The Longitudinal Index Theorem for Foliations

1 The Longitudinal Index Theorem for FoliationsA. Connes and G. Skandalis Institut des Hautes Etudes Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France Universit e Pierre et Marie Curie, Tour 45-46, HUER 48, 4 place Jussieu, Paris, FranceIntroductionIn this paper1, we use the bivariantKtheory of Kasparov ([19]) as a basic tool to prove theK-theoreticalversion of the Index Theorem for Longitudinal elliptic differential operators for Foliations which is stated asa problem in [10], Section 10. When the foliation is by the fibers of a fibration, this Theorem reduces tothe Atiyah-Singer Index Theorem for families ([2], ). It implies the Index Theorem for measuredfoliations ([9], Theorem , p. 136) and unlike the latter makes sense for arbitrary Foliations , not necessarilygifted with a holonomy invariant transverse Index in the Atiyah Singer Theorem for families ([2]) is an element of theK-theoryK0(B) of the basespace of the fibration.

2 In the case of Foliations the baseBis the space of leaves of the foliation (V,F). Thisspace of leaves, as a topological space, is often degenerate(if the foliation is minimal there are no nontrivialopen sets inV/F). The algebraC(B) of continuous functions onBis replaced by a canonically definedC -algebra:C (V,F), cf. [9], [10]. TheK-theoryK0(C (V,F)) of thisC -algebra plays the role ofK0(B).In the case of a fibrationC (V,F) is (Morita) equivalent toC(B) so thatK0(C (V,F)) =K0(B).LetDbe an elliptic differential operator along the leaves of the foliation (V,F). SinceDis elliptic it hasan inverse moduloC (V,F) hence it gives an element Inda(D) ofK0(C (V,F)). Let us now describe thetopological Index . Letibe an auxiliary imbedding of the manifoldVinR2n. LetNbe the total space of thenormal bundle to the leaves:Nx= (i (Fx)) R2n.

3 Let us foliate V=V R2xby F, F(x,t)=Fx {0},so that the leaves of ( V , F) are just L=L {t}, whereLis a leaf of (V,F) andt R2n. The map(x, ) (x,i(x) + ) turns an open neighborhood of the 0-section inNinto an open transversalTof thefoliation ( V , F). For a suitable open neighborhood ofTin V, theC -algebraC ( , F) of the restrictionof Fto is (Morita) equivalent toC0(T), hence the inclusionC ( , F) C ( V , F) yields aK-theory map:K0(N) K0(C ( V , F)). SinceC ( V , F) =C (V,F) C0(R2n), one has, by Bott periodicity, the equalityK0(C ( V , F)) =K0(C (V,F)).Using the Thom isomorphismK0(F ) is identified withK0(N) so that one gets by the above construction,the topological Index :Indt:K0(F ) K0(C (V,F)).Our main result is the equality: Inda(D) = Indt([ D]) where Dis the Longitudinal symbol ofDand [ D] isits class inK0(F ).

4 In the first section, we formalize the elliptic pseudo-differential calculus for families of operators onXindexedbyY, in terms of the bivariant Kasparov theory. This gives a map of theK-theory with compact supportK (T X Y) to the bivariant groupKK (X,Y). We then compute directly the Kasparov product of twosuch the second section we first recall the definition of [10] of the analytical elementf! KK(X,Y) corre-sponding to aK-oriented mapffromXtoY. We then prove that (idX)! is the unit of the ringKK(X,X).Using the computation of Section 1, we then prove the equality: (f g)! =g! f!. Computingf! in the case1 Communicated by H. Araki, August 16, an immersion, one gets that in all cases the mapK(X) K(Y) given byf! coincides with the classicalwrong way map inK-theory. This statement is an Index Theorem for Section 3 we show that in the context of smooth manifolds the elements of the bivariant KasparovgroupKK(C(V),C(W)) have a natural interpretation in terms of correspondences Kasparov product is then given by a simple fibered productformula whose existence relies on thetransversality Theorem .

5 It follows then that, in this context, all Kasparov products can be computed inpurely geometric term. For instance the Poincar e duality in analyticalKtheory, is easily derived at the endof Chapter III from simple geometric considerations. As applications we also derive the odd Index Theorem ofBaum-Douglas [6] and Kasparov [17]. As another example we exhibit geometrically the correspondence froma submanifoldW(of the manifoldV) to the complement ofWinVwhose associated analytical element inKK1(W,V/W) is given by the exact sequence ofC algebras: 0 C0(V/W) C0(V) C0(W) 0. Inparticular, the connecting map from theKhomology ofV/Wto theKhomology ofWhas again a simplegeometric Section 4 we prove the above-mentioned Longitudinal Index Theorem and at the same time the existenceof a map :K , (BG) K (V/F) from the geometric group [5], [10] of a foliation (V,F) with graphGtothe analytical group.

6 We then illustrate it by a simple , in the appendix we introduce in the general theory of Kasparov a notion of connection whichallows to compute Kasparov products without modifying firsttheC modules involved. It gives an implicitcharacterization of the Kasparov product which we use in a crucial manner in our computations throughoutthe paper. We also give a detailed description of the notion ofK-orientation for microbundles which is thenatural framework of Section 2. All the results of this paperhave been announced in [11].Table of contents1 The Kasparov product of pseudo-differential families2 Wrong way functoriality3 Composition of correspondences and applications4 The Longitudinal Index theorem5 Appendix and implicit characterization of Kasparov product6 Appendix of microbundles1 The Kasparov product of pseudo-differential familiesLetXbe a smooth manifold andYa locally compact parameter space.

7 In this section we shall first interpretthe construction of continuous families, indexed byY, of pseudo-differential operators onX, as yielding amap :K(T X Y) KK(X,Y). Then we compute the Kasparov product of two such families from aformula at the symbol the simplicity of the statements that follow we will assume that the manifoldXis compact. We willindicate in Remarks ( (a)) and ( (a)) the minor changes needed in the non compact byC0(Y) theC -algebra of continuous functions vanishing at be an open subset ofX YandEa Hermitian vector bundle over . LetCc( ,E 1/2) be the spaceof continuous (1/2 density) sections ofEwith compact support. LetH=H( ,E) denote the completionofCc( ,E 1/2) with respect to theC0(Y) valued inner producth , i(y) =Rh (x,y), (x,y)i. ThenHis a HilbertC -module ([18], Definition 2) overC0(Y).

8 We denote here by 1/2the bundle of half densities in theXdirection. The scalar producth (x,y), (x,y)iis a density in theXdirection. Hence its integral over y={x|(x,y) }is well all this paper we will use half densities without mentioning the bundle they are attached to. Theiruse will be to give formulae which do not depend upon the choice of a measure in the Lebesgue A HilbertC -moduleEoverC0(Y) is the space of continuous sections vanishing at of a continuousfleld (Ey)y Yof Hilbert spaces (in the sense of [12], 10). Here we haveH( ,E)y=L2( y,Ey) whereEyis the restriction ofEto y={x|(x,y) }. Note that the scalar product inL2( y,Ey) iscanonically defined using half Following [18] and [19], we take all scalar products to be linear in the second variable, antilinear in next define the algebra of order 0 pseudo-differential families.

9 It is a subalgebra of theC -algebraL(H( ,E)) of endomorphisms of the HilbertC0(Y) moduleH( ,E) ([18], Definition 3).We shall first see the pseudo-differential families as actingon theC ,0sections with compact support of ([2]) that a functionfonX Yis of classC ,0if the mapy f( ,y) fromYtoC (X) is notion of bundle of classC ,0over an open subset X Yis defined analogously. IfEis such abundle,C ,0c(E) denotes the space ofC ,0sections ofE, with compact support in (cf. [2], p. 121-124).SetT ={(x, ,y)/(x, ) T X,(x,y) }. A symbol of order 0 is a functiona C ,0(T ,L(E)) which,uniformly on compact subsets ofY, has an asymptotic expansiona Pm=0 mwhere mis homogeneous ofdegree m. To such a symbol one associates a continuous linear mapf PffromC ,0c( ,E 1/2) toC ,0( ,E 1/2) by the usual formula:(Pf)(x,y) =Zexp(i (x,x , ))a(x,y, ) (x,x ,y)f(x ,y)dx d the integral is an oscillating integral, is a phase function and is a cut off function associated withthe diagonal ofX X( (x,x ,y) L(Ex ,y 1/2x ,Ex,y 1/2x), (x,x,y) = 1Ex,y, cf.)

10 For instance [7] or[16]).As (Pf)(x,y) only depends upon the restriction offtoX {y},Pis a family (Py)y Yof pseudo-differentialoperators onX. We shall say that the support ofPis contained in a closed subsetKof , when, for eachy Y, the distribution kernel ofPyhas support inKy Ky(Ky={x X,(x,y) K}).It is clear (cf. [24], Theorem 1, p. 243) that a familyPas above, with compact supportK , extends toan endomorphism (still calledP) of theC0(Y) moduleH( ,E). Let 0( ,E) denote the norm closure inL(H( ,E)) of the -algebra generated by the aboveP s and the idealR(H( ,E)) of compact endomorphismsofH( ,E) ([18], Definition 4).A bounded sectionf Cb( ,L(E)) determines an endomorphism still notedf ofH( ,E) by the formula(f )(x,y) =f(x,y) (x,y)( Cc( ,E 1/2)). ForP 0( ,E) andf Cb( ,L(E)) bothfPandPf 0( ,E). Hence theclosed subspace ofL(H( ,E)) generated by 0( ,E) andCb( ,L(E)) is aC -algebra ( ,E).