set T = √ (cid:18) P + Y ) ∗ P + Y (cid:19) . Since y is in the connected component of the identity in Aut( S ± ) , we have [ T ± ] F = [ P ± ] F . If ∆ ∈ Ψ G ( S ) denotes a laplacian, Equation (18) gives: T = (1 + ∆) − / ( D ⊕ D ′ ) + R with R ∈ C ∗ ( G , End( S )) . (19)We know that the class σ F ( T ) of T in the C ∗ -algebra Ψ G ( S ) /C ∗ ( G O , End( S )) is invert-ible. By density of C ∞ c ( G , End( S )) into C ∗ ( G , End( S )) , we can choose R ∈ C ∞ c ( G , End( S )) such that σ F ( T )) remains invertible, where T = T − R + R . We write R =(1 + ∆) − / (1 + ∆) / R and we approximate (1 + ∆) / R ∈ C ∗ ( G , End( S )) by anelement R ∈ C ∞ c ( G , End( S )) close enough so that σ F ( T )) remains invertible, where T = (1 + ∆) − / ( D ⊕ D ′ ) + (1 + ∆) − / R . Thus, B := ( D ⊕ D ′ ) + R is a F -fully elliptic operator and R ∈ Ψ −∞G ( S ) . Withoutloss of generality, we can assume that R, R , R are close enough so that T and T arehomotopic among F -fully elliptic operators. We conclude: [ B + ] F = [ T + ] F = [ P + ] F = − [ P − ] F = − [ B − ] F . -HOMOLOGY 9 In the odd case now, we consider the map τ ∗ : KK ( C , C ∗ ( T F M )) −→ KK ( C , C ∗ ( A )) .Let P be a fully elliptic lift of P and p ∈ M N ( C ( M )) a projector such that E = p ( M × C N ) . We have τ ∗ [ pC ∗ ( T F M ) N , P| T F X ] = [ pC ∗ ( A ) N , σ pr ( P )] , therefore by as-sumption [ pC ∗ ( A ) N , σ pr ( P )] = [ pC ∗ ( A ) N , σ pr ( D )] ∈ KK ( C , C ∗ ( A )) . This meansthat after the addition of degenerate Kasparov modules we have a homotopy between σ pr ( P ) and σ pr ( D ) within the automorphisms of the C ∗ -algebra L ( τ E ) / K ( τ E ) where E = C ∗ ( T F M, E ) and τ E = C ∗ ( A , E ) = E ⊗ τ C ∗ ( A ) . Now by lifting homotopiesthrough the epimorphism of C ∗ -algebras: e τ : L ( E ) / K ( E ) −→ L ( τ E ) / K ( τ E ) as in the even case, and using again a density argument, we conclude the proof. (cid:3) Lie manifolds. If M is a compact manifold with embedded corners, it is understoodthat a decomposition into closed faces is given, which are themselves compact manifoldswith embedded corners inheriting the induced decomposition into faces. The set ofcodimension k faces of M is denoted by F k M .We will consider Lie manifold structures on manifolds with embedded corners [1]. Anysuch structure on M is given by an almost injective Lie algebroid A , the latter beingintegrable by [18] into a unique, up to isomorphism, s -connected Lie groupoid G . By aslight abuse of notation, we will continue to call Lie manifold any pair ( M, G ) such that(1) G ⇒ M is a Lie groupoid over M ;(2) M = M \ ∂M is saturated and G M = M × M .Let ( M, G ) be a Lie manifold. We denote by Ψ ∗V ( M ; E , E ) the algebra of pseudodif-ferential operators of Lie type on M = M \ ∂M acting between the sections of vectorbundles E j → M [1]. It coincides with the image of Ψ ∗G ( E , E ) by the vector represen-tation r when G is s -connected [1]. By a slight abuse of notation, we continue to set Ψ ∗V = r Ψ ∗G in the general case. Equivalently, Ψ ∗V is isomorphic to the space obtainedby restricting elements of Ψ ∗G to the fiber of G over any arbitrary interior point x . Theisomorphism comes then from the diffeomorphism r : G x → M .The ∂M -indicial symbols and ∂M -joint symbols can be related with their analoguesfor any F ∈ F ( M ) in the obvious way, and we can summarize this in the followingcommutative diagram in which the outer rectangle is made of fibered products: Σ m∂M ( E ) π (cid:15) (cid:15) π / / C ∞ ( S ( A ∗ ) , π ∗ End( E )) ( r F ) F ∈F M ) (cid:15) (cid:15) Ψ m G ( E ) σ pr ⊕ I ∂M h h h h ◗◗◗◗◗◗◗◗◗◗◗◗◗ I ∂M v v v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ σ pr ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ) ) ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Q F ∈F ( M ) Ψ m G F ( E | F ) ⊕ F σ pr ,F / / / / Q F C ∞ ( S ( A ∗| F ) , π ∗ End( E | F )) We observe that with F = ∂M , we have O = M \ ∂M =: M and thus G O = M × M . It follows that the ideal in (5) is the ideal of compact operators and that ind ∂M takes integer values. Since r : C ∗ ( G O , End( E )) ≃ → K ( L V ( M, E )) and r : Ψ G ( E ) ֒ →B ( L V ( M, E )) [1], the sequence (5) also shows that r ( P ) : H s V ( M ; E ) → H s − m V ( M ; E ) isFredholm, when P ∈ Ψ m G ( E ) is ∂M -fully elliptic , and that ind ∂M computes its Fredholmindex. Here H s V ( M, E ) are the Sobolev spaces of sections of E associated with the Liestructure and L V = H V . From now on, whenever we take F = ∂M as a closed saturated subspace of a Liemanifold ( M, G ) , we will omit it in the notation, for instance: fully means ∂M -fully, G F denotes G F ∂M , etc ... and we set V F Ell( M ) = F Ell ∂M ( G ) . Pullbacks.
Let ϕ : Σ −→ M be a surjective tame submersion, that is, a surjectivesubmersion satisfying (2). Then the decomposition into faces F ∗ ( M ) can be lifted to adecomposition into faces of Σ . An interesting example occurs in the so-called clutchingspaces and vector bundle modification: Example 3.4.
Let M be a manifold with corners and π : V → M be a real vectorbundle. Consider the sphere bundle ϕ : Z = S ( V ⊕ R ) → M . Then ϕ is a tamesurjective submersion. We say that Z is the clutching space of π : V → M .We recall the notion of pull-backs. Definition 3.5.
Let ( A , ̺ ) be a Lie algebroid over M and G ⇒ M a Lie groupoid over M . Let ϕ : Σ → M be a surjective submersion. (1) The
Lie agebroid pullback of ( A , ̺ ) over Σ is the Lie-algebroid ( ϕ A , ϕ ̺ ) given by: ϕ A = { ( v, w ) ∈ ϕ ∗ A × T Σ : ̺ ( v ) = dϕ ( w ) } ⊂ ϕ ∗ A ⊕ T Σ and ϕ ̺ = pr . (2) The
Lie groupoid pullback of G over Σ is the Lie groupoid ϕ G ⇒ Σ given by ϕ G = { ( x, γ, y ) ∈ Σ × G × Σ : ϕ ( x ) = r ( γ ) , ϕ ( y ) = s ( γ ) } , with structure maps: ϕ r( x, γ, y ) = x , ϕ s( x, γ, y ) = y , ( x, γ, y )( y, γ ′ , z ) = ( x, γγ ′ , z ) .Note that the Lie groupoid pullback of the Lie algebroid A also makes sense and willbe denoted by ϕ A if a risk of confusion occurs.Both notions are consistent since: Theorem 3.6.
Let ( M, G ) be a Lie manifold and ϕ : Σ → M be a tame surjectivesubmersion. Then (Σ , ϕ G ) is a Lie manifold such that A ( ϕ G ) ∼ = ϕ A .Moreover, ϕ G is Morita equivalent to G and the ( V -related ) Lie structure ϕ V is givenby: ϕ V = { e V ∈ Γ( T Σ) ; ∃ V ∈ V , dϕ ( e V ) = V ◦ ϕ } . This is a compilation of known facts: the isomorphism A ( ϕ G ) ∼ = ϕ A is proved in [32],the Morita equivalence is proved in [22], that ϕ G gives a Lie structure on Σ is done in[39] and further developments can be found in [46].Let ( M, G ) be a Lie manifold and ϕ : Σ → M a tame surjective submersion. Theoperations consisting of taking the adiabatic deformation and taking a pullback donot commute, however there are natural deformation groupoids relating them and it isexploited in [46] in order to obtain pushforward maps (which are there occurences ofwrong way maps). We need to recall the results of [46] in our notation. Firstly, thereis a deformation Lie groupoid: L ϕ = ( ϕ G ) ad × { u = 0 } ∪ ϕ ( G ad ) × (0 , u ⇒ Σ × [0 , t × [0 , u , (20)where ϕ = ϕ × Id : Σ × [0 , t → M × [0 , t . Let z be the lift by ϕ of a defining functionfor the boundary of M and consider the following saturated subgroupoids of L ϕ : L F ϕ := L ϕ | ( z,t ) =(0 , = ( ϕ G ) F × { u = 0 } ∪ ϕ ( G F ) × (0 , u , (21) L ncϕ := L ϕ | tz =0 = T Σ × { u = 0 } ∪ ϕ T M × (0 , u ⇒ Σ ∂ Σ × [0 , u . (22) L ϕ := L ϕ | t =0 = ϕ A × { u = 0 } ∪ ϕ A × (0 , u ⇒ M × [0 , u . (23) -HOMOLOGY 11 Using the restrictions homomorphisms ev u = i , i = 0 , , and the natural Morita equiva-lence M we get homomorphisms (see [46] for more details): ϕ ad ! = M ◦ [ev u =1 ] ◦ [ev u =0 ] − : K ∗ ( C ∗ (( ϕ G ) ad )) −→ K ∗ ( C ∗ ( G ad ))) . (24)Considering the variations (21), (22) and (23) of (20) leads to similar homomorphisms: ϕ F ! : K ∗ ( C ∗ (( ϕ G ) F )) −→ K ∗ ( C ∗ ( G F ))) , (25) ϕ nc ! : K ∗ ( C ∗ ( T Σ)) −→ K ∗ ( C ∗ ( T M )) . (26) ϕ : K ∗ ( C ∗ ( ϕ A )) −→ K ∗ ( C ∗ ( A )) . (27)Then the following result is essentially a rephrasing of [46]: Theorem 3.7.
Let ( M, G ) be a Lie manifold and ϕ : Σ → M a tame surjective sub-mersion. Then the map ϕ nc ! commutes with Fredholm index: ind F ∂M ◦ ϕ nc ! = ind F ∂ Σ (28) Here the target group of the index maps ind F• is replaced by Z after applying the obviousMorita equivalences.In other words, if B ∈ Ψ ∗ ϕ G ( e E + , e E − ) and P ∈ Ψ ∗G ( E + , E − ) are fully elliptic operatorsand satisfy ϕ nc ! [ B ] ∂ Σ , ev = [ P ] ∂ Σ , ev then B and P have the same Fredholm index.Proof. Using the groupoid L ϕ and the definition of ind F we get a commutative diagram: K ( C ∗ ( T Σ)) K ( C ∗ (( ϕ G ) F ∂ Σ )) K ( C ∗ (Σ × Σ )) Z K ( C ∗ ( L nc ϕ )) K ( C ∗ ( L F ϕ )) K ( C ∗ (Σ × Σ × [0 , Z K ( C ∗ ( ϕ T M )) K ( C ∗ ( ϕ ( G F ∂M ))) K ( C ∗ (Σ × Σ )) Z K ( C ∗ ( T M )) K ( C ∗ ( G F ∂M )) K ( C ∗ ( M × M )) Z ϕ nc! ind F ∂ Σ ev tz =0 ev t =1 M ev u =0 ev u =1 ev tz =0 ev u =0 ev u =1 ev t =1 ev u =0 ev u =1 M× p ∗ M ev tz =0 M ev t =1 M M ind F ∂M ev tx =0 ev t =1 M (29)Above p = ev u = α for arbitrary α ∈ [0 , . The result follows immediately. (cid:3) Theorem 3.8.
Let ( M, G ) be a Lie manifold, π : V → M be a real vector bundle anddenote by ϕ : Σ = S ( V ⊕ R ) → M the associated clutching and i : V ֒ → Σ the embeddingas an open hemisphere. Then(1) π nc ! and π are isomorphisms, the latter being the inverse of the Thom isomor-phism of the complex bundle π A → A .(2) We have the identities: ϕ • ! ◦ (cid:0) i ∗ ◦ ( π • ! ) − ) = Id , with • = nc or . (30) Proof.
Considering the groupoids L π , L ϕ and the embeddings provided by i , we getcommutative diagrams: · · · K ( C ∗ ( ϕ G| ∂ Σ × (0 , K ( C ∗ ( T Σ)) K ( C ∗ ( ϕ A )) · · ·· · · K ( C ∗ ( G| ∂M × (0 , K ( C ∗ ( T M )) K ( C ∗ ( A )) · · · M ev t =0 ϕ nc ! ϕ ev t =0 (31) · · · K ( C ∗ ( π G| ∂V × (0 , K ( C ∗ ( T V )) K ( C ∗ ( π A )) · · ·· · · K ( C ∗ ( G| ∂M × (0 , K ( C ∗ ( T M )) K ( C ∗ ( A )) · · · M ev t =0 π nc ! π ev t =0 (32) K ( C ∗ ( T Σ)) ϕ nc ! / / K ( C ∗ ( T M )) K ( C ∗ ( T V )) i ∗ O O π / / K ( C ∗ ( T M )) Id O O (33) K ( C ∗ ( ϕ A )) ϕ / / K ( C ∗ ( A )) K ( C ∗ ( π A )) i ∗ O O π / / K ( C ∗ ( A )) Id O O (34)Since L π is the Thom groupoid of the complex bundle π A → A , we get that π is theinverse of the Thom isomorphism of this complex bundle [20]. By Diagram (32) andthe five lemma, π nc ! is an isomorphism too. Now Diagrams (34) and (33) give: ϕ • ! ◦ (cid:0) i ∗ ◦ ( π • ! ) − ) = Id , with • = nc or . (35)This proves that ϕ and ϕ nc ! are surjective and gives explicit sections. (cid:3) Finally, we denote by map ϕ ∗ : K ( µ ∂ Σ ) −→ K ( µ ∂M ) the map induced by ϕ nc ! throughthe isomorphism of Theorem 2.4. Note that the homomorphisms q ∂ Σ and q ∂M are thenexchanged with the homomorphisms ev t =0 .4. Geometric K -homology Geometric cycles.
Let ( M, G ) be a Lie manifold. Definition 4.1.
A even (odd) geometric cycle over ( M, G ) is a 4-tuple x = (Σ , ϕ, E, B ) consisting of(1) an even (odd) dimensional compact manifold with corners M ;(2) a tame surjective submersion ϕ : Σ −→ M ;(3) an even (odd) Dirac bundle ( E, D ) on (Σ , ϕ G ) ;(4) an self-adjoint even (odd) Dirac ∂ Σ -taming B of D .The set of geometric cycles of parity j is denoted by V E geoj ( M ) .If the geometric cycle x = (Σ , ϕ, E, B ) is even, we get a class [ B ] := [ B + ] ∂ Σ , ev ∈ K ( C ∗ ( T Σ)) -HOMOLOGY 13 and if it is odd, we get a class [ B ] := [ B ] ∂ Σ , odd ∈ KK ( C , C ∗ ( T Σ)) ≃ K ( C ∗ ( T Σ)) . Applying the pushforward map ϕ nc! , we get classes in K ∗ ( C ∗ ( T M )) as well. Definition 4.2.
Let x = (Σ , ϕ, E, B ) be a geometric cycle of parity j . A geometriccycle x ′ = (Σ ′ , ϕ ′ , E ′ , B ′ ) is isomorphic to x if there exists a diffeomorphism κ : Σ → Σ ′ such that(1) ϕ ′ ◦ κ = ϕ and ( E, D ) ≃ κ ∗ ( E ′ , D ′ ) is an isomorphism of Dirac bundles.(2) [ B ] j = κ ∗ [ B ′ ] j ∈ K j ( C ∗ ( T Σ)) . Definition 4.3. (1) The disjoint union of x i = (Σ i , ϕ i , E i , B i ) ∈ V E geo ∗ ( M ) , i = 0 , , is x ∪ x = (Σ ∪ Σ , ϕ ∪ ϕ , E ∪ E , B ∪ B ) ∈ V E geo ∗ ( M ) . (36)(2) The direct sum of x i = (Σ , ϕ, E i , B i ) ∈ V E geo ∗ ( X ) , i = 0 , , is x ⊕ x = (Σ , ϕ, E ⊕ E , B ⊕ B ) ∈ V E geo ∗ ( M ) . (37)(3) If x = (Σ , ϕ, E, B ) ∈ V E geoi ( M ) then the opposite cycle is − x = (Σ , ϕ, E op , B op ) . (38)Here E op is the same bundle, with the opposite grading in the even case, and B op = − B . We have [ B op ] j = − [ B ] j for j = 0 , . Cobordisms.
To define cobordism of geometric cycles, we introduce:
Definition 4.4.
A cobordism over M is a triple ( W, Y, Φ) such that(1) Φ : W −→ M is a surjective submersion,(2) Y = H ∪ · · · ∪ H k is a union of boundary faces of W called relative boundaryfaces, the remaining boundary faces being called absolute and absolute faces areassumed to be pairwise disjoint,(3) Φ − ( ∂M ) = Y and for all relative boundary faces H , Φ( H ) ∈ F ( M ) ,(4) If H ∈ F ( W ) is absolute, then Φ | H : H → M is tame.The submersion Φ : W → M is no more tame. As a consequence, if ( M, G ) is a Liemanifold, the absolute faces of W are not saturated for the pullback groupoid Φ G . Wewill replace the latter by [34, 22] : Φ b G ⇒ W (39)defined by blowing up successively all the absolute faces H , . . . , H ℓ of W . Firstly, set: G = Φ G , β = Id , H = Φ | H G (40)and then for any i ≥ : G i +1 = Sblup r,s ( G i ; H i ) β i +1 −→ G i , H i +1 = ( β ◦ . . . β i +1 ) − ( Φ | Hi +1 G ( M )) (41)The required groupoid is Φ b G = G ℓ and it does not depend on the order of blows-upfor the initial submanifolds Φ | Hi G ( M ) are pairwise disjoint. This is true for any i forthe blowups manifolds Sblup( G i ; H i ) = [ G i , H i ] by [33], and since the structural mapscoincide over the open dense subset ( W \ ∂W ) , the result follows. We now have byconstruction:(1) all faces H ∈ F ( W ) are saturated for Φ b G . (2) For any absolute H ∈ F ( W ) , we consider an open neighborhood H ⊂ U ≃ H × [0 , + ∞ ) and then we have by [22, Paragraph 5.3.5] (after rearranging ifnecessary the order of blowups in order to have H at the end): ( Φ b G ) U ≃ (cid:0) Sblup r,s ( W , H ) × W G ℓ − (cid:1) | U ≃ (Sblup r,s ( W , H ) | U × U Φ | U ( G ) ≃ ( H × [0 , + ∞ ) ⋊ R ) × U Φ | U ( G ) , using U ≃ H × [0 , + ∞ ) . (42)In particular ( Φ b G ) H ≃ R × Φ | H ( G ) (43)The last isomorphism will be called a boundary decomposition of Φ b G and the choiceof collar diffeomorphism U ≃ H × [0 , + ∞ ) (coming from the choice of a defining func-tion) for the absolute faces will be considered as part of the data in the sequel. Theidentifications above provide an isomorphism ( Φ b A ) U ≃ ρ ∗ ( R × Φ | H ( A )) where ρ : U → H corresponds to the first projection through the collar diffeomorphism.A Dirac bundle ( E, D ) over ( W, Φ b G ) is locally of product type if it satisfies [10, Defini-tion 3.4] with T H × T N replaced here by T [0 , × Φ | H ( A ) and using the isomorphismabove. In such a case, we can apply the boundary reduction of [10, Paragraph 3.2],replacing H and T N there respectively by [0 , + ∞ ) and Φ | H ( A ) here, and we get a Diracbundle (or rather an isomorphism class of Dirac bundles) over ( H, Φ | H ( G )) of the op-posite parity denoted by ( E H , D H ) and called the boundary reduction of ( E, D ) to theabsolute face H . Definition 4.5.
A (even, odd) cobordism over ( M, G ) is a 5-tuple w = ( W, Y, Φ , E, B ) where:(1) ( W, Y, Φ) is a (even, odd dimensional) manifold cobordism over M ;(2) A (even, odd) Dirac bundle ( E, D ) on ( W, Φ b G ) is given;(3) a self-adjoint (even, odd) Dirac Y -taming B of D .A null-cobordism is a cobordism with exactly one absolute face.If j is the parity of the cobordism, we get a class [ B ] j ∈ K j ( C ∗ ( Φ b G F Y ))) ≃ K j ( C ∗ ( T Y W )) .Let w = ( W, Y, Φ , E, B ) be a odd cobordism over ( M, G ) , let ϕ : Σ → M where Σ isan absolute face of W . Denote by D the underlying Dirac operator of w and pick up aboundary reduction ( E Σ , D Σ ) of ( E, D ) . We consider the commutative diagram: K ( C ∗ ( T Y W )) K ( C ∗ ( R × T Σ)) K ( C ∗ ( T Σ)) K ( Φ b A ) K ( R × ϕ A ) K c ( ϕ A ) ρ Σ ev t =0 Bott ev t =0 ev t =0 ρ Σ Bott (44)where ρ Σ is the composition of the natural restriction homomorphism C ∗ ( T Y W ) −→ C ∗ (( Φ b G ) F | Σ ) with the isomorphism induced by (43) at the level of adiabatic and Fredholm groupoids: C ∗ (( Φ b G ) F | Σ ) ≃ C ∗ ( R × ( ϕ G ) F ) -HOMOLOGY 15 and then with the morphism ev zt =0 : C ∗ ( ϕ G ) F ) → C ∗ ( T Σ) (that is, the one whichgives the K -equivalence C ∗ ( R × ( ϕ G ) F ) K ∼ C ∗ ( R × T Σ) ). It follows immediately fromthe diagram (44) that ev t =0 (cid:0) Bott ◦ ev zt =0 ◦ ρ Σ ([ B ] Y, odd ) (cid:1) = [ D Σ ] pr , ev . (45)In particular, D Σ is ∂ Σ -tameable and we can pick up a taming B Σ using Theorem 3.3such that: [ B Σ ] ∂ Σ , ev = Bott ◦ ρ Σ ([ B ] Y, odd ) . (46)We get an even geometric cycle (cid:0) Σ , ϕ, E Σ , B Σ (cid:1) ∈ V E geo ( M ) . (47)Picking up a different representative ( E ′ Σ , D ′ Σ ) and a different taming B ′ Σ satisfying (46)provides isomorphic geometric cycles. Definition 4.6.
With the notation above, the isomorphism class of (47) is called theboundary reduction of w to Σ .The case of the other parity is similar. Definition 4.7. • Two geometric cycles x i = (Σ i , ϕ i , E i , B i ) over ( M, G ) , i = 0 , , are cobordant if thereexists a cobordism w = ( W, Y, Φ , E, B ) such that(1) The absolute faces of W are exactly Σ and Σ ;(2) Φ | Σ i = ϕ i (3) The boundary reduction of w with respect to Σ i is the isomorphism class of ( − i x i . • A geometric cycle x = (Σ , ϕ, E, Ψ) over ( M, G ) is null-cobordant if there exists anull-cobordism with Σ as unique absolute face. Example 4.8. (1) Let x i = (Σ , ϕ, E, B i ) ∈ V E geo ∗ ( M ) , i = 0 , , be two even cycles connected by a tame homotopy . This means that there exists a C ∞ homotopy of Dirac bundles ( E, D t ) (i.e., C ∞ homotopies of Clifford homomorphisms and of connections)and a family ( B t ) t ∈ [0 , of tamings of ( D t ) t ∈ [0 , connecting B to B and suchthat: σ ∂ Σ (Λ − B t ) ∈ Σ ∂ Σ ( E ) × is a continuous path. In particular: [ B , + ] ∂ Σ , ev = [ B , + ] ∂ Σ , ev ∈ K ( C ∗ ( T Σ)) .The cycles x and x are then cobordant. A cobordism is given by w = (cid:0) Σ × [0 , t , Y, ϕ ′ , E ′ , B ′ (cid:1) where:(a) Y = ∂ Σ × [0 , , ϕ ′ = pr ◦ ϕ and E ′ = pr ∗ E ⊗ C .(b) ( E ′ , D ′ ) is the product of ( E, D t ) , with ([0 , × C , it (1 − t ) ∂∂t ) . Thus: D ′ = D t + it (1 − t ) ∂∂t . (48)(c) The absolute faces are Σ = Σ × { } and Σ = Σ × { } . (d) We then consider the commutative diagram: K ( C ∗ ( T Y (Σ × [0 , τ / / ev t =0 (cid:15) (cid:15) K ( ϕ A × b T [0 , ev t =0 (cid:15) (cid:15) K ( C ∗ (( T Σ) × R )) τ / / K ( ϕ A × R ) K ( C ∗ ( T Σ)) β O O τ / / K ( ϕ A ) β O O (49)The vertical arrows are isomorphisms. Since ev t =0 [ D ′ ] pr , odd = [ D , + ] pr , ev ⊗ β and [ D , + ] pr , ev belongs to im τ ∗ , we get that [ D ′ ] pr , odd belongs to im τ ∗ too and we apply Theorem 3.3 to choose a Y -taming B ′ of D ′ such that ev t =0 [ B ′ ] Y, odd = [ B , + ] ∂ Σ , ev ⊗ β .(2) For any x ∈ V E geo ( M ) , the geometric cycle x ∪ ( − x ) is null-cobordant. A null-cobordism is given by the previous cobordism in which Σ × { , } is consideredas the unique absolute face.(3) Let x = (Σ , ϕ, E, B ) ∈ V E geo ( M ) with underlying Dirac operator denoted by D .Let ( E ′ , D ′ ) be the product of ( E, D ) with the spin Dirac bundle ( E , D ) of D . Let ( E , D ) be the boundary reduction of ( E , D ) , that is the spin Diracbundle associated with the spin structure of S that bounds the one of ( E , D ) of D . Let also ( E ′′ , D ′′ ) be the product of ( E, D ) with ( E , D ) . Consider thecommutative diagram: K ( C ∗ ( T ( ∂ Σ) × D (Σ × D ))) τ / / ρ (cid:15) (cid:15) K ( ϕ A × b T D ) ρ (cid:15) (cid:15) K ( C ∗ ( T ∂ Σ × S (Σ × S ) × R )) τ / / K ( ϕ A × T S × R ) K ( C ∗ ( T ∂ Σ × S (Σ × S ))) S O O τ / / K ( ϕ A × T S ) S O O K ( C ∗ ( T ∂ Σ (Σ))) ⊗ D O O τ / / K ( ϕ A ) ⊗ D O O (50)The map ρ corresponds to the restriction to the boundary of D and S is thesuspension isomorphism. All the vertical arrows are isomorphisms.Using the lower part of the diagram we first see that there exists a ∂ Σ × S -taming B ′′ of D ′′ such that [ B ′′ ] ∂ Σ × S , odd = [ B ] ∂ Σ , ev ⊗ [ D ] odd . We then set: x = (cid:0) Σ × S , pr ◦ ϕ, E ′′ , B ′′ (cid:1) ∈ V E geo ( M ) . Now using the upper part of the diagram, we obtain a ∂ Σ × D -taming B ′ of D ′ such that ρ [ B ′ ] ( ∂ Σ) × D , ev = S [ B ′′ ] ∂ Σ × S , odd . This provides a null-cobordism for x as follows: w = (cid:0) Σ × D , ( ∂ Σ) × D , pr ◦ ϕ, E ′ , B ′ (cid:1) . (51) -HOMOLOGY 17 Starting with an odd x , we get a null-cobordant cycle x in the same way.(4) Combining the previous constructions, we see that for any x, y ∈ V E geo ( M ) , thegeometric cycles y and y ∪ x are cobordant. Vector bundle modification.
Let x = (Σ , ϕ, E, B ) be an even geometric cycle over ( M, G ) and π : V → Σ an even rank Spin c -vector bundle. Consider the sphere bundle Z = S ( V ⊕ R ) of pr ◦ π : V ⊕ R → Σ . The total space Z is a compact manifold withcorners and the projection π m : Z → Σ is tame, as well as ϕ m = ϕ ◦ π m : Z −→ M .Note that ϕ m ( G ) = π m ( ϕ G ) and ϕ m A ≃ π ∗ m ( ϕ A ) ⊕ V Z (52)where
V Z = ker dπ m . The Spin c -structure of V induces a Spin c -structure on the bundle V Z → Σ and therefore on the bundle A Z := ϕ m A −→ ϕ A =: A Σ . (53)Using L π m , we get a commutative diagram: K ( C ∗ ( G ( Z ) ad )) K ( C ∗ ( A Z )) K ( C ∗ ( G (Σ) ad )) K ( C ∗ ( A Σ )) ev t =0 ( π m ) ad ! ( π m ) ev t =0 (54)Here G ( Z ) = ϕ m ( G ) and G (Σ) = ϕ G . The horizontal maps are isomorphisms. Proceedingas in [20], we prove that ( π m ) is the Thom isomorphism of the Spin c -bundle A Z −→ A Σ .Then ( π m ) ad ! is an isomorphism too.Let D Σ be the Dirac operator of the cycle x , let S Z be the complex spinor bundle of V Z and set E Z = E ⊗ S Z . By [7, Propositions 3.6 and 3.11] we can choose a Diracbundle ( E Z , D Z ) such that ( π m ) [ D Z )] pr , ev = [ D Σ ] pr , ev . We then deduce from the previous diagram that ( π m ) ad ! ([ D Z ] ad , ev ) = [ D Σ ] ad , ev ∈ K ( C ∗ ( G (Σ) ad )) . Now, since D Σ is tameable, we obtain by using the commutative diagram: K ( C ∗ ( G ( Z ) | ∂Z )) K ( C ∗ ( G ( Z ) F )) K ( C ∗ ( G ( Z ) ad )) K ( C ∗ ( G ( Z ) | ∂Z )) K ( C ∗ ( G (Σ) | ∂ Σ )) K ( C ∗ ( G (Σ) F )) K ( C ∗ ( G (Σ) ad )) K ( C ∗ ( G (Σ) | ∂ Σ )) . ∂ M≃ ( π m ) F ! ≃ ev ∂Z × ( π m ) ad ! ≃ M≃ ∂ ev ∂ Σ × (55) that D Z is ∂Z -tameable too and we pick up a ∂Z -taming B Z such that ( π m ) F ! [ B Z ] = [ B ] The constructions are similar for odd geometric cycles.
Definition 4.9.
A vector bundle modification of a geometric cycle x = (Σ , ϕ, E, B ) over ( M, G ) by an even rank Spin c -vector bundle π : V → Σ is a geometric cycle m ( x, V ) = ( Z, ϕ m , E m , B Z ) over G ( M ) such that(1) Z = S ( V ⊕ R ) is the sphere bundle;(2) ϕ m = ϕ ◦ π m : Z −→ M ;(3) π nc! [ B Z ] ∂Z, ∗ = [ B ] ∂ Σ , ∗ ∈ K ∗ ( C ∗ T Σ)) . Geometric K -homology.Definition 4.10. The equivalence relation ∼ on V E geo ∗ ( M ) is the one generated by thefollowing operations:(i) Isomorphisms of geometric cycles;(ii) Direct sums: if x i = (Σ , ϕ, E i , B i ) , i = 1 , are geometric cycles then x ∪ x ∼ (Σ , ϕ, E ⊕ E , B ⊕ B ); (56)(iii) Cobordisms;(iv) Vector bundle modifications.The quotient set V K geo ∗ ( M ) := V E geo ∗ ( M ) / ∼ is called geometric K -homology of ( M, G ) . Theorem 4.11.
The following formula: ∀ x , x ∈ V E geo ( M ) , [ x ] + [ x ] = [ x ∪ x ] . (57) turns ( V K geo ( M ) , +) into an abelian group.Proof. If the equivalence x ∼ x ′ is given by one of the four elementary operations ofDefinition 4.10 then x ∪ x ∼ x ′ ∪ x . Therefore (57) is well defined. Commutativity andassociativity of + is obvious. The neutral element is represented by x and − [ x ] = [ − x ] for any x ∈ V E geo ( M ) : details are provided in Example 4.8. (cid:3) Comparison map.Theorem 4.12.
The map (Σ , ϕ, E, B ) ∈ V E geo ∗ ( M ) ϕ nc ! [ B ] ∂ Σ , ∗ gives rise to a welldefined homomorphism: λ : V K geo ∗ ( M ) → K ∗ ( C ∗ ( T M )) . (58) Proof.
We prove the even case, the odd one is similar.
Invariance under cobordism:
Let w = ( W, Y, Φ , E, B ) be a cobordism between two evengeometric cycles x i = (Σ i , ϕ i , E i , B i ) , i = 0 , over ( M, G ) . We denote by:(1) G ( W ) = Φ b G the groupoid associated with the cobordism w ;(2) G i = ϕ i G the groupoid associated with x i , i = 0 , ;(3) ρ i := ρ Σ i : C ∗ ( T Y W ) → C ∗ ( T Σ i × R ) the homomorphism defined just afterDiagram (44). • Recall that we have by definition of a cobordism of geometric cycles: ρ i [ B ] Y, odd = ( − i [ B i ] ∂ Σ i , ev ⊗ β where β is the chosen Bott generator of K ( C ∗ ( R )) . • We are going to prove that ( ϕ ) nc! ([ B ] ∂ Σ , ev ) = ( ϕ ) nc! ([ B ] ∂ Σ , ev ) (59)where the homomorphisms are defined in (26).Denote by J i the kernel of ρ i and observe that J + J = C ∗ ( T Y W ) . Set J = J ∩ J and consider the exact sequences: −→ J −→ J i ρ i −→ C ∗ ( T Σ i × R ) −→ . (60)for i = 0 , . We denote by ∂ i ∈ KK ( C ∗ ( T Σ i × R ) , J i ) the associated boundaryelements. Consider also: −→ J −→ C ∗ ( T Y W ) ρ ⊕ ρ −→ C ∗ ( T Σ × R ) × C ∗ ( T Σ × R ) −→ (61) -HOMOLOGY 19 whose ideal is given by J = J ∩ J and boundary element denoted by ∂ . By [46, Lemma3.5]) we have: ∂ ( x , x ) = ∂ ( x ) + ∂ ( x ) ∈ KK ( C , J ) , x i ∈ KK ( C , C ∗ ( T Σ i × R )) . (62)We know that ρ ⊕ ρ ([ B ] Y, odd ) = ([ B ] ∂ Σ , ev ⊗ β, − [ B ] ∂ Σ , ev ⊗ β ) , (63)therefore formula (62) and exactness imply: ∂ ([ B ] ∂ Σ , ev ⊗ β ) = ∂ ([ B ] ∂ Σ , ev ⊗ β ) ∈ K ( J ) . (64)It is now time to extend the deformation (20): L w = ( G ( W )) ad × { u = 0 } ∪ (Φ ) b ( G ad ) × (0 , u ⇒ W × [0 , t × [0 , u . (65)Here Φ = Φ ◦ pr : W × [0 , t → M . We can consider various saturated sub-groupoidsof L w . For instance: L nc Y ( w ) = L w | ( W ×{ t =0 }∪ Y × (0 , t ) × [0 , u . (66)If we consider in L nc Y ( w ) the faces corresponding to Σ i , we recover the groupoids L nc ϕ i × R .Continuing in this way and denoting by Φ ′ : W \ (Σ ∪ Σ ) → M the restriction of Φ (which is again a surjective submersion), we get the following commutative diagram: K ( C ∗ ( T Σ × R )) ⊕ K ( C ∗ ( T Σ × R )) K ( J ) K ( C ∗ ( L nc ϕ × R )) ⊕ K ( C ∗ ( L nc ϕ × R )) K ( ... ) K ( C ∗ ( ϕ ( T M × R ))) ⊕ K ( C ∗ ( ϕ ( T M × R ))) K ( C ∗ ( Φ ′ ( T M ))) K ( C ∗ ( T M × R )) ⊕ K ( C ∗ ( T M × R )) K ( C ∗ T M )) ∂ ≃ u =0 u =1 ∂ ′ ≃ u =0 u =1 M≃ ∂ ′′ M≃ ∂ ′′′ (67)The map in the bottom line is given by addition and Bott periodicity, in particular, if ∂ ′′′ ( u ⊕ v ) = 0 then u = − v .The map obtained from top to bottom in the left column is equal to ( ϕ ) nc! ⊗ Id ⊕ ( ϕ ) nc! ⊗ Id .Therefore, using the equality (64) together with the commutativity of the previousdiagram and the remarks just above, we conclude that the equality (59) holds true. Invariance under vector bundle modification:
Consider x = (Σ , ϕ, E, Ψ) ∈ V E geo ( M ) and a vector bundle modification m ( x, V ) =( Z, ϕ m , E m , B Z ) . One one hand, we know that ( π m ) F ! ([ B Z ]) = [ B ] . On the other hand, we have ϕ m = ϕ ◦ π m , therefore: ( ϕ m ) F ! ([ B Z ]) = ( ϕ ◦ π m ) F ! = ( ϕ F ! ◦ ( π m ) F ! )([ B Z ]) = ϕ F ! ([ B ]) and the equality ( ϕ m ) F ! ([ B Z ]) = ϕ F ! ([ B ]) proves the invariance under vector bundlemodification. (cid:3) Reduction to Callias
Let ( M, G ) be a Lie manifold. We are going to define a clutching map : e c : Γ( µ ∂M ) → V K geo ( M ) (68)and show that it descends to a clutching homomorphism c : V F Ell( M ) → V K geo ( M ) .We set Σ A = S ( A ⊕ R ) for the clutching space of A . We denote by ϕ the tamesubmersion Σ A → M . We can write, Σ A = B ( A ) + ∪ B ( A ) − , where B ( A ) denotes theball bundle in A and B ( A ) ± denote the upper and lower hemispheres respectively. Let b A = A ∪ S ( A ) be the radial compactification of A and b π : b A → M the correspondingprojection map.Let P ∈ Ψ m V ( M ; E , E ) be a fully elliptic operator. By ellipticity of P we have anisomorphism σ pr ( P ) : π ∗ E ∼ −→ π ∗ E . The clutched bundle E σ → Σ A is defined by the glueing of pullbacks of E and E ,along the boundary stratum S ( A ) , using σ pr ( P ) : E σ = b π ∗ E ∪ S ( A ) b π ∗ E . To define a geometric cycle e c ( E , E , σ f (Λ − P )) associated with ( E , E , σ f (Λ − P )) ∈ Γ( µ ∂M ) , we consider the Dirac operator D Σ A on the Lie manifold (Σ A , ϕ G ) associatedwith the Spin c -structure of ϕ A → Σ A , and we first observe: Lemma 5.1. ( [8] ) Let ( M, G ) be a Lie manifold and P ∈ Ψ G ( E , E ) be an ellipticoperator. Then: ( i ◦ Thom)([ P ] pr , ev ) = [ D Σ A ] pr , ev ⊗ ([ E σ ] − [ ϕ ∗ E ]) ∈ K ( ϕ A ) . (69)This lemma is proved in [8] in the case A = T M . The proof is the same here. Secondly:
Proposition 5.2.
Let ( M, G ) be a Lie manifold and P ∈ Ψ m G ( E , E ) be a fully ellipticoperator. Then there exists an even Dirac bundle ( E, D ) on (Σ A , ϕ G ) with boundarytaming B such that ϕ nc ! ([ B ] ∂ Σ A , ev ) = [ P ] ∂M, ev ∈ K ( C ∗ ( T M )) . (70) In particular, B + and P have the same Fredholm index.Proof. Rewriting the proof of Theorem 3.8, we get a commutative diagram: K c ( A ∗ ) K c ( π A ∗ ) K c ( ϕ A ∗ ) K ( C ∗ ( T M )) K ( C ∗ ( T A )) K ( C ∗ ( T Σ A )) Thom −⊗ [ i ]ev t =0 ( π nc ! ) − ev t =0 −⊗ [ i ] ev t =0 (71) We only treat here the clutching map in the even case, the odd case is similar. -HOMOLOGY 21
By the previous lemma and this diagram, we obtain the existence of a fully elliptic Q on ϕ G such that [ Q ] pr , ev = [ D Σ A ] pr , ev ⊗ ([ E σ ] − [ ϕ ∗ E ]) = [ D ] pr , ev (72)where ( D , E σ ⊕ ϕ ∗ E ) is the even Dirac bundle with: D = D Σ A ⊗ E σ ⊕ D op Σ A ⊗ ϕ ∗ E and D Σ A ⊗ E σ is the Dirac operator on Σ twisted by E σ and D op Σ A = − D Σ A . Now, usingTheorem 3.3, we get an even Dirac bundle ( e E, e D ) with boundary taming B = e D + R such that we still have [ B ] ∂ Σ A , ev = [ Q ] ∂ Σ A , ev ∈ K ( C ∗ ( T Σ A )) , from which we conclude that (70) holds true using Theorem 3.8, (2). The last assertioncomes from Theorem 3.7. (cid:3) Now, using Proposition 5.2 to pick up suitable ( e E, e D ) and B , we define: e c ( E , E , σ ∂M (Λ − m P )) = [(Σ A , e E, ϕ, B )] iso ∈ V K geo ( M ) . The clutching map e c sends a relative cycle to an isomorphism class of geometric cycles(note that two choices of B yield isomorphic cycles). Theorem 5.3.
The map e c induces a well defined homomorphism c : V F Ell( M ) → V K geo ( M ) . Proof.
Let ( P ) rel := ( E , F , p ) and ( P ) rel := ( E , F , p ) be elements of Γ ( µ M ) which are equivalent. Let α, β ∈ Γ ( µ M ) be elementary elements such that ( E , F , p ) ⊕ α ≃ ( E , F , p ) ⊕ β. Without loss of generality, we can consider that E = E = E , F = F = F and p , p connected by a smooth homotopy p t of fully elliptic symbols. Denote by σ t : π ∗ E −→ π ∗ F the map obtained by extending by homogeneity σ pr ( P t ) over A \ { } andmultiplying it with a function χ ∈ C ∞ ( A ) such that χ ( ξ ) = 0 in a neighborhood of the -section and χ ( ξ ) = 1 near infinity.Define b E via the glueing diffeomorphism [0 , × S ( A ) × E → [0 , × S ( A ) × F, ( t, x, v ) ( t, x, σ t ( v )) along the cylinder [0 , × S ( A ) . This furnishes the vector bundle b E over thecylinder clutching space b Σ := [0 , × Σ . Denote by b ϕ : b Σ −→ M the natural projection.Setting Y = [0 , × ∂ Σ , we already get a cobordism over M , namely ( b Σ , b ϕ, Y ) .Let D j be the Dirac operator underlying the geometric cycle e c (( P j ) rel ) . We consider on b Σ the Spin c -structure that coincides with the one of Σ at Σ × { } and to the oppositeone at Σ × { } .Lut us consider a Dirac operator b D on the Lie manifold ( b Σ , b ϕ b G ) whose boundary re-ductions at Σ × { j } coincide with D j . The corresponding Clifford vector bundle over b Σ is: e E = S b Σ ⊗ b E ⊕ ( − S b Σ ) ⊗ b ϕ ∗ F. Arguing as in Example 4.8, we obtain a Y -taming b B such that ( b Σ , b ϕ, Y, e E, b B ) is acobordism between the geometric cycles e c (( P j ) rel ) . (cid:3) Theorem 5.4.
The diagram K ∗ ( µ ) V K geo ( M ) K ( C ∗ ( T M )) c pd λ commutes. In particular, if P : C ∞ ( M, E ) → C ∞ ( M, E ) denotes a fully elliptic pseu-dodifferential operator on the Lie manifold ( M, G ) , then there is a taming C on the Liemanifold (Σ A , ϕ G ) such that ind( C ) = ind( P ) . The proof follows from Proposition 5.2.
Appendix A. Cobordisms
Theorem A.1.
The cobordism relation is transitive up to isomorphisms of geometriccycles.Proof.
Let x i = (Σ i , ϕ i , E i , B i ) , i = 1 , , , be geometric cycles over ( M, G ) such that x is cobordant to x and x cobordant to x . Let w i = ( W i , Y i , Φ i , F i , C i ) , i = 1 , berespective cobordisms. We set Φ = Φ ∪ Φ : W = W ∪ Σ W −→ M , Y = Y ∪ Y . Thetriple ( W, Φ , Y ) is a manifold cobordism over M between Σ and Σ . We set : e G ( W ) = G ( W ) [ G (Σ ) × R G ( W ) . (73)Also, setting F = F ∪ Σ F and using the point (3) in Definition 4.7, we get a Diracbundle ( F, e D ) over ( W, e G ( W )) that restricts to the one of w j on W j , j = 1 , . Inparticular, we have by assumption that ( F, e D ) is Y -tameable.Note that e G ( W ) = SBlup r,s ( G ( W ) , G ( W ) Σ Σ ) where G ( W ) = b Φ G ( M ) . (74)Choosing a collar decomposition around Σ into W , we get a deformation Lie groupoid: H ( W ) = { t = 0 } × e G ( W ) ∪ (0 , t × G ( W ) ⇒ [0 , × W and it is clear that the Dirac bundle ( F, e D ) can be lifted to a Dirac bundle ( F, C ) over ( W × [0 , , H ( W )) and we thus end with a Dirac bundle ( F, D ) over ( W, G ( W )) byrestriction at t = 1 .Since e D is Y -tameable, we get using the following diagram (where we use this timeFredholm groupoids rather than their K -equivalent non commutative tangent spacecounterparts) that C is Y × [0 , -tameable too and finally D is Y -tameable. K ( C ∗ ( e G ( W ) F Y )) K ( C ∗ ( e G ( W ) ad )) K ( C ∗ ( H ( W ) F [0 , × Y )) K ( C ∗ ( H ( W ) ad )) K ( C ∗ ( G ( W ) F Y )) K ( C ∗ ( G ( W ) ad )) . | t =1 | t =0 | t =1 | t =0 ≃ (75) -HOMOLOGY 23 Fixing a Y -taming B for D then provides the required cobordism ( W, Y, Φ , F, B ) be-tween x and x . (cid:3) Acknowledgements
For useful discussions we thank Bernd Ammann, Ulrich Bunke, Paulo Carrillo Rouse,Claire Debord, Magnus Goffeng, Victor Nistor, Elmar Schrohe, Georges Skandalis andGeorg Tamme. The first author was supported by the DFG-SPP 2026 ‘Geometryat Infinity’. The second author was supported by the Grant ANR-14-CE25-0012-01SINGSTAR.
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