The Twisted Higher Harmonic Signature for Foliations

THE TWISTED HIGHER HARM ONIC SIGNA TURE F OR F OLIA TIONS MOULA Y-T AHAR BENAMEUR AND JAM ES L. HEITSCH Abstract. W e prov e that the higher harmonic si gnature of an even dimensional oriented Riemannian foliation F of a compact Riemannian manifold M with coeﬃcient s in a leafwi se U ( p, q )-ﬂat complex bundle is a leafwise homotop y inv ariant. W e also pro v e the leafwise homotop y inv ariance of the twisted higher Betti classes. Consequences f or the Novik o v conjecture for f ol iations and for groups are i nv estigated. 1. Introduction In this pap er, we prov e that the hig her har monic sig nature, σ ( F, E ), of a 2 ℓ dimensiona l orie nted Rie- mannian foliation F o f a compac t Riemannian manifold M , t wisted by a leafwise ﬂa t complex bundle E over M , is a le afwise homotopy inv ariant. W e a lso de r ive impo rtant conseque nc e s for the Novik ov conjecture for foliations a nd for groups. W e assume that E admits a non-deg enerate p ossibly indeﬁnite Her mitian metric which is pres erved by the leafwise ﬂat structure. As explained in [G96 ], this includes the leafwise O ( p, q )-ﬂat and the leafwise symplectic-ﬂat ca ses. W e assume that the pro jection o nto the t wisted le afwise har monic forms in dimension ℓ is transversely smoo th. This is true whenever the leafwise parallel transla tion on E deﬁned by the ﬂat structure is a bo unded map, in particular whenever the pr e s erved metric on E is p os itive deﬁnite. It is satisﬁed for imp o rtant exa mples , e.g., the examples of Lusztig [Lu72] which proved the No viko v conjecture for free a be lia n gr o ups, and it is alwa ys true whenever E is a bundle as so ciated to the norma l bundle o f the foliation. In pa rticular, the smo othness assumption is fulﬁlled for the (unt w is ted) leafwise signature oper ator. An y metr ic on M determines a metric on each lea f L of F , so also on all cov ers o f L . The bundle E | L c an b e pulled back to a ﬂat bundle (als o denoted E ) on a n y cov er o f L . These le a fwise metrics a nd the leafwise ﬂat bundle E determine leafwise La placians ∆ E and Ho dge ∗ o p erators on the diﬀerential forms o n L with co eﬃcients in E | L , as well as on all c overs of L . The Ho dge op erato r determines an inv olution whic h commutes with ∆ E , so ∆ E splits as a s um ∆ E = ∆ E , + + ∆ E , − , in particular in dimension ℓ , ∆ E ℓ = ∆ E , + ℓ + ∆ E , − ℓ . T o ea ch leaf L of F , we asso cia te the formal diﬀerence of the (in gener al, inﬁnite dimensional) spaces Ker(∆ E , + ℓ ) and Ker(∆ E , − ℓ ) o n e L , the simply c o nnected cover of L . W e assume that the Sch wartz kernel of the pro jection onto Ker(∆ E ℓ ) = Ker(∆ E , + ℓ ) ⊕ Ker(∆ E , − ℓ ) v ar ie s smo o thly tr a nsversely . Roughly sp eaking, transverse smo othness means that the Ker(∆ E , ± ℓ ) are “smo oth bundles ov er the leaf space o f F ”. W e deﬁne a Cher n-Connes character ch a for such bundles whic h takes v alues in the Haeﬂiger cohomolog y of F . The higher harmonic signature of F is deﬁned as σ ( F, E ) = ch a (Ker(∆ E , + ℓ )) − ch a (Ker(∆ E , − ℓ )) . Our main theorem is the following. Theorem 9.1. Supp ose that M is a c omp act Riema nnian manifold, with oriente d Ri emannian foliatio n F of dimension 2 ℓ , and that E is a le afwise ﬂat c omplex bu nd le over M with a ( p ossibly indeﬁnite) non- de gener ate Hermitian metric which is pr eserve d by the le afwise ﬂat struct ur e. Assume t hat the pr oje ction onto K er(∆ E ℓ ) for t he asso ciate d foliation F s of the homotopy gr oup oid of F is tr ansversely sm o oth. Then σ ( F, E ) is a le afwise homotopy invariant. In particular, supp ose that M ′ , F ′ , and E ′ satisfy the hypothesis of Theorem 9.1, and that f : M → M ′ is a lea fwise homo topy e q uiv alence, which is leafwise oriented. Set E = f ∗ ( E ′ ) with the induced lea fwise ﬂa t structure and pres erved metric. Then f induces an isomorphism f ∗ from the Haeﬂig er cohomo logy of F ′ to 1 2 M.-T. BENAMEUR AND J. L. HEITSCH that of F , a nd f ∗ ( σ ( F ′ , E ′ )) = σ ( F , E ) . A priori, σ ( F, E ) depends on the metric on M . How e ver, it is an immediate corollary of Theor em 9 .1 tha t it is indep endent of this metr ic since the ident it y map is a leafwise ho motopy equiv alence b etw een ( M , F ; g 0 ) and ( M , F ; g 1 ). In general, σ ( F, E ) dep ends on the ﬂat structure and the metric on E , in particular on the splitting of E = E + ⊕ E − int o positive (resp. negative) deﬁnite sub bundles. Our tec hniques also give the lea fwise homotopy inv ariance of the twisted hig her Betti classes. When the t wisting bundle E is trivial, this extends (in the Riema nnia n case) the main theor em of [HL91]. Theorem 10.6 Supp ose that M is a c omp act R iemannian manifold, with oriente d Riemannian foliation F of dimension p . L et E b e a le afwise ﬂat c omplex bund le over M with a (p ossibly indeﬁnite) non-de gener ate Hermitian metric which is pr eserve d by the le afwise ﬂat stru ctur e. Assu me that the pr oje ct ion onto Ker(∆ E ) for the asso ciate d foli ation F s of the homo topy gr oup oid of F is tra nsversely smo oth. Then t he twiste d high er Betti classes β j ( F, E ) , 0 ≤ j ≤ p , ar e le afwise homotopy invariants. W e now giv e some background to place the results of this paper in context. Let M and M ′ be closed oriented manifolds with oriented foliations F and F ′ . L e t ϕ : ( M ′ , F ′ ) → ( M , F ) be an or iented, leafwise oriented, leafwise ho motopy eq uiv alence. Denote the homo topy group oid o f F b y G , and let f : M → B G b e a classifying map for F . The BC No viko v conjecture predicts that for every x ∈ H ∗ ( B G ; R ), Z M L ( T F ) ∪ f ∗ x = Z M ′ L ( T F ′ ) ∪ ( f ◦ ϕ ) ∗ x. It is ea sy to check that this co njecture reduces to the case where the leav es hav e even dimension. In the case of a folia tion with a single closed leaf with fundamen ta l group Γ and denoting by f : M → B Γ a cla ssifying map for the univ ersal c ov er of M , the BC Noviko v c onjecture reduces to the classical Noviko v conjecture Z M L ( T M ) ∪ f ∗ x = Z M ′ L ( T M ′ ) ∪ ( f ◦ ϕ ) ∗ x, ∀ x ∈ H ∗ ( B Γ; R ) . A powerful approach to the No viko v conjecture was initiated by Kaspa r ov in [K8 8]. He actually prov es a stronger v ersion of the Noviko v co njecture, namely the ratio na l injectivity o f the famo us Baum-Co nnes map [K S03, HgK01, La02]. See [T99] fo r a pro of of this injectivit y for a la r ge cla ss of folia tio ns, including hyperb olic fo liations. Note that it is still an op en question whether the B aum-Connes map is ra tionally injectiv e for Riemannian foliations. A second approach to the Novik ov c onjecture was initiated b y Connes and his collab ora tors [CM9 0] and uses cyclic co homology and the homo topy inv ariance of the Miscenko symmetric signature in the K -theor y of the r educed group C ∗ -algebra [K88, M7 8]. This metho d proved succe s sful, [CGM93], for the la rgest known cla ss of gro ups, including Gromov-hyperb olic g r oups. F or folia tions, the homotopy inv ariance o f the corresp o nding Miscenko class in the K -theor y of the C ∗ -algebra of G was explained in [BC00, BC85] and prov ed indep endently in [KaM85] and [HiS92]. It reduces the BC Noviko v conjecture to a n extension problem in the K - theory of foliations , together with a coho mo logical lo ngitudinal index for m ula. The extension pr oblem was ﬁrst solved b y Connes for cer tain co cy c les in [C86], by using a highly no n trivial analytic break through. F or general co cycles , the extension problem is a serio us o bstacle a nd many eﬀorts hav e been ma de in this dir ection [Cu04, CuQ97, LMN05 , N97, P 95, Me]. See also the recent [Ca] for an alternative appro ach. The pres ent pap er was inspired by a thir d method mainly due to Lusztig [L u72], and to idea s o f Gromov [G96]. It relies on the fact that for discrete gro ups having en ou gh ﬁnite dimensional U ( p, q ) r epr esentations , the even c ohomolog y of the classifying space B Γ is gener ated by U ( p, q ) ﬂat K -theory classe s . The main theorem needed in this approach is the o riented homo topy inv ariance of the twisted signature by s uch K - theory classes. This approach has b een extended in [CGM90, CGM93] to co ver all the known cases, using the concept of groups having enough almost r epr esentations and almost ﬂat K -theory classes. Recall tha t in no n-commutativ e geometry , the index of an elliptic op erator is usually deﬁned as a certain C ∗ -algebra K theory class co nstructed o ut of the op e rator its e lf, witho ut r eference to its kernel or cokernel. TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 3 In the sp ecia l (co mm utative) sub-case of a ﬁbra tio n, the Chern c haracter o f this op erato r K theory cla ss coincides with the Che r n c haracter of th e index bundle determined by the operato r . In the (non-commutativ e ) case of foliatio ns , this equality is not known in genera l. See [BH08], wher e conditions are given for it to hold, as well as [N97] and the r e cent [AGS]. F or the signature op erator , and its twists by leafwise almo s t ﬂat K -theor y classes, the C ∗ -algebra K -theory index is well known to be a leafwise homotopy in v ariant of the foliation [HiS92]. Ho wev er , in or der to deduce explicit r esults on the BC Noviko v conjecture for foliations, one needs to deﬁne a Chern-Connes character of this C ∗ -algebra K -theor y clas s and to co mpute it. O ur approach to this pro blem is to use the index bundle of the twisted leafwise signature op erator , whose Chern- Connes character in Haeﬂiger coho mology is well deﬁned as so on as the bundle is. It is ther e fore a na tur al problem to prov e direc tly the homotopy inv ariance of the Chern-Co nnes c haracter of the leafwise signature index bundle and its twists b y lea fwise (almost) ﬂat K -theory classes. Our progra m to attac k the BC Novik ov conjecture for foliations consists of three steps. • Given a K -theory class y = [ E + ] − [ E − ] ov e r B G , prove that the character istic num b er Z M L ( T F ) ∪ f ∗ ch( y ) equals the higher leafwise harmonic s ignature t wisted b y f ∗ y . • Prove that the higher leafwise harmonic signa tur e twisted by leafwise almost ﬂa t K -theory cla sses of the a mbian t manifold is a lea fwise oriented, leafwise homotopy inv ariant. • Prove that complex bundles E = E + ⊕ E − , such that [ f ∗ E + ] − [ f ∗ E − ] is a leafwise almos t ﬂat K -theor y class, generate the K -theory of B G . It is clea r that solving these thre e pr oblems for a clas s o f foliations implies the BC Noviko v c o njecture for that class. The ﬁrst step was partially co mpleted in o ur previous pa pe r s [BH04, BH08], where we pr ov e d this eq uality under cer tain as s umptions, whic h were subsequently removed in [A GS], provided the bundle E + ⊕ E − is glob al ly ﬂat. W e conjecture that the result is still true under the far less res trictive assumption that E + ⊕ E − is only le afwise ﬂa t. The s econd step is the goal of the present pap er, when the co eﬃcient bundle E has a leafwise ﬂa t str uctur e and the foliatio n is Riema nnian. See [BH09] for further r esults o n this question. Our res ults so far o n the third step rely on deep but now classical results of Gromov [G96], and a llow us, (assuming our c o njecture above), to prove, for instance, the BC No viko v conjecture, without extr a assump- tions, for the subring of H ∗ ( B G ; R ) g enerated by H 1 ( B G ; R ) and H 2 ( B G ; R ). Again se e the forthcoming pap er [BH09]. Finally , we conjecture that the Riemannian assumption can b e remov ed, and that the only s erious o bstacle now lies in the third step. Contents 1. Int ro duction 1 2. Notation and review 4 3. Chern-Connes character for transversely smo o th idemp otents 6 4. The t wisted higher harmonic signature 10 5. Connections, curv ature, and the Chern-Connes c haracter 16 6. Leafwise maps 21 7. Induced bundles 30 8. Induced connections 36 9. Leafwise homotopy inv ariance of the twisted hig her harmonic signature 37 10. The t wisted leafwise signature oper ator and the twisted higher Betti cla sses 49 11. Consequences of the Main The o rem 51 References 57 W e now brieﬂy describ e the c o nten ts o f each section. Section 2 contains notation and some review. In Section 3, we construct the Chern- Connes character for transversely smo oth ide mp otents, which takes v alues in the Haeﬂiger cohomolo gy of the folia tio n. In Section 4, we deﬁne the t wisted higher harmonic signature, 4 M.-T. BENAMEUR AND J. L. HEITSCH and prove that if the paralle l tr anslation using the ﬂat str ucture on E is b ounded, then the pro jection to the twisted harmonic forms is tr ansversely smoo th. Section 5 contains t wo importa n t concepts essential to the pro of of o ur main theor em, namely the no tio n of a “smo o th bundle ov er the spa ce of leaves of F ” , a nd the extension to suc h bundles of the classical Chern-W eil theory of characteristic classes. This allows us to compare the c haracteristic classe s of such bundles on diﬀere nt manifolds. Section 6 is concerned with the study o f leafwise ho motopy equiv alences, and their induced maps on Haeﬂiger cohomolo g y and on leafwise Sob olev cohomologies. In general, leafwise homotopy equiv a lences do not b ehave well on So b o lev fo r ms and cohomolog ies. T o overcome these diﬃculties, w e use tw o diﬀerent construc tio ns. The ﬁrst, due to Hilsum- Sk andalis [HiS92], g ives smo oth bo unded maps b et ween So bo lev forms. The second, which uses the Whitney isomorphism b etw een simplicial a nd smo oth cohomolo g y , gives us control of the leafwise cohomolog ies. In Section 7, we prov e that the pull- backs under leafwise homo topy equiv ale nce s of cer tain smo oth bundles ov er the space of leaves are still smo oth bundles. Section 8 extends the notion of pulled-back connections. Section 9 contains the pro of of the main theorem. In Sectio n 10, w e prov e the equality be tw een the twisted higher harmonic signa ture and the Cher n- Connes character of the index bundle of the twisted leafwise s ignature op erator . W e explain how o ur methods extend to prove Theo rem 10.6. W e also conjecture a cohomolo g ical formula for the twisted higher ha rmonic signa tur e, which is a lready know to b e true in some cases. See [H95, HL99, BH08] a nd the forthcoming [AGS]. Finally , in Se c tion 11 we show how o ur results lead to impo rtant conseque nc e s for the Novik ov conjecture for foliations and for groups. A cknow le dgments. W e are indebted to J . Alv a rez-Lop ez, A. Connes, J. Cuntz, Y. Kordyuko v, J. Renault, J. Ro e, G. Sk andalis , D. Sulliv an, and K. Whyte for many useful discussions. Part of this work was done while the ﬁrst author was visiting the Universit y of Illinois at Chicago, the second author was visiting the Univ ersity of Metz, and b o th a uthors were visiting the Ins titut Henri Poincar´ e in Paris, and the Mathematisches F o r sch ungs institut Ob erwolfach. Both authors are most grateful for the w arm hospitality a nd generous suppo rt of their hosts. 2. Not a tion and review Throughout this pap er M denotes a smoo th compac t Riemannian manifold of dimension n , and F denotes an oriented Riemannian foliation of M o f dimension p = 2 ℓ and codimensio n q . So n = p + q . The tang ent bundle of F is denoted by T F , its normal bundle by ν , and its dua l normal bundle by ν ∗ . W e as sume that the metric on M , when restricted to ν , is bundle like, s o the holonomy maps of ν and ν ∗ are isometrie s . A leaf of F is denoted L . W e denote b y U a ﬁnite go o d cov e r of M by foliation charts a s deﬁned in [HL90]. If V → N is a vector bundle ov er a manifold N , we deno te the space of smo o th sections by C ∞ ( V ) or by C ∞ ( N ; V ) if we wan t to emphasiz e the base space of the bundle. The compactly supp o rted sections are denoted b y C ∞ c ( V ) or C ∞ c ( N ; V ). The space of diﬀerential k forms on N is deno ted A k ( N ), and w e set A ∗ ( N ) = ⊕ k ≥ 0 A k ( N ). The spac e of compac tly s uppo rted k forms is denoted A k c ( N ), and A ∗ c ( N ) = ⊕ k ≥ 0 A k c ( N ). The de Rham exterior deriv ative is deno ted d or d N . The tangent and co tangent bundles of N will be deno ted T N and T ∗ N . The (reduced) Haeﬂiger coho mology o f F , [Ha80], [BH08], is g iven as follows. F or e ach U i ∈ U , let T i ⊂ U i be a tra ns versal and set T = S T i . W e may a ssume that the closur es of the T i are disjoint. Le t H be the holonomy pseudo group induced b y F on T . Denote the exterior deriv ative by d T : A k c ( T ) → A k +1 c ( T ). The usual Haeﬂig e r cohomolog y is deﬁned using the quotient of A k c ( T ) by the vector subspace L k generated by elements of the for m α − h ∗ α where h ∈ H and α ∈ A k c ( T ) has supp ort co n tained in the range of h . The (reduced) Haeﬂiger cohomo lo gy uses the quotient of A k c ( T ) by the closure L k of L k . W e take this closure in the following sense. (The re a der should note that in previous pap er s, we said that we used the C ∞ top ology to take this closure, but in fact we used the one given her e.) L k consists of all elements in ω ∈ A k c ( T ), s o that there are sequences { ω n } , { b ω n } ⊂ L k with || ω − ω n || → 0 and || d T ( ω ) − b ω n ) || → 0. The nor m || · || is the sup norm, that is || ω || = sup x ∈ T || ω ( x ) || x , wher e || · || x is the nor m on ( ∧ k T ∗ T ) x . Set A k c ( M / F ) = A k c ( T ) / L k . The ex terior deriv ative d T induces a contin uous diﬀerential d H : A k c ( M / F ) → A k +1 c ( M / F ). Note that A k c ( M / F ) and d H are indepe nden t of the choice of cov er U . In this pa per , the complex {A ∗ c ( M / F ) , d H } and its cohomo logy H ∗ c ( M / F ) will b e called, resp ectively , the Hae ﬂig er for ms and Haeﬂiger cohomology o f TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 5 F . The reader should no te that this cohomo logy app ea rs as a quotient in the general computation of cyclic homology for foliations carried o ut in [BN94]. As the bundle T F is or ient ed, there is a contin uous op en surjective linear map, calle d integration over the leav es, Z F : A p + k ( M ) − → A k c ( M / F ) which co mmu tes with the e x terior deriv atives d M and d H . Given ω ∈ A p + k ( M ), write ω = P ω i where ω i ∈ A p + k c ( U i ). Integrate ω i along the ﬁbe rs of the submersio n π i : U i → T i to obtain Z U i ω i ∈ A k c ( T i ). Deﬁne Z F ω ∈ A k c ( M / F ) to b e the c la ss of X i Z U i ω i . It is indep endent of the choice of the ω i and of the cov er U . As Z F commutes with d M and d H , it induces the map Z F : H p + k ( M ; R ) → H k c ( M / F ). F o r convenience w e will b e working o n the homotopy gr oup oids (als o called the mono drom y gro upo ids) of our foliations, but our r esults extend to the holonomy group oid, as well as any group oids b etw een these t wo extremes. Recall that the homotopy gr oup oid G of F co nsists o f equiv alence class es of paths γ : [0 , 1] → M such that the image of γ is contained in a lea f of F . Two such paths γ 1 and γ 2 are eq uiv alent if they ar e in the same leaf and homotopy equiv alent (with endpo int s ﬁxed) in that lea f. Two cla sses may b e comp os ed if one ends where the second b egins and the comp osition is just the juxtap osition of the t wo pa ths. This makes G a group oid. The spac e G (0) of units o f G consists of the e q uiv alence class e s o f the co ns tant paths, a nd we ident ify G (0) with M . F o r Riemannian foliations, G is a Hausdorﬀ dimension 2 p + q manifold, in fact a ﬁbration. The basic op en sets deﬁning its manifold structure are given a s follows. Given U, V ∈ U and a leafwise path γ starting in U and ending in V , deﬁne ( U, γ , V ) to be the set o f equiv alence class es of leafwise paths starting in U and ending in V which are homotopic to γ throug h a homotopy of leafwise paths whos e end p oints remain in U and V respectively . It is easy to see, using the holonomy deﬁned b y γ from a trans versal in U to a transversal in V , that if U, V ≃ R p × R q , then ( U, γ , V ) ≃ R p × R p × R q . The source and ra ng e maps of the g roup oid G are the tw o natural maps s , r : G → M given by s  [ γ ]  = γ (0), r  [ γ ]  = γ (1). G has t w o natural transverse foliatio ns F s and F r whose leav es ar e resp ectively e L x = s − 1 ( x ), and e L x = r − 1 ( x ), for each x ∈ M . Note that r : e L x → L is the simply co nnected cov er ing of L . W e will work with the foliation F s . No te that the in tersection of any le a f e L x and any basic op en set ( U, γ , V ) consists of a t most o ne placque of the foliation F s in ( U, γ , V ), i.e. ea ch e L x passes through a ny ( U, γ , V ) at most once. There is a c anonical lift of the normal bundle ν of F to a bundle ν G ⊂ T G so that T G = T F s ⊕ T F r ⊕ ν G , and r ∗ ν G = ν a nd s ∗ ν G = ν . It is given as follows. Let [ γ ] ∈ G w ith s  [ γ ]  = x , r  [ γ ]  = y . Denote b y exp : ν → M the exp onential map. Given X ∈ ν x and t ∈ R suﬃciently small, there is a unique leafwise path γ t : [0 , 1] → M so that i) γ t (0) = exp( tX ) ii) γ t ( s ) ∈ exp( ν γ ( s ) ). In particular γ 0 = γ . Thus the family [ γ t ] in G deﬁnes a tange nt vector b X ∈ T G [ γ ] . It is ea sy to chec k that s ∗ ( b X ) = X and r ∗ ( b X ) is the parallel translate of X a long γ to ν y . The metric g 0 on M induces a ca nonical metric g 0 on G as follows. T G = T F s ⊕ T F r ⊕ ν G and these bundles are m utually orthogona l. So the normal bundle ν s of T F s is ν s = T F r ⊕ ν G . On T F r , g 0 is s ∗  g 0 | T F  , on T F s it is r ∗  g 0 | T F  , a nd on ν G it is r ∗  g 0 | ν  , which, since F is Riemannian and the metric on ν is bundle-like, is the same a s s ∗  g 0 | ν  . W e denote b y E a le afwise ﬂat co mplex bundle over M . This means that there is a co nnec tion ∇ E on E over M whic h, when restr icted to any leaf L of F , is a ﬂat connection, i.e. its curv ature ( ∇ E ) 2 | L = ( ∇ E | L ) 2 = 0. This is equiv alent to the condition that the para lle l transla tion deﬁned by ∇ E | L , when 6 M.-T. BENAMEUR AND J. L. HEITSCH restricted to contractible lo ops in L , is the identit y . W e assume that E admits a (p oss ibly indeﬁnite) non- degerera te Hermitian metric, deno ted { · , ·} , which is pr eserved b y the leafwise ﬂat structure. This means that if φ 1 and φ 2 are local leafwise ﬂat se c tions o f E , then their inner pro duct { φ 1 , φ 2 } is a lo cally constan t function on ea ch leaf. Mo re gener ally , it is c ha racterized by the fact that for general se ctions φ 1 and φ 2 , and for an y v ector ﬁeld X ta ngent to F , X { φ 1 , φ 2 } = {∇ E ,X φ 1 , φ 2 } + { φ 1 , ∇ E ,X φ 2 } . W e denote also by E its pull ba ck by r to a leafwise (for the foliatio n F s ) ﬂat bundle on G along with its inv a riant metric and leafwise ﬂat connection. The context should make it cle ar which bundle w e are using. A splitting of E is a decomp osition E = E + ⊕ E − (of E on M ) into an or thogonal sum o f t w o sub-bundles so that the metric is ± deﬁnite on E ± . Splittings always exist a nd any tw o are homotopic. The splitting deﬁnes an inv o lution γ o f E . If φ is a lo ca l section of E with φ = φ + + φ − where φ ± is a loc a l section of E ± , then γ φ = φ + − φ − . If w e change the sign of the metric on E − , we obtain a positive deﬁnite Hermitian metric on E − and so also on E o ver b oth M and G . In genera l, this new metric on E , denoted ( · , · ), is not preserved by the ﬂat structure . Example 2.1. As s ume that the c o dimension of F is even, say q = 2 k . Set E = ∧ k ν ∗ ⊗ C . The bund les ν and ν ∗ have natura l ﬂat structur es along the le aves given by the holonomy maps (which deﬁne ﬂat lo c al se ctions). Sinc e the metric on ν is bund le-like, t he induc e d volume form on ν ∗ is invariant u n der the holonomy of F . Denote by ∗ ν the Ho dge ∗ op er ator on ∧ ∗ ν ∗ , and also its extension t o E . Given t wo elements φ 1 and φ 2 of E x , set { φ 1 , φ 2 } = √ − 1 k 2 ∗ ν ( φ 1 ∧ ν φ 2 ) , wher e ∧ ν : E ⊗ E → ∧ 2 k ν ∗ ⊗ C . We le ave it to the r e ader to che ck that E and {· , ·} satisfy the hyp othesis of The or em 9.1. Denote by A ∗ c ( F s , E ) the g r aded alge bra of leafwise (for F s ) diﬀer ent ial fo r ms o n G with co eﬃcients in E which hav e compact s uppo rt when r estricted to a ny leaf of F s . A Riemannian s tructure on F induces one on F s . As usual there is the leafwise Riema nnian Ho dge op erato r ∗ , which gives an inner pr o duct o n ea ch A k c ( F s , E ). In particular, if α 1 and α 2 are leafwise R v alued k forms and φ 1 and φ 2 are sections of E , then < α 1 ⊗ φ 1 , α 2 ⊗ φ 2 > ( x ) = Z e L x ( φ 1 , φ 2 ) α 1 ∧ ∗ α 2 = Z e L x { φ 1 , γ φ 2 } α 1 ∧ ∗ α 2 . W e denote by A ∗ (2) ( F s , E ) the ﬁeld of Hilb e rt spaces ov er M which is the le a fwise L 2 completion o f these diﬀerential forms under this inner pro duct, i.e. A ∗ (2) ( F s , E ) x = L 2 ( e L x ; ∧ T ∗ F s ⊗ E ) . This is a contin uo us ﬁeld of Hilber t spaces, s e e [C79]. Beca use M is compact, the s paces L 2 ( e L x ; ∧ k T ∗ F s ⊗ E ) do not dep end on our choice of metrics. Ho wev er , the inner pro ducts o n these spa c e s do dep end on the metrics, as do the Hilbert norms, denoted k · k 0 . If E is the one dimensio nal trivia l bundle with the trivial ﬂat structure, then A ∗ (2) ( F s , E ) is just the leafwise L 2 forms (no w with co eﬃcien ts in C ) for the foliation F s and is denoted A ∗ (2) ( F s , C ). 3. Chern-Connes character for transversel y smoo th id empotents Since we need the “transverse diﬀerential” and g raded trace used in [BH04] to deﬁne the Chern-Connes character, we now brieﬂy recall that co nstruction. Consider the algebr a C ∞ c ( G ; ∧ F s ⊗ E ) of s mo oth compactly supp orted s ections ov er G of the bundle, here denoted ∧ F s ⊗ E , whose ﬁb er at γ ∈ G is ( ∧ F s ⊗ E ) γ = Hom(( ∧ T ∗ F ⊗ E ) s ( γ ) , ( ∧ T ∗ F ⊗ E ) r ( γ ) ) . If α ∈ C ∞ c ( G ; ∧ F s ⊗ E ), it deﬁnes the leafwise ope r ator A which acts on φ ∈ A ∗ (2) ( F s , E ) x by ( Aφ )( γ ) = Z e L x α ( γ γ − 1 1 ) φ ( γ 1 ) d γ 1 , TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 7 where γ , γ 1 ∈ e L x , and we identify ( T ∗ F s ) γ with T ∗ F r ( γ ) . In [BH0 4], w e deﬁned a Chern-Connes c haracter from the K − theory of this algebra to the Haeﬂiger cohomolog y of the fo liation, ch a : K 0 ( C ∞ c ( G ; ∧ F s ⊗ E )) − → H ∗ c ( M / F ) , given a s follows. C o nsider the co nnection ∇ on ∧ T ∗ F s ⊗ E g iven by ∇ = r ∗ ( ∇ F ⊗ ∇ E ) wher e ∇ F is a connection on ∧ T ∗ F deﬁned by a co nnec tio n on T ∗ F . Then ∇ : C ∞ ( ∧ T ∗ F s ⊗ E ) → C ∞ ( T ∗ G ⊗ ∧ T ∗ F s ⊗ E ), and we ma y ex tend ∇ to an op erator of degre e one on C ∞ ( ∧ T ∗ G ⊗ ∧ T ∗ F s ⊗ E ), where on decomp osa ble sections ω ⊗ φ , with ω ∈ C ∞ ( ∧ k T ∗ G ), ∇ ( ω ⊗ φ ) = dω ⊗ φ + ( − 1 ) k ω ∧ ∇ φ. The foliation F s has dual normal bundle ν ∗ s = s ∗ ( T ∗ M ), a nd ∇ deﬁnes a quasi-c onne ction ∇ ν acting o n C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ) by the compo sition C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ) i − → C ∞ ( ∧ T ∗ G ⊗ ∧ T ∗ F s ⊗ E ) ∇ − → C ∞ ( ∧ T ∗ G ⊗ ∧ T ∗ F s ⊗ E ) p ν − → C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ) , where i is the inclusio n and p ν is induced by the pro jectio n p ν : T ∗ G → ν ∗ s determined by the deco mp os ition T G = T F s ⊕ ν s . Denote by ∂ ν : End( C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E )) → E nd( C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E )) the linear op era tor g iven by the graded commutator ∂ ν ( T ) = [ ∇ ν , T ] . Recall: that ( ∂ ν ) 2 is given by the comm utator with the curv ature θ ν = ( ∇ ν ) 2 of ∇ ν ; that θ ν is a leafwise diﬀerential o pe r ator whic h is at worst o rder one; and that the deriv atives of all orders of its co eﬃcients ar e uniformly bounded, with the bound p ossibly depe nding on the order of the deriv a tive. See [BH08]. W e may conside r the algebr a A ∗ ( M ) b ⊗ C ∞ ( M ) C ∞ c ( G ; ∧ F s ⊗ E ) as a subspace of the space of A ∗ ( M )-equiv ar iant endomorphisms of C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ) b y using the A ∗ ( M ) mo dule structure of C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ), where for φ ∈ C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ), and ω ∈ A ∗ ( M ), we set ω · φ = s ∗ ( ω ) ∧ φ . The op erator ∂ ν maps A ∗ ( M ) b ⊗ C ∞ ( M ) C ∞ c ( G ; ∧ F s ⊗ E ) to itself. Denote by C ∞ ( G ; ∧ F s ⊗ E ) the space of all smo oth sections ov er G o f ∧ F s ⊗ E . F o r T an element of A ∗ ( M ) b ⊗ C ∞ ( M ) C ∞ ( G ; ∧ F s ⊗ E ), deﬁne the trace of T to be the Haeﬂiger form T r( T ) given by T r ( T ) = Z F tr( T ( x )) dx = Z F i ∗ (tr( T | i ( M ))) dx, where x is the class o f the constant pa th at x , tr ( T ( x )) is the A ∗ ( M )-equiv ar iant trace of the Sch wartz kernel of T at x , a nd so b elongs to ∧ T ∗ M x , and dx is the leafwise volume form asso cia ted with the ﬁxed or ientation of the foliation F . When restricted to the subspa ce A ∗ ( M ) b ⊗ C ∞ ( M ) C ∞ c ( G ; ∧ F s ⊗ E ), the map T r : A ∗ ( M ) b ⊗ C ∞ ( M ) C ∞ c ( G ; ∧ F s ⊗ E ) − → A ∗ c ( M / F ) is a g r aded trace which satisﬁes T r ◦ ∂ ν = d H ◦ T r , see [BH04], and Lemma 6.3 of [BH08]. Moreover, the equality (T r ◦ ∂ ν )( T ) = ( d H ◦ T r)( T ) extends to all tr ansversely smo oth op erator s T . See Deﬁnition 3 .2 b elow and [BH08]. Since ∂ 2 ν is not necessarily zero , we used Connes’ X − trick to cons truct a new graded diﬀerential a lgebra ( e B , δ ) out of the graded quasi-diﬀerential alg ebra B = ( A ∗ ( M ) b ⊗ C ∞ ( M ) C ∞ c ( G ; ∧ F s ⊗ E ) , ∂ ν ). See [C94], p. 229 for the deﬁnition o f the gr ading, the extension o f ∂ ν to the diﬀerential δ , and the pro duct structure on e B . As a vector s pace, e B is M 2 ( B ), the space of 2 × 2 ma trices with co e ﬃcie nts in B , and B embeds as a subalgebra of e B b y using the ma p T ֒ →  T 0 0 0  . F o r homog eneous e T ∈ e B of degree k , Connes deﬁnes Φ( e T ) = T r( T 11 ) − ( − 1) k T r ( T 22 θ ν ) , 8 M.-T. BENAMEUR AND J. L. HEITSCH and extends to a rbitrary e lement s b y linea rity . The map Φ : e B → A ∗ c ( M / F ) is then a gr aded tra c e, and again w e ha ve Φ ◦ δ = d H ◦ Φ, see [BH04]. The (algebraic) Chern-Connes c ha racter in the even c ase is then the morphism ch a : K 0 ( C ∞ c ( G ; ∧ F s ⊗ E )) − → H ∗ c ( M / F ) deﬁned as follows. Let B = [ ˜ e 1 ] − [ ˜ e 2 ] b e an element of K 0 ( C ∞ c ( G ; ∧ F s ⊗ E )), where ˜ e 1 = ( e 1 , λ 1 ) a nd ˜ e 2 = ( e 2 , λ 2 ). The λ i are N × N matrices of complex n um ber s, and the e i are in M N ( C ∞ c ( G ; ∧ F s ⊗ E )), the N × N matrices ov er C ∞ c ( G ; ∧ F s ⊗ E ), which w e may consider as elements of M N ( e B ). Denote by tr : M N ( e B ) → e B the usual tr ace. Then the Haeﬂiger form (Φ ◦ tr)  e 1 exp  − ( δ e 1 ) 2 2 iπ   − (Φ ◦ tr)  e 2 exp  − ( δ e 2 ) 2 2 iπ   is closed and its Haeﬂiger cohomology class depe nds o nly on B , [BH0 4]. This Hae ﬂig er cohomology class is precisely the Chern-Connes c haracter of B . So, 3.1. ch a ( B ) =  (Φ ◦ tr)  e 1 exp  − ( δ e 1 ) 2 2 iπ   − (Φ ◦ tr)  e 2 exp  − ( δ e 2 ) 2 2 iπ    . W e wan t to co nsider the Chern-Connes characters o f idemp otents, such as the pr o jectio n onto the twisted leafwise ha rmonic for ms, which in g eneral do not deﬁne elemen ts of K 0 ( C ∞ c ( G ; ∧ F s ⊗ E )). The idempotents we a re in terested in are b ounded leafwise smo othing op erato r s on ∧ T ∗ F s ⊗ E . In order to deﬁne the Chern-Connes character of such idemp otents, we need the concept of “transverse smo othness” for A ∗ ( M ) equiv aria nt b ounded leafwise s mo othing op era tors on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E . If H is such an op erator, w e can write it as H = H [0] + H [1] + · · · + H [ n ] , where H [ k ] is homogeneous of degree k , that is, for all j , H [ k ] : C ∞ ( ∧ j ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ) → C ∞ ( ∧ j + k ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ) . Recall that for any [ γ ] ∈ G , s ∗ : ν s, [ γ ] → T M s ( γ ) is an isomor phis m. Th us any X ∈ C ∞ ( ∧ k T M ) deﬁnes a section, denoted b X , of ∧ k ν s . F or such X , i b X H [ k ] is a b o unded leafwise smo o thing op erator on ∧ T ∗ F s ⊗ E . F o r any v e c to r ﬁeld Y on M , set ∂ Y ν ( i X H [ k ] ) = i b Y ( ∂ ν ( i b X H [ k ] )) , which (if it exists) is an o p erator on ∧ T ∗ F s ⊗ E . Deﬁnition 3.2. An A ∗ ( M ) e quivaria nt b ounde d le afwise smo othing op er ator H on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E is tr ansversely s m o oth pr ovide d that for any X ∈ C ∞ ( ∧ k T M ) , and any ve ctor ﬁelds Y 1 , ..., Y m on M , the op er ator ∂ Y 1 ν ...∂ Y m ν ( i X H [ k ] ) is a b ounde d le afwise smo othing op er ator on ∧ T ∗ F s ⊗ E . An y element of C ∞ c ( G ; ∧ F s ⊗ E ) is transversely smo o th. If the le a fwise parallel tr anslation along E is a bo unded map, then the pro jection ont o the leafwise harmo nic forms with coeﬃcients in E (for the foliation F s ) is trans versely smo o th. See Theorem 4.4 b elow. Since ∂ ν is a deriv a tio n, it is immediate that the comp osition of transversely smo oth op era tors is tr ansversely smo oth. It is also ea sy to prove that the Sch wartz kernel of any transversely s mo oth op erator is a smo oth section in all v aria bles, se e [BH08]. If K is a b ounded leafwise smoothing op er a tor on ∧ T ∗ F s ⊗ E , we may extend it to an A ∗ ( M ) equiv ar iant bo unded leafwise smo o thing op erator on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E by using the A ∗ ( M ) mo dule struc tur e of C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ). Mo re speciﬁc a lly , given φ ∈ C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ), write it as φ = X j s ∗ ( ω j ) ⊗ φ j , TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 9 where the ω j ∈ A ∗ ( M ), and the φ j ∈ C ∞ ( ∧ T ∗ F s ⊗ E ). The n K ( φ ) = X j s ∗ ( ω j ) ⊗ K ( φ j ) . It is easy to check tha t this is w ell deﬁned. The proo f o f Lemma 4.5 o f [BH08] ex tends easily to giv e the following. Lemma 3.3. Su pp ose t hat A is an A ∗ ( M ) -e quivariant le afwise di ﬀer ential op er ator of ﬁnite or der on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E , and t hat the derivatives of al l or ders of its c o eﬃcients ar e u niformly b ounde d, with the b ou n d p ossibly dep ending on the or der of the derivative. Supp ose that K is a b ounde d le afwise smo othing op er ator on ∧ T ∗ F s ⊗ E , and ex tend it to an A ∗ ( M ) -e quivariant b ounde d le afwise sm o othing op er ator on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E . Then AK and K A ar e A ∗ ( M ) -e quivariant b ounde d le afwise smo othing op er ators on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E . If K is tra nsversely smo oth, so ar e AK and K A . F o r the conv enience of the r eader, we recall wha t the conditions on A mean. W rite A = P n 0 A [ k ] , wher e A [ k ] is homog eneous of degree k . Let X ∈ C ∞ ( ∧ k T M ) b e a loca l section of nor m one, and consider i b X A [ k ] which is a diﬀerential op erator (sa y of order d ) on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E . Let ( U, γ , V ) b e a basic op en set for G where U, V ∈ U , the ﬁxed go o d co ver. Then U and V come w ith ﬁxed co or dinates x 1 , ..., x p , w 1 , ..., w q and y 1 , ..., y p , z 1 , ..., z q . The x i and y i are the leaf co ordina tes for F , and the w i and z i are the normal co ordinates. The co o rdinates for ( U, γ , V ) a re then x 1 , ..., x p , y 1 , ..., y p , z 1 , ..., z q , and the y 1 , ..., y p are the le a f co ordinates for F s . With res pe ct to an orthonorma l basis o f ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E on ( U, γ , V ), i b X A [ k ] may b e written as a matrix o f oper ators of the for m d X | α | =0 a α ( x, y , z ) ∂ | α | ∂ y α 1 1 ...∂ y α p p , where the a α are lo cally deﬁned smo oth functions . Then each deriv a tive of the a α with re sp ect to the v ariables x, y, and z is assumed to b e globally bo unded ov er all basic op en sets for G , and the b ound may depe nd o n how man y der iv atives are taken. Note that o p erators A which are the pull backs of op erators on M , such as θ ν and τ , sa tisfy the h yp othesis of Lemma 3.3. Using L e mma 3.3, it is easy to show that being transversely smo o th is independent of the choice of ∇ ν . Finally , we need the concept o f G inv ariant A ∗ ( M )-equiv ar iant op erator s. Supp ose that H = H [0] + H [1] + · · · + H [ n ] is an A ∗ ( M )-equiv ar iant bounded lea fwise smo othing op erator acting on the sections of ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E . Then H is G inv ariant pr ovided it satisﬁes tw o require men ts. (1) F or any X = X 1 ∧ · · · ∧ X k ∈ C ∞ ( ∧ k T M ) with some X j ∈ C ∞ ( T F ), i b X H [ k ] = 0 . This means that H [ k ] deﬁnes a n op era tor H [ k ] : C ∞ ( ∧ j ν ∗ G ⊗ ∧ T ∗ F s ⊗ E ) → C ∞ ( ∧ j + k ν ∗ G ⊗ ∧ T ∗ F s ⊗ E ) , and that for k > q , H [ k ] = 0. Each γ ∈ e L y x ≡ e L x ∩ e L y , deﬁnes an action W γ : C ∞ ( e L x , ∧ ∗ ν ∗ G ⊗ ∧ T ∗ F s ⊗ E ) → C ∞ ( e L y , ∧ ∗ ν ∗ G ⊗ ∧ T ∗ F s ⊗ E ) , given by [ W γ ξ ]( γ ′ ) = ξ ( γ ′ γ ) , γ ′ ∈ e L y . Let y ′ = r ( γ ′ ), and no te that [ W γ ξ ]( γ ′ ) ∈ ( ∧ ∗ ν ∗ G ⊗ ∧ T ∗ F s ⊗ E ) γ ′ , which we iden tify with ∧ ∗ ν ∗ y ⊗ ( ∧ T ∗ F ⊗ E ) y ′ , while ξ ( γ ′ γ ) ∈ ( ∧ ∗ ν ∗ G ⊗ ∧ T ∗ F s ⊗ E ) γ ′ γ , whic h w e identify with ∧ ∗ ν ∗ x ⊗ ( ∧ T ∗ F ⊗ E ) y ′ . T o eﬀect this action, we identify ∧ ∗ ν ∗ x with ∧ ∗ ν ∗ y using the holonomy a long γ . The second require ment of H is: (2) F or any γ ∈ e L y x , ( γ · H ) y ≡ W γ ◦ H x ◦ W − 1 γ = H y , where H x is the action o f H on ∧ ∗ ν ∗ G ⊗ ∧ T ∗ F s ⊗ E | e L x . Essentially then, H is G inv ariant means that it deﬁnes the s ame op er ator on each e L ⊂ s − 1 ( L ) for each leaf L of F . Note that ∂ ν preserves G in v ariant A ∗ ( M )-equiv ar iant tr ansversely smo o th opera tors. In [BH08], we extended our Chern-Connes character to G inv ariant transversely smo o th idempo tent s. The essential res ult ne e ded w as L e mma 4.13 of that pap er, which we state for further re fer ence. 10 M.-T. BENAMEUR AND J. L. HEITSCH Lemma 3.4. Supp ose that H and K ar e G invariant A ∗ ( M ) -e quivariant t r ansversely smo oth op er ators acting on the se ctions of ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E . Then (Φ ◦ tr )([ H, K ]) = 0 . Lemma 3.4 and the pr o of of Theo rem 4.1 of [BH04] immediately imply Theorem 3.5 . L et e b e a G invaria nt tra nsversely smo oth idemp otent acting on the se ctions of ∧ T ∗ F s ⊗ E . Then (Φ ◦ tr)  e exp  − ( δ e ) 2 2 iπ   is a close d Haeﬂiger form whose Haeﬂiger c ohomolo gy class, denote d ch a ( e ) , dep ends only on e . In addi tion, if e t , 0 ≤ t ≤ 1 , is a smo oth family of such idemp otent s, then ch a ( e 0 ) = ch a ( e 1 ) . The Haeﬂiger class ch a ( e ) is the Chern-Connes c ha racter of e . Lemma 3.6. Two G invariant t ra nsversely smo oth idemp otents which have the same image, have t he same Chern-Connes char acter. Pr o of. Supp os e that e 0 and e 1 are tw o such idemp otents. T he n e 0 ◦ e 1 = e 1 and e 1 ◦ e 0 = e 0 , and the family e t = te 1 + (1 − t ) e 0 is a smoo th family o f G inv ariant transversely s mo oth idempo tent s connecting e 0 to e 1 . Theorem 3.5 then giv es the res ult.  4. The twisted higher harmo nic signa ture W e now deﬁne the twisted higher harmonic sig nature σ ( F , E ). The leafwise de Rha m diﬀerential on G extends to a closed o per ator on A ∗ (2) ( F s , C ) which coincides with the lifted one fro m the foliation ( M , F ) a nd it is denoted by d s . The leafwise formal adjoint of d s with resp ect to the Hilb ert s tr ucture is well deﬁned and is denoted by δ s , and δ s = − ∗ d s ∗ . Denote b y ∆ the Laplacian given by ∆ = ( d s + δ s ) 2 = d s δ s + δ s d s , and denote by ∆ k its action on A k (2) ( F s , C ). The leafwise ∗ op era tor also gives the lea fwise inv olution τ on A ∗ (2) ( F s , C ), where as usual, τ = √ − 1 k ( k − 1)+ ℓ ∗ on A k (2) ( F s , C ), a nd it is easy to c hec k that δ s = − τ d s τ , so τ ( d s + δ s ) = − ( d s + δ s ) τ , and ∆ τ = τ ∆. These o p erators extend to A ∗ (2) ( F s , E ) as follows. Since the o p er ators are all leafwise, lo ca l and linear, we need o nly deﬁne them for lo cal sectio ns of the for m α ⊗ φ where α is a lo cal k form o n e L , and φ is a loca l section of E | e L . Then d s ( α ⊗ φ ) = d s α ⊗ φ + ( − 1) k α ∧ ∇ e L E φ, b ∗ ( α ⊗ φ ) = ∗ α ⊗ γ φ, b τ ( α ⊗ φ ) = τ α ⊗ γ φ, where ∇ e L E is ∇ E restricted to e L , s o ∇ e L E φ is a lo c al section of T ∗ e L ⊗ E . W e deﬁne the wedge pr o duct α ∧ ∇ e L E φ (as a lo cal s e ction of ∧ k +1 T ∗ e L ⊗ E ) in the ob vious w ay . Lemma 4.1 . We have δ s = − b ∗ d s b ∗ = − b τ d s b τ . Note that d 2 s = 0, so also δ 2 s = 0 since b ∗ 2 = ± 1. Pr o of. C o nsider t w o sections α 1 ⊗ φ 1 and α 2 ⊗ φ 2 , and set Q ( α 1 ⊗ φ 1 , α 2 ⊗ φ 2 )( x ) = Z e L x { φ 1 , φ 2 } α 1 ∧ α 2 , (and extend to all of A ∗ (2) ( F s , E ) by linea rity). Then < α 1 ⊗ φ 1 , α 2 ⊗ φ 2 > = Q ( α 1 ⊗ φ 1 , b ∗ ( α 2 ⊗ φ 2 )) . TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 11 Now suppo se that α 1 is a lo ca l k − 1 fo r m o n e L , α 2 is a lo cal k form on e L , a nd φ 1 is ﬂa t. If φ 2 is a n ar bitrary section of E , set { φ 1 , α 2 ⊗ φ 2 } = α 2 { φ 1 , φ 2 } . (Note that α 2 is C v alued). As {· , ·} is preser ved b y the ﬂat structure and φ 1 is ﬂa t, it follows that on e L , ∇ e L E { φ 1 , φ 2 } = { φ 1 , ∇ e L E φ 2 } . Acting on functions on e L , d s = ∇ e L E , so d s { φ 1 , φ 2 } = { φ 1 , ∇ e L E φ 2 } . Then < d s ( α 1 ⊗ φ 1 ) , α 2 ⊗ φ 2 > = Z e L x { φ 1 , γ φ 2 } d s α 1 ∧ ∗ α 2 = ( − 1 ) k Z e L x α 1 ∧ d s ( { φ 1 , γ φ 2 } ∗ α 2 ) , while < α 1 ⊗ φ 1 , − b ∗ d s b ∗ ( α 2 ⊗ φ 2 ) > = ( − 1) Q ( α 1 ⊗ φ 1 , b ∗ 2 d s b ∗ ( α 2 ⊗ φ 2 )) = ( − 1) k Q ( α 1 ⊗ φ 1 , d s b ∗ ( α 2 ⊗ φ 2 )) = ( − 1) k Q ( α 1 ⊗ φ 1 , ( d s ∗ α 2 ) ⊗ γ φ 2 + ( − 1) k ∗ α 2 ∧ ∇ e L E γ φ 2 ) = ( − 1) k Z e L x α 1 ∧ ( d s ∗ α 2 ) { φ 1 , γ φ 2 } + ( − 1) k α 1 ∧ ∗ α 2 ∧ { φ 1 , ∇ e L E γ φ 2 } = ( − 1) k Z e L x α 1 ∧ d s ( ∗ α 2 { φ 1 , γ φ 2 } ) = ( − 1) k Z e L x α 1 ∧ d s ( { φ 1 , γ φ 2 } ∗ α 2 ) .  Denote by ∆ E the La placian given by ∆ E = ( d s + δ s ) 2 = d s δ s + δ s d s , and denote by ∆ E k its action on A k (2) ( F s , E ). Note that b τ is still an in v olution ev en at the bundle lev el, and that b τ ( d s + δ s ) = − ( d s + δ s ) b τ and ∆ E b τ = b τ ∆ E still hold. As usua l, the space of twisted har monic forms Ker(∆ E ) is rela ted to the leafwise co homology o f the t wisted forms. The s pace of closed L 2 forms in A ∗ (2) ( F s , E ) is deno ted by Z ∗ (2) ( F s , E ) a nd it is a Hilbe r t subspace. The space of exa ct L 2 forms in A ∗ (2) ( F s , E ) is denoted b y B ∗ (2) ( F s , E ), and we denote its closure by B ∗ (2) ( F s , E ). W e deno te b y H ∗ (2) ( F s , E ) the lea fwise reduced twisted L 2 cohomolog y o f the foliation, that is H ∗ (2) ( F s , E ) = Z ∗ (2) ( F s , E ) / B ∗ (2) ( F s , E ) . Here is a well kno wn Hodge result that we state for further use. See the Appendix o f [HL9 0] Lemma 4.2. The ﬁeld Ker(∆ E k ) is a subﬁeld of Z k (2) ( F s , E ) , and Z k (2) ( F s , E ) = K er(∆ E k ) ⊕ B k (2) ( F s , E ) . Thus the natur al pr oje ction Z k (2) ( F s , E ) → H k (2) ( F s , E ) induc es by r estr iction an isomorphism Ker(∆ E k ) ≃ H k (2) ( F s , E ) . In addition A ∗ (2) ( F s , E ) = Ker( d s + ( d s ) ∗ ) ⊕ Im( d s ) ⊕ Im( δ s ) . That is, for e ach x ∈ M , L 2 ( e L x ; ∧ T ∗ e L x ⊗ ( E | e L x )) = Ker( d x s + δ x s ) ⊕ Im( d x s ) ⊕ Im( δ x s ) . W e assume that the pro jection P ℓ onto K er(∆ E ℓ ) is transversely smo o th. It is a cla s sical result that this pro jection is a b ounded leafwise smo othing op er ator, so what we are really assuming is a form of smo othness in transverse dir ections. This condition holds in man y impo rtant cas es, see the comments b elow a fter the statement of Theorem 4.4. Denote by A ∗ ± ( F s , E ) the ± 1 eigenspaces of b τ , and by Ker(∆ E ± ℓ ) the in tersections A ∗ ± ( F s , E ) ∩ Ker(∆ E ℓ ). Denote by π ± = 1 2 ( P ℓ ± b τ ◦ P ℓ ), and note that these are the pro jections onto Ker(∆ E ± ℓ ), resp ectively . Since the ope rator b τ sa tis ﬁe s the hypothesis of Lemma 3.3, b oth π ± are tr ansversely smoo th, and their Chern-Connes c ha racters c h a ( π ± ) are w ell deﬁned Hae ﬂiger cohomology classes. Deﬁnition 4.3 . Supp ose that the pr oje ction P ℓ onto Ker(∆ E ℓ ) is tr ansversely smo oth. The higher twiste d harmonic signatur e σ ( F , E ) is the diﬀer enc e σ ( F, E ) = ch a ( π + ) − c h a ( π − ) . 12 M.-T. BENAMEUR AND J. L. HEITSCH T o justify our cla im that our as sumption of transverse smo othness for P ℓ holds in imp o rtant cases, we hav e the follo wing whic h is a n extension o f a result due to Gong and Rothenberg, [GR97]. Theorem 4.4. If the le afwise p ar al lel tr anslation along E is a b oun de d map, then the pr oje ction P onto Ker(∆ E ) is tr ansversely smo oth. The conclusion of Gong-Rothenberg is that the Sch wartz k er nel o f P is smo o th in a ll its v aria bles. F o r Riemannian foliations, P is alwa ys transversely smo oth for the class ical sig nature op er ator (that is, with co eﬃcients in the trivial one dimensiona l bundle) using either the holonomy or the homotopy g roup oid. P is transversely smo o th whenever the preserved metric on E is po sitive deﬁnite. It is smo oth in impo rtant examples, e.g. Lusztig [Lu72]. If the lea fwise parallel transla tion along E is a b ounded map, P is also transversely smo oth using the holonomy g roup oid, provided that the ﬂat structure o n E ov er each holonomy cov er ing ha s no holonomy (so using the ﬂat structure to translate a frame of a sing le ﬁb er o f E | e L to all of e L trivializes E | e L ). It is an op en question whether the pro jection to the leafwise har monic forms has tr ansversely smo o th Sch wartz kernel when F is not Riemannian. It is satisﬁed for all foliatio ns with compact leav es and Hausdorﬀ group oid [EMS76, Ep76]. Since the pap er [GR97] has no t bee n publishe d, we g ive their pro of here that P dep ends smo othly on x ∈ M , and then show how to g et transverse smo othness from it. Pr o of. L e t U ⊂ M b e a foliation chart and choos e x 0 ∈ U . Then there a diﬀeomor phism ϕ U : U × e L x 0 ≃ s − 1 ( U ), a nd a bundle isomorphism ψ U : U × ( E | e L x 0 ) ≃ E | s − 1 ( U ), covering ϕ U and preser ving the leafwise ﬂat structure. They a re constructed as follo ws. The normal bundle ν s = T F r ⊕ ν G ≃ s ∗ ( T M ) deﬁnes a lo c al transverse translation for the leaves of the foliation F s . See [Hu93, W83]. W e may as sume that U is the diﬀeomorphic imag e under exp x 0 of a neigh bo rho o d b U of 0 ∈ T M x 0 . Then for all x ∈ U , there is a unique X ∈ b U so that x = exp x 0 ( X ). Deﬁne γ x : [0 , 1] → M to b e γ x ( t ) = ex p x 0 ( tX ). Given x suﬃcient ly close to x 0 , for an y z ∈ e L x 0 there is a unique path b γ z ( t ) in G so that b γ z (0) = z , b γ z ( t ) ∈ e L γ x ( t ) , and b γ ′ z ( t ) ∈ ( ν s ) b γ z ( t ) . The tra nsverse transla te Φ x ( z ) of z to e L x is just b γ z (1). Φ x is a smooth diﬀeomorphism fro m e L x 0 to e L x , and we set ϕ U ( x, z ) = Φ x ( z ), which is a smo oth diﬀeomor phism from U × e L x 0 to s − 1 ( U ). Since we are using the ho motopy group oid, each e L is simply co nnected, so E | e L x is a trivial bundle for each x ∈ M , a nd using the ﬂat s tructure to transla te a frame of a single ﬁb er of E | e L x to a ll of e L x trivializes E | e L x . Cho ose a lo cal orthonorma l framing e 1 , ..., e k of E | U (on M ). This framing is a lso a lo cal framing of E | i ( U ) (o n G ). Using the leafwise ﬂat structure of E to tra nslate it a long the e L , we get a leafwise ﬂat framing e s 1 , ..., e s k of E | s − 1 ( U ). F or ( x, P j a j e s j ( z )) ∈ U × ( E | e L x 0 ), set ψ U ( x, X j a j e s j ( z )) = X j a j e s j ( ϕ U ( x, z )) . That is, the image of φ ∈ E z (where z ∈ e L x 0 ) under ψ U ( x, · ) is obtained by ﬁr st par allel transla ting φ along e L x 0 to E i ( x 0 ) , obtaining P j a j e j ( i ( x 0 )), and then parallel tr anslating P j a j e j ( i ( x )) along e L x to E ϕ U ( x,z ) . It is clear that ψ U cov er s ϕ U and preserves the leafwise ﬂat structure. There is a natura lly deﬁned bundle ma p Ψ U ( x ) : ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 ) → ∧ T ∗ e L x ⊗ ( E | e L x ) for each x ∈ U , which on a deco mpo sable elemen t α ⊗ φ ∈ ( ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) z is giv en b y Ψ U ( x )( α ⊗ φ ) = ((Φ − 1 x ) ∗ α ) ⊗ ψ U ( x, φ ) . W e als o denote by Ψ U the induced map Ψ U ( x ) : C ∞ c ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) → C ∞ c ( e L x ; ∧ T ∗ e L x ⊗ ( E | e L x )) . Ψ U ( x ) is inv er tible, commutes with the extended de Rham op era to rs, and dep ends smo o thly on x . Note that Φ − 1 x is a diﬀeomorphism of uniformly bounded dilation (as is Φ x ). If the le a fwise parallel translatio n along TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 13 E is a b o unded map, then the map ψ U is a b ounded map, a nd Ψ U extends to the following commutativ e diagram, L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) ❄ L 2 ( e L x ; ∧ T ∗ e L x ⊗ ( E | e L x )) Ψ U ( x ) d x 0 s ✲ ✲ d x s L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) ❄ L 2 ( e L x ; ∧ T ∗ e L x ⊗ ( E | e L x )) . Ψ U ( x ) 4.5. So Ψ U ( x )(Ker( d x 0 s )) ⊂ K er( d x s ) and Ψ U ( x )( Im( d x 0 s )) ⊂ Im( d x s ) . By Lemma 4.2, we have L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) = Ker( d x 0 s + δ x 0 s ) ⊕ Im( d x 0 s ) ⊕ Im( δ x 0 s ) , and L 2 ( e L x ; ∧ T ∗ e L x ⊗ ( E | e L x )) = Ker( d x s + δ x s ) ⊕ Im( d x s ) ⊕ Im( δ x s ) . With respe ct to these decompositio ns , w e may wr ite Ψ U ( x ) =       Ψ 11 ( x ) 0 Ψ 13 ( x ) Ψ 21 ( x ) Ψ 22 ( x ) Ψ 23 ( x ) 0 0 Ψ 33 ( x )       . It follows immediately that Ψ 22 ( x ) : Im( d x 0 s ) → Im( d x s ) is an in vertible ma p which depends smo othly on x . Let R x 0 : L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) → Im( d x 0 s ) be the ortho g onal pro jection. Deﬁne e R x = Ψ 22 ( x ) R x 0 Ψ − 1 U ( x ) , which equals Ψ U ( x ) R x 0 Ψ − 1 U ( x ) , since Ψ 22 ( x ) R x 0 = Ψ U ( x ) R x 0 . The n e R x is a n idemp otent which v ar ies smo othly in x , a nd has imag e Im( d x s ). How ever, it might not b e an orthogo nal pro jection. Set Q x = I + ( e R x − e R ∗ x )( e R ∗ x − e R x ) . Then Q x is an in v ertible self adjoin t o per ator whic h depends smo o thly on x , a nd the orthogonal pr o jection R x : L 2 ( e L x ; ∧ T ∗ e L x ⊗ ( E | e L x )) → Im( d x s ) is just R x = e R x e R ∗ x Q − 1 x , so R x depe nds s mo othly on x . Let τ x be the Ho dg e type op era tor such that δ x s = ± τ − 1 x b d x s τ x , where b d x s is the diﬀerential asso cia ted with the antidual bundle E ∗ of E . The op erato r τ − 1 x maps Im( b d x s ) onto Im( δ x s ). Set b S x = τ − 1 x e S x τ x , wher e e S x is the op er ator for b d x s corres p o nding to the op erato r e R x for d x s . The a rgument above, with E replaced b y its antidual E ∗ , shows that e S x , so also b S x , is an idempotent dep ending smoothly o n x . No te that b S x has image Im( δ x s ). As above, w e get that the or thogonal pro jection S x : L 2 ( e L x ; ∧ T ∗ e L x ⊗ ( E | e L x )) → Im( δ x s ) dep ends smo othly on x . Thus the orthogo nal pr o jection P = I − ( R x + S x ) depends smo o thly on x . W e now show that P is transversely smoo th. T o do this, we view everything on U × e L x 0 , using ϕ U , and ψ U . Thanks to Diagr a m 4.5, we are reduced to considering the op erato r d x 0 s : L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) → L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) acting ov e r each p oint x ∈ U , that is the twisted leafwise de Rham o pe rator on the fo liation U × e L x 0 . W e use φ U and ψ U to pull ba ck the s tr uctures on s − 1 ( U ) and w e use the s ame notation to deno te these pull ba cks. In particular, we have the connectio n ∇ and the no rmal bundle ν s used to deﬁne ∂ ν . The leafwise pro jection P x onto the twisted leafwise harmonics depends on the leafwise metrics on e L x 0 and E | e L x 0 , whic h v a ry with x ∈ U . First we prov e that we ma y assume that the normal bundle ν s is the bundle T U ⊂ T ( U × e L x 0 ). Deno te the op erato r corresp onding to ∂ ν constructed using T U by ∂ U . Given a (bounded) vector ﬁeld Y on U , w e 14 M.-T. BENAMEUR AND J. L. HEITSCH hav e t wo lifts, b Y to ν s and b Y 0 to T U . The diﬀerence b Y − b Y 0 is tang ent to the ﬁb ers e L x 0 , so the diﬀerence of the ope r ators ∂ Y ν − ∂ Y U = [ ∇ b Y − ∇ b Y 0 , · ] = [ ∇ b Y − b Y 0 , · ] is the commutator with a leafwise diﬀeren tial op erator of order one, whose co eﬃcients and a ll their deriv atives are uniformly bo unded, with the b ound p ossibly depe nding on the or der of the deriv a tive. F or s ∈ Z , we denote b y W s = W ∗ s ( e L x 0 , E ) the usual s -th Sob ole v space which is the completion of C ∞ c ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) under the usual s -th Sob ole v nor m. Then Υ( Y ) := ∇ b Y − ∇ b Y 0 deﬁnes a bo unded leafwise ope r ator from a ny W ∗ s to W ∗ s − 1 , and b oth Υ ( Y ) P x and P x Υ( Y ) are b ounded leafwise smo othing op erator s s ince P x is leafwise smo o thing. As ∂ Y ν P x = ∂ Y U P x + [Υ( Y ) , P x ] , ∂ Y ν P x is a bounded leafwise smo othing op erator if and only ∂ Y U P x is. Now assume tha t for all Y 1 , Y 2 , ∂ Y 1 U P x and ∂ Y 2 U ∂ Y 1 U P x are bo unded le a fwise smo o thing oper ators. Again using the fact that ∂ Y ν = ∂ Y U + [Υ( Y ) , · ], we hav e ∂ Y 2 ν ∂ Y 1 ν P x = ∂ Y 2 U ∂ Y 1 U P x + [ ∂ Y 2 U Υ( Y 1 ) , P x ] + [Υ( Y 1 ) , ∂ Y 2 U P x ] + [Υ( Y 2 ) , ∂ Y 1 U P x ] + [Υ( Y 2 ) , [Υ( Y 1 ) , P x ]] . which is a b ounded leafwise smo othing op er ator since ∂ Y 2 U Υ( Y 1 ) has the same prop erties a s Υ( Y 1 ), namely it is a leafwise diﬀerential oper ator of order one, whose co eﬃcie nts and all their deriv atives are uniformly bo unded, with the bo und p ossibly dep ending on the order of the deriv ative. As the a rguement is s ymmetric in ∂ ν and ∂ U , ∂ Y 1 ν P x and ∂ Y 2 ν ∂ Y 1 ν P x are b ounded leafwise smo othing op erator s if and only if ∂ Y 1 U P x and ∂ Y 2 U ∂ Y 1 U P x are. Con tinuing in this manner , we hav e that ∂ Y 1 ν P x , ∂ Y 2 ν ∂ Y 1 ν P x , ..., a nd ∂ Y m ν ...∂ Y 1 ν P x are bo unded leafwise smo othing o pe rators if a nd only if ∂ Y 1 U P x , ∂ Y 2 U ∂ Y 1 U P x , ..., and ∂ Y m U ...∂ Y 1 U P x are. Thus we may assume that ν s = T U . Next we show that we may use any co nnection w e pleas e, pr ovided it is in the s a me b ounded geometr y class as ∇ . Supp ose that ∂ 0 is ano ther der iv ation constructed from the connection ∇ 0 in the sa me b ounded geometry class a s ∇ . Then ∂ Y ν − ∂ Y 0 = [ ∇ ν Y − ∇ 0 ,ν Y , · ], a nd ∇ ν Y − ∇ 0 ,ν Y is a lea fwise diﬀerential o p er ator o f or der zero, whose co eﬃcients and all their deriv atives are uniformly bounded, with the b ound p ossibly depe nding on the or de r of the deriv ative. So ∇ ν Y − ∇ 0 ,ν Y deﬁnes a b ounded o p er ator from any Sob ole v space W s to itself. Pr o ceeding just as we did ab ov e, we hav e that ∂ Y 1 ν P x , ∂ Y 2 ν ∂ Y 1 ν P x , ..., and ∂ Y m ν ...∂ Y 1 ν P x are b o unded leafwise smo othing opera tors if and only if ∂ Y 1 0 P x , ∂ Y 2 0 ∂ Y 1 0 P x , ..., and ∂ Y m 0 ...∂ Y 1 0 P x are. Thu s, we are reduced to sho wing that ∂ Y m 0 ...∂ Y 1 0 ( P ) is a bounded leafwise smo othing op erator. The connectio n we choose is that pulled back from L x 0 under the obvious map U × e L x 0 → L x 0 . W e leav e it to the reader to show that this is in the same b ounded geometry class a s ∇ . Now w e can choo se co ordinates on U so w e ma y think of U = D n with co o rdinates, x 1 , ..., x n , and x 0 = (0 , ..., 0). When we do, ∂ ∂ /∂ x i m 0 ...∂ ∂ /∂ x i 1 0 P x = ∂ m P x /∂ x i m ...∂ x i 1 . Thu s w e a re r e duced to considering a smo oth fa mily o f smo o thing o p erators P x acting on the space of sections o f ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 ). The parameter x determines the metric g x we use on this space, and P x is the as so ciated pro jection o nto the twisted harmonic sections. Note that the asso ciated Sob olev spaces W ∗ s are the same for all the g x since these metrics a re all in the same bounded geo metry class. The norms on W ∗ s do dep end on the parameter x . Howev er they ar e all co mpa rable, so we may assume that we hav e a single norm || · || s on each W ∗ s , whic h is indep endent of x . Denote ∂ m /∂ x i m ...∂ x i 1 by ∂ m i m ...i 1 . W e need to prov e that for all s and k ≥ 0 , ∂ m i m ...i 1 P x deﬁnes a b ounded map from W ∗ s to W ∗ s + k . Given K : W ∗ s → W ∗ s + k , denote the s, s + k norm of K b y || K || s + k,s . Then || K || s + k,s = || (1 + ∆) ( s + k ) / 2 K (1 + ∆) − s/ 2 || 0 , 0 , where ∆ is the La placian a sso ciated to the metric o n ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 ). Since the norms asso ciated to diﬀerent metr ics ar e compara ble, we may use any metric g x with asso c iated Laplacian ∆ x we like. No w P x = (1 + ∆ x ) ( s + k ) / 2 P x (1 + ∆ x ) − s/ 2 , so ∂ i P x = ∂ i ((1 + ∆ x ) ( s + k ) / 2 P x (1 + ∆ x ) − s/ 2 ) = ∂ i (1 + ∆ x ) ( s + k ) / 2 P x (1 + ∆ x ) − s/ 2 + (1 + ∆ x ) ( s + k ) / 2 ∂ i P x (1 + ∆ x ) − s/ 2 + (1 + ∆ x ) ( s + k ) / 2 P x ∂ i (1 + ∆ x ) − s/ 2 , TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 15 which gives (1 + ∆ x ) ( s + k ) / 2 ∂ i P x (1 + ∆ x ) − s/ 2 = ∂ i P x − ∂ i (1 + ∆ x ) ( s + k ) / 2 P x (1 + ∆ x ) − s/ 2 − (1 + ∆ x ) ( s + k ) / 2 P x ∂ i (1 + ∆ x ) − s/ 2 . So, || ∂ i P x || s + k,s = || (1 + ∆ x ) ( s + k ) / 2 ∂ i P x (1 + ∆ x ) − s/ 2 || 0 , 0 = || ∂ i P x − ∂ i (1 + ∆ x ) ( s + k ) / 2 P x (1 + ∆ x ) − s/ 2 − (1 + ∆ x ) ( s + k ) / 2 P x ∂ i (1 + ∆ x ) − s/ 2 || 0 , 0 ≤ || ∂ i P x || 0 , 0 + || ∂ i (1 + ∆ x ) ( s + k ) / 2 P x (1 + ∆ x ) − s/ 2 || 0 , 0 + || (1 + ∆ x ) ( s + k ) / 2 P x ∂ i (1 + ∆ x ) − s/ 2 || 0 , 0 . Now for a ny r , (1 + ∆ x ) r / 2 and ∂ i (1 + ∆ x ) r / 2 are leafwise diﬀerential op er ators of o rder r , whose co eﬃcients and all their deriv atives ar e uniformly b ounded, with the b ound p oss ibly dep ending on the order o f the deriv ative, but indep endent of x . So they deﬁne b ounded op era tors fro m W ∗ s to W ∗ s − r , for any s , with b o und independent of x . Since P x is leafswise s mo othing, it deﬁnes a b ounded op er ator from any W ∗ r to any W ∗ s , whose bound is also indep endent of x , since || P x || s,r = || (1 + ∆ x ) − s/ 2 P x (1 + ∆ x ) − r / 2 || 0 , 0 = || P x || 0 , 0 ≤ 1 , Thu s w e ha ve || ∂ i (1 + ∆ x ) ( s + k ) / 2 P x (1 + ∆ x ) − s/ 2 || 0 , 0 ≤ || ∂ i (1 + ∆ x ) ( s + k ) / 2 || 0 ,s + k || P x || s + k, − s || (1 + ∆ x ) − s/ 2 || − s, 0 is b o unded independently of x . Similar ly || (1 + ∆ x ) ( s + k ) / 2 P x ∂ i (1 + ∆ x ) − s/ 2 || 0 , 0 is b o unded independently of x . Thus ∂ i P x : W ∗ s → W ∗ s + k is bounded if and only if ∂ i P x : W ∗ 0 → W ∗ 0 is. Now for an y m a nd and an y r , ∂ m i m ...i 1 (1 + ∆ x ) r / 2 is als o a leafwise diﬀerential op erator of order r , whose co eﬃcients and all their deriv ativ es a re uniformly bounded, with the b ound p oss ibly depending o n the or der of the deriv a tive, but indep endent of x . Using this fact, a straig ht forward induction argument shows that ∂ m i m ...i 1 P x : W ∗ s → W ∗ s + k is bounded if and only if ∂ m i m ...i 1 P x : W ∗ 0 → W ∗ 0 is. Now we hav e (working on W ∗ 0 = L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 ))) that P x = I − ( R x + S x ), where R x is the orthogo nal pr o jectio n R x : L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) → Im( d x 0 s ) ⊂ L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) obtained using the metric g x . At the p oint x , R x 0 also has image Im( d x 0 s ), but R x 0 might not b e an o rthogona l pro jection using the metric g x . As above R x is given by R x = R x 0 R ∗ x x 0 Q − 1 x , where Q x = I + ( R x 0 − R ∗ x x 0 )( R ∗ x x 0 − R x 0 ) . and R ∗ x x 0 is the adjoin t of R x 0 constructed using the metric g x . Since I = Q x Q − 1 x , w e have tha t 0 = ∂ i I = ∂ i ( Q x Q − 1 x ) = ( ∂ i Q x ) Q − 1 x + Q x ( ∂ i Q − 1 x ) . So ∂ i Q − 1 x = − Q − 1 x ( ∂ i Q x ) Q − 1 x , and a b o ot-stra pping argument shows that ∂ m i m ...i 1 ( Q − 1 x ) is bo unded if ∂ m i m ...i 1 Q x is. It follows that ∂ m i m ...x i 1 R x is bo unded if ∂ m i m ...i 1 R x 0 and ∂ m i m ...i 1 R ∗ x x 0 are bounded. As ∂ i R x 0 = 0 for all i , w e are r e duced to considering R ∗ x x 0 . W e may write the metric g x as g x ( u, v ) = g x 0 ( G x u, v ) where G x is a nonnegative s elf-adjoint (inv er tible) op erator with r esp ect to g x 0 , as is its in verse. Since g x is the pull back o f a family of metr ics deﬁned on the compact manifold M , G x is smoo th in all its v ariable s , and it and all its der iv atives a re unifor mly b ounded, and the s ame is true for the inv ers e G − 1 x . Thus for all m , both ∂ m i m ...i 1 G x and ∂ m i m ...i 1 G − 1 x deﬁne b ounded op erator s o n L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )) (s inc e they a re or der zero diﬀerential op erato rs). F or any b ounded op erator A on L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )), the adjoint of A with resp ect to g x is A ∗ x = G − 1 x A ∗ G x , 16 M.-T. BENAMEUR AND J. L. HEITSCH where A ∗ is the a djoint with resp ect to g x 0 . It follo ws immedia tely that fo r all m , ∂ m i m ...i 1 R ∗ x x 0 = ∂ m i m ...i 1 ( G − 1 x R ∗ x 0 G x ) is a b ounded op erato r on W ∗ 0 = L 2 ( e L x 0 ; ∧ T ∗ e L x 0 ⊗ ( E | e L x 0 )), since ∂ i R ∗ x 0 = 0 fo r all i . Thus for a ll m , ∂ m i m ...i 1 R x is a bounded op era tor on W ∗ 0 . It remains to show that for all m , ∂ m i m ...i 1 S x is a bounded o p er ator on W ∗ 0 . T o do this we may pr o ceed as we did ab ove, using the o pe r ators e S x and b S x . W e need only observe that the Ho dg e type op er a tor τ x has the same prop erties tha t G x do es. Th us for all m , ∂ m i m ...i 1 P x = − ( ∂ m i m ...i 1 R x + ∂ m i m ...i 1 S x ) is a bo unded op erator on W ∗ 0 , and w e co nclude that P x is transversely smo oth.  Prop ositi o n 4. 6. If P is tr ansversely smo oth, then the pr oje ctions onto A ∗ ± ( F s , E ) ∩ (Ker(∆ E k ) ⊕ K er(∆ E p − k )) , k 6 = ℓ , and Ker(∆ E ± ℓ ) ar e tr ansversely smo oth. Pr o of. Deno te by P k the pro jection onto Ker(∆ E k ). It is immediate that P is transversely smo oth if and o nly if all the P k are tra nsversely smo o th. F or k 6 = ℓ , the pro jection onto A ∗ ± ( F s , E ) ∩ (Ker(∆ E k ) ⊕ Ker(∆ E p − k )) is given by π ± k = P k ± τ ◦ P k , (since P k ◦ τ ◦ P k = 0 in those cases), a nd the pro jection onto K er(∆ E ± ℓ ) is giv en by π ± = 1 2 ( P ℓ ± τ ◦ P ℓ ). As the opera tor τ satisﬁes the hypo thesis o f Lemma 3 .3, and each P k is tr a nsversely smo oth, so is eac h τ ◦ P k , so all o f the pr o jections are als o transversely smo oth.  5. Connections, cur v a ture, and the Chern-Connes character W e now give an alternate construction of the Chern-Connes characters ch a ( π + ) a nd c h a ( π − ) us ing “con- nections” and “curv atures” deﬁned on “smoo th sub-bundles” of A ∗ (2) ( F s , E ). Deﬁnition 5.1. A smo oth subbun d le of A ∗ (2) ( F s , E ) over M /F is a G invariant tr ansversely smo oth idem- p otent π 0 acting on A ∗ (2) ( F s , E ) . Example 5.2. (1) Any idemp otent in the alg ebr a of su p er ex p e onential ly de c aying op er ators on ∧ T ∗ F s ⊗ E , deﬁne d in [BH08] , is a smo oth subbund le of A ∗ (2) ( F s , E ) over M /F . So, any smo oth c omp actly supp orte d idemp otent is a smo oth subbund le of A ∗ (2) ( F s , E ) over M /F . (2) The Wassermann idemp otent of the le afwise signatur e op er ator, as deﬁne d for instanc e in [BH08] , is a very imp ortant sp e cial c ase of (1) ab ove. In this c ase we take E = M × C . (3) A p ar adigm for such a smo oth subbund le is given by pr oje ction onto the kernel of a le afwise el liptic op er ator acting on A ∗ (2) ( F s , E ) (induc e d fr om a le afwise el liptic op er ator on F ). In p articular, the pr oje ctions π + and π − . Deﬁnition 5.3. The sp ac e C ∞ 2 ( ∧ T ∗ F s ⊗ E ) c onsists of al l elements ξ ∈ C ∞ ( G ; ∧ T ∗ F s ⊗ E ) ∩ A ∗ (2) ( F s , E ) such t hat for any quasi-c onne ction ∇ ν , and any ve ctor ﬁelds Y 1 , ..., Y m on M , ∇ ν Y 1 ... ∇ ν Y m ( ξ ) ∈ C ∞ ( G ; ∧ T ∗ F s ⊗ E ) ∩ A ∗ (2) ( F s , E ) , wher e ∇ ν Y i = i b Y i ∇ ν . Note that if ξ ∈ C ∞ ( G ; ∧ T ∗ F s ⊗ E ), ∇ ν Y 1 ... ∇ ν Y m ( ξ ) is automatica lly in C ∞ ( G ; ∧ T ∗ F s ⊗ E ), and that if ξ ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ), then ∇ ν Y 1 ... ∇ ν Y m ( ξ ) ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ). Note also that C ∞ c ( G ; ∧ T ∗ F s ⊗ E ) ⊂ C ∞ 2 ( ∧ T ∗ F s ⊗ E ). Prop ositi o n 5.4. If H is a tr ansversely smo oth op er ator on ∧ T ∗ F s ⊗ E , then H maps C ∞ 2 ( ∧ T ∗ F s ⊗ E ) to itself. Pr o of. L e t ξ ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ). As H is transversely smo o th, it follows easily that H ξ ∈ C ∞ ( G ; ∧ T ∗ F s ⊗ E ) ∩ A ∗ (2) ( F s , E ). Fix a quasi-co nnection ∇ ν , and let Y be a vector ﬁeld on M . Then ∇ ν Y ( H ξ ) = ∇ ν Y H ξ − H ∇ ν Y ξ + H ∇ ν Y ξ = ( ∂ Y ν H ) ξ + H ( ∇ ν Y ξ ) , which is in C ∞ ( G ; ∧ T ∗ F s ⊗ E ) ∩ A ∗ (2) ( F s , E ), s ince H and ∂ Y ν H are transversely smo o th, and ξ and ∇ ν Y ξ are in C ∞ 2 ( ∧ T ∗ F s ⊗ E ). An obvious inductio n argument now shows that H ξ ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ).  TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 17 Let π 0 be a smo oth subbundle of A ∗ (2) ( F s , E ) ov er M /F . Deﬁnition 5.5. A smo oth se ction of π 0 is an element ξ ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ) which satisﬁes π 0 ξ = ξ . The set of al l smo oth se ctions is denote d C ∞ ( π 0 ) . The space C ∞ ( π 0 ) is a C ∞ ( M ) mo dule, where ( f · ξ )([ γ ]) = f ( s ( γ )) ξ ([ γ ]). In addition, C ∞ ( π 0 ) = π 0 ( C ∞ 2 ( ∧ T ∗ F s ⊗ E )) ⊃ π 0 ( C ∞ c ( G ; ∧ T ∗ F s ⊗ E )). Deﬁnition 5.6. Denote by C ∞ ( ∧ T ∗ M ; π 0 ) the c ol le ction of al l sm o oth se ctions of ∧ T ∗ M with c o eﬃcients in C ∞ ( π 0 ) , and by C ∞ c ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ) the c ol le ction of al l smo oth se ctions of ∧ T ∗ M with c o eﬃcients in C ∞ c ( G ; ∧ T ∗ F s ⊗ E ) . There a r e natural ac tio ns of A ∗ ( M ) on C ∞ ( ∧ T ∗ M ; π 0 ) a nd C ∞ c ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ), and under these actions C ∞ ( ∧ T ∗ M ; π 0 ) ≃ A ∗ ( M ) ˆ ⊗ C ∞ ( M ) C ∞ ( π 0 ) , and C ∞ c ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ) ≃ A ∗ ( M ) ˆ ⊗ C ∞ ( M ) C ∞ c ( G ; ∧ T ∗ F s ⊗ E ) , with the rig ht completio ns . Thus π 0 : C ∞ c ( G ; ∧ T ∗ F s ⊗ E ) → C ∞ ( π 0 ) extends to the A ∗ ( M ) e q uiv ariant map π 0 : C ∞ c ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ) → C ∞ ( ∧ T ∗ M ; π 0 ) . A lo c al invariant element is a lo cal section ξ of A ∗ (2) ( F s , E ) deﬁned on an open subset U ⊂ M s o that for any le afwise path γ 1 in U , ξ ([ γ ]) = ξ ([ γ γ 1 ]) for all γ with s ( γ ) = r ( γ 1 ). Lo cal inv ariant element s are common. In par ticula r, any lo cally deﬁned element ξ ∈ A ∗ (2) ( F s , E ) deﬁnes lo cal inv ariant elements. Suppose that ξ is deﬁned o n a foliation chart U ⊂ M for F , a nd let P x be the placque in U containing the po int x . Given y ∈ P x , let γ y be a path in P x starting a t x and ending at y . Deﬁne e ξ y ∈ L 2 ( e L y ; ∧ T ∗ F s ⊗ E ) by e ξ y ([ γ ]) = ξ x ([ γ γ y ]). Then e ξ is a lo c al inv a riant element of A ∗ (2) ( F s , E ) deﬁned along P x . By re s tricting ξ to a tr a nsversal T in a foliatio n chart U and then extending inv ariantly to e ξ we obtain lo c al inv ariant elements of A ∗ (2) ( F s , E ) deﬁned ov e r U . One can of course extend this constr uction from c hart to c hart as far as one likes, for e x ample along any path γ : [0 , 1] → L in a le af L . If γ is a clo sed loo p, the section at 1 will not agree in general with the section at 0, so one does not in general obtain global in v a riant sections this w ay . Deﬁnition 5.7 . A c onne ct ion ∇ on π 0 is a line ar map ∇ : C ∞ ( ∧ T ∗ M ; π 0 ) → C ∞ ( ∧ T ∗ M ; π 0 ) of de gr e e one, so that (1) for ω ∈ A k ( M ) and ξ ∈ C ∞ ( π 0 ) , ∇ ( ω ⊗ ξ ) = d M ω ⊗ ξ + ( − 1) k ω ∧ ∇ ξ ; (2) for lo c al invariant ξ ∈ C ∞ ( π 0 ) , and X ∈ C ∞ ( T F ) , ∇ X ξ = 0 , i. e. ∇ is ﬂ at along F ; (3) ∇ is invariant under the right action of G ; (4) the le afwise op er ator ∇ π 0 − π 0 ∇ ν π 0 : C ∞ c ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ) → C ∞ ( ∧ T ∗ M ; π 0 ) is t r ansversely smo oth. The usual pr o of sho ws that since ∇ satisﬁes (1), it is lo cal in the sense that ∇ ξ ( x ) dep ends only on ξ | U where U is a ny op en set in M with x ∈ U . F o r ∇ to b e inv ariant under the right action of G means the following. Let γ b e a leafwise path in M from x = s ( γ ) to y = r ( γ ). Let ξ b e a lo cal in v ariant sectio n of π 0 deﬁned on a neighborho o d of the path γ . F or X ∈ ν x , w e may use the natural ﬂat structure on ν to parallel translate X to γ ∗ ( X ) ∈ ν y . Then w e require, ∇ X ξ = ( R γ ) − 1 ∇ γ ∗ ( X ) ξ = R γ − 1 ∇ γ ∗ ( X ) ξ , where the isomorphism R γ : L 2 ( e L s ( γ ) ; ∧ T ∗ F s ⊗ E ) → L 2 ( e L r ( γ ) ; ∧ T ∗ F s ⊗ E ) is given b y R γ ( ξ )([ γ 1 ]) = ξ [( γ 1 γ ]). Note that this condition do es not dep end on the choice of normal bundle ν b ecaus e the amb iguity involv es things of the form ∇ Y ξ where Y ∈ T F . But this is zero b ecause ∇ is ﬂa t along F . T o see that ∇ π 0 − π 0 ∇ ν π 0 is a leafwise opera tor, let ξ ∈ C ∞ c ( G ; ∧ T ∗ F s ⊗ E ), and ω ∈ A k ( M ) . Then ( ∇ π 0 − π 0 ∇ ν π 0 )( ω ⊗ ξ ) = π 0 ( ∇ − ∇ ν ) π 0 ( ω ⊗ ξ ) = π 0 ( ∇ − ∇ ν )( ω ⊗ π 0 ( ξ )) = 18 M.-T. BENAMEUR AND J. L. HEITSCH π 0  d M ω ⊗ π 0 ( ξ ) + ( − 1) k ω ∧ ∇ π 0 ( ξ ) − d M ω ⊗ π 0 ( ξ ) − ( − 1) k ω ∧ ∇ ν π 0 ( ξ )  = ( − 1) k π 0  ω ∧ ( ∇ − ∇ ν ) π 0 ( ξ )  = ( − 1) k ω ∧ ( ∇ π 0 − π 0 ∇ ν π 0 ) ξ , so ∇ π 0 − π 0 ∇ ν π 0 is a leafwise opera tor. Next w e show that C ∞ c ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ) is in the domain of π 0 ∇ ν π 0 . W e iden tify C ∞ c ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ) with the subspa c e C ∞ c ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E )) o f C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E )). Now ∂ ν ( π 0 ) = [ ∇ ν , π 0 ], and (b y assumption) it is transversely smo oth. Thus we ha v e ∇ ν π 0 = π 0 ∇ ν + ∂ ν ( π 0 ) , so π 0 ∇ ν π 0 = π 0 ∇ ν + π 0 ∂ ν ( π 0 ) . The domain of the op er a tor on the r ight c ontains C ∞ c ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ). Lemma 5.8 . π 0 ∇ ν is a c onne ction on π 0 . Pr o of. Since π 0 commutes with the action o f A ∗ ( M ) o n C ∞ ( ∧ T ∗ M ; π 0 ), to sho w that π 0 ∇ ν maps the s pace C ∞ ( ∧ T ∗ M ; π 0 ) to itself, we need only show that for any lo cal section ξ ∈ C ∞ ( π 0 ), and any lo cal vector ﬁeld X on M , ( π 0 ∇ ν ξ )( X ) = π 0 ( i b X ∇ ν ξ ) = π 0 ( ∇ ν X ξ ) is in C ∞ ( π 0 ), wher e b X is the lift of X to ν s . As ξ ∈ C ∞ ( π 0 ), it is in C ∞ 2 ( ∧ T ∗ F s ⊗ E ) a nd π 0 ( ξ ) = ξ , so ∇ ν X π 0 ξ = ∇ ν X ξ ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ). As π 0 is trans versely smo oth, i b X ∂ ν ( π 0 )( ξ ) ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ). Since π 0 ∇ ν = ∇ ν π 0 + ∂ ν ( π 0 ), we have ( π 0 ∇ ν ξ )( X ) ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ). Finally , as π 2 0 = π 0 , π 0 ( π 0 ( ∇ ν X ξ )) = π 0 ( ∇ ν X ξ ). Thus ( π 0 ∇ ν ξ )( X ) ∈ C ∞ ( π 0 ), and π 0 ∇ ν maps C ∞ ( ∧ T ∗ M ; π 0 ) to itself. The o pe r ator π 0 ∇ ν satisﬁes (1 ) beca use π 0 commutes with the actio n o f A ∗ ( M ) on C ∞ ( ∧ T ∗ M ; π 0 ). In particular, for ω ∈ A k ( M ) and ξ ∈ C ∞ ( π 0 ), w e ha v e π 0 ∇ ν ( s ∗ ω ⊗ ξ ) = π 0 ρ ν ( r ∗ ( ∇ F ⊗ ∇ E )( s ∗ ω ⊗ ξ )) = π 0 ρ ν  d G ( s ∗ ω ) ⊗ ξ + ( − 1) k s ∗ ω ∧ r ∗ ( ∇ F ⊗ ∇ E ) ξ  = π 0 ρ ν ( s ∗ d M ω ⊗ ξ ) + ( − 1) k π 0 ρ ν ( s ∗ ω ∧ r ∗ ( ∇ F ⊗ ∇ E ) ξ ) = s ∗ d M ω ⊗ π 0 ξ + ( − 1) k s ∗ ω ∧ π 0 ρ ν r ∗ ( ∇ F ⊗ ∇ E ) ξ = d M ω ⊗ ξ + ( − 1) k ω ∧ π 0 ∇ ν ξ . T o show that π 0 ∇ ν satisﬁes (2 ), let X ∈ T F x , ξ b e a lo cal inv ariant section of π 0 deﬁned nea r x , and [ γ ] ∈ e L x . The fact that ξ is inv ariant means that there is a section b ξ of ∧ T ∗ F ⊗ E deﬁned in a neighborho od of r ( γ ) so that ξ = r ∗ b ξ in a neighborho o d of [ γ ]. Recall that ν s = T F r ⊕ ν G . Since X ∈ T F x , b X ∈ T F r and r ∗ ( b X ) = 0 . Now π 0 ∇ ν X ξ = π 0 ( r ∗ ( ∇ F ⊗ ∇ E ) b X ( ξ )) , but at [ γ ],  r ∗ ( ∇ F ⊗ ∇ E ) b X ( ξ )  [ γ ] = ( ∇ F ⊗ ∇ E ) r ∗ ( b X [ γ ]) b ξ = ( ∇ F ⊗ ∇ E ) 0 b ξ = 0 , so π 0 ∇ ν X ξ = 0 . W e leav e it to the reader to chec k that π 0 ∇ ν satisﬁes (3 ) of Deﬁnition 5 .7, which is a straight forward computation, using the fact that for X ∈ ν x and [ γ ] ∈ e L x , r ∗ ( b X [ γ ] ) = γ ∗ ( X ), the parallel translate o f X along γ to ν r ( γ ) . π 0 ∇ ν obviously s atisﬁes (4).  Remark 5.9. If b ∇ ν is another p artial c onne ction, then the diﬀer enc e b ∇ ν − ∇ ν is a le afwise op er ator which satisﬁes the hyp othesis of L emma 3.3 , so π 0 b ∇ ν π 0 − π 0 ∇ ν π 0 = π 0 ( b ∇ ν − ∇ ν ) π 0 is tr ansversely smo oth and π 0 b ∇ ν is also a c onne ct ion on π 0 . So, as in the classic al c ase, the sp ac e of c onne ct ions is an aﬃne sp ac e whose line ar p art is c omp ose d of tr ansversely sm o oth op era tors. TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 19 Now supp ose that ∇ is a ny connection on π 0 . Deﬁne the curv ature θ of ∇ to b e θ = ∇ 2 . The usual computation sho ws that θ is a leafwise opera tor, that is Lemma 5.1 0. F or any ω ∈ A ∗ ( M ) and any ξ ∈ C ∞ ( π 0 ) , ∇ 2 ( ω ⊗ ξ ) = ω ∧ ∇ 2 ( ξ ) . Denote b y C ∞ ( ∧ T ∗ M ; A ∗ (2) ( F s ⊗ E )) the space of all smo o th sections o f ∧ T ∗ M with co eﬃcie nts in A ∗ (2) ( F s ⊗ E ). Smo o thness means that the section is smo o th when vie wed as a section o f ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ov er G . Extend ∇ to an op er ator on C ∞ ( ∧ T ∗ M ; A ∗ (2) ( F s ⊗ E )), by comp osing it with the obvious extension of π 0 to C ∞ ( ∧ T ∗ M ; A ∗ (2) ( F s ⊗ E )). The curv ature of ∇◦ π 0 , is given b y ( ∇◦ π 0 ) 2 = ∇ ◦ π 0 ◦ ∇ ◦ π 0 = ∇ ◦ ∇ ◦ π 0 = θ ◦ π 0 , since π 0 ◦ ∇ = ∇ . W e will a ls o de no te these new op erator s by ∇ and θ . Note that although ∇ is an op er ator which diﬀerent iates tra nsversely to the foliation F s , the op erator θ is a purely leafwise op era tor, thanks to Lemma 5.10. Also note that θ = π 0 θ = θ π 0 . Lemma 5.1 1. θ is tr ansversely smo oth. Pr o of. Se t A = π 0 ∇ π 0 − π 0 ∇ ν π 0 , a tra nsversely smo oth op erato r . Then θ = ( π 0 ∇ π 0 ) 2 = π 0 ∇ ν π 0 ∇ ν π 0 + π 0 ∇ ν π 0 Aπ 0 + π 0 Aπ 0 ∇ ν π 0 + A 2 . As A is tr a nsversely smo oth, so is A 2 . Since π 0 A = Aπ 0 = A , the terms π 0 ∇ ν π 0 Aπ 0 + π 0 Aπ 0 ∇ ν π 0 = π 0 ∇ ν Aπ 0 + π 0 A ∇ ν π 0 = π 0 [ ∇ ν , A ] π 0 = π 0 ∂ ν ( A ) π 0 , which is transversely smo o th. Now ∇ ν π 0 = π 0 ∇ ν + ∂ ν ( π 0 ), so π 0 ∇ ν π 0 ∇ ν π 0 = π 0 ( ∇ ν ) 2 π 0 + π 0 ∂ ν ( π 0 ) ∇ ν π 0 = π 0 θ ν π 0 + π 0 ∂ ν ( π 0 ) π 0 ∇ ν + π 0 ∂ ν ( π 0 ) ∂ ν ( π 0 ) . The curv ature θ ν = ( ∇ ν ) 2 satisﬁes the hyp o thesis o f of Lemma 3.3. As π 0 is transversely smo oth, it follows from Lemma 3 .3 that π 0 θ ν π 0 is trans versely smo o th. Using the fa c ts that ∂ ν is a deriv ation and π 0 is an idempo tent , it is a simple exer cise to show that π 0 ∂ ν ( π 0 ) π 0 = 0. Finally , π 0 ∂ ν ( π 0 ) ∂ ν ( π 0 ) is the comp osition of transversely smo oth op erator s, so trans versely s mo oth. Thus θ is transversely smo o th.  Set π 0 e − θ / 2 iπ = π 0 + [ n/ 2] X k =1 ( − 1) k θ k (2 iπ ) k k ! , and consider the Haeﬂiger for m T r( π 0 e − θ / 2 iπ ). (Note that 2 iπ is the co mplex n umber .) Theorem 5. 1 2. The Haeﬂiger form T r ( π 0 e − θ / 2 iπ ) is close d and its c ohomolo gy class do es not dep end on the c onne ction use d to deﬁne it. Pr o of. T he zer o -th or de r term of T r( π 0 e − θ / 2 iπ ) is T r( π 0 ), a nd s inc e π 0 is a unifor mly b o unded leafwise smo othing op e rator, w e ha v e (see [BH08]), d H T r( π 0 ) = T r( ∂ ν ( π 0 )) = T r( ∂ ν ( π 2 0 )) = 2 T r( π 0 ∂ ν ( π 0 )) = 2 T r( π 0 ∂ ν ( π 0 ) π 0 ) = 0 . since π 0 is a ( G inv ariant transversely smo oth) idemp otent. Lemma 5.1 3. F or k > 0 , d H T r ( θ k ) = 0 . Pr o of. Fir st note that for k > 0, [ ∇ , θ k ] = [ ∇ , ∇ 2 k ] = ∇ ◦ ∇ 2 k − ∇ 2 k ◦ ∇ = 0 . Also note that ∇ = π 0 ∇ ν π 0 + A, wher e A satisﬁes the hypo thesis of Lemma 3 .4, a s does θ k . Thus 0 = T r([ ∇ , θ k ]) = T r([ π 0 ∇ ν π 0 + A, θ k ]) = T r ([ π 0 ∇ ν π 0 , θ k ]) = T r ( π 0 ∇ ν θ k − θ k ∇ ν π 0 ) = T r(( π 0 − 1) ∇ ν θ k + ∇ ν θ k − θ k ∇ ν − θ k ∇ ν ( π 0 − 1)) = T r (( π 0 − 1) ∇ ν θ k ) − T r( θ k ∇ ν ( π 0 − 1)) + T r([ ∇ ν , θ k ]) . 20 M.-T. BENAMEUR AND J. L. HEITSCH Note that the three terms are well deﬁned since the three op erators a re A ∗ ( M )-equiv ar iant. As θ = π 0 θ = θπ 0 , θ k = π 0 θ k π 0 , and w e hav e T r (( π 0 − 1) ∇ ν θ k ) = T r(( π 0 − 1) ∇ ν π 0 θ k π 0 ) = T r(( π 0 − 1) π 0 ∇ ν θ k π 0 ) + T r(( π 0 − 1) ∂ ν ( π 0 ) θ k π 0 ) = 0 , since b oth terms are zero. The ﬁrst ter m is zero b eca use ( π 0 − 1) π 0 = 0 . The second term is zer o b ecause bo th ( π 0 − 1) ∂ ν ( π 0 ) θ k and π 0 are G inv ariant and transversely smo oth, so b y Lemma 3.4, T r (( π 0 − 1) ∂ ν ( π 0 ) θ k π 0 ) = T r ( π 0 ( π 0 − 1) ∂ ν ( π 0 ) θ k ) = 0 . Similarly , T r ( θ k ∇ ν ( π 0 − 1)) = 0 . Thu s, 0 = T r([ ∇ ν , θ k ]) = T r( ∂ ν ( θ k )) . It follows easily from Lemma 6.3 of [BH08] that d H T r ( θ k ) = T r( ∂ ν ( θ k )), so w e hav e the Lemma.  T o complete the pr o of o f Theor em 5.12, we note that a standard a rgument in the theory of c haracteristic classes shows that Lemma 5.1 4. The Haeﬂiger class of T r( π 0 e − θ / 2 iπ ) do es not dep end on the choic e of c onne ction ∇ on π 0 .  Deﬁnition 5.15. Th e Chern-Connes char acter ch a ( π 0 ) of the tr ansversely s m o oth idemp otent π 0 is the c ohomolo gy class of the H aeﬂliger form T r( π 0 e − θ / 2 iπ ) , that is ch a ( π 0 ) = [T r( π 0 e − θ / 2 iπ )] . Remark 5.16. In [ H9 5] , [BH04] , and [BH08] we deﬁne d Chern-Connes char acters for various obje cts. It is cle ar fr om t he r esults of those p ap ers that the deﬁnition given her e is c onsistent with t hose deﬁnitions. In p articular, if ∇ = π 0 ∇ ν is a c onne ction on π 0 c onstruct e d fr om a c onne ction ∇ F ⊗ ∇ E on ∧ T ∗ F ⊗ E , then the material in Se ction 5 of [BH08] (which shows that the deﬁnitions of [H95] and [BH04 ] c oincid e) along with the c omment after D eﬁnition 3.11 of [BH08] shows that t he Chern-Connes char acter given her e for π 0 and the Chern-Connes char acter for π 0 given in [BH08] ar e the same. Thus al l thr e e c onstructions of ch a ( π 0 ) yield the same Haeﬂiger class. Remark 5.17. Note that in Se ctions 3 and 5 we may r eplac e t he bund le ∧ T ∗ F s ⊗ E by any bu nd le on G induc e d by r fr om a bund le on M , and the r esults ar e stil l valid. Before leaving this section, we reco rd so me facts w e will ne e d later. In particular , we show that a ny connection ∇ is loc al in the sense that for X tr ansverse to F and any lo cal inv ariant section ξ o f π 0 , ∇ X ξ depe nds o nly on ξ r e stricted to an y transversal T with X tangent to T . See Coro llary 5 .21 below. Lemma 5.1 8. Le t U b e a c o or dinate chart for F . Ther e is a c ount able c ol le ction of smo oth lo c al invariant se ctions of π 0 on U which sp ans C ∞ ( π 0 ) | U as a mo dule over C ∞ ( U ) . Pr o of. L e t T b e a transversal in U . The set s − 1 ( T ) is cov er ed b y a countable collection of co ordinate charts of the form ( U, γ , V ). In ea ch chart, cho ose a co un table co llection of smo o th sectio ns { ξ V ,γ i } o f ∧ T ∗ F s ⊗ E with suppo rt in ( U, γ , V ) ∩ s − 1 ( T ) so that for any section ξ of A ∗ (2) ( F s , E ), ξ | ( U, γ , V ) ∩ s − 1 ( T ) may be written as a line a r combination (ov er the functions on s ( U, γ , V ) ∩ T ) of the { ξ V ,γ i } . Now extend the elements of this set to local inv ariant sections ov e r U , als o deno ted { ξ V ,γ i } . The collection of sections of C ∞ ( π 0 ) | U S = [ V ,γ ,i π 0 ( ξ V ,γ i ) , then spans C ∞ ( π 0 ) | U as a module over C ∞ ( U ), and the π 0 ( ξ V ,γ i ) , are loca lly inv ariant sec tio ns o ver U .  As a co ns equence, w e deduce the following. TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 21 Corollary 5.1 9. If two c onne ctions ∇ and b ∇ on π 0 agr e e on lo c al invariant se ctions, then t hey ar e the same. Note that the bundle E = r ∗ E is ﬂat (in fact tr ivial) a long the leav es of the other foliation F r of G , since its leav es are just r − 1 ( x ) for x ∈ M . Denote b y d r the obvious diﬀerential a sso ciated to ∧ T ∗ F r ⊗ E . Giv en a lo cal section ξ ∈ C ∞ 2 ( ∧ T ∗ F s ⊗ E ), we may view d r ξ a s a loca l element of C ∞ ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ). Note that d 2 r ξ = 0, and ξ is locally inv ariant if and only if d r ξ = 0. Note that for ξ ∈ C ∞ ( π 0 ) and X ∈ C ∞ ( T F ), ∇ X ξ = d r ξ ( X ). T o see this, write ξ = P j g j ξ j , where ξ j ∈ A k (2) ( F s , E ) are lo cal in v ariant elements, and the g j are smo oth loca l functions on M . Then Conditions (1) and (2 ) of Deﬁnition 5.7 give ∇ X ξ = X j d M g j ( X ) ξ j = X j d F g j ( X ) ξ j = X j d F g j ( X ) ξ j + g j d r ξ j ( X ) = d r ξ ( X ) . Let U be a foliation c ha rt for F with transversal T , and ∇ a connection on π 0 . The n on U , ∇ is the pull back of ∇ restricted to π 0 | T . More sp eciﬁcally , for X tangent to T and ξ ∈ C ∞ ( π 0 | T ), with lo cal in v ariant extension e ξ to C ∞ ( π 0 | U ), deﬁne ∇ T X ξ ≡ ∇ X e ξ . W e may assume tha t U ≃ R p × T with co o r dinates ( x, t ) and plaques R p × t . Denote b y ρ : U → T the pro jection. Let x ∈ U and X ∈ T M x , a nd set T x = x × T . W rite X = X F + ρ ∗ ( X ) wher e X F ∈ T F x and ρ ∗ ( X ) is tangent to T x . Let ξ ∈ C ∞ ( π 0 | U ), and deﬁne the pull back connection ρ ∗ ( ∇ T ) b y ρ ∗ ( ∇ T ) X ξ = d r ξ ( X F ) + ∇ T ρ ∗ ( X ) ( ξ | T x ) = d r ξ ( X F ) + ∇ ρ ∗ ( X ) ^ ( ξ | T x ) , and extend to C ∞ ( ∧ T ∗ U ; π 0 ) by using (1) of Deﬁnition 5.7 and the fact that C ∞ ( ∧ T ∗ U ; π 0 ) ≃ A ∗ ( U ) ⊗ C ∞ ( U ) C ∞ ( π 0 | U ). Denote the c ur v ature ( ∇ T ) 2 of ∇ T by θ T . Prop ositi o n 5. 20. ∇ | U = ρ ∗ ( ∇ T ) , and θ | U = ρ ∗ ( θ T ) . Pr o of. L e t ξ ∈ C ∞ ( π 0 | U ) and supp ose that X ∈ T F , so X F = X and ρ ∗ ( X ) = 0 . Then ρ ∗ ( ∇ T ) X ξ = d r ξ ( X ) = ∇ X ξ . Next suppo se that ξ is lo cal inv ariant, and X is tangent to T x , so X F = 0 and ρ ∗ ( X ) = X . Then ρ ∗ ( ∇ T ) X ξ = ∇ ρ ∗ ( X ) ^ ( ξ | T x ) = ∇ X ξ , since ^ ( ξ | T x ) = ξ , as ξ is lo ca l inv ariant. Th us ∇ | U a nd ρ ∗ ( ∇ T ) agree on lo c a l inv ariant sections, so they are equal. F o r the second equation, writing ρ ∗ ( ∇ T ) = d r + ∇ T , w e have θξ = d 2 r ξ + ∇ T d r ξ + d r ∇ T ξ + ( ∇ T ) 2 ξ = ( ∇ T ) 2 ξ , since d 2 r = 0 and ∇ T ◦ d r = − d r ◦ ∇ T . But, with the notation ρ ∗ ( ∇ T ) = d r + ∇ T , ( ∇ T ) 2 ξ = ρ ∗ ( θ T ) ξ .  The follo wing is immediate. Corollary 5.2 1. ∇ is lo c al in the sense that for X tr ansverse to F and any lo c al invariant se ction ξ of π 0 , ∇ X ξ dep ends only on ξ | T wher e T is any tr ansversal with X t angent to it. 6. Leafwise maps Let M and M ′ be compa c t Riemannian ma nifolds with oriented fo lia tions F a nd F ′ . The r esults of this s e ction do not r equire F or F ′ to be Riemannian. Let f : M → M ′ be a smo oth le a fwise homoto py equiv alence which preserves the leafwise o rientations. (W e need only a ssume transverse smo othness, and leafwise co nt inu ity . A standard ar gument then allows f to be approximated b y a smo oth map.) Suppo se that E ′ → M ′ is a leafwise ﬂat complex bundle o v er M ′ which sa tis ﬁes the hypo thesis of Theorem 9.1, and 22 M.-T. BENAMEUR AND J. L. HEITSCH set E = f ∗ ( E ′ ). L e t g : M ′ → M be a leafwise homotopy in verse of f . Then there ar e leafwise homotopies h : M × I → M and h ′ : M ′ × I → M ′ with I = [0 , 1], so that for all x ∈ M , x ′ ∈ M ′ h ( x, 0 ) = x, h ( x, 1) = g ◦ f ( x ) , h ′ ( x ′ , 0) = x ′ , and h ′ ( x ′ , 1) = f ◦ g ( x ′ ) . W e b egin by r ecalling t wo results on such le a fwise maps from [HL91]. Lemma 6.1 (Lemma 3.17 of [HL91 ]) . Given ﬁ n ite c overings of M and M ′ by foliation charts, ther e is a numb er N such that for e ach plac que Q of M ′ , ther e ar e at most N plaques P of M su ch that f ( P ) ∩ Q 6 = ∅ . Thu s f is lea fwise uniformly prop er and so induces a well deﬁned map f ∗ : H ∗ c ( L ′ f ( x ) ; R ) → H ∗ c ( L x ; R ). In general this map does not extend to the leafwise L 2 forms, as sho wn b y simple examples. Lemma 6.2 (Lemma 3 .16 of [HL91]) . F or any ﬁnite c over of M by foliation charts ther e is a numb er N such t hat for e ach plac que P of M , ther e ar e at most N plaques Q such that h ( Q × I ) ∩ P 6 = ∅ . Note that this lemma implies that there is a globa l bo und on the leafwise distance that h mov es po int s, i. e. there is a global b ound on the leafwise lengths of all the curves { γ x | x ∈ M } , where γ x ( t ) = h ( x, t ). W e remark that since f is a homotopy equiv a le nc e betw een M and M ′ , the dimensions of M and M ′ are the same. Theorem 6.3. f induc es an isomorphism f ∗ : H ∗ c ( M ′ /F ′ ) → H ∗ c ( M / F ) on Haeﬂiger c ohomolo gy with inverse g ∗ . Pr o of. T he map f induces a map b f on transversals. In particular , suppo se that U , a nd U ′ are foliation charts o f M a nd M ′ resp ectively , a nd tha t f ( U ) ⊂ U ′ . If T and T ′ are tr ansversals of U and U ′ , then f induces the map b f : T → T ′ . Lemma 6.4 . b f : T → T ′ is an immersion. Pr o of. B eing an immer sion is a lo ca l prop erty , so by reducing the size of our charts if necessa ry , we may assume that g ( U ′ ) ⊂ U 1 where U 1 is a foliation c hart for F , with transversal T 1 . Then b g : T ′ → T 1 . The leafwise homoto py h induces a map b h : T → T 1 . In particular this is the map induced on tra ns versals by the map x → h ( x, 1). Since h is con tin uous and leafwise, it is easy to see tha t b h = h γ where h γ is the holonomy along the leafwise path γ x ( t ) = h ( x, t ), where x ∈ T . Thus, b h is locally inv ertible. Since h is a homotopy of g f to the ident it y , the comp os ition, b h − 1 b g b f : T → T is the iden tit y , so b f must b e an immersion.  Since b g must als o b e an immersion, it follows immediately that the co dimensio ns of F and F ′ are the same, and so the dimensions o f F and F ′ are also the s ame. T o c o nstruct the map f ∗ : H ∗ c ( M ′ /F ′ ) → H ∗ c ( M / F ), we pro ceed as follows. Let U and U ′ be ﬁnite go o d cov er s of M and M ′ resp ectively . W e may assume that for each U ∈ U , we hav e chosen a U ′ ∈ U ′ so that f ( U ) ⊂ U ′ and that the induced map o n tra nsversals b f : T → T ′ is a diﬀeomorphism onto its image. Let α ′ ∈ H ∗ c ( M ′ /F ′ ). Since f is onto, we may choos e a Haeﬂiger form φ ′ = X U ∈U φ ′ U in α ′ , so that φ ′ U has suppo rt in b f ( T ) where T is a transversal in U . W e then deﬁne b f ∗ ( α ′ ) to b e the class of the Haeﬂiger form X U ∈U b f ∗ ( φ ′ U ) . The question of whether b f ∗ is w ell deﬁned reduces to showing the follo wing. Lemma 6 . 5. Supp ose that U 1 and U 2 ar e foliation charts on M with tr ansversals T 1 and T 2 . Supp ose further that φ ′ is a Haeﬂiger form on M ′ with supp ort c ont aine d in b f ( T 1 ) ∩ b f ( T 2 ) . Then as Haeﬂiger forms on M , [ b f | T 1 ] ∗ ( φ ′ ) = [ b f | T 2 ] ∗ ( φ ′ ) . Pr o of. Se t b f i = b f | T i . By writing φ ′ as a sum of Haeﬂiger forms a nd reducing the size of their supp orts, w e may assume that the s upp or t of φ ′ is contained in a transversal T ′ , that b g ( T ′ ) is contained in a transversal T of M a nd that the holono my maps h i : T i → T determined b y the paths γ i ( t ) = h ( x i , t ), for x i ∈ T i are deﬁned on the supp orts of b f ∗ i ( φ ′ ), r esp ectively . F urther, we may supp ose that all the maps b f 1 , b f 2 , h 1 , h 2 and TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 23 b g | T ′ are diﬀeomorphisms onto their ima ges. Since h is a homotopy o f g f to the ident it y , b f 1 = b g − 1 ◦ h 1 and b f 2 = b g − 1 ◦ h 2 , so b f ∗ 1 ( φ ′ ) = h ∗ 1 ◦ ( b g − 1 ) ∗ ( φ ′ ) and b f ∗ 2 ( φ ′ ) = h ∗ 2 ◦ ( b g − 1 ) ∗ ( φ ′ ). Thus, b f ∗ 1 ( φ ′ ) = h ∗ 1 ◦ ( h − 1 2 ) ∗ ( b f ∗ 2 ( φ ′ )) so as Haeﬂiger for ms, b f ∗ 1 ( φ ′ ) = b f ∗ 2 ( φ ′ ).  It now follows eas ily that the induced map on Hae ﬂig er co homology f ∗ : H ∗ c ( M ′ /F ′ ) → H ∗ c ( M / F ) is an isomorphism with in verse g ∗ .  Lemma 6.6 . f induc es a wel l deﬁne d smo oth le afwise map ˇ f : G → G ′ , which is le afwise uniformly pr op er. Pr o of. Se t ˇ f ([ γ ]) = [ f ◦ γ ]. That ˇ f is well deﬁned and smoo th is clear. Simila rly , set ˇ g ([ γ ′ ]) = [ g ◦ γ ′ ]. Let U b e a ﬁnite go o d cov er o f M . Since M is c o mpact, there is a b o und m ( P ) on the diameter of a ny plaque in the cov er U . Then m ( P ) is also a bo und for any plaque of F s in the corr esp onding cov er of G . Let U ′ be a ﬁnite g o o d cov er of M ′ , such that for each U ′ ∈ U ′ there is U ∈ U so that g ( U ′ ) ⊂ U . Given ( U ′ , γ ′ , V ′ ) in the c ov er of G ′ corres p o nding to U ′ , choose U, V ∈ U with g ( U ′ ) ⊂ U and g ( V ′ ) ⊂ V . If we set γ = g ◦ γ ′ , then ˇ g ( U ′ , γ ′ , V ′ ) ⊂ ( U, γ , V ) . B e c ause U ′ is a go o d cov er , there is ǫ > 0 so that if z ′ 0 , z ′ 1 ∈ e L ′ with d e L ′ ( z ′ 0 , z ′ 1 ) < ǫ , then there is a ( U ′ , γ ′ , V ′ ) with z ′ 0 , z ′ 1 ∈ ( U ′ , γ ′ , V ′ ), so ˇ g ( z ′ 0 ) , ˇ g ( z ′ 1 ) ∈ ( U, γ , V ) . Since ˇ g ( e L ′ ) ∩ ( U, γ , V ) consists of at mos t one plaque of ˇ g ( e L ′ ), it follows tha t d e L ( ˇ g ( z ′ 0 ) , ˇ g ( z ′ 1 )) < m ( P ). Thus, if z ′ t is a path in e L ′ of length less than C , then ˇ g ◦ z ′ t is a path in ˇ g ( e L ′ ) of length less than m ( P ) C /ǫ . Suppo se that f ( x ) = x ′ and let A ′ ⊂ e L ′ x ′ hav e diameter dia( A ′ ) ≤ C . Let z 0 , z 1 ∈ e L x with ˇ f ( z i ) = z ′ i ∈ A ′ , and cho ose a path z ′ t in e L ′ x ′ of length less than C be tw een z ′ 0 and z ′ 1 . The n ˇ g ◦ z ′ t is a path in e L gf ( x ) of leng th less than m ( P ) C /ǫ . Compo sition on the r ight by the path γ x ( t ) = h ( x, t ) is an isometry from e L gf ( x ) to e L x . So (ˇ g ◦ z ′ t ) · γ x is a path in e L x of length less than m ( P ) C /ǫ . Thus d e L x ([( ˇ g ◦ z ′ 0 ) · γ x ] , [(ˇ g ◦ z ′ 1 ) · γ x ]) ≤ m ( P ) C /ǫ. By Lemma 6.2, the path γ y has length bo unded b y say B , for all y ∈ M . Set y i = r ( z i ), a nd no te tha t [ γ − 1 y i · ( ˇ g ◦ z ′ i ) · γ x ] = z i , since h is a lea fwise homotop y equiv alence betw e e n g ◦ f and the identit y . As d e L x ( z i , [(ˇ g ◦ z ′ i ) · γ x ]) = d e L x ([ γ − 1 y i · (ˇ g ◦ z ′ i ) · γ x ] , [(ˇ g ◦ z ′ t ) · γ x ]) ≤ length( γ y i ) ≤ B , we hav e d e L x ( z 0 , z 1 ) ≤ 2 B + m ( P ) C /ǫ . Thu s dia( ˇ f − 1 ( A ′ )) ≤ 2 B + m ( P ) dia( A ′ ) /ǫ , and ˇ f is leafwise uniformly proper .  Thu s ˇ f induces a well deﬁned map ˇ f ∗ : H ∗ c ( e L ′ f ( x ) ; R ) → H ∗ c ( e L x ; R ). As noted ab ove, in genera l this map do es not induce a well deﬁned map on lea fwise L 2 forms. W e will use tw o diﬀerent cons tr uctions to deal with this problem. First we adapt the co nstruction of the L 2 pull-back map of Hilsum-Sk anda lis in [HiS92] to our s etting. This has the adv antage that it is transversely s mo oth. Ho wev er , it is no t obvious that its action on leafwise L 2 cohomolog y resp ects the wedge pro duct, so we will also use the cons tr uction in [HL91], which is based o n res ults of Do dziuk, [D77]. W e assume the r eader is familiar w ith Sobo lov theor y o f spaces of sections of a vector bundle o ver a manifold. F o r s ∈ Z , deno te by W ∗ s ( F s , E ) the ﬁeld o f Hilb e r t spaces over M given by W ∗ s ( F s , E ) x = W ∗ s ( e L x , E ), the s -th Sob olev space of diﬀerential fo rms on e L x with co e ﬃcient s in E | e L x . Just as it do es for the leafwise L 2 forms, the co mpactness of M implies that these space s do not depe nd on our choice o f Riemannian structure. Note that W ∗ s ⊂ W ∗ s 1 if s ≥ s 1 , and set W ∗ ∞ ( F s , E ) = \ s ∈ Z W ∗ s ( F s , E ) and W ∗ −∞ ( F s , E ) = [ s ∈ Z W ∗ s ( F s , E ) . Equip W ∗ ∞ ( F s , E ) with the induced loca lly con vex to po logy . Let i : M ′ ֒ → R k be a n imbedding of the compac t ma nifold M ′ in some E uclidean s pa ce R k , and identify M ′ with its image. W e ass ume for conv enience that k is e ven. F o r x ′ ∈ M ′ and t ∈ R k , deﬁne p ( x ′ , t ) to be the pro jection of the tangent vector X t = d ds | s =0 ( x ′ + st ) at x ′ determined by t , to the le af L ′ x ′ in ( M ′ , F ′ ) ⊂ R k . In par ticular, ﬁrst pro ject X t to T F ′ x ′ and then expo nent iate it to L ′ x ′ , thinking of L ′ x ′ as a 24 M.-T. BENAMEUR AND J. L. HEITSCH Riemannian manifold in its own right. Since M ′ is co mpact, we ma y choos e a ball B k ⊂ R k so small that the restriction o f the smo oth map p f = p ◦ ( f , i d ) : M × B k → M ′ to any p f : L x × B k → L ′ f ( x ) is a submersion. Lifting this map to the gr oup oids, w e get p f : G × B k − → G ′ , which is a le a fwise map if G × B k is endow ed with the foliation F s × B k . Note that p f : e L x × B k → e L ′ f ( x ) is the map induced o n the co verings by p f : L x × B k → L ′ f ( x ) . In particular , p f ([ γ ] , t ) is the comp ositio n of leafwise paths P f ( γ , t ) and f ◦ γ , p f ([ γ ] , t ) = [ P f ( γ , t ) · ( f ◦ γ )] , where P f ( γ , t ) : [0 , 1] → L ′ f ( r ( γ )) is P f ( γ , t )( s ) = p f ( r ( γ ) , st ) . T o see that this is a smo oth map, let ( U, γ , V ) × B k and ( U ′ , f ◦ γ , V ′ ) b e lo c a l co o rdinate charts on G × B k and G ′ , resp ectively , with coo r dinates ( w, y, z , t ) and ( w ′ , y ′ , z ′ ). Then in these co ordinates, p f ( w, y , z , t ) = ( w ′ ( f ( w , y )) , y ′ ( f ( w , y )) , z ′ ( p f ( y , z , t ))) , where the second p f is the map p f : V × B k → V ′ . The crucial fact ab out p f is that it has all the sa me essential pr op erties o f the pro jectio n π 1 : G × B k → G . First note that, b ecause f and ˇ f are leafwise uniformly prop er and M × B k is compa c t, b oth the maps denoted p f are a lso lea fwise uniformly prop er . Second, w e may assume that the metric on each L x × B k (resp ectively e L x × B k ) is the pro duct of a ﬁb er wis e metric for the submers io n p f and the pull-back under p f of the metric on L ′ f ( x ) (resp ectively e L ′ f ( x ) ). T o see this, g ive L × B k the pro duct metric, using the s tandard metric on B k . The induced metric o n e L × B k is then the pro duct metr ic. The ﬁb ers of b oth submersions p f inherit a Riemannian metric , and we denote b y dv ol ver t the canonica l k form o n b oth L × B k and e L × B k whose restrictio n to the oriented ﬁb ers of p f is the volume form. Denote b y ∗ the Hodge op erator on b o th L × B k and e L × B k , and similarly fo r ∗ ′ on L ′ and e L ′ . Co nsider the sub-bundle p ∗ f T ∗ F ′ ⊂ T ∗ ( F × B k ), a nd its orthogo nal complement p ∗ f T ∗ F ′ ⊥ . Deﬁne a new metric o n T ∗ ( F × B k ) = p ∗ f T ∗ F ′ ⊕ p ∗ f T ∗ F ′ ⊥ (and so a ls o on T ∗ ( F s × B k )) by declaring that these sub-bundles are still orthogona l, and the new metric o n p ∗ f T ∗ F ′ ⊥ is the same as the original, while the new metric on p ∗ f T ∗ F ′ is the pullback of the metric on T ∗ F ′ . Denote the leafwise Ho dge op era tor of the new metr ic by b ∗ . As r emarked a bove, this change of metric do es not alter any of our Sobo lev spaces. In particular, note that for an y no n-zero α ∈ ∧ ℓ T ∗ ( F × B k ) and any c ∈ R ∗ + , 0 < cα ∧ b ∗ cα cα ∧ ∗ cα = α ∧ b ∗ α α ∧ ∗ α , so the compactness of the sphere bundle ( ∧ ℓ T ∗ ( F × B k ) − { 0 } ) / R ∗ + implies that there are 0 < C 1 < C 2 , so that for all α ∈ ∧ ℓ T ∗ ( F × B k ), C 1 α ∧ ∗ α ≤ α ∧ b ∗ α ≤ C 2 α ∧ ∗ α, where we identify the o riented volume elements of L × B k at a p oint with R ∗ + . This prop erty is inherited by the tw o induced metrics on T ∗ ( F s × B k ), s o the tw o no rms used to deﬁne the Sobo lev spa ces W ℓ s ( F s , E ) a re comparable. Thus, we can substitute the second metric for the ﬁrs t, or what is mor e notationally conv enient, assume that the ﬁrst metr ic s atisﬁes the sa me pull bac k pr op erty as the second. Simple computations give t wo immediate consequences of this assumption. Namely , for any α 1 , α 2 ∈ ∧ ℓ T ∗ F ′ s , 6.7. p ∗ f α 1 ∧ ∗ p ∗ f α 2 = dv ol ver t ∧ p ∗ f ( α 1 ∧ ∗ ′ α 2 ) , and 6.8. dv ol ver t ∧ p ∗ f α 1 ∧ ∗ ( dvol ver t ∧ p ∗ f α 2 ) = dv ol ver t ∧ p ∗ f ( α 1 ∧ ∗ ′ α 2 ) . TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 25 Denote by π 2 : G × B k → B k the pro jection, and choose a smo oth compactly s uppo rted k -form ω o n B k whose integral is 1. W e shall refer to such a form as a Bott form on B k . Deno te b y e ω the exterior m ultiplication by the diﬀerential k − form π ∗ 2 ω o n G × B k . F or ξ ∈ A ∗ c ( F ′ s , E ′ ), we deﬁne f ( i,ω ) ( ξ ) ∈ A ∗ c ( F s , E ) as f ( i,ω ) ( ξ ) = ( π 1 , ∗ ◦ e ω ◦ p ∗ f )( ξ ) . The map p f : G × B k − → G ′ is a lea fwise (for F s × B k ) submersion extending ˇ f , so p ∗ f ( ξ ) is a leafwise form on G × B k with co e ﬃcient s in the bundle p ∗ f E ′ . The map π 1 , ∗ is int egration ov er the ﬁb er of the pro jection π 1 : G × B k → G of such for ms . In ge ne r al, the ﬁb er of p ∗ f E ′ is not constant on ﬁber s of the ﬁbration π 1 : G × B k → G . T o correct for this, w e use the parallel translation given by the ﬂat structure o f p ∗ f E ′ to ident ify all the ﬁb ers of p ∗ f E ′ | z × B k with ( p ∗ f E ′ ) ( z , 0) = ( ˇ f ∗ E ′ ) z = ( f ∗ E ′ ) r ( z ) . This is well deﬁned b ecause the ball B k ⊂ R k is co n tractible, so parallel tra nslation is indep endent o f the path taken fr om ( z , 0) to ( z , t ) in z × B k . Prop ositi o n 6. 9. F or any s ∈ Z , f ( i,ω ) extends to a b ounde d op er ator fr om W ∗ s ( F ′ s , E ′ ) to W ∗ s ( F s , E ) . Pr o of. F or this pro of only , for α 1 ⊗ φ 2 and α 2 ⊗ φ 2 ∈ A ∗ c ( F s , E ), we set ( α 1 ⊗ φ 1 ) ∧ ( α 2 ⊗ φ 2 ) = ( φ 1 , φ 2 ) α 1 ∧ α 2 and ( α 1 ⊗ φ 1 ) ∧ ∗ ( α 2 ⊗ φ 2 ) = ( φ 1 , φ 2 ) α 1 ∧ ∗ α 2 , where ( · , · ) is the p ositive deﬁnite metric on E . Similarly for A ∗ c ( F ′ s , E ′ ). Since p f is leafwise uniformly proper , C = sup [ γ ′ ] ∈G ′ Z p − 1 f ([ γ ′ ]) dv ol ver t < + ∞ . Thanks to 6.7, we then ha ve for any α ⊗ φ ∈ A ℓ c ( e L ′ f ( x ) , E ′ ) = C ∞ c ( e L ′ ; ∧ ℓ T ∗ e L ′ f ( x ) ⊗ E ′ ), k p ∗ f (( α ⊗ φ ) f ( x ) ) k 2 0 = Z e L x × B k ( p ∗ f φ, p ∗ f φ ) p ∗ f α ∧ ∗ p ∗ f α = Z e L x × B k ( p ∗ f φ, p ∗ f φ ) dvol ver t ∧ p ∗ f ( α ∧ ∗ ′ α ) Z e L ′ f ( x ) h Z p − 1 f ([ γ ′ ]) dv ol ver t i ( φ, φ ) α ∧ ∗ ′ α ≤ C Z e L ′ f ( x ) ( φ, φ ) α ∧ ∗ ′ α = C k α ⊗ φ k 2 0 . This inequality extends to all ξ ∈ A ℓ (2) ( e L ′ f ( x ) , E ′ ) = W ℓ 0 ( e L ′ f ( x ) , E ′ ), so p ∗ f extends to a unifor mly b ounded (i.e. indep endent of x ) op erato r from W ℓ 0 ( e L ′ f ( x ) , E ′ ) to W ℓ 0 ( e L x × B k , p ∗ f E ′ ), that is p ∗ f deﬁnes a b ounded op erator from W ℓ 0 ( F ′ s , E ′ ) to W ℓ 0 ( F s × B k , p ∗ f E ′ ). Cho ose a sub-bundle b H ⊂ T F ⊕ T B k so that for ea ch L x , it is a hor izontal distribution for the submersion p f : L x × B k → L ′ f ( x ) . The map ( r × id ) ∗ : T F s ⊕ T B k → T F ⊕ T B k is an isomorphism on each ﬁb er, so b H determines a sub- bundle H o f T F s ⊕ T B k , a nd H | e L x × B k is a hor izontal distribution for the submer s ion p f : e L x × B k → e L ′ f ( x ) . Cho o se a ﬁnite collection of leafwise vector ﬁelds b Y 1 , . . . , b Y N on M ′ which generate C ∞ ( T F ′ ) ov er C ∞ ( M ′ ). Lift these to lea fwise (for F ′ s ) vector ﬁelds Y 1 , . . . , Y N on G ′ , and lift these latter to sections of H , denoted X 1 , . . . , X N . If X ver t is a vertical vector ﬁeld on e L × B k with resp ect to p f , then i X vert ◦ p ∗ f = 0. Modulo such vector ﬁelds, the X i generate T e L ⊕ T B k ov er C ∞ ( e L × B k ). In addition, i X j ◦ p ∗ f = p ∗ f ◦ i Y j . Thus, for any ξ ∈ A ℓ c ( e L ′ f ( x ) , E ′ ), any Y K = Y k 1 ∧ · · · ∧ Y k ℓ , and any j 1 , . . . , j m , with j i ∈ { 1 , . . . , N } , k i X j 1 d · · · i X j m d ( p ∗ f ( ξ )( Y K )) k 0 = k p ∗ f ( i Y j 1 d · · · i Y j m d ( ξ ( Y K )) k 0 ≤ √ C k i Y j 1 d · · · i Y j m d ( ξ ( Y K )) k 0 . A classica l ar g ument then shows that for any s ≥ 1, p ∗ f extends to a uniformly b ounded op era tor fr om W ℓ s ( e L ′ f ( x ) , E ′ ) to W ℓ s ( e L x × B k , p ∗ f E ′ ), that is a b o unded oper ator fr o m W ℓ s ( F ′ s , E ′ ) to W ℓ s ( F s × B k , p ∗ f E ′ ). 26 M.-T. BENAMEUR AND J. L. HEITSCH The op era tor e ω maps W ℓ s ( e L x × B k , p ∗ f E ′ ) to W k + ℓ s ( e L x × B k , p ∗ f E ′ ) a nd is uniformly b ounded, since ω and all its deriv ativ es ar e b o unded. Th us fo r s ≥ 0, e ω ◦ p ∗ f is a b ounded op era tor fr om W ℓ s ( F ′ s , E ′ ) to W k + ℓ s ( F s × B k , p ∗ f E ′ ). F o r the case o f s < 0, w e dualize the argument above. Denote by p f , ∗ int egration o f ﬁb er compactly suppo rted forms along the ﬁbers of the submersion p f . W e c la im that for an y α ∈ A k + ℓ c ( e L x × B k ), 6.10. p f , ∗ α ∧ ∗ ′ p f , ∗ α ≤ C p f , ∗ ( α ∧ ∗ α ) , where, as ab ove, we identif y the or ient ed v olume elements o f e L ′ f ( x ) at a p oint with R ∗ + . Any such α may b e written as α = α 1 + α 2 , where p f , ∗ ( α 2 ) = 0, and α 1 = dv ol ver t ∧ α 3 , with α 3 ∈ C ∞ c ( p ∗ f ( ∧ ℓ T ∗ e L ′ f ( x ) )). Then p f , ∗ ( α ∧ ∗ α ) = p f , ∗ ( α 1 ∧ ∗ α 1 ) + p f , ∗ ( α 2 ∧ ∗ α 2 ) + p f , ∗ ( α 1 ∧ ∗ α 2 ) + p f , ∗ ( α 2 ∧ ∗ α 1 ) . The las t tw o terms are zer o, since α 1 ∧ ∗ α 2 = 0 a s dv ol ver t ∧ ∗ α 2 = 0, and p f , ∗ ( α 2 ∧ ∗ α 1 ) = 0 since α 2 ∧ ∗ α 1 do es not con tain dvol ver t . Thus p f , ∗ ( α ∧ ∗ α ) = p f , ∗ ( α 1 ∧ ∗ α 1 ) + p f , ∗ ( α 2 ∧ ∗ α 2 ) ≥ p f , ∗ ( α 1 ∧ ∗ α 1 ) . But, p f , ∗ α 1 ∧ ∗ ′ p f , ∗ α 1 = p f , ∗ α ∧ ∗ ′ p f , ∗ α, so w e need only prove 6.10 for α = dv ol ver t ∧ α 3 , with α 3 ∈ C ∞ c ( p ∗ f ( ∧ ℓ T ∗ e L ′ f ( x ) )). Cho ose a ﬁnite collectio n of sections β 1 , . . . , β r of ∧ ℓ T ∗ F ′ , so that β i ∧ ∗ ′ β j = 0 if i 6 = j , and the β i generate C ∞ ( ∧ ℓ T ∗ F ′ ) ov er C ∞ ( M ′ ). Denote als o by β i the lift o f thes e s ections to se ctions of ∧ ℓ T ∗ F ′ s . Then, α = dv ol ver t ∧ α 3 , ma y b e written as α = X i g i dv ol ver t ∧ p ∗ f β i , where the g i are smo oth compactly supported functions on e L x × B k . Now, p f , ∗ α ∧ ∗ ′ p f , ∗ α = X i p f , ∗ ( g i dv ol ver t ) β i ∧ ∗ ′ X j p f , ∗ ( g j dv ol ver t ) β j = X i [ p f , ∗ ( g i dv ol ver t )] 2 β i ∧ ∗ ′ β i . Thanks to 6.8, p f , ∗ ( α ∧ ∗ α ) = p f , ∗ ( X i ( g i dv ol ver t ∧ p ∗ f β i ) ∧ ∗ X j ( g j dv ol ver t ∧ p ∗ f β j )) = p f , ∗ ( X i,j g i g j dv ol ver t ∧ p ∗ f ( β i ∧ ∗ ′ β j )) = X i p f , ∗ ( g 2 i dv ol ver t ) β i ∧ ∗ ′ β i ≥ X i [ p f , ∗ ( g i · 1 dv ol ver t )] 2 p f , ∗ (1 dvol ver t ) β i ∧ ∗ ′ β i ≥ 1 C X i [ p f , ∗ ( g i dv ol ver t )] 2 β i ∧ ∗ ′ β i = 1 C p f , ∗ α ∧ ∗ ′ p f , ∗ α, proving 6.10. Note tha t the second to last inequa lit y is just Ca uch y-Schw ar tz. Thu s, for all α ∈ A k + ℓ c ( e L x × B k ), k p f , ∗ α k 2 0 = Z e L ′ f ( x ) p f , ∗ α ∧ ∗ ′ p f , ∗ α ≤ C Z e L ′ f ( x ) p f , ∗ ( α ∧ ∗ α ) = C Z e L x × B k α ∧ ∗ α = C k α k 2 0 . Using the facts that p f , ∗ commutes with the de Rham diﬀerentials, p f , ∗ ◦ i X vert = 0 a nd i Y j ◦ p f , ∗ = p f , ∗ ◦ i X j , it is easy to deduce, just as for p ∗ f , that for any s ≥ 0 , p f , ∗ ◦ e ω extends to a uniformly b ounded oper ator (say with b o und C s ) from W ℓ s ( e L x × B k , p ∗ f E ′ ) to W ℓ s ( e L ′ f ( x ) , E ′ ). Now s upp o s e tha t ξ ′ ∈ W ℓ s ( e L ′ f ( x ) , E ′ ) for some s < 0, and recall that k ( e ω ◦ p ∗ f )( ξ ′ ) k s is giv en b y k ( e ω ◦ p ∗ f )( ξ ′ ) k s = sup ξ | < ξ ′ , ( p f , ∗ ◦ e ω )( ξ ) > ) | k ξ k − s ≤ s up ξ k ξ ′ k s k ( p f , ∗ ◦ e ω )( ξ ) k − s k ξ k − s ≤ C s k ξ ′ k s , TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 27 where the supremums are ta ken ov er a ll ξ ∈ W ℓ − s ( e L x × B k , p ∗ f E ′ ). Thus, for any s < 0 (and so for all s ∈ Z ), e ω ◦ p ∗ f is a uniformly b ounded o p e rator from W ℓ s ( e L ′ f ( x ) , E ′ ) to W k + ℓ s ( e L x × B k , p ∗ f E ′ ), so e ω ◦ p ∗ f is a b ounded op erator from W ℓ s ( F ′ s , E ′ ) to W ℓ s ( F s × B k , p ∗ f E ′ ). F o r all s ∈ Z , the image of e ω ◦ p ∗ f consists of π 1 -ﬁb er compactly supp or ted distributional forms. T he argument a bove for p f , ∗ applied to π 1 , ∗ shows that it is uniformly b ounded a s a ma p from Im( e ω ◦ p ∗ f ) ⊂ W k + ℓ s ( e L x × B k , p ∗ f E ′ ) to W ℓ s ( e L x , E ). Thus, for a ll s ∈ Z , f ( i,ω ) extends to a b ounded op erato r from W ℓ s ( F ′ s , E ′ ) to W ℓ s ( F s , E ).  As ω is clos ed, e ω commutes with de Rham diﬀerentials. The image o f e ω ◦ p ∗ f is con tained in the π 1 - ﬁber compactly supp orted forms, so f ( i,ω ) = π 1 , ∗ ◦ e ω ◦ p ∗ f commutes with de Rham diﬀerentials. It follows immediately that the extension o f f ( i,ω ) to the L 2 forms also commut es with the closur es of the de Rha m diﬀerentials, so f ( i,ω ) induces a well deﬁned map e f ∗ : H ∗ (2) ( F ′ s , E ′ ) − → H ∗ (2) ( F s , E ) o n lea fwise reduced L 2 cohomolog y . As remarked ab ove, the pr op erties of this map (using this deﬁnition) ar e not immediately obvious. T o deal with this problem, w e now switc h our p oint of view to tha t in [HL91], and give another construction of the map e f ∗ . Let K = [ e L K e L be a bounded leafwise triangulation o f F s , (see [HL91]) induced fr om a bo unded le afwise triangulation to F . Then K e L is a bounded triangulation o f the leaf e L . A simplicial k -co chain ϕ on K e L with co eﬃcients in E assigns to each k -simplex σ o f K e L an element ϕ ( σ ) ∈ E σ , the ﬁb er o f E over the baryce n ter of σ . T o deﬁne the co-b oundary map δ , we ident ify E σ with the ﬁb ers of E over the barycenters o f the simplices in the b oundar y of σ us ing the ﬂat structur e of E . This is well deﬁned since σ is contractible. Denote b y C k ( p ) ( K e L , E ) the space of simplicial k -cochains ϕ on K e L with co e ﬃcient s in E suc h that X σ k -simplex of K e L ( ϕ ( σ ) , ϕ ( σ )) p/ 2 < + ∞ . The homology of the complex ( C ∗ ( p ) ( K e L , E ) , δ ) is the ℓ p cohomolog y of the simplicial co mplex K e L with co eﬃcients in E . It is denoted H ∗ △ ,p ( e L, E ). The classica l Whitney a nd de Rham maps extend to well deﬁned chain morphisms W : C ∗ ( p ) ( K e L , E ) → A ∗ ( p ) ( e L, E ) and I : A ∗ ( p ) ( e L, E ) → C ∗ ( p ) ( K e L , E ) , which induce bo unded iso morphisms in c o homology (which are inv erses of each other), with b ounds inde- pendent o f e L , for p = 1 , 2. See [HL91] for p = 2, and [GKS88] for p = 1. As ab ov e, to deﬁne these maps, we use the classical deﬁnitions coupled with the fact that for any p o int x ∈ σ , the ﬂa t s tructure of E | σ gives a natural isomorphism betw een E x and E σ . Let f K,K ′ : K e L → K ′ e L ′ be an orient ed leafwise simplicial approximation of ˇ f as in [HL91]. It is unifor mly prop er, so it deﬁnes a pull-back map f ∗ △ on ℓ p co chains with co eﬃcients in E ′ , which co mm utes with the cob oundaries. The induced map on cohomology is also denoted f ∗ △ . Set f ∗ D = W ◦ f ∗ △ ◦ H Prop ositi o n 6. 11. e f ∗ = f ∗ D : H ∗ (2) ( F ′ s , E ′ ) − → H ∗ (2) ( F s , E ) . Pr o of. As B k is a ﬁnite CW-complex, the map p f induces the w ell deﬁned ma p p ∗ f , △ : H ∗ ∆ , 2 ( e L ′ , E ′ ) → H ∗ △ , 2 ( e L × B k , p ∗ f E ′ ) . Denote by β the s implicial k coc ycle H ω on B k , and b y π 2 : e L × B k → B k a simplicia l appr oximation (after suitable sub divisio ns ) of the pro jection. W e choose the sub division ﬁne enough so that the cup pro duct by the bounded k co cycle π ∗ 2 β induces the well deﬁned map [ β ] ∪ : H ∗ ∆ , 2 ( e L × B k , p ∗ f E ′ ) → H ∗ + k △ , 2 ,c ( e L × B k , p ∗ f E ′ ) , 28 M.-T. BENAMEUR AND J. L. HEITSCH where H ∗ ∆ , 2 ,c ( e L × B k , p ∗ f E ′ ) denotes the ℓ 2 simplicial c o homology of cochains which are zero on an y simplex that intersects the b o unda ry of e L × B k , that is “ ﬁber co mpactly supp orted” co cy cles. Cap product with the fundamen tal cycle [ B k ] of B k gives the map ∩ [ B k ] : H ∗ + k ∆ , 2 ,c ( e L × B k , p ∗ f E ′ ) → H ∗ △ , 2 ( e L, E ) . Denote by H ∗ (2) ,c ( e L × B k , p ∗ f E ′ ) the cohomolo gy of L 2 forms which are zero on some neig hborho o d of the bo undary e L × B k . Note that H ∗ △ , 2 ( e L × B k , p ∗ f E ′ ) is a mo dule over H ∗ △ , 2 ( e L × B k ), H ∗ (2) ( e L × B k , p ∗ f E ′ ) is a mo dule ov er H ∗ (2) ( e L × B k ), and ∩ [ B k ] : H ∗ + k ∆ , 2 ,c ( e L × B k , p ∗ f E ′ ) → H ∗ △ , 2 ( e L, E ) is deﬁned. Then, the following diagram comm utes. H ∗ ∆ , 2 ( e L ′ , E ′ ) ❄ H ∗ (2) ( e L ′ , E ′ ) W p ∗ f , △ ✲ ✲ p ∗ f H ∗ ∆ , 2 ( e L × B k , p ∗ f E ′ ) ❄ H ∗ (2) ( e L × B k , p ∗ f E ′ ) W [ β ] ∪ ✲ ✲ [ ω ] ∧ H ∗ + k ∆ , 2 ,c ( e L × B k , p ∗ f E ′ ) ❄ H ∗ + k (2) ,c ( e L × B k , p ∗ f E ′ ) W ∩ [ B k ] ✲ ✲ π 1 , ∗ H ∗ ∆ , 2 ( e L, E ) ❄ H ∗ (2) ( e L, E ) . W Since p f is a smo oth submersio n, it deﬁnes the b ounded op erator p ∗ f : H ∗ (2) ( e L ′ , E ′ ) → H ∗ (2) ( e L × B k , p ∗ f E ′ ), and W ◦ p ∗ f , △ = p ∗ f ◦ W by the na turality of the Whitney map. The square in the middle commutes b ecause W is compatible with cup and wedge pro ducts in cohomolog y and W [ β ] = [ ω ]. Finally the RHS squa r e is commutativ e beca use W is compa tible with cap pr o ducts, and integration o ver the ﬁb ers of π 1 is exactly cap pro duct b y the fundamental class in ho mo logy of B k . The bottom line of this diagram is e f ∗ , so we need only sho w that W ◦ ∩ [ B k ] ◦ [ β ] ∪ ◦ p ∗ f , △ ◦ W − 1 = f ∗ D = W ◦ f ∗ △ ◦ I . As W − 1 = H , this r educes to showing that ∩ [ B k ] ◦ [ β ] ∪ ◦ p ∗ f , △ = f ∗ △ . The zero section i : e L ֒ → e L × B k induces i ∗ △ : H ∗ ∆ , 2 ( e L × B k , p ∗ f E ′ ) → H ∗ △ , 2 ( e L, E ) , and the pro jection π 1 : e L × B k → e L induces π ∗ 1 , △ : H ∗ ∆ , 2 ( e L, E ) → H ∗ ∆ , 2 ( e L × B k , p ∗ f E ′ ) . These maps satisfy π ∗ 1 , △ ◦ i ∗ △ = i d H ∗ ∆ , 2 ( e L × B k ,p ∗ f E ′ ) . Thu s w e ha ve ([ β ] ∪ ) ◦ p ∗ f , △ = ([ β ] ∪ ) ◦ π ∗ 1 , △ ◦ i ∗ △ ◦ p ∗ f , △ = ([ β ] ∪ ) ◦ π ∗ 1 , △ ◦ f ∗ △ . By the Thom Iso morphism Theorem, ([ β ] ∪ ) ◦ π ∗ 1 , △ : H ∗ ∆ , 2 ( e L, E ) → H ∗ + k ∆ , 2 ,c ( e L × B k , p ∗ f E ′ ) is an isomo rphism whose in verse is precisely ∩ [ B k ].  Corollary 6. 1 2. The map e f ∗ : H ∗ (2) ( F ′ s , E ′ ) − → H ∗ (2) ( F s , E ) on le afwise r e duc e d L 2 c ohomolo gy induc e d by f ( i,ω ) do es not dep end on the choic es of i and ω . If f 1 and f 2 ar e le afwise homotopy e quivalent, then e f ∗ 1 = e f ∗ 2 . If g : ( M ′ , F ′ ) → ( M , F ) is a le afwise homotopy inverse for f , then e g ∗ ◦ e f ∗ = i d and e f ∗ ◦ e g ∗ = i d , so e f ∗ is an isomorphism, with inverse e g ∗ . Pr o of. F or any choice of i and ω , e f ∗ = f ∗ D , so they a r e a ll the s a me. The other prop erties of e f ∗ follow from these same prop erties for f ∗ D which are easy to prov e using classical arguments.  The following result will be needed for the pro o f of the main theorem. Reca ll the deﬁnition o f the pairing Q from the proo f of Lemma 4.1. TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 29 Prop ositi o n 6. 13. If ξ ′ 1 and ξ ′ 2 ar e close d L 2 se ctions of ∧ ℓ e L ′ f ( x ) ⊗ E ′ , then Q x ( e f ∗ ( ξ ′ 1 ) , e f ∗ ( ξ ′ 2 )) = Q ′ f ( x ) ( ξ ′ 1 , ξ ′ 2 ) . Pr o of. F or this, w e need the cup pr o duct for simplicial co chains with co eﬃcients in E (and E ′ ). No te that since E has t w o (p ossibly ) diﬀerent metr ics o n it, we hav e tw o (po ssibly) diﬀeren t w ays of deﬁning this cup pro duct, dep ending on which metric we use. W e w ill use the (poss ibly indeﬁnite) metric { · , ·} . The deﬁnition we wan t to extend is that of [LS0 3], Equa tion (3.30). F or or dinary degree ℓ co c ha ins ϕ 1 and ϕ 2 , this is ( ϕ 1 ∪ ϕ 2 )( σ ) = 1 (2 ℓ + 1)! X i,j ϕ 1 ( σ i ) ϕ 2 ( σ j ) , where σ i and σ j are cer tain faces of the 2 ℓ s implex σ , and ϕ 1 ( σ i ) and ϕ 2 ( σ j ) ar e re a l num b ers. If ϕ 1 and ϕ 2 are co chains with co eﬃcients in E , then ϕ 1 ( σ i ) a nd ϕ 2 ( σ j ) a re elements of E σ i and E σ j resp ectively (which we identify with E σ ), and their cup pro duct is an ordina ry ( C v alued) co chain which is giv en by the formula ( ϕ 1 ∪ ϕ 2 )( σ ) = 1 (2 ℓ + 1)! X i,j { ϕ 1 ( σ i ) , ϕ 2 ( σ j ) } . In [LS03][3.3 0] L ¨ uck and Sc hic k show that for their deﬁnition of the cup pro duct, the Whitney map satisﬁes Prop ositi o n 6. 14. F or any ℓ 2 simplicia l c o chains ϕ 1 and ϕ 2 on e L with c o eﬃcients in E , Q ( W ( ϕ 1 ) , W ( ϕ 2 )) = Z e L W ( ϕ 1 ∪ ϕ 2 ) . Actually , they prove it when E is the o ne dimensional trivial bundle. The pr o of extends immediately to our case, s ince it is a lo ca l statemen t, and lo ca lly E is trivial w ith the metric the pull-back from the metric on a s ingle ﬁbe r. The re a son w e use the metric {· , ·} in the cup pro duct, and not the metric ( · , · ), is so that this result will pass to simplicial ℓ 2 cohomolog y cla sses Ξ 1 and Ξ 2 . In particular , we have Q ( W (Ξ 1 ) , W (Ξ 2 )) = Z e L W (Ξ 1 ⊔ Ξ 2 ) = < [ e L ] , Ξ 1 ⊔ Ξ 2 > where ⊔ is the cup pro duct of ℓ 2 cohomolog y classes with co eﬃcients in E , whic h takes v alues in the usual ℓ 1 cohomolog y (no coeﬃcients), and [ e L ] is the fundamen tal c lass in bo unded simplicial ho mo logy . As H and W are inverses of each other on co homology , we immediately hav e for any L 2 cohomolog y classe s Ψ ′ 1 and Ψ ′ 2 on e L ′ with co e ﬃcient s in E ′ , < [ e L ′ ] , ( I Ψ ′ 1 ) ⊔ ( I Ψ ′ 2 ) > = Z e L ′ Ψ ′ 1 ∧ Ψ ′ 2 . It is clear from the deﬁnitio ns of the cup pro duct a nd of f ∗ △ that, for any class es Ξ ′ 1 , Ξ ′ 2 ∈ H ∗ ∆ , 2 ( e L ′ , E ′ ), the following equa lit y holds in H ∗ ∆ , 1 ( e L, E ), f ∗ △ Ξ ′ 1 ⊔ f ∗ △ Ξ ′ 2 = f ∗ △ (Ξ ′ 1 ⊔ Ξ ′ 2 ) . W e need only prov e the prop o s ition for f ∗ D . Reca ll that if ξ ′ 1 = α ′ 1 ⊗ φ ′ 1 and ξ ′ 2 = α ′ 2 ⊗ φ ′ 2 , then ξ ′ 1 ∧ ξ ′ 2 = { φ ′ 1 , φ ′ 2 } α ′ 1 ∧ α ′ 2 , a nd we extend to all ξ ′ 1 and ξ ′ 2 by line a rity . Let Ψ ′ 1 and Ψ ′ 2 be the cohomolo gy classes determined by ξ ′ 1 and ξ ′ 2 . Then Q x ( e f ∗ ( ξ ′ 1 ) , e f ∗ ( ξ ′ 2 )) = Z e L x e f ∗ D ( ξ ′ 1 ) ∧ e f ∗ D ( ξ ′ 2 ) = Z e L x e f ∗ D (Ψ ′ 1 ) ∧ e f ∗ D (Ψ ′ 2 ) = Z e L x ( W ◦ f ∗ △ ◦ I Ψ ′ 1 ) ∧ ( W ◦ f ∗ △ ◦ I Ψ ′ 2 ) = Z e L x W (( f ∗ △ ◦ I Ψ ′ 1 ) ⊔ ( f ∗ △ ◦ I Ψ ′ 2 )) = < [ e L ] , ( f ∗ △ ◦ I Ψ ′ 1 ) ⊔ ( f ∗ △ ◦ I Ψ ′ 2 ) > = < [ e L ] , f ∗ △ ( I Ψ ′ 1 ⊔ I Ψ ′ 2 ) > = < [ f △ , ∗ e L ] , ( I Ψ ′ 1 ⊔ I Ψ ′ 2 ) > = 30 M.-T. BENAMEUR AND J. L. HEITSCH < [ e L ′ ] , I Ψ ′ 1 ⊔ I Ψ ′ 2 > = Z e L ′ f ( x ) Ψ ′ 1 ∧ Ψ ′ 2 = Z e L ′ f ( x ) ξ ′ 1 ∧ ξ ′ 2 = Q ′ f ( x ) ( ξ ′ 1 , ξ ′ 2 ) .  7. Induced bundles W e as s ume ag ain that F and F ′ are Riemannian foliations, and in this section take e f ∗ to be e f ∗ = f ( i,ω ) = π 1 , ∗ ◦ e ω ◦ p ∗ f : W ∗ −∞ ( F ′ , E ′ ) → W ∗ −∞ ( F, E ) . The restriction o f e f ∗ gives isomor phis ms from Ker(∆ E ′ ℓ ), Ker(∆ E ′ + ℓ ), a nd Ker(∆ E ′ − ℓ ) to their images which we denote by Im e f ∗ = e f ∗ (Ker(∆ E ′ ℓ )) , Im e f ∗ + = e f ∗ (Ker(∆ E ′ + ℓ )) , and Im e f ∗ − = e f ∗ (Ker(∆ E ′ − ℓ )) , resp ectively . W e us e similar notation fo r the map e g ∗ : W ∗ −∞ ( F, E ) → W ∗ −∞ ( F ′ , E ′ ). Note that for x ∈ M , g f ( x ) 6 = x in gener al, which crea tes technical problems. T o deal with this, c ho ose a leafwise homotopy equiv a le nc e h : M × I → M betw een the identit y map on M and g f . Recall the smo oth leafwise path γ x from x to g f ( x ), g iven by γ x ( t ) = h ( x, t ). It deter mines the isometry R x : e L gf ( x ) → e L x , given by R x ([ γ ]) = [ γ · γ x ]. F or a ny Sob olev space W ∗ s ( e L x , E ), R x determines the isometry R ∗ x : W ∗ s ( e L x , E ) → W ∗ s ( e L gf ( x ) , E ) . In particular for s = 0, it giv es the isometr y , R ∗ x : L 2 ( e L x ; ∧ T ∗ F s ⊗ E ) → L 2 ( e L gf ( x ) ; ∧ T ∗ F s ⊗ E ) . W e sha ll a lso co nsider the smo oth leafwise paths γ ′ x ′ from x ′ ∈ M ′ to f g ( x ′ ) given by γ ′ x ′ ( t ) = h ′ ( x ′ , t ) where h ′ is a ﬁxed leafwise homotop y b etw een the iden tity of M ′ and f g . Given x ∈ M , deﬁne the iso metry R ′ x : e L ′ f ( x ) → e L ′ f ( x ) to be R ′ x [ γ ′ ] = [ γ ′ · f ( γ x ) − 1 · γ ′ f ( x ) ] . This induces the isometry R ′∗ : L 2 ( e L ′ f ( x ) ; ∧ T ∗ F ′ s ⊗ E ′ ) → L 2 ( e L ′ f ( x ) ; ∧ T ∗ F ′ s ⊗ E ′ ) . Note that the compos ition R ′ x ◦ ˇ f ◦ R x ◦ ˇ g : e L ′ f ( x ) → e L ′ f ( x ) is homotopic to the identit y map, since for [ γ ′ ] ∈ e L ′ f ( x ) , R ′ x ◦ ˇ f ◦ R x ◦ ˇ g ([ γ ′ ]) = [ f g ( γ ′ ) · f ( γ x ) · f ( γ x ) − 1 · γ ′ f ( x ) ] = [ f g ( γ ′ ) · γ ′ f ( x ) ] . Set L t ( γ ′ ) = ( γ ′ − 1 r ( γ ′ ) | [0 ,t ] ) · f g ( γ ′ ) · γ ′ f ( x ) . Then L 0 ( γ ′ ) = f g ( γ ′ ) · γ ′ f ( x ) , and L 1 ( γ ′ ) = γ ′ − 1 r ( γ ′ ) · f g ( γ ′ ) · γ ′ f ( x ) . Now s ( L 1 ( γ ′ )) = s ( γ ′ ) and r ( L 1 ( γ ′ )) = r ( γ ′ ), and h ′ provides a leafwise homoto py b etw een L 1 ( γ ′ ) a nd γ ′ , so they deﬁne the s ame element in e L ′ f ( x ) . Thus L t induces a homotopy from R ′ x ◦ ˇ f ◦ R x ◦ ˇ g to the iden tit y map. F or x ∈ M , consider the compo sition ( P ′ ℓ e g ∗ R ∗ x P ℓ e f ∗ R ′∗ x P ′ ℓ ) f ( x ) : L 2 ( e L ′ f ( x ) ; ∧ T ∗ F ′ s ⊗ E ′ ) → L 2 ( e L ′ f ( x ) ; ∧ T ∗ F ′ s ⊗ E ′ ) . Since R ′ x ◦ ˇ f ◦ R x ◦ ˇ g : e L ′ f ( x ) → e L ′ f ( x ) is homotopic to the identit y and P ∗ ℓ is the iden tit y on cohomo logy , it follows that ( P ′ ℓ e g ∗ R ∗ x P ℓ e f ∗ R ′∗ x P ′ ℓ ) f ( x ) induces the identit y on cohomolo g y , whic h is naturally isomorphic to Ker(∆ E ′ ℓ ) f ( x ) = Im( P ′ ℓ ) f ( x ) . So its restriction ( P ′ ℓ e g ∗ R ∗ x P ℓ e f ∗ R ′∗ x P ′ ℓ ) f ( x ) : Ker(∆ E ′ ℓ ) f ( x ) → Ker(∆ E ′ ℓ ) f ( x ) is the iden tity . TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 31 W e now sho w that Im e f ∗ + determines a smooth subbundle of A ℓ (2) ( F s , E ) ov er M /F . Set π f + = e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ . Then for each x ∈ M , ( π f + ) x : L 2 ( e L x ; ∧ T ∗ F s ⊗ E ) → L 2 ( e L x ; ∧ T ∗ F s ⊗ E ) is bounded and leafwise smo othing since π ′ + and P ℓ are, and R ′∗ x , R ∗ x , e f ∗ and e g ∗ are b o unded maps. W e leav e it to the reader to show tha t π f + is G inv ariant using the equalit y [ g f ( γ ) · γ x ] = [ γ y · γ ] for any γ ∈ G with s ( γ ) = x and r ( γ ) = y . As ab ove, this equality ho lds since the tw o paths star t and end at the same p oints and a leafwise ho motopy be tween them can be co nstructed using the lea fwise homotopy equiv alence h . W e extend π f + to an A ∗ ( M ) equiv aria nt op er a tor on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E in the usual w a y . Prop ositi o n 7. 1. π f + : A ℓ (2) ( F s , E ) → Im e f ∗ + is a tr ansversely smo oth idemp otent. Pr o of. Fir st we hav e, ( π f + ) 2 = e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ = e f ∗ R ′∗ π ′ + P ′ ℓ e g ∗ R ∗ P ℓ e f ∗ R ′∗ P ′ ℓ π ′ + e g ∗ R ∗ P ℓ = e f ∗ R ′∗ ( π ′ + ) 2 e g ∗ R ∗ P ℓ = e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ = π f + , since π ′ + = π ′ + P ′ ℓ = P ′ ℓ π ′ + , and for ea ch x ∈ M , ( P ′ ℓ e g ∗ R ∗ P ℓ e f ∗ R ′∗ P ′ ℓ ) f ( x ) : Ker(∆ E ′ ℓ ) f ( x ) → Ker(∆ E ′ ℓ ) f ( x ) is the iden tit y map, and Ker(∆ E ′ ℓ ) ⊃ Im( π ′ + ). As P ℓ is transversely smo oth, we need only show that e f ∗ R ′∗ π ′ + e g ∗ R ∗ is transversely smo oth. Let ∇ E and ∇ E ′ be the lea fwise ﬂat connectio ns on E and E ′ and ∇ F ′ and ∇ F be the Riemannian connections on T ∗ F ′ and T ∗ F , resp ectively . Denote b y ∇ ν and ∇ ′ ν the quasi-connectio ns on C ∞ ( ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E ) and C ∞ ( ∧ ν ′∗ s ⊗ ∧ T ∗ F ′ s ⊗ E ′ ) constructed from ∇ F ⊗ ∇ E , and ∇ F ′ ⊗ ∇ E ′ , respectively . Now supp ose H is any G inv ariant op e rator of degree zero on ∧ T ∗ F s ⊗ E , e.g. H = e f ∗ R ′∗ π ′ + e g ∗ R ∗ . If X ∈ C ∞ ( T F ), then since H and ∇ ν are G in v ariant, ∂ X ν ( H ) = 0. A v ector ﬁeld Y on M is a Γ vector ﬁeld provided that for any X ∈ C ∞ ( T F ), [ X, Y ] ∈ C ∞ ( T F ). If Y ∈ C ∞ ( ν ) is a Γ vector ﬁeld, it is inv ariant under the parallel tra nslation deﬁned by F , so ∂ Y ν ( H ) is G inv ariant. Globally deﬁned Γ vector ﬁelds ra rely exist. The r estriction of a global vector ﬁeld to an op en subset will b e called a lo c a l extendable vector ﬁeld. Such lo cal vector ﬁelds have all their deriv atives b ounded. Any loca l Γ vector ﬁeld may , after a slight reduction in its domain o f deﬁnition, b e ex tended to a g lobal vector ﬁeld. Finally , a bounded function (on M ) times a b ounded leafwise smo othing op erator yields a bo unded leafwise smo othing op era to r. With this in mind, the problem of showing that such an H is tr ansversely smoo th ma y be recast as follows (with the pro of left to the r eader). Lemma 7.2. Supp ose H is a de gr e e zer o G invariant A ∗ ( M ) e quivariant (homo gene ous of de gr e e 0 ) b ounde d le afwise s m o othing op er ator on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E . Then H is tr ansversely smo oth if and only if for al l lo c al extendable Γ ve ctor ﬁelds Y 1 , ..., Y m ∈ C ∞ ( ν ) , the op er ator ∂ Y 1 ν ...∂ Y m ν ( H ) is a b ounde d le afwise smo othing op er ator on ∧ T ∗ F s ⊗ E . Note that the expres sion e f ∗ R ′∗ π ′ + e g ∗ R ∗ ∇ ν makes sense a s e f ∗ R ′∗ π ′ + e g ∗ R ∗ is a well deﬁned A ∗ ( M ) e q uiv ari- ant op er ator on ∧ ν ∗ s ⊗ ∧ T ∗ F s ⊗ E . Note further that the expressio n R ∗ ∇ ν do es not make sense in genera l. How ever, r estricted to a ny suﬃciently small tra nsverse submanifold, g f is a diﬀeomorphism onto its imag e , so ( g f ) − 1 is well deﬁned on this image. This makes it p ossible to prov e the follo wing. Lemma 7.3. Supp ose Y ∈ ν x , then e f ∗ R ′∗ π ′ + e g ∗ R ∗ ∇ ν Y = e f ∗ R ′∗ π ′ + e g ∗ ∇ ν h ∗ ( Y ) R ∗ , wher e h ∗ ( Y ) ∈ ν gf ( x ) is the p ar al lel tr anslate of Y along γ x . If Y ′ ∈ ν ′ f ( x ) , then e f ∗ R ′∗ ∇ ′ ν Y ′ π ′ + e g ∗ R ∗ = e f ∗ ∇ ′ ν h ′ ∗ ( Y ′ ) R ′∗ π ′ + e g ∗ R ∗ , wher e h ′ ∗ ( Y ′ ) ∈ ν ′ f ( x ) is the p ar al lel tr ans- late of Y ′ along f ( γ x ) − 1 · γ ′ f ( x ) . 32 M.-T. BENAMEUR AND J. L. HEITSCH Pr o of. L e t ( U x , γ , V ) be a lo ca l chart containing [ γ ] ∈ e L x , and ( U gf ( x ) , γ γ − 1 x , V ) a lo cal chart ab out [ γ γ − 1 x ] ∈ e L gf ( x ) . T o compute e f ∗ π ′ + e g ∗ R ∗ ∇ ν Y , w e may restric t o ur atten tion to s − 1 ( T ), where T is any submanifold o f M whic h has Y tangen t to it. W e may ass ume that T ⊂ U x , and g f restricted to T is a diﬀeo morphism onto its imag e g f ( T ), which is also a trans verse submanifold, with g f ( T ) ⊂ U gf ( x ) . Now s − 1 ( T ) ∩ ( U x , γ , V ) ≃ V and s − 1 ( g f ( T )) ∩ ( U gf ( x ) , γ γ − 1 x , V ) ≃ V , and the diﬀeomorphisms with V a r e just given by the r estriction of the target ma p r . In a ddition,  ∇ ν Y | s − 1 ( T )  ◦ r ∗ = r ∗ ◦  ( ∇ F ⊗ ∇ E ) ν Y γ  and  ∇ ν h ∗ ( Y ) | s − 1 ( g f ( T ))  ◦ r ∗ = r ∗ ◦  ( ∇ F ⊗ ∇ E ) ν h ∗ ( Y ) γ γ − 1 x  , where ( ∇ F ⊗ ∇ E ) ν is the quasi-connection o n ∧ T ∗ F ⊗ E ov er M , constr ucted using the normal bundle ν of F , Y γ is the pa rallel tra nslation of Y a lo ng γ , and h ∗ ( Y ) γ γ − 1 x is the pa rallel tra nslation o f h ∗ ( Y ) along γ γ − 1 x . So Y γ = h ∗ ( Y ) γ γ − 1 x . The restrictio n of R , R T : s − 1 ( g f ( T )) → s − 1 ( T ) is well deﬁned, since ( g f ) − 1 is well deﬁned o n g f ( T ). In fact, it is a diﬀeomor phism whic h lo cally is just r − 1 ◦ r . R T induces the map on leafwise diﬀere ntial forms R ∗ T : C ∞ ( s − 1 ( T ); ∧ T ∗ F s ⊗ E ) → C ∞ ( s − 1 ( g f ( T )); ∧ T ∗ F s ⊗ E ) , which extends to the op erator R ∗ T : C ∞ ( s − 1 ( T ); ∧ T ∗ ( s − 1 ( T )) ⊗ E ) → C ∞ ( s − 1 ( g f ( T )); ∧ T ∗ ( s − 1 ( g f ( T ))) ⊗ E ) , It is clear that R ∗ T ∇ ν Y is a well deﬁned ma p, a nd since lo cally R T = r − 1 ◦ r , we hav e R ∗ T ∇ ν Y = ∇ ν h ∗ ( Y ) R ∗ T . But R ∗ T is just the restriction o f R ∗ to s − 1 ( T ), so R ∗ ∇ ν Y = ∇ ν h ∗ ( Y ) R ∗ . The second s tatement is prov ed in the same w ay .  Prop ositi o n 7. 4. The op er ators e f ∗ ∇ ′ ν − ∇ ν e f ∗ and e g ∗ ∇ ν − ∇ ′ ν e g ∗ ar e le afwise diﬀe r ential op er ators 1 , whose c omp osition with a b oun de d le afwise smo othing op er ator is again a b ounde d le afwise smo othing op er ator. Pr o of. W e will only do the pro o f fo r e f ∗ as the proo f for e g ∗ is the same. Let ω ⊗ α ⊗ φ ∈ C ∞ c ( ∧ ν ′∗ s ⊗ ∧ T ∗ F ′ s ⊗ E ′ ), with ω ∈ s ∗ A k ( M ′ ), α ∈ C ∞ c ( G ′ ; ∧ T ∗ F ′ s ), a nd φ ∈ C ∞ c ( G ′ ; E ′ ). Then d ′ s ( ω ⊗ α ⊗ φ ) = ( − 1 ) k ω ⊗ d ′ s ( α ⊗ φ ) . Now e f ∗ ∇ ′ ν ( ω ⊗ α ⊗ φ ) = e f ∗ ( d M ′ ω ⊗ α ⊗ φ + ( − 1) k ω ⊗ ∇ ν F ′ α ⊗ φ + ( − 1) k ω ⊗ α ⊗ ∇ ν E ′ φ ) = d M f ∗ ω ⊗ e f ∗ α ⊗ e f ∗ φ + ( − 1) k f ∗ ω ⊗ e f ∗ ∇ ν F ′ α ⊗ e f ∗ φ + ( − 1) k f ∗ ω ⊗ e f ∗ α ⊗ e f ∗ ∇ ν E ′ φ. On the other ha nd, ∇ ν e f ∗ ( ω ⊗ α ⊗ φ ) = d M f ∗ ω ⊗ e f ∗ α ⊗ e f ∗ φ + ( − 1) k f ∗ ω ⊗ ∇ ν F e f ∗ α ⊗ e f ∗ φ + ( − 1) k f ∗ ω ⊗ e f ∗ α ⊗ ∇ ν E e f ∗ φ. Thu s ( e f ∗ ∇ ν ′ − ∇ ν e f ∗ )( ω ⊗ α ⊗ φ ) = ( − 1 ) k f ∗ ω ⊗  ( e f ∗ ∇ ν F ′ − ∇ ν F e f ∗ ) α ⊗ e f ∗ φ + e f ∗ α ⊗ ( e f ∗ ∇ ν E ′ − ∇ ν E e f ∗ ) φ  , which contains no diﬀerentiation of ω , so e f ∗ ∇ ′ ν − ∇ ν e f ∗ is indee d a leafwise o per ator, as are its individua l comp onents e f ∗ ∇ ν F ′ − ∇ ν F e f ∗ and e f ∗ ∇ ν E ′ − ∇ ν E e f ∗ . Next consider the lea fwise oper ator e f ∗ ∇ ν F ′ − ∇ ν F e f ∗ acting on C ∞ ( ∧ T ∗ F ′ s ). Set d ν = p ν d G and d ′ ν = p ν ′ d G ′ . In lo ca l coor dinates, we may wr ite ∇ ν F ′ and ∇ ν F as p ν ′ ( d G ′ + Θ F ′ ) and p ν ( d G + Θ F ), r esp ectively , where Θ F ′ and Θ F are leafwise diﬀeren tial opera tors (of or der zero) with co e ﬃcie n ts in T ∗ G ′ and T ∗ G . T he n we ha ve e f ∗ ∇ ν F ′ − ∇ ν F e f ∗ = e f ∗ p ν ′ ( d G ′ + Θ F ′ ) − p ν ( d G + Θ F ) e f ∗ = 1 By a leafwi se diﬀeren tial op erator, it is sometimes meant, here and in the sequel, oper ators generat ed lo cally by ρ 7→ f ∗ ∂ ρ ∂ x i where the x i s are l eafwise v ari ables. TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 33 e f ∗ d ′ ν − d ν e f ∗ + e f ∗ p ν ′ Θ F ′ − p ν Θ F e f ∗ . Lemma 7.5 . e f ∗ d ′ ν − d ν e f ∗ and e g ∗ d ν − d ′ ν e g ∗ ar e le afwise op er ators, with e f ∗ d ′ ν − d ν e f ∗ = − e f ∗ d ′ s + d s e f ∗ , and e g ∗ d ν − d ′ ν e g ∗ = − e g ∗ d s + d ′ s e g ∗ . Pr o of. Ag ain we only prove this only fo r e f ∗ d ′ ν − d ν e f ∗ . As e f ∗ ∇ ν F ′ − ∇ ν F e f ∗ and e f ∗ p ν ′ Θ F ′ − p ν Θ F e f ∗ are lea fwise op erator s , so is e f ∗ d ′ ν − d ν e f ∗ . On G × B k we hav e the foliatio n F s × B k , with all its baggage. In par ticula r, w e use the pr o duct metric on G × B k , and we hav e the transverse der iv ative d B ν . Lo cal charts o n G × B k are g iven by subsets of the form ( U, γ , V ) × B k , where ( U, γ , V ) is a lo cal chart for G . It is clear that in these lo ca l co o rdinates, d ν and d B ν hav e exactly the same fo r m. It is then ob vious from the deﬁnitions o f π 1 , ∗ and e ω , that d ν ( π 1 , ∗ ◦ e ω ) = ( π 1 , ∗ ◦ e ω ) d B ν and d s ( π 1 , ∗ ◦ e ω ) = ( π 1 , ∗ ◦ e ω ) d B s , where d B s is the leafwise deriv ative asso cia ted to the foliation F s × B k . As e f ∗ = π 1 , ∗ ◦ e ω ◦ p ∗ f , to prove that e f ∗ d ′ ν − d ν e f ∗ = − e f ∗ d ′ s + d s e f ∗ , w e need only prove that p ∗ f d ′ ν − d B ν p ∗ f = − p ∗ f d ′ s + d B s p ∗ f . This is purely a lo cal question, and the usual pro of shows that we need only prove it for compactly suppo rted functions on G ′ . Denote by p ′ s the pro jection p ′ s : T G ′ → T F ′ s determined by the splitting T G ′ = ν ′ s ⊕ T F ′ s , and by p B F : T ( G × B k ) → T ( F s × B k ) a nd p B ν : T ( G × B k ) → ν B , the pro jections determined by the s plitting T ( G × B k ) = ν B ⊕ T ( F s × B k ). Let φ ∈ C ∞ c ( G ′ ). If X ∈ T ( F s × B k ), then p B ν ( X ) = 0, and p f ∗ X ∈ T F ′ s , so p ′ ν p f ∗ ( X ) = 0. Thus ( p ∗ f d ′ ν φ − d B ν p ∗ f φ )( X ) = p ∗ f (( d ′ ν φ ) p f ∗ ( X )) − ( d G × B k p ∗ f φ ) p B ν ( X ) = p ∗ f (( d G ′ φ ) p ′ ν p f ∗ ( X )) = 0 . Next, suppo se X ∈ ν B , the normal bundle to F s × B k , and note that p f ∗ X is not necessarily in ν ′ s . Then ( p ∗ f d ′ ν φ )( X ) = p ∗ f (( d ′ ν φ )( p f ∗ X )) = p ∗ f (( d G ′ φ )( p ′ ν p f ∗ X )) = p ∗ f (( d G ′ φ )( p f ∗ X )) − p ∗ f (( d G ′ φ )( p ′ s p f ∗ X )) = ( d G × B k p ∗ f φ )( X ) − p ∗ f (( d ′ s φ )( p f ∗ X )) = ( d G × B k p ∗ f φ )( p B ν X ) − p ∗ f (( d ′ s φ )( p f ∗ X )) = ( d B ν p ∗ f φ − p ∗ f d ′ s φ )( X ) . So ( p ∗ f d ′ ν − d B ν p ∗ f ) φ = ( − p ∗ f d ′ s φ ) p B ν = ( − p ∗ f d ′ s φ )( I − p B F ) = − p ∗ f d ′ s φ + ( p ∗ f d ′ s φ ) p B F = − p ∗ f d ′ s φ + d B s p ∗ f φ, since, restricted to T ( F s × B k ), p ∗ f d ′ s φ = d B s p ∗ f φ . Thu s e f ∗ d ′ ν − d ν e f ∗ = − e f ∗ d ′ s + d s e f ∗ .  So e f ∗ ∇ ν F ′ − ∇ ν F e f ∗ = d s e f ∗ − e f ∗ d ′ s + e f ∗ p ν ′ Θ F ′ − p ν Θ F e f ∗ , a leafwise diﬀeren tia l op erator (of o rder at most one). Finally , consider e f ∗ ∇ ν E ′ − ∇ ν E e f ∗ acting on C ∞ c ( E ′ ). In lo cal co o rdinates, and with r esp ect to lo ca l framings of E ′ and E , we may write ∇ E ′ = d G ′ + Θ E ′ and ∇ E = d G + Θ E , where Θ E ′ and Θ E are leafwise diﬀerential op erator s (of order zero) with c o eﬃcien ts in T ∗ G ′ and T ∗ G . T he n e f ∗ ∇ ν E ′ − ∇ ν E e f ∗ = e f ∗ p ν ′ ∇ E ′ − p ν ∇ E e f ∗ = e f ∗ p ν ′ ( d G ′ + Θ E ′ ) − p ν ( d G + Θ E ) e f ∗ = e f ∗ d ′ ν − d ν e f ∗ + e f ∗ p ν ′ Θ E ′ − p ν Θ E e f ∗ = − e f ∗ d ′ s + d s e f ∗ + e f ∗ p ν ′ Θ E ′ − p ν Θ E e f ∗ , since the proo f of Lemma 7.5 abov e extends to show tha t e f ∗ d ′ ν − d ν e f ∗ = − e f ∗ d ′ s + d s e f ∗ , with r esp ect to the lo cal framings. So e f ∗ ∇ ν E ′ − ∇ ν E e f ∗ = d s e f ∗ − e f ∗ d ′ s + e f ∗ p ν ′ Θ E ′ − p ν Θ E e f ∗ , also a leafwise diﬀeren tia l opera tor (of o rder at most o ne). Now obs erve that if we use co ordina tes on G ′ and G and fra mings of E ′ and E coming from co ordiantes on M ′ and M , and fr a mings of E ′ and E over M ′ and M , all of who se deriv atives a re uniformly bo unded, 34 M.-T. BENAMEUR AND J. L. HEITSCH then d s e f ∗ − e f ∗ d ′ s + e f ∗ p ν ′ Θ F ′ − p ν Θ F e f ∗ and d s e f ∗ − e f ∗ d ′ s + e f ∗ p ν ′ Θ E ′ − p ν Θ E e f ∗ are (a t w orst) o rder one diﬀerential op era to rs which have a ll of their deriv atives unifor mly bounded. Th us e f ∗ ∇ ν ′ − ∇ ν e f ∗ and all its deriv atives deﬁne b ounded o p erators from W ∗ s ( F ′ , E ′ ) to W ∗ s − 1 ( F, E ) for e a ch s , and so their comp o s itions with a bounded leafwise smo othing op era tor ar e aga in b ounded leafwise smo o thing oper a tors.  Note that the pr o of ab ov e als o proves that the comp ositio n of Υ f = e f ∗ ∇ ′ ν − ∇ ν e f ∗ or Υ g = e g ∗ ∇ ν − ∇ ′ ν e g ∗ with a transversely smo oth ope rator is a gain a transversely smo oth o p er ator. By virtue of Lemma 7.2, we will be us ing only lo cal extendable Γ vector ﬁelds Y 1 , ..., Y m in proving that e f ∗ R ′∗ π ′ + e g ∗ R ∗ is transversely smo oth. Thus we ma y r ewrite Lemma 7.3 a s e f ∗ R ′∗ π ′ + e g ∗ R ∗ ∇ ν = e f ∗ R ′∗ π ′ + e g ∗ ∇ ν R ∗ and e f ∗ ∇ ′ ν R ′∗ π ′ + e g ∗ R ∗ = e f ∗ R ′∗ ∇ ′ ν π ′ + e g ∗ R ∗ . Then ∂ ν ( e f ∗ R ′∗ π ′ + e g ∗ R ∗ ) = [ ∇ ν , e f ∗ R ′∗ π ′ + e g ∗ R ∗ ] = ∇ ν e f ∗ R ′∗ π ′ + e g ∗ R ∗ − e f ∗ R ′∗ π ′ + e g ∗ R ∗ ∇ ν = e f ∗ R ′∗ ∇ ′ ν π ′ + e g ∗ R ∗ − e f ∗ R ′∗ π ′ + ∇ ′ ν e g ∗ R ∗ − Υ f R ′∗ π ′ + e g ∗ − e f ∗ R ′∗ π ′ + Υ g R ∗ . So, 7.6. ∂ Y 1 ν ( e f ∗ R ′∗ π ′ + e g ∗ R ∗ ) = i b Y 1 e f ∗ R ′∗ ∂ ν ′ ( π ′ + ) e g ∗ R ∗ − ( i b Y 1 Υ f ) R ′∗ π ′ + e g ∗ R ∗ − e f ∗ R ′∗ π ′ + ( i b Y 1 Υ g ) R ∗ . By assumption, ∂ ν ′ ( π ′ + ) is a bo unded leafwise smoo thing op erator , so i b Y 1 e f ∗ R ′∗ ∂ ν ′ ( π ′ + ) e g ∗ R ∗ is also. The op erator s i b Y 1 Υ f , a nd i b Y 1 Υ g are leafwise o per ators which hav e all their deriv atives b ounded, so their com- po sition with a b ounded leafwise smo othing ope r ator (e.g . R ′∗ π ′ + e g ∗ ) is aga in a b ounded leafwise smo othing op erator . Th us for any lo cal extenda ble Γ vector ﬁeld Y 1 on M , ∂ Y 1 ν ( e f ∗ R ′∗ π ′ + e g ∗ R ∗ ) is a b ounded leafwise smo othing op e rator. T o contin ue the induction ar gument, we need the following. Lemma 7.7. L et Y ∈ C ∞ ( ν ) b e a lo c al exten dable Γ ve ctor ﬁeld, then ther e is a b ounde d ve ctor ﬁeld Z ′ on G ′ so that for any ([ γ ] , t ) ∈ G × B k , i b Y ([ γ ] ,t ) p ∗ f = p ∗ f i Z ′ ( p f ([ γ ] ,t )) . Given this, then at ([ γ ] , t ) ∈ G × B k we hav e i b Y 1 p ∗ f R ′∗ ∂ ν ′ ( π ′ + )([ γ ] , t ) = i b Y 1 ([ γ ] ,t ) p ∗ f R ′∗ ∂ ν ′ ( π ′ + ) = p ∗ f ( R ′∗ i Z ′ 1 ( p f ([ γ ] ,t )) ∂ ν ′ ( π ′ + )) = p ∗ f ( R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + ) p f ([ γ ] , t )) . That is, i b Y 1 p ∗ f R ′∗ ∂ ν ′ ( π ′ + ) = p ∗ f R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + ) so i b Y 1 e f ∗ R ′∗ ∂ ν ′ ( π ′ + ) e g ∗ R ∗ = e f ∗ R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + ) e g ∗ R ∗ . Lemma 7.8 . If ρ is a t r ansversely smo oth op er ator on A ∗ (2) ( F ′ s , E ′ ) and Z ′ is a b ounde d ve ctor ﬁeld on G ′ , then i Z ′ ∂ ν ′ ( ρ ) is a tr ansversely sm o oth op er ator. Pr o of. Since i Z ′ ∂ ν ′ ( ρ ) = i p ν ′ ( Z ′ ) ∂ ν ′ ( ρ ), w e may assume that Z ′ = P j g j b X ′ j , wher e X ′ j is a ﬁnite lo cal basis for the v ector ﬁelds o n M ′ , and the g j are smo o th functions whic h are g lobally b ounded along with all their deriv atives. Then i Z ′ ∂ ν ′ ( ρ ) = P j g j i b X ′ j ∂ ν ′ ( ρ ) = P j g j ∂ X ′ j ν ′ ( ρ ), which is cle a rly tra nsversely smo o th s ince the g j and all their deriv atives are globally bo unded.  Using Equation 7.6, we ha ve ∂ Y 2 ν ∂ Y 1 ν ( e f ∗ R ′∗ π ′ + e g ∗ R ∗ ) = ∂ Y 2 ν  e f ∗ R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + ) e g ∗ R ∗ − ( i b Y 1 Υ f ) R ′∗ π ′ + e g ∗ R ∗ − e f ∗ R ′∗ π ′ + ( i b Y 1 Υ g ) R ∗  . Repe a ting the argument ab ov e we get ∂ Y 2 ν ( e f ∗ R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + ) e g ∗ R ∗ ) = i b Y 2 e f ∗ R ′∗ ∂ ν ′ ( i Z ′ 1 ∂ ν ′ ( π ′ + )) e g ∗ R ∗ − ( i b Y 2 Υ f ) R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + ) e g ∗ R ∗ − e f ∗ R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + )( i b Y 2 Υ g ) R ∗ = e f ∗ R ′∗ i Z ′ 2 ∂ ν ′ ( i Z ′ 1 ∂ ν ′ ( π ′ + )) e g ∗ R ∗ − ( i b Y 2 Υ f ) R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + ) e g ∗ R ∗ − e f ∗ R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + )( i b Y 2 Υ g ) R ∗ , TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 35 which is b ounded and leafwise smo othing since i Z ′ 1 ∂ ν ′ ( π ′ + ) is transv ersely smoo th. As ∂ Y 2 ν is a deriv ation, w e ha v e ∂ Y 2 ν (( i b Y 1 Υ f ) R ′∗ π ′ + e g ∗ R ∗ ) = ∂ Y 2 ν ( i b Y 1 Υ f )( R ′∗ π ′ + e g ∗ R ∗ ) + ( i b Y 1 Υ f ) ∂ Y 2 ν ( R ′∗ π ′ + e g ∗ R ∗ ) . The op er ators ∂ Y 2 ν ( i b Y 1 Υ f ) and i b Y 1 Υ f comp osed with bounded leafwise smo othing op era tors pro duce b o unded leafwise smo othing op er ators. As R ′∗ π ′ + e g ∗ R ∗ and ∂ Y 2 ν ( R ′∗ π ′ + e g ∗ R ∗ ) ar e b ounded leafwise smo othing op er a- tors, this term is a b ounded leafwise smo othing op erator . Similarly for the third term. Now, a straight forward induction ar gument ﬁnishes the pro o f, modulo the pro of of Lemmas 7.7. Pr o of. T o prove Lemma 7.7, we “factor thro ugh the graph” . In particular , consider the map p f ,G : G × B k → G × B k × G ′ given b y p f ,G ( γ , t ) = ( γ , t, p f ( γ , t )) which is a diﬀeomor phism onto its image. Denote by F ′ G,s the foliation of G × B k × G ′ whose leaves are of the form e L × B k × e L ′ , and denote b y E ′ G the pull ba ck of E ′ under the pr o jection G × B k × G ′ → G ′ . W e wan t to co nstruct a tra nsversely smoo th idemp otent π ′ + ,G which will play the role of π ′ + . Howev er , π ′ + ,G will not be acting on A ∗ (2) ( F ′ G,s , E ′ G ) ov er M × M ′ , but ra ther on the space denoted A ∗ (2) ( F ′ G,s , ∧ T ∗ F ′ s ⊗ E ′ G ) ov er M × M ′ , which ass o ciates to each ( x, x ′ ) the Hilbert s pace L 2 ( e L ′ x ′ ; ∧ T ∗ F ′ s ⊗ E ′ ). Then ( π ′ + ,G ) ( x,x ′ ) := ( π ′ + ) x ′ : L 2 ( e L ′ x ′ ; ∧ T ∗ F ′ s ⊗ E ′ ) → L 2 ( e L ′ x ′ ; ∧ T ∗ F ′ s ⊗ E ′ ) is w ell deﬁned, and it is o bvious that π ′ + ,G is a tr a nsversely smo oth idempotent, and ha s imag e Ker(∆ E ′ + ℓ ). T o deﬁne the action e p ∗ f ,G of e p f ,G on A ∗ (2) ( F ′ G,s , ∧ T ∗ F ′ s ⊗ E ′ G ), we ma y consider this spac e as a subspace of all the forms on G × B k × G ′ by using the pull bac k of the pro jection G × B k × G ′ → G ′ . When w e do so, e p ∗ f ,G is just the usual induced map, and on each ﬁb er L 2 ( e L ′ f ( x ) ; ∧ T ∗ F ′ s ⊗ E ′ ) it equals p ∗ f . Next deﬁne e g ∗ G : A ∗ (2) ( F s , E ) → A ∗ (2) ( F ′ G,s , ∧ T ∗ F ′ s ⊗ E ′ G ) to be ( e g ∗ G ) g ( x ′ ) := ( e g ∗ ) g ( x ′ ) : L 2 ( e L g ( x ′ ) ; ∧ T ∗ F s ⊗ E ) → L 2 ( e L ′ x ′ ; ∧ T ∗ F ′ s ⊗ E ′ ) , for each x ′ ∈ M ′ . Finally , the action of R ′∗ on A ∗ (2) ( F ′ s , E ′ ) extends easily to an actio n on A ∗ (2) ( F ′ G,s , ∧ T ∗ F ′ s ⊗ E ′ G ). Then p ∗ f ,G R ′∗ π ′ + ,G e g ∗ G R ∗ = p ∗ f R ′∗ π ′ + e g ∗ R ∗ , a nd we may w ork w ith G × B k × G ′ , F ′ G,s , p ∗ f ,G , e g ∗ G , and π ′ + ,G in place of G ′ , F ′ , p ∗ f , e g ∗ , a nd π ′ + , resp ectively . As p f ,G is a diﬀeomorphism onto its image, we may push forward vector ﬁelds such a s the b Y i on G (whic h are b ounded b ecause F is Riemannian) to b ounded vector ﬁelds Z ′ i on G × B k × G ′ . Note that these vector ﬁelds are only deﬁned a long the image of p f ,G , but this is suﬃcient for our purp oses, since things of the for m e f ∗ R ′∗ i Z ′ 2 ∂ ν ′ ( i Z ′ 1 ∂ ν ′ ( π ′ + )) e g ∗ R ∗ − ( i b Y 2 Υ f ) R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + ) e g ∗ R ∗ − e f ∗ R ′∗ i Z ′ 1 ∂ ν ′ ( π ′ + )( i b Y 2 Υ g ) R ∗ , are still w ell deﬁned.  This completes the pro of that π f + : A ℓ (2) ( F s , E ) → Im e f ∗ + is a transversely smo oth idemp otent.  The same a rgument shows that Im e f ∗ − , and Im e f ∗ determine smoo th bundles over M /F , denoted π f − and π f resp ectively . In fact, we may use the pro of ab ov e to prov e . Prop ositi o n 7.9. If ρ is a tr ansversely smo oth op er ator on ∧ ν ′∗ s ⊗ F ′ s ⊗ E ′ , t hen e f ∗ R ′∗ ρ e g ∗ R ∗ is a tr ansversely smo oth op er ator on ∧ ν ∗ s ⊗ F s ⊗ E . 36 M.-T. BENAMEUR AND J. L. HEITSCH 8. Induced connections Let ∇ ′ = π ′ + ∇ ′ ν π ′ + be the connection o n the sub-bundle π ′ + = Ker(∆ E ′ + ℓ ), determined by the quasi- connection ∇ ′ ν on ∧ ℓ T ∗ F ′ s ⊗ E ′ . W e now prov e that ∇ ′ induces a connection ∇ o n π f + . Lemma 8.1 . If ξ ′ is a lo c al invariant se ction of π ′ + , then e f ∗ ( ξ ′ ) is a lo c al invariant se ction of π f + . Pr o of. Rec all that for ([ γ ] , t ) ∈ G × B k , p f ([ γ ] , t ) = [ P f ( γ , t ) · ( f ◦ γ )], the co mp o sition of the lea fwise paths P f ( γ , t ) and f ◦ γ , where P f ( γ , t ) : [0 , 1] → L ′ f ( s ( γ )) is the leafwise path giv en b y P f ( γ , t )( s ) = p f ( r ( γ ) , st ) . Then e f ∗ ( ξ ′ )([ γ γ 1 ]) = π 1 , ∗ ◦ e ω (( p ∗ f ξ ′ )([ γ γ 1 ] , t )) = π 1 , ∗ ◦ e ω ( p ∗ f ( ξ ′ ( P f ( γ γ 1 , t ) · ( f ◦ γ γ 1 )))) = π 1 , ∗ ◦ e ω ( p ∗ f ( ξ ′ ( P f ( γ , t ) · ( f ◦ γ ) · ( f ◦ γ 1 )))) = π 1 , ∗ ◦ e ω ( p ∗ f ( ξ ′ ( P f ( γ , t ) · ( f ◦ γ )))) , since ξ ′ is loca l inv ariant. But this last equals π 1 , ∗ ◦ e ω ( p ∗ f ξ ′ ([ γ ] , t )) = e f ∗ ( ξ ′ )([ γ ]) .  Lemma 8.2 . Any lo c al invariant se ction ξ of π f + induc es a lo c al invariant se ction e f −∗ ξ of π ′ + . Pr o of. L e t T be a transversal in M on which ξ is deﬁned. W e may assume that T is so small tha t f | T is a diﬀeomor phism o n to its ima ge T ′ . Then ( e f ∗ ) − 1 : Im e f ∗ + → Ker(∆ E ′ + ℓ ) is well deﬁned over T , and in fact is given by the map R ′∗ P ′ ℓ e g ∗ R ∗ | T . T o s ee this, note that over T ′ , the map R ′∗ P ′ ℓ e g ∗ R ∗ e f ∗ : Ker(∆ E ′ ℓ ) → Ker(∆ E ′ ℓ ) is the iden tit y map, since it induces the identit y map on co homology , and that Ker(∆ E ′ + ℓ ) ⊂ Ker(∆ E ′ ℓ ). F or s implicity , we s hall denote R ′∗ P ′ ℓ e g ∗ R ∗ | T by e f −∗ . F or x ′ ∈ T ′ , deﬁne ( e f −∗ ξ )( x ′ ) ≡ e f −∗ ( ξ ( f − 1 ( x ′ ))) . This gives a w ell deﬁned smooth section on T ′ . Extend it to a loca l in v ariant s ection on a neighborho od of T ′ . W e leave it to the rea der to show that this construc tio n is well deﬁned, that is it doe s not dep e nd on the c hoice of T .  In order to deﬁne the induced c onnection ∇ , we need only deﬁne it on lo cal inv ariant sections, and then extend it using (1) of Deﬁnition 5.7. Deﬁnition 8.3 . L et ξ b e a lo c al invaria nt se ction of π f + . Given X ∈ T M , set X ′ = f ∗ ( X ) . Deﬁne ∇ X ( ξ ) = e f ∗ ( ∇ ′ X ′ ( e f −∗ ξ )) . Extend to ξ ∈ C ∞ ( ∧ T ∗ M ; π f + ) by using ( 1) of Deﬁn ition 5.7. Prop ositi o n 8. 4. ∇ is a c onne ction on π f + . Pr o of. W e need to c heck that the four conditions of Deﬁnition 5.7 are satisﬁed. 5.7(1): F or diﬀerential forms, this is satisﬁed by deﬁnition, so we need to check it for functions . Sp eciﬁca lly , we need that for any lo cal function ω on M which is co nstant o n plaque s of F (i.e. lo cal inv ariant functions), and for an y X ∈ T M , and any local inv ariant sectio n ξ of π f + , ∇ X ( ω ξ ) = d M ω ( X ) ξ + ω ∇ X ξ . If X ∈ T F , this is trivia lly true since bo th sides are zero. Now supp ose that X is transverse to F , with X ′ = f ∗ ( X ), and let T be a tr ansversal of F with X tangent to T . W e may assume that T is so small that f res tricted to T is a diﬀeomo rphism onto its image T ′ , a transversal of F ′ , with in verse f − 1 : T ′ → T . The vector X ′ is tangen t to T ′ , and thanks to Co rollar y 5.21, we hav e ∇ X ( ω ξ ) = e f ∗ ( ∇ ′ X ′ ( e f −∗ ( ω ξ ))) = e f ∗ ( ∇ ′ X ′ (( ω ◦ f − 1 ) e f −∗ ξ )) = e f ∗ h X ′ ( ω ◦ f − 1 ) e f −∗ ξ + ( ω ◦ f − 1 ) ∇ ′ X ′ e f −∗ ξ i = TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 37 ( X ′ ( ω ◦ f − 1 ) ◦ f ) e f ∗ e f −∗ ξ + ω e f ∗ ( ∇ ′ X ′ e f −∗ ξ ) = ( X ω ) ξ + ω ∇ X ξ = d M ω ( X ) ξ + ω ∇ X ξ . 5.7(2): If X ∈ T F , then X ′ ∈ T F ′ , and as e f −∗ ξ is loca l inv ariant, ∇ ′ X ′ ( e f −∗ ξ ) = 0, so ∇ X ( ξ ) = e f ∗ ( ∇ ′ X ′ ( e f −∗ ξ )) = 0 and ∇ is ﬂa t alo ng F . 5.7(3): The fact that ∇ is G − inv ariant is a simple exercise whic h is left to the reader. 5.7(4): W e need to show that A = ∇ π f + − π f + ∇ ν π f + : C ∞ c ( ∧ T ∗ M ; ∧ T ∗ F s ⊗ E ) → C ∞ ( ∧ T ∗ M ; π f + ) is trans- versely s mo oth. Now π f + = e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ and ∇ = e f ∗ ∇ ′ e f −∗ = e f ∗ ∇ ′ R ′∗ P ′ ℓ e g ∗ R ∗ = e f ∗ π ′ + ∇ ′ ν π ′ + R ′∗ P ′ ℓ e g ∗ R ∗ . Using the proo f o f Prop osition 7.4, w e have that, modulo transversely smo oth oper ators, A = e f ∗ π ′ + ∇ ′ ν π ′ + R ′∗ P ′ ℓ e g ∗ R ∗ e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ − e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ ∇ ν e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ = e f ∗ π ′ + ∇ ′ ν π ′ + R ′∗ P ′ ℓ e g ∗ R ∗ e f ∗ R ′∗ P ′ ℓ π ′ + e g ∗ R ∗ P ℓ − e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ e f ∗ ∇ ′ ν R ′∗ π ′ + e g ∗ R ∗ P ℓ = e f ∗ π ′ + ∇ ′ ν π ′ + R ′∗ π ′ + e g ∗ R ∗ P ℓ − e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ e f ∗ ∇ ′ ν R ′∗ π ′ + e g ∗ R ∗ P ℓ , since P ′ ℓ e g ∗ R ∗ e f ∗ R ′∗ P ′ ℓ is the identit y on Im( P ′ ℓ ) = Ker(∆ E ′ ℓ ) ⊃ Im( π ′ + ). Now R ′∗ π ′ + = π ′ + R ′∗ , a nd ∇ ′ ν π ′ + = ( ∇ ′ ν π ′ + ) π ′ + = π ′ + ∇ ′ ν π ′ + + [ ∇ ′ ν , π ′ + ] π ′ + , and [ ∇ ′ ν , π ′ + ] is transversely smo oth since π ′ + is. So using P rop ositio n 7.9, w e ha v e that mo dulo tra nsversely smo oth oper ators, e f ∗ R ′∗ π ′ + e g ∗ R ∗ P ℓ e f ∗ ∇ ′ ν R ′∗ π ′ + e g ∗ R ∗ P ℓ = e f ∗ R ′∗ π ′ + P ′ ℓ e g ∗ R ∗ P ℓ e f ∗ π ′ + ∇ ′ ν π ′ + R ′∗ e g ∗ R ∗ P ℓ = e f ∗ π ′ + R ′∗ P ′ ℓ e g ∗ R ∗ P ℓ e f ∗ P ′ ℓ π ′ + ∇ ′ ν π ′ + R ′∗ e g ∗ R ∗ P ℓ = e f ∗ π ′ + ∇ ′ ν π ′ + R ′∗ e g ∗ R ∗ P ℓ , since R ′∗ P ′ ℓ e g ∗ R ∗ P ℓ e f ∗ P ′ ℓ is also the identit y on Im( P ′ ℓ ). As π ′ + R ′∗ = π ′ + π ′ + R ′∗ = π ′ + R ′∗ π ′ + , A = 0 modulo transversely smo oth op era tors, that is, A is transversely smo oth.  9. Leafwise homo to py inv ariance of the twisted higher harmonic signa ture In this s ection we prove our main theore m that the twisted higher harmonic signature is a lea fwise homotopy inv ariant. Theorem 9 .1. Supp ose that M is a c omp act Riemannian manifold, with oriente d Riema nnian foliation F of dimension 2 ℓ , and that E is a le afwise ﬂat c omplex bund le over M with a ( p ossibly indeﬁnite) non-de gener ate Hermitian metric which is pr eserve d by the le afwise ﬂat stru ctur e. Assu me that the pr oje ct ion onto Ker(∆ E ℓ ) for the asso ciate d foliation F s of t he homotopy gr oup oid of F is tr ans versely smo oth. Then σ ( F, E ) is a le afwise homotopy invariant. Recall that the pro jection onto Ker(∆ E ℓ ) is transversely smo oth: for the (unt wisted) leafwise s ignature op erator ; whenever E is a bundle asso cia ted to the nor mal bundle of the fo lia tion; a nd whenever the leafwise parallel tr anslation on E deﬁned by the ﬂat structure is a b ounded map, in particular whenev er the pre s erved metric on E is positive deﬁnite. Note also that these co nditio ns are preserved under pull-back by a leafwise homotopy equiv alence. Suppo se that M ′ , F ′ , and E ′ satisfy the hypothesis o f Theo rem 9.1, and that f : M → M ′ is a leafwise homotopy equiv alence, which prese rves the leafwise orientations. Set E = f ∗ ( E ′ ) with the induced leafwise ﬂat str ucture a nd pre s erved metric. Ass ume that the pr o jections to Ker(∆ E ℓ ) and K er(∆ E ′ ℓ ) are tra nsversely smo oth. Then we need to show that ch a ( π ± ) = f ∗ (ch a ( π ′ ± )) . W e do this in t wo stages . The ﬁrst is to prov e Theorem 9.2. ch a ( π ± ) = ch a ( π f ± ) . Pr o of. Rec all that π f ± = e f ∗ R ′∗ π ′ ± e g ∗ R ∗ P ℓ , and set b π f ,t ± = tπ f ± + (1 − t ) P ℓ π f ± . A simple computatio n, using the fact that π f ± P ℓ = π f ± , shows tha t the b π f ,t ± are idemp otents, and as P ℓ and the π f ± are tra nsversely smo oth, the b π f ,t ± are smooth families of transversely smo oth idemp otents. It follo ws 38 M.-T. BENAMEUR AND J. L. HEITSCH from Theor em 3.5 that ch a ( b π f , 0 ± ) = ch a ( b π f , 1 ± ). Since b π f , 1 ± = π f ± , we need to show that ch a ( b π f , 0 ± ) = ch a ( π ± ). W e will do only the + case a s the other ca se is the sa me. Set b π f ± = b π f , 0 ± . Consider the pairing s < , >, a nd Q deﬁned in Sectio n 4. Note that Q ( d s α 1 , α 2 ) = ( − 1) ℓ +1 Q ( α 1 , d s α 2 ). Using a pa rtition o f unit y and linearity , this reduces to considering sections of compact suppor t of the form α = ω ⊗ φ , where ω ∈ C ∞ c ( e L ; ∧ T ∗ e L ) a nd φ is a ﬂa t section of E , wher e it is immediate. So B ℓ (2) ( F s , E ) is totally isotropic under the pa iring Q , and it is orthogona l to Ker(∆ E ℓ ) under the pairing < , > . In addition, this equation implies that Q induces a well deﬁned pair ing Q : H ℓ (2) ( F s , E ) ⊗ H ℓ (2) ( F s , E ) → B ( M ) , where B ( M ) denotes the Bor e l C v alued functions on M . It further implies that P ℓ restricted to the co cycles Z ℓ (2) ( F s , E ) pr eserves Q . The s ubspaces Ker(∆ E + ℓ ) and Ker(∆ E − ℓ ) are orthogo na l under b oth of the pair ings, since Q ( b τ α 1 , α 2 ) = Q ( α 1 , b τ α 2 ). As Ker(∆ E ℓ ) = Ker(∆ E + ℓ ) ⊕ Ker(∆ E − ℓ ), so also Z ℓ (2) ( F s , E ) = Ker(∆ E + ℓ ) ⊕ Ker(∆ E − ℓ ) ⊕ B ℓ (2) ( F s , E ) . The kernels of b oth b π f + and π + contain Ker( P ℓ ), so we may restrict our a tten tion to Im( P ℓ ) = Ker(∆ E ℓ ). The image of b π f + is P ℓ (Im( e f ∗ + )). Lemma 9.3 . π + : P ℓ (Im( e f ∗ + )) → Ke r(∆ E + ℓ ) is an isomorphism with b ounde d inverse. Pr o of. B y Prop ositio n 6.13, e f ∗ restricted to K er(∆ E ′ ℓ ) takes the pairing Q ′ to the pairing Q . (Note that Q is ± deﬁnite on the Im( π ± ) if ℓ is even, while it is √ − 1 Q , whic h is ± deﬁnite on Im( π ± ) is ℓ is odd. W e will ﬁnesse this p oint.) Since P ℓ (restricted to the co cycle s ) pres erves the pa iring Q , Q is p ositive deﬁnite on P ℓ (Im( e f ∗ + )). Given 0 6 = α ∈ P ℓ (Im( e f ∗ + )), write it (uniquely) as α = α + + α − , where α ± ∈ Ker(∆ E ± ℓ ). Then 0 < Q ( α, α ) = < α + , α + > − < α − , α − > ≤ < α + , α + >, so π + ( α ) = α + 6 = 0 and π + : P ℓ (Im( e f ∗ + )) → Ke r(∆ E + ℓ ) is one-to-one. The ab ov e inequality also implies that π − 1 + is b ounded, with b ound √ 2. The element α = π − 1 + ( α + ) and || α || 2 = < α, α > = < α + , α + > + < α − , α − > = || α + || 2 + || α − || 2 . Since 0 < Q ( α, α ), || α − || 2 < || α + || 2 , so || π − 1 + ( α + ) || 2 = || α || 2 ≤ 2 || α + || 2 . Next we show that π + is onto. Choose α ∈ Ker(∆ E + ℓ ) which is or thogonal to π + ( P ℓ (Im e f ∗ + )). The subspaces P ℓ (Im e f ∗ + ) and P ℓ (Im e f ∗ − ) are o rthogonal under Q . T he ir direct sum is the space K er(∆ ℓ ) o f a ll harmonic for ms, since π f + + π f − induces the identit y on cohomology . W rite α = β + + β − , with β ± ∈ P ℓ (Im e f ∗ ± ). Then k α k 2 = Q ( α, α ) = Q ( α, β + ) + Q ( α, β − ) = Q ( α, π + β + ) + Q ( α, π − β + ) + Q ( α, β − ) = Q ( α, β − ) . The las t equality is a conseq ue nc e of the facts that α is Q or thogonal to π + ( P ℓ (Im e f ∗ + )) a nd that Ker(∆ E + ℓ ) and Ker(∆ E − ℓ ) are Q orthogonal. Hence, we ha ve, 0 ≤ k α k 2 = Q ( β + , β − ) + Q ( β − , β − ) = Q ( β − , β − ) ≤ 0 . So, α = 0, and π + is onto.  The ma p π − 1 + is deﬁned on π + ( P ℓ (Im e f ∗ + )). Deﬁne ρ + to be orthogo nal pro jection o nt o π + ( P ℓ (Im e f ∗ + )) comp osed with π − 1 + , i.e., ρ + = π − 1 + ◦ π + : A ℓ (2) ( F s , E ) → P ℓ (Im( e f ∗ + )) . Then ρ + is an idemp otent and has image P ℓ (Im e f ∗ + ), which equals b π f + . W e cla im that ρ + is transversely smo oth. If so, then ch a ( ρ + ) is deﬁned and c h a ( ρ + ) = ch a ( b π f + ), since they have the same image. Note tha t ρ + is pro jection to P ℓ (Im( e f ∗ + )) along Ker( π + ). With this description, it is immediate that ρ + ◦ π + = ρ + and π + ◦ ρ + = π + since π + is pro jection to Ker(∆ E + ℓ ) alo ng Ker( π + ). As ab ove, we may form the smo oth family of transversely smo oth idempotents tρ + + (1 − t ) π + which connects ρ + to π + . Again, it follows from TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 39 Theorem 3.5 that ch a ( ρ + ) = ch a ( π + ), and we have ch a ( π + ) = ch a ( π f + ). So to ﬁnish the pro o f w e need only show that ρ + is transversely smo oth. Now b π f ± = P ℓ π f ± = P ℓ e f ∗ R ′∗ π ′ ± e g ∗ R ∗ P ℓ , and recalling that P ′ ℓ e g ∗ R ∗ P ℓ e f ∗ R ′∗ P ′ ℓ = P ′ ℓ , and π ′ ± = π ′ ± P ′ ℓ = P ′ ℓ π ′ ± , w e ha v e ( b π f ± ) 2 = b π f ± and b π f ± b π f ∓ = 0 . These idemp otents are transversely smo oth, since P ℓ and the π f ± are tra nsversely smo o th. They also satisfy b π f + + b π f − = P ℓ , a nd their kernels both contain Ker( P ℓ ). Finally , no te that the Im( b π f ± ) = P ℓ (Im( e f ∗ ± )). Next set A = π + + b π f − . Lemma 9.4. The op er ator A and its adjoint A t ar e tr ansversely smo oth, and A is an isomorphism when r estricte d to Ker(∆ E ℓ ) . Pr o of. A is tra nsversely smooth beca us e bo th π + and b π f − are. As A t = ( π + + b π f − ) t = π + + ( b π f − ) t , we need only show that ( b π f − ) t = P ℓ R ∗ t e g ∗ t π ′ − R ′∗ t e f ∗ t P ℓ is trans versely smo o th. The op era tors P ℓ and π ′ − are transversely smo oth, and R ∗ t = ( R ∗ ) − 1 and R ′∗ t = ( R ′∗ ) − 1 , s inc e they are b oth is ometries. No w consider e f ∗ t and e g ∗ t , restricted to the harmo nic forms. Let α ′ ∈ Im( P ′ ℓ ) and α ∈ Im( π + ). Then < α ′ , e f ∗ t α > = < e f ∗ α ′ , α > = Q ( e f ∗ α ′ , b ∗ α ) = Q ( e f ∗ α ′ , b τ α ) = Q ( e f ∗ α ′ , α ) = Q ( e f ∗ α ′ , e f ∗ R ′∗ e g ∗ R ∗ α ) = Q ′ ( α ′ , R ′∗ e g ∗ R ∗ α ) = Q ′ ( α ′ , π ′ + R ′∗ e g ∗ R ∗ α + π ′ − R ′∗ e g ∗ R ∗ α ) = Q ′ ( α ′ , b τ π ′ + R ′∗ e g ∗ R ∗ α − b τ π ′ − R ′∗ e g ∗ R ∗ α ) = Q ′ ( α ′ , b ∗ π ′ + R ′∗ e g ∗ R ∗ α − b ∗ π ′ − R ′∗ e g ∗ R ∗ α ) = < α ′ , ( π ′ + R ′∗ e g ∗ R ∗ − π ′ − R ′∗ e g ∗ R ∗ ) α > . So on Im( π + ), e f ∗ t = π ′ + R ′∗ e g ∗ R ∗ − π ′ − R ′∗ e g ∗ R ∗ . Similarly , on Im( π − ), e f ∗ t = − π ′ + R ′∗ e g ∗ R ∗ + π ′ − R ′∗ e g ∗ R ∗ . Thu s on Im( P ℓ ), e f ∗ t = ( π ′ + R ′∗ e g ∗ R ∗ − π ′ − R ′∗ e g ∗ R ∗ ) π + − ( π ′ + R ′∗ e g ∗ R ∗ − π ′ − R ′∗ e g ∗ R ∗ ) π − = ( π ′ + − π ′ − ) R ′∗ e g ∗ R ∗ ( π + − π − ) . Similarly , e g ∗ t = ( π + − π − ) R ∗ e f ∗ R ′∗ ( π ′ + − π ′ − ). As ( π ′ + − π ′ − ) π ′ − ( π ′ + − π ′ − ) = π ′ − , R ∗ commutes with π ± , R ′∗ commutes with π ′ ± , and P ℓ π ± = π ± , w e ha v e ( b π f − ) t = ( π + − π − ) e f ∗ R ′∗ π ′ − e g ∗ R ∗ ( π + − π − ) , which is transversely smo o th. Next, note tha t Q is p ositive deﬁnite on Im( π + ) and I m( b π f + ), and is negative deﬁnite o n Im( π − ) and Im( b π f − ). So Im( π ± ) ∩ Im( b π f ∓ ) = { 0 } . Let α ∈ Ker(∆ E ℓ ) with A ( α ) = 0. Then π + ( α ) = − b π f − ( α ) and π + ( α ) , b π f − ( α ) ∈ Im( π + ) ∩ Im( b π f − ) = { 0 } . Thus α ∈ Ke r( π + ) ∩ Ker( b π f − ) ∩ Ker(∆ E ℓ ) = Im( π − ) ∩ Im( b π f + ) = { 0 } , so α = 0, and A is one-to-one. Now A (Im( b π f + )) = π + ( P ℓ (Im( e f ∗ + ))) = Im( π + ), so Im( π + ) ⊂ Im( A ). Just as π + maps Im( b π f + ) isomor - phically to Im( π + ), π − maps Im( b π f − ) isomor phically to Im( π − ). Giv en α ∈ Im( π − ), let β ∈ Im( b π f − ), with π − ( β ) = α , so β = π − ( β ) + π + ( β ) = α + π + ( β ), tha t is α = β − π + ( β ). Now A ( β ) = π + ( β ) + b π f − ( β ) = π + ( β ) + β , since β ∈ Im( b π f − ). So β ∈ Im( A ), since π + ( β ) ∈ Im( π + ) ⊂ Im( A ). Thus α = β − π + ( β ) ∈ Im( A ), a nd w e hav e Im ( π − ) ⊂ Im( A ). As A is linear a nd co n tains Im( π ± ), it also con tains Im( π + ) ⊕ Im( π − ) = Ker(∆ E ℓ ), and A is on to.  Lemma 9.5 . A − 1 , the inverse of A r estricte d to Ker(∆ E ℓ ) , is a b ounde d isomorph ism of Ke r(∆ E ℓ ) . 40 M.-T. BENAMEUR AND J. L. HEITSCH Pr o of. A − 1 is b o unded if and only if ther e is a cons tant C > 0, so tha t || A ( α ) || ≥ C for a ll x ∈ M and a ll α ∈ K e r(∆ E ℓ ) x with || α || = 1 . If not, there are s equences x j ∈ M and α j ∈ Ker(∆ E ℓ ) x j with || α j || = 1, a nd lim j →∞ || A ( α j ) || = lim j →∞ || π + ( α j ) + b π f − ( α j ) || = 0 , that is, 0 = lim j →∞ π + ( α j ) + b π f − ( α j ) = lim j →∞ π + ( α j ) + π + ( b π f − ( α j )) + π − ( b π f − ( α j )) = lim j →∞ π + ( α j + b π f − ( α j )) + π − ( b π f − ( α j )) . This implies that lim j →∞ π − ( b π f − ( α j )) = 0. Now 0 ≥ Q ( b π f − ( α j ) , b π f − ( α j )) = || π + ( b π f − ( α j )) || 2 − || π − ( b π f − ( α j )) || 2 , so lim j →∞ π + ( b π f − ( α j )) = 0, whic h gives that lim j →∞ b π f − ( α j ) = 0, so also lim j →∞ π + ( α j ) = 0. Since α j = π + ( α j ) + π − ( α j ), w e ha v e lim j →∞ ( π − ( α j ) − α j ) = 0 , in particular, lim j →∞ || π − ( α j ) || = lim j →∞ || α j || = 1. Now Q ( π − ( α j ) , π − ( α j )) = − || π − ( α j ) || 2 , so lim j →∞ Q ( π − ( α j ) , π − ( α j )) = − 1. Since Q is cont inuous, lim j →∞ Q ( α j , α j ) = lim j →∞ Q ( π − ( α j ) , π − ( α j )) = − 1. The fact that lim j →∞ b π f − ( α j ) = 0 and α j = b π f + ( α j ) + b π f − ( α j ) implies that lim j →∞ ( b π f + ( α j ) − α j ) = 0 , and as abov e, the fa c t tha t Q ( b π f + ( α j ) , b π f + ( α j )) ≥ 0 implies that lim inf j Q ( α j , α j ) ≥ 0 , which contradicts that fac t that lim j →∞ Q ( α j , α j ) = − 1 .  Now consider the map B = A t A , which is tr ansversely s mo oth, and is an iso morphism when restricted to Ker(∆ E ℓ ). Denote by B − 1 the compo sition of ma ps: B − 1 : A ℓ (2) ( F s , E ) P ℓ − → Ker(∆ E ℓ ) B − 1 ℓ − → Ker(∆ E ℓ ) , where B − 1 ℓ is the in verse of B restricted to Ker(∆ E ℓ ). Since ρ + takes v alues in P ℓ (Im( e f ∗ + )) = Im( b π f + ), Aρ + = π + , so B ρ + = A t π + , and ρ + = B − 1 A t π + . Thus w e are reduced to showing that B − 1 is transversely smo oth. Restricting once again to Ker(∆ E ℓ ), we have that the o p erator B is p ositive, and A and A − 1 are b ounded op erator s , so there are constants 0 < C 0 < C 1 < ∞ so that for all α ∈ Ker(∆ E ℓ ), α 6 = 0, C 0 ≤ < α, α > ≤ C 1 . Thu s the sp ectrum of B on K e r(∆ E ℓ ), σ ( B ) ⊂ [ C 0 , C 1 ], and for λ > 0, σ ( B λ ) ⊂ [ C 0 λ , C 1 λ ], and σ ( P ℓ − B λ ) ⊂ [1 − C 1 λ , 1 − C 0 λ ]. In particular , for λ > C 1 we have 0 < 1 − C 1 λ ≤ || P ℓ − B λ || ≤ 1 − C 0 λ < 1 . Since B = P ℓ B P ℓ , this estimate actually ho lds on A ℓ (2) ( F s , E ), and for all Sob olev no rms a sso ciated to A ℓ (2) ( F s , E ). This estimate alo ng with the fact that x − 1 = 1 λ P ∞ n = o (1 − x λ ) n , pr ovided that | 1 − x λ | < 1, implies that for λ > C 1 , B − 1 = 1 λ ∞ X n =0  P ℓ − B λ  n , TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 41 where w e set  P ℓ − B λ  0 = P ℓ . F or N ∈ Z + , set D N = 1 λ N X n =0  P ℓ − B λ  n , where again  P ℓ − B λ  0 = P ℓ . Then D N is a unifor mly bo unded (o ver all N ) transversely s mo oth op er a tor, and it con v erges to B − 1 in all Sobolev norms. Thus B − 1 is a bounded leafwise smo othing op erator . Let Y b e a vector ﬁeld on M , and c o nsider ∂ Y ν D N = 1 λ P N n =0 ∂ Y ν  P ℓ − B λ  n  . F or any in teg ers k 1 , k 2 , and for N > 1, || 1 λ ∞ X n = N +1 ∂ Y ν  P ℓ − B λ  n  || k 1 ,k 2 ≤ 1 λ ∞ X n = N +1 || ∂ Y ν  P ℓ − B λ  n  || k 1 ,k 2 ≤ 1 λ ∞ X n = N +1 n − 1 X r =0 || P ℓ − B λ || r k 1 ,k 1 || ∂ Y ν  P ℓ − B λ  || k 1 ,k 2 || P ℓ − B λ || n − r − 1 k 2 ,k 2 = 1 λ || ∂ Y ν  P ℓ − B λ  || k 1 ,k 2 ∞ X n = N +1 n || P ℓ − B λ || n − 1 ≤ 1 λ || ∂ Y ν  P ℓ − B λ  || k 1 ,k 2 ∞ X n = N +1 n  1 − C 0 λ  n − 1 . This conv erges to 0 as N → ∞ , as || ∂ Y ν  P ℓ − B λ  || k 1 ,k 2 is ﬁnite since P ℓ − B λ is transversely smo oth. Thus the trans verse der iv ative ∂ Y ν D N conv er ges in a ll Sob olev no r ms, so lim N →∞ ∂ Y ν D N exists, and it is bounded and leafwise smo othing. Prop ositi o n 9. 6 . ∂ Y ν B − 1 exists, in p articular, ∂ Y ν D N c onver ges in al l Sob olev norms to ∂ Y ν B − 1 , so ∂ Y ν B − 1 is a b ounde d le afwise smo othing op er ator. Pr o of. As ∂ Y ν D N conv er ges in all Sob o lev norms, we only need prov e that ∂ Y ν B − 1 exists and that it eq ua ls lim N →∞ ∂ Y ν D N . Recall the situation in the pro of of Theor em 4.4. F o r y close to x in M , we hav e the smoo th diﬀeomorphism Φ y : e L x → e L y . Given Y ∈ T M x , set γ ( t ) = e x p x ( tY ). F or z ∈ e L x and t suﬃciently small, say | t | ≤ ǫ , we hav e the path t → b γ z ( t ), which co vers γ ( t ) and has tang ent vector in ν s . So for | t | ≤ ǫ , the diﬀeomorphism Φ γ ( t ) : e L x → e L γ ( t ) exists. The vector Y deﬁnes the tra nsverse vector ﬁeld b Y along e L x , i. e. a smo oth section of ν s | e L x , by requiring s ∗ ( b Y ) = Y . Then, the op erator ∂ Y ν ( · ) = [ ∇ ν b Y , · ] can be realized as ∂ /∂ t ( · ) as follows. W e may parallel transla te all o b jects on e L x to e L γ ( t ) (and vice-versa) a long the paths b γ z ( t ), using the connection ∇ . W e will denote this parallel transla tion by Φ t (and the reverse by Φ − 1 t ). Thus any section of ξ ∈ C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ) deﬁnes a section Φ t ( ξ ) = ξ t of C ∞ c ( e L γ ( t ) ; ∧ ℓ T ∗ F s ⊗ E ) given by ξ t ( z ) = Φ t ( ξ (Φ − 1 γ ( t ) ( z ))) , and ξ t is smoo th in t . Note that for such a lo cal sec tion, ∇ b Y ξ t = ∇ ν b Y ξ t (as b Y ∈ ν s ) is deﬁned and equals 0, since ξ t is paralle l translatio n along integral curves of b Y for the co nnection ∇ . In fac t, if we set Y ( t ) = γ ′ ( t ), then, ∇ b Y ( t ) ξ t = ∇ ν b Y ( t ) ξ t = 0. F urther note that Φ γ ( t ) is a diﬀeomor phism o f b ounded dilatio n and the induced actio n on E is also b ounded, so the lo cal op er ators Φ t and Φ − 1 t are b ounded when acting o n sections of C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ), (resp ectively C ∞ c ( e L γ ( t ) ; ∧ ℓ T ∗ F s ⊗ E ). W e denote these b ounds by || Φ t || and || Φ − 1 t || resp ectively . The b ounds are uniform in t for | t | ≤ ǫ . Similarly , we may par allel translate op erator s such as D N from nearby leav es to e L x as follows. Given ξ 1 , ξ 2 ∈ C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ), deﬁne the op erator D N ,t on e L x by < D N ,t ( ξ 1 ) , ξ 2 > = < Φ − 1 t  D N ,γ ( t ) ( ξ 1 ,t )  , ξ 2 > . 42 M.-T. BENAMEUR AND J. L. HEITSCH This is well deﬁned and smo oth in t provided | t | ≤ ǫ . Th us, the op erator ∂ ( D N ,t ) ∂ t    t =0 is well deﬁned as a map from C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ) to C ∞ ( e L x ; ∧ ℓ T ∗ F s ⊗ E ). Lik ewise, ∇ b Y ( D N ,γ ( t ) ( ξ t )) is well deﬁned for all ξ ∈ C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ), and takes v a lues in C ∞ ( e L x ; ∧ ℓ T ∗ F s ⊗ E ). The fundamen tal relationship b etw een parallel translation and the co nnection ∇ translates to the equa tion 9.7.  ∂ ( D N ,t ) ∂ t    t =0  ( ξ ) = ∇ b Y ( D N ,γ ( t ) ( ξ t )) . In fact, for all t 0 ∈ [ − ǫ, ǫ ],  ∂ ( D N ,t ) ∂ t    t = t 0  ( ξ ) = Φ − 1 t 0  ∇ b Y ( t 0 ) ( D N ,γ ( t ) ( ξ t ))  , since Φ − 1 t = Φ − 1 t 0 ◦ Φ − 1 t,t 0 , where Φ − 1 t,t 0 is parallel translation from e L γ ( t ) to e L γ ( t 0 ) . F o r ξ ∈ C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ) we have ∂ Y ν D N ξ = [ ∇ ν b Y , D N ] ξ = [ ∇ b Y , D N ] ξ = ∇ b Y D N ,γ ( t ) ( ξ t ) − D N ∇ b Y ( ξ t ) = ∇ b Y D N ,γ ( t ) ( ξ t ) , since ∇ b Y ( ξ t ) = 0. So by Equation 9.7 we ha v e ∂ Y ν D N = ∂ ( D N ,t ) ∂ t    t =0 . As ab ove, this extends to 9.8. ∂ ( D N ,t ) ∂ t = Φ − 1 t  ∂ Y ( t ) ν D N ,γ ( t )  , provided | t | ≤ ǫ . F o r t ∈ [ − ǫ, ǫ ], s et D ′ N ,t = Φ − 1 t  ∂ Y ( t ) ν D N  = ∂ ( D N ,t ) /∂ t , and D ′ t = Φ − 1 t  lim N →∞ ∂ Y ( t ) ν D N  , and B − 1 t = Φ − 1 t ( B − 1 ). Note carefully that the following co mputatio n takes place on the leaf e L x . F or ξ 1 , ξ 2 ∈ C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ), and h ∈ (0 , ǫ ), w e ha ve that    − − Z h 0 < D ′ t ( ξ 1 ) , ξ 2 > dt    ≤    − < D N ,h ( ξ 1 ) , ξ 2 >    +    < D N ,h ( ξ 1 ) , ξ 2 > − < D N , 0 ( ξ 1 ) , ξ 2 > − Z h 0 < D ′ N ,t ( ξ 1 ) , ξ 2 > dt    +    < D N , 0 ( ξ 1 ) , ξ 2 > −    +    Z h 0 < ( D ′ N ,t − D ′ t )( ξ 1 ) , ξ 2 > dt    . The term < D N ,h ( ξ 1 ) , ξ 2 > − < D N , 0 ( ξ 1 ) , ξ 2 > − Z h 0 < D ′ N ,t ( ξ 1 ) , ξ 2 > dt = 0 , since D ′ N ,t = ∂ ( D N ,t ) /∂ t . The term    < D N , 0 ( ξ 1 ) , ξ 2 > −    ≤ || D N , 0 − B − 1 0 || || ξ 1 || || ξ 2 || , which go e s to 0 as N → ∞ , since D N → B − 1 in norm. Likewise, the term    − < D N ,h ( ξ 1 ) , ξ 2 >    =    < Φ − 1 h ( B − 1 γ ( h ) − D N ,γ ( h ) )Φ h ( ξ 1 ) , ξ 2 >    ≤ || Φ − 1 h || || B − 1 γ ( h ) − D N ,γ ( h ) || || Φ h || || ξ 1 || || ξ 2 || , which go e s to 0 as N → ∞ , since D N → B − 1 in norm and || Φ − 1 h || and || Φ h || are bounded. TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 43 The term    Z h 0 < ( D ′ N ,t − D ′ t )( ξ 1 ) , ξ 2 > dt    ≤ Z h 0 || Φ − 1 t || || ∂ Y ( t ) ν D N ,γ ( t ) − lim b N →∞ ∂ Y ( t ) ν D b N ,γ ( t ) || || Φ t || || ξ 1 || || ξ 2 || dt, which go es to 0 as N → ∞ , since || Φ − 1 t || and || Φ t || are uniformly b ounded for t ∈ [0 , h ], and ∂ Y ν D N conv er ges in norm. Thu s lim N →∞    − − Z h 0 < D ′ t ( ξ 1 ) , ξ 2 > dt    = 0 , and as the expr ession inside the limit is indep endent of N , it actua lly equals 0. This implies that < lim h → 0 1 h  B − 1 h − B − 1 0 − Z h 0 D ′ t dt  ( ξ 1 ) , ξ 2 > = 0 , for all ξ 1 , ξ 2 ∈ C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ), so lim h → 0 1 h  B − 1 h − B − 1 0 − Z h 0 D ′ t dt  = 0 as a map from C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ) to C ∞ ( e L x ; ∧ ℓ T ∗ F s ⊗ E ). Next w e ha v e    < lim t → 0 D ′ t ( ξ 1 ) , ξ 2 > − < D ′ 0 ( ξ 1 ) , ξ 2 >    ≤    < lim t → 0 ( D ′ t − D ′ N ,t )( ξ 1 ) , ξ 2 >    +    < ( lim t → 0 D ′ N ,t − D ′ N , 0 )( ξ 1 ) , ξ 2 >    +    < ( D ′ N , 0 − D ′ 0 )( ξ 1 ) , ξ 2 >    ≤ lim t → 0 || Φ − 1 t || || lim b N →∞ ∂ Y ( t ) ν D b N ,γ ( t ) − ∂ Y ( t ) ν D N ,γ ( t ) || || Φ t || || ξ 1 || || ξ 2 || +    lim t → 0 < D ′ N ,t ( ξ 1 ) , ξ 2 > − < D ′ N , 0 ( ξ 1 ) , ξ 2 >    + || ∂ Y (0) ν D N ,γ (0) − lim b N →∞ ∂ Y (0) ν D b N ,γ (0) || || ξ 1 || || ξ 2 || . The ﬁrst and last terms ca n b e made arbitrarily small (for t ∈ [0 , h ]) by choos ing N suﬃciently large. The middle term equals zero since < D ′ N ,t ( ξ 1 ) , ξ 2 > is contin uous in t , whic h follows immedia tely from Equatio n 9.8 and the fac t that D N is transversely smo oth. Th us, 0 = < lim t → 0 D ′ t ( ξ 1 ) , ξ 2 > − < D ′ 0 ( ξ 1 ) , ξ 2 > = < ( lim t → 0 D ′ t − D ′ 0 )( ξ 1 ) , ξ 2 >, which holds for a ll ξ 1 , ξ 2 ∈ C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ), so lim t → 0 D ′ t − D ′ 0 = 0 , that is D ′ t is contin uous at zero . The op erator lim h → 0 1 h  Z h 0 D ′ t dt  is also well deﬁned as a ma p fro m C ∞ c ( e L x ; ∧ ℓ T ∗ F s ⊗ E ) to C ∞ ( e L x ; ∧ ℓ T ∗ F s ⊗ E ), and as D ′ t is con tin uous at zero, we ha v e lim h → 0 1 h  Z h 0 D ′ t dt  = D ′ 0 . Again b y the fundamental relationship betw een parallel translation and ∇ , we ha v e lim h → 0 B − 1 h − B − 1 0 h = ∂ Y ν B − 1 , so ∂ Y ν B − 1 = lim h → 0 B − 1 h − B − 1 0 h = lim h → 0 1 h  Z h 0 D ′ t dt  = D ′ 0 = lim N →∞ ∂ Y ν D N , and ∂ Y ν B − 1 is a b ounded lea fwis e smo othing op erato r.  44 M.-T. BENAMEUR AND J. L. HEITSCH A b o o t strapping ar gument now ﬁnishes the pro of. Let Y 1 , Y 2 be vector ﬁelds on M . As B − 1 B = P ℓ and the ∂ Y i ν are deriv ations, w e ha v e ( ∂ Y 2 ν B − 1 ) B + B − 1 ( ∂ Y 2 ν B ) = ∂ Y 2 ν P ℓ , so ∂ Y 2 ν B − 1 = − B − 1 ( ∂ Y 2 ν B ) B − 1 + ( ∂ Y 2 ν P ℓ ) B − 1 , which is in the domain of ∂ Y 1 ν . Applying it, we obtain ∂ Y 1 ν ∂ Y 2 ν B − 1 = −  ( ∂ Y 1 ν B − 1 )( ∂ Y 2 ν B ) B − 1 + B − 1 ( ∂ Y 1 ν ∂ Y 2 ν B ) B − 1 + B − 1 ( ∂ Y 2 ν B )( ∂ Y 1 ν B − 1 )  + ( ∂ Y 1 ν ∂ Y 2 ν P ℓ ) B − 1 + ( ∂ Y 2 ν P ℓ )( ∂ Y 1 ν B − 1 ) , which is a b ounded leafwise smo othing map, since B and P are transversely smo oth and ∂ Y 1 ν B − 1 is b ounded and leafwise smo othing. P ro ceeding by induction, we hav e that for all vector ﬁelds Y 1 , ..., Y m on M , the op erator ∂ Y 1 ν · · · ∂ Y m ν B − 1 is bounded and leafwise smo othing, so B − 1 is transversely smo oth. This completes the pro of Theor em 9.2.  Finally , we pro ve Theore m 9.1, that is w e prov e Theorem 9.9. ch a ( π f ± ) = f ∗ (ch a ( π ′ ± )) . Pr o of. W e will o nly pr ove that ch a ( π f + ) = f ∗ (ch a ( π ′ + )), as the other pro of is the same. W e beg in by constructing specia l c overs of M and M ′ . Let { b U ′ } be a ﬁnite op e n cover o f M ′ by foliation charts with transversals b T ′ . Cho ose the b U ′ so sma ll tha t g | b T ′ is a diﬀeomorphism. Denote by ρ ′ b U ′ : b U ′ → b T ′ the pro jection. Let { U } b e a ﬁnite open cov er of M by foliation c ha rts with tr ansversals T . Since the collection of op en sets f − 1 ( b U ′ ) cov er M , we may choo se the U sma ll enough so that for ea ch U , there is a b U ′ U with f ( U ) ⊂ b U ′ U . W e may further assume that the U ar e so s mall that f | T is a diﬀeomorphism. Set U ′ = ( ρ ′ b U ′ U ) − 1 ( ρ ′ b U ′ U ( f ( U ))) . Then the set { U ′ } is a ﬁnite op en cover of M ′ by foliation charts, f ( U ) ⊂ U ′ , and T ′ = f ( T ) is a trans versal of U ′ . Denote the pro jection ρ ′ b U ′ U | U ′ → T ′ by ρ ′ . Set V = f − 1 ( U ′ ), and note that V is not necess arily connec ted. How ever, V ⊃ U whose transversal T is taken diﬀeomorphically o nt o T ′ by f . There is a well deﬁned pro jection ρ : V → T , given b y ρ = ( f | T ) − 1 ◦ ρ ′ ◦ f . Reca ll the co nnection ∇ o n π f + , (induced from the co nnection ∇ ′ on π ′ + ) which we will use to construct c h a ( π f + ), and set ∇ T = ∇ | T with curv a ture θ T . Then just as in Prop osition 5.20, we ha v e Lemma 9.1 0. ∇ | V = ρ ∗ ( ∇ T ) and θ | V = ρ ∗ ( θ T ) . Pr o of. T he pr o of is essentially the same. T o eﬀect it, we nee d to b e a ble to deﬁne lo cal in v ariant sections ov er V , a nd to do this, w e need families o f leafwise paths, such that moving along them gives the pro jection ρ . Given y ∈ V , cho ose a leafwise path γ ′ y : [1 , 2] → U ′ from ρ ′ ( f ( y )) to f ( y ). Le t h : M × I → M b e a leafwise homotopy b e t ween the identit y map and g ◦ f . In particula r, h ( x, 0) = x and h ( x, 1) = g f ( x ). Deﬁne the leafwise path γ y from ρ ( y ) to y a s follows: γ y ( t ) = h ( ρ ( y ) , t ) for 0 ≤ t ≤ 1 ; γ y ( t ) = g ( γ ′ y ( t )) for 1 ≤ t ≤ 2 ; and γ y ( t ) = h ( y , 3 − t ) for 2 ≤ t ≤ 3 . Since f ( ρ ( y )) = ρ ′ ( f ( y )), this do e s give a pa th from ρ ( y ) to y . Using the γ y , we may extend a ny lo ca l section deﬁned on T to a lo cal inv ariant section on all of V , and then pro ceed just a s in the pro of of Prop osition 5.20.  The co nnection ∇ T ′ (whic h is ∇ ′ restricted to π ′ + | T ′ ), and its curv a ture θ T ′ satisfy ∇ ′ | U ′ = ρ ′ ∗ ( ∇ T ′ ) and θ ′ | U ′ = ρ ′ ∗ ( θ T ′ ). Set b f = f | T , and deﬁne b f ∗ ( ∇ T ′ ) and b f ∗ ( θ T ′ ) as follows. Let ξ ∈ C ∞ ( π f + | T ), a nd suppo se that X and Y are tangent to T . Set X ′ = b f ∗ ( X ) = f ∗ ( X ), and Y ′ = b f ∗ ( Y ) = f ∗ ( Y ), both of which are tangent to T ′ . Deﬁne b f ∗ ( ∇ T ′ ) X ξ = e f ∗ ( ∇ T ′ X ′ ( e f −∗ ξ | T ′ )) and  b f ∗ ( θ T ′ )( X, Y )  ξ = e f ∗ ( θ T ′ ( X ′ , Y ′ )( e f −∗ ξ | T ′ )) . TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 45 Lemma 9.1 1. b f ∗ ( ∇ T ′ ) = ∇ T and b f ∗ ( θ T ′ ) = θ T . Pr o of. T he element ξ ∈ C ∞ ( π f + | T ) deter mines the lo ca l in v ariant s ections e ξ of π f + and e f −∗ ξ of π ′ + . Then b f ∗ ( ∇ T ′ ) X ξ = e f ∗ ( ∇ T ′ X ′ ( e f −∗ ξ | T ′ )) = e f ∗ ( ∇ ′ X ′ e f −∗ ξ ) = ∇ X e ξ = ∇ T X ξ . Next, using loca l spanning sets of π f + | V , and π ′ + | U ′ it is not diﬃcult to s how that θ T ( X, Y ) = ∇ T X ∇ T Y − ∇ T Y ∇ T X − ∇ T [ X,Y ] . and similarly for θ T ′ ( X ′ , Y ′ ). Then ∇ T X ∇ T Y ξ = e f ∗ ∇ T ′ X ′ e f −∗ e f ∗ ∇ T ′ Y ′ e f −∗ ξ = e f ∗ ∇ T ′ X ′ ∇ T ′ Y ′ e f −∗ ξ and ∇ T Y ∇ T X ξ = e f ∗ ∇ T ′ Y ′ ∇ T ′ X ′ e f −∗ ξ . Since b f is a diﬀeomorphism, b f ∗ ([ X, Y ]) = [ X ′ , Y ′ ], so ∇ T [ X,Y ] ξ = e f ∗ ∇ T ′ [ X ′ ,Y ′ ] e f −∗ ξ . It follows immediately that  b f ∗ ( θ T ′ )( X, Y )  ξ = e f ∗ θ T ′ ( X ′ , Y ′ ) e f −∗ ξ = θ T ( X, Y ) ξ .  Now consider the cur v ature op erator θ ′ of ∇ ′ ov er U ′ . W e may as s ume that U ′ ≃ R p × R q with co ordinates x ′ 1 , ..., x ′ n , and that T ′ = { 0 } × R q . Cho os e a lo c al inv ariant spanning set { ξ ′ i } of π ′ + | U ′ . Recall that for α ′ 1 ⊗ φ ′ 1 , α ′ 2 ⊗ φ ′ 2 sections of ∧ T ∗ e L ′ ⊗ E ′ , Q ′ ( α ′ 1 ⊗ φ ′ 1 , α ′ 2 ⊗ φ ′ 2 ) = Z e L ′ { φ ′ 1 , φ ′ 2 } α ′ 1 ∧ α ′ 2 = Z e L ′ ( α ′ 1 ⊗ φ ′ 1 ) ∧ ( α ′ 2 ⊗ φ ′ 2 ) . There are functions a ′ i,j,k,l on T ′ (thanks to Pr op osition 5.20) so that the action of θ ′ on a sectio n ξ ′ of π ′ + is giv en b y θ ′ ( ξ ′ ) = n X k,l = p +1 X i,j a ′ i,j,k,l Q ′ ( ξ ′ j , ξ ′ ) ξ ′ i dx ′ k ∧ dx ′ l = X i,j,k,l a ′ i,j,k,l h Z e L ′ ξ ′ j ∧ ξ ′ i ξ ′ i dx ′ k ∧ dx ′ l . The reason that we ca n represent θ ′ this wa y is b ecause for any ξ ′ ∈ Ker( π ′ + ) and any b ξ ′ ∈ Im( π ′ + ), Q ′ ( ξ ′ , b ξ ′ ) = 0. This follows fro m the facts that < ξ ′ , b ξ ′ > = 0, Q ′ ( ξ ′ , b ∗ b ξ ′ ) = < ξ ′ , b ξ ′ > , and b ξ ′ = b τ b ξ ′ = √ − 1 ℓ 2 b ∗ b ξ ′ . Let x ′ ∈ U ′ and y ′ , z ′ ∈ e L x ′ . With resp ect to the spanning set { ξ ′ i } and the lo cal co o rdinates on U ′ , the Sch wartz kernel Θ ′ x ′ ( y ′ , z ′ ) of θ ′ | U ′ is giv en b y Θ ′ x ′ ( y ′ , z ′ ) = n X k,l = p +1 X i,j a ′ i,j,k,l ( ρ ′ ( x ′ )) ξ ′ i ( y ′ ) ⊗ ξ ′ j ( z ′ ) dx ′ k ∧ dx ′ l . W e write this more succinctly as Θ ′ | U ′ = X i,j,k,l a ′ i,j,k,l ξ ′ i ⊗ ξ ′ j dx ′ k ∧ dx ′ l . Recall that x ′ ∈ e L x ′ is the class of the cons tant path at x ′ , that we identif y M ′ with its image under x ′ → x ′ , and that Z U ′ is integration over the ﬁbration U ′ → T ′ . Let { ψ ′ U ′ } b e a partition of unit y sub ordinate to the sp ecial cov er { U ′ } of M ′ . Then T r ( θ ′ ) | T ′ = Z U ′ ψ ′ U ′ ( x ′ ) X i,j,k,l a ′ i,j,k,l ( ρ ′ ( x ′ )) ξ ′ i ( x ′ ) ∧ ξ ′ j ( x ′ ) dx ′ k ∧ dx ′ l . 46 M.-T. BENAMEUR AND J. L. HEITSCH Note that we do not mu ltiply the in tegrand by the lea fwise volume form dx ′ , since this is alrea dy incorp orated in it b y our use o f the leafwise diﬀerential forms ξ ′ i in the Sc hw ar tz kernel Θ ′ of θ ′ . In particular , b eing very precise, Θ ′ x ′ ( y ′ , z ′ ) = X i,j,k,l a ′ i,j,k,l ( ρ ′ ( x ′ )) ξ ′ i ( y ′ ) ⊗ i vol ( z ′ ) [ ξ ′ j ( z ′ ) ∧ ( · )] dx ′ k ∧ dx ′ l , where v ol( z ′ ) is the o r iented unit length vector in ( ∧ 2 ℓ T F s ) z ′ . Then tr(Θ ′ x ′ ( x ′ , x ′ )) dx ′ = X i,j,k,l a ′ i,j,k,l ( ρ ′ ( x ′ ))( i vol( x ′ ) [ ξ ′ i ( x ′ ) ∧ ξ ′ j ( x ′ )]) dx ′ dx ′ k ∧ dx ′ l = X i,j,k,l a ′ i,j,k,l ( ρ ′ ( x ′ )) ξ ′ i ( x ′ ) ∧ ξ ′ j ( x ′ ) dx ′ k ∧ dx ′ l . T o av oid no ta tional ov erlo ad, we will not b e this precise. The G ′ inv a riance o f θ ′ allows us to compute T r( θ ′ ) a s follows. Denote the plaque of x ′ in U ′ by P x ′ . Let j ′ : P x ′ → e L x ′ be the map given by: j ′ ( w ′ ) is the class of any leafwise path in P x ′ from x ′ to w ′ . Then the v alue of T r( θ ′ ) at ρ ′ ( x ′ ) ∈ T ′ is giv en b y T r( θ ′ )( ρ ′ ( x ′ )) = Z j ′ ( P x ′ ) ψ ′ U ′ ( j ′ − 1 ( y ′ )) X i,j,k,l a ′ i,j,k,l ( ρ ′ ( x ′ )) ξ ′ i ( y ′ ) ∧ ξ ′ j ( y ′ ) | j ′ ( P x ′ ) dx ′ k ∧ dx ′ l . Abusing notation once again by iden tifying P x ′ with its ima ge under j ′ , w e ha v e that at ρ ′ ( x ′ ) ∈ T ′ , T r( θ ′ )( ρ ′ ( x ′ )) = Z P x ′ ψ ′ U ′ ( y ′ ) X i,j,k,l a ′ i,j,k,l ( ρ ′ ( x ′ )) ξ ′ i ( y ′ ) ∧ ξ ′ j ( y ′ ) dx ′ k ∧ dx ′ l = X i,j,k,l a ′ i,j,k,l ( ρ ′ ( x ′ )) h Z P x ′ ψ ′ U ′ ( y ′ ) ξ ′ i ( y ′ ) ∧ ξ ′ j ( y ′ ) i dx ′ k ∧ dx ′ l . Similar remarks apply to all p ow ers of θ ′ . W e now return to our analys is o n V = f − 1 ( U ′ ), w he r e w e have the normal co ordinates x p +1 , ..., x n given by x i = x ′ i ◦ f ◦ ρ , so dx i = f ∗ ( dx ′ i ). If we set ξ i = e f ∗ ( ξ ′ i ), then the ξ i are a spanning set of π f + | V . Set a i,j,k,l = a ′ i,j,k,l ◦ f ◦ ρ , where ρ : V → T . Using Lemma 9.11 alo ng with Prop os ition 6 .13, the Sch w artz kernel Θ x ( y , z ) of θ | V is given by Θ x ( y , z ) = X i,j,k,l a i,j,k,l ( ρ ( x )) ξ i ( y ) ⊗ ξ j ( z ) dx k ∧ dx l , and the action θ | V is θ ( ξ ) = n X k,l = p +1 X i,j a i,j,k,l Q ( ξ j , ξ ) ξ i dx k ∧ dx l = X i,j,k,l a i,j,k,l h Z e L ξ j ∧ ξ i ξ i dx k ∧ dx l . That is Θ = e f ∗ Θ ′ . W e ar e interested in the Sch wartz kernels Θ ′ k and Θ k of the o p erators θ ′ k and θ k . These are given by Θ ′ k x ′ ( y ′ , z ′ ) = Z e L x ′ Z e L x ′ . . . Z e L x ′ Θ ′ x ′ ( y ′ , w ′ 1 ) ∧ Θ ′ x ′ ( w ′ 1 , w ′ 2 ) ∧ . . . ∧ Θ ′ x ′ ( w ′ k − 1 , z ′ ) and Θ k x ( y , z ) = Z e L x Z e L x . . . Z e L x Θ x ( y , w 1 ) ∧ Θ x ( w 1 , w 2 ) ∧ . . . ∧ Θ x ( w k − 1 , z ) , where the integration is done ov er rep eated v a riables. Using Prop ositio n 6 .13 again, we have immediately that Θ k = e f ∗ (Θ ′ k ) . TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 47 F o r each ψ ′ U ′ in the partition o f unity sub ordinate to { U ′ } , set ψ V = ψ ′ U ′ ◦ f , whic h g ives a partition o f unit y sub ordinate to the open cov er { V } of M . Denote by Z V int egration ov er the ﬁbr ation ρ : V → T . Recall the map i : M → G given by i ( x ) = x , the class of the constant pa th at x . Lemma 9.1 2. T r ( θ k ) = X V Z V ψ V i ∗ tr(Θ k ) . Pr o of. It suﬃces to sho w that for an y diﬀeren tial form ω on M , Z F ψ V ω and Z V ψ V ω deﬁne the same Haeﬂiger form. Let W 0 , ..., W k , W k +1 , ..., W m be an op en cov er of M by foliation c harts, with transversals S 0 , ..., S m . W e ma y assume that W 0 , ..., W k are the only elements which intersect the suppor t of ψ V non-trivially , and that these s e ts a re subsets of V . Let b ψ 0 , ..., b ψ m be a par tition o f unity sub ordina te to the W j . W e r e quire that W 0 = U and S 0 = T . Rec a ll that ρ ′ : U ′ → T ′ is the pro jection. F or j = 1 , ..., k , c ho ose a p oint y j ∈ S j . Then ρ ′ ( f ( y j )) = f ( ρ ( y j )), and as in the pr o of of Lemma 9.10, we deﬁne the leafwise path γ j from ρ ( y j ) to y j . By construction, the ho lo nomy map h j induced by the leafwise path γ j (whic h has doma in po ssibly a prop er subset of S 0 ) has range all of S j . In addition, for each S j , the map h − 1 j : S j → S 0 = T is just the restriction to S j of the pr o jection ρ : V → T . Then the Haeﬂiger classes Z F ψ V ω ≡ k X j =0 Z W j b ψ j ψ V ω = Z W 0 b ψ 0 ψ V ω + k X j =1 h ∗ j  Z W j b ψ j ψ V ω  = Z W 0 b ψ 0 ψ V ω + k X j =1 h ∗ j  Z W j b ψ j ψ V ω  . The Hae ﬂiger form Z W 0 b ψ 0 ψ V ω + k X j =1 h ∗ j  Z W j b ψ j ψ V ω  is supp orted on S 0 = T , and it fo llows immediately from the fact that h − 1 j : S j → S 0 is just ρ : S j → T , that it e q uals Z V ψ V ω .  Now c h a ( π f + ) = h T r  π f + + [ n/ 2] X k =1 ( − 1) k θ k (2 iπ ) k k ! i , and by Theore m 9.2, this equals ch a ( π + ), which is indep endent of the Bott form ω used to construct e f ∗ . Let φ b e a smo oth even function on R , decr e asing on [0 , 1], with φ (0) = 1 and φ ( x ) = 0 for | x | ≥ 1, and let ω be the Bott form whic h is a multiple o f φ ( x 1 ) ...φ ( x k ) dx 1 ...dx k . F o r t > 0 , let q t : R k → R k be the diﬀeomo rphism q t ( x ) = x/t . Denote by ω t the smo oth family of B ott forms given by ω t = q ∗ t ω , and denote b y e f ∗ t the map constructed us ing ω t . Then for all t > 0 and k ≥ 1, w e hav e h T r ( θ k ) i = h X V Z V ψ V i ∗ tr(Θ k ) i = h X V Z V f ◦ ψ ′ U ′ i ∗ tr( e f ∗ t (Θ ′ k )) i = h X V lim t → 0 Z V f ∗ ( ψ ′ U ′ ) i ∗ e f ∗ t (tr Θ ′ k ) i . W e may use the ω t to construct the family of maps f ∗ t (analogo us to the family e f ∗ t ), deﬁned o n the o riginal foliation F . As b o th e f ∗ t and f ∗ t are lo cally constructed, and tr Θ ′ k is G ′ inv a riant, it is clear that i ∗ e f ∗ t (tr Θ ′ k ) = f ∗ t ( i ′ ∗ tr Θ ′ k ) . Thu s, h X V lim t → 0 Z V f ∗ ( ψ ′ U ′ ) i ∗ e f ∗ t (tr Θ ′ k ) i = h X V lim t → 0 Z V f ∗ ( ψ ′ U ′ ) f ∗ t ( i ′ ∗ tr Θ ′ k ) i . It is a classical result that on ea ch plaque in V , the compactly suppo rted forms f ∗ ( ψ ′ U ′ ) f ∗ t ( i ′ ∗ tr Θ ′ k ) are bo unded independently o f t ∈ [0 , 1], and con verge p o in t wise to f ∗ ( ψ ′ U ′ ) f ∗ ( i ′ ∗ tr Θ ′ k ) = f ∗ ( ψ ′ U ′ i ′ ∗ tr Θ ′ k ). By 48 M.-T. BENAMEUR AND J. L. HEITSCH the Dominated Conv erg ence Theore m, we hav e h T r ( θ k ) i = h X V Z V lim t → 0 f ∗ ( ψ ′ U ′ ) f ∗ t ( i ′ ∗ tr Θ ′ k ) i = h X U ′ Z f − 1 ( U ′ ) f ∗ ( ψ ′ U ′ i ′ ∗ tr(Θ ′ k )) i = h f ∗ X U ′ Z U ′ ψ ′ U ′ i ′ ∗ tr(Θ ′ k ) i = f ∗ h T r( θ ′ k ) i . As ch a ( π ′ + ) = h T r  π ′ + + [ n/ 2] X k =1 ( − 1) k θ ′ k (2 iπ ) k k ! i , to ﬁnish the pro o f that ch a ( π f + ) = f ∗ (ch a ( π ′ + )), w e need only show that h T r( π f + ) i = f ∗ h T r( π ′ + ) i . Just as w e did with θ ′ , w e ma y w r ite the Sc hw artz k er nel of π ′ + | U ′ as ( π ′ + ) x ′ ( y ′ , z ′ ) = X i,j b ′ i,j ( ρ ′ ( x ′ )) ξ ′ i ( y ′ ) ⊗ ξ ′ j ( z ′ ) , where the b ′ i,j are functions on T ′ , and the a ction of π ′ + on a s ection ξ ′ is given by π ′ + ( ξ ′ ) = X i,j b ′ i,j Q ′ ( ξ ′ j , ξ ′ ) ξ ′ i . Set b i,j = e f ∗ b ′ i,j = b ′ i,j ◦ f ◦ ρ a nd ξ i = e f ∗ ( ξ ′ i ), a nd consider the ope r ator e π f + on A ℓ (2) ( F s , E ), where e π f + | V = P i,j b i,j ξ i ⊗ ξ j , whic h acts by e π f + ( ξ ) = X i,j b i,j Q ( ξ j , ξ ) ξ i . Then e π f + is a G in v ariant idempo tent , has image equa l to Im( π f + ), and ha s a smo oth Sc h w artz kernel. In general e π f + 6 = π f + bec ause for ms of the type δ s β , which are in the k ernel of π f + , are not necessar ily in the kernel of e π f + . Howev er , since e π f + has smo oth Sch wartz kernel, T r( e π f + ) is well deﬁned, and its Sch w artz kernel is just e f ∗ of the Sc h wartz kernel of π ′ + . Arguing as we did for θ k , w e get h T r( e π f + ) i = f ∗ h T r ( π ′ + ) i . Lemma 9.1 3. h T r ( π f + ) i = h T r ( e π f + ) i . Pr o of. Since Im( π f + ) = Im( e π f + ), and both ar e idempo ten ts, w e need only show that e π f + is transversely smooth, and then apply Lemma 3.6. W e will b e using the no ta tion of Section 6. Supp ose the K ′ is the Sch wartz kernel o f a G ′ inv a riant bo unded lea fwise smo o thing op era tor on A ℓ (2) ( F ′ s , E ′ ), which is given lo ca lly , with resp ect to a lo cal inv ariant spanning set { ξ ′ i } of A ℓ (2) ( F ′ s , E ′ ), b y K ′ = P i,j b ′ i,j ξ ′ i ⊗ ξ ′ j , with the actio n given by K ′ ( ξ ′ ) = X i,j b ′ i,j Q ′ ( ξ ′ j , ξ ′ ) ξ ′ i . Now consider the op era tors e f ∗ K ′ on A ℓ (2) ( F s , E ) and e p ∗ f K ′ on A ℓ (2) ( F s × B k , p ∗ f E ′ ), with lo cal Sch w artz kernels e f ∗ K ′ = X i,j e f ∗ b ′ i,j e f ∗ ξ ′ i ⊗ e f ∗ ξ ′ j , and e p ∗ f K ′ = X i,j p ∗ f b ′ i,j ( p ∗ f ξ ′ i ∧ ω ) ⊗ ( p ∗ f ξ ′ j ∧ ω ) , where ω is a Bott form on B k . Rec a ll that π 1 , ∗ is in tegration ov er the ﬁb er of the pro jection π 1 : G × B k → G , and p f , ∗ is int egration ov er the ﬁb er of the submersion p f : G × B k → G ′ . Straight forward computations show that for ξ ∈ A ℓ (2) ( F s , E ) and e ξ ∈ A ℓ (2) ( F s × B k , p ∗ f E ′ ), e f ∗ K ′ ( ξ ) = π 1 , ∗  e p ∗ f K ′ ( π ∗ 1 ξ )  and e p ∗ f K ′ ( e ξ ) = p ∗ f  K ′ ( p f , ∗ ( ω ∧ e ξ ))  ∧ ω . TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 49 The maps π 1 , ∗ , π ∗ 1 , p ∗ f , p f , ∗ , and ∧ ω are all b ounded ma ps , and K ′ is b ounded and lea fwise smo othing. Thus e f ∗ K ′ is a bounded leafwise s mo othing op er ator. Applying this to K ′ = π ′ + , we hav e that e π f + is a b ounded leafwise smo o thing oper ator. Using Pr op osition 7.4, it is easy to show that ∂ Y ν e π f + = [ A ( Y ) , e π f + ] + e f ∗ ( i Z ′ ∂ ν ′ π ′ + ) , wher e Y and Z ′ are as in Lemma 7 .8, and A ( Y ) is a lea fwise op er a tor whose comp osition with a b o unded lea fwise s mo othing op erator is again a b o unded leafwise smo othing op era tor. Applying the a rgument ab ov e to i Z ′ ∂ ν ′ π ′ + , we hav e that ∂ Y ν e π f + is also a b ounded leafwise smo othing oper a tor. An o bvious induction argument ﬁnishes the pro of.  Thu s h T r ( π f + ) i = h T r( e π f + ) i = f ∗ h T r( π ′ + ) i , and w e are done.  10. The twisted leafwise signa ture opera tor and the twisted higher Betti classes In this section we give some immediate conse quences of our results. In particular, we show that the t wisted hig her harmonic sig nature eq ua ls the (gra ded) Cher n- Connes character in Haeﬂiger cohomology o f the “index bundle” of the twisted lea fwise signature op erator , that is the (graded) Chern-Connes c haracter ch a ( P ) of the pro jection P onto all the t wisted leafwise harmo nic forms. W e co njecture a cohomolo gical formula for this Chern- Connes character, which has already b een proven in some cases. W e also indicate how our metho ds prov e that the t wisted higher Betti n umber s are leafwise ho motopy inv a riants. Consider the ﬁrst o rder lea fwise op erator D E = d s + δ s , whic h is formally self adjoint and satisﬁe s ( D E ) 2 = ∆ E . Because of this, the k ernel of D E is the same as the kernel o f ∆ E . Recall the ± 1 eigenspaces A ∗ ± ( F s , E ) of the in volution b τ of A ∗ (2) ( F s , E ), and that D E b τ = − b τ D E , so we hav e the o pe rators D E ± : A ∗ ± ( F s , E ) → A ∗ ∓ ( F s , E ), and D E + is designated the twisted lea fwise signature oper ator. Denote by P ± the pro jectio ns onto the Ker( D E ± ). W e assume that the pro jection P to Ker(∆ E ) is transversely s mo oth, so the P ± are also. Then the (graded) Chern-Co nnes character of the index bundle of the t wisted leafwise signature o p e r ator, c h a ( P ), is deﬁned, and is g iven by ch a ( P ) = ch a ( P + ) − c h a ( P − ) = h ch a ( ℓ − 1 X j =0 P j + τ P j ) + ch a ( 1 2 ( P ℓ + τ P ℓ )) i − h ch a ( ℓ − 1 X j =0 P j − τ P j ) + c h a ( 1 2 ( P ℓ − τ P ℓ )) i . As in the case of compac t ma nifolds, w e ha ve Theorem 10. 1. Supp ose that M is a c omp act Rie mannian manifold, with oriente d Riemannian foliatio n F of dimension 2 ℓ , and that E is a le afwise ﬂat c omplex bu nd le over M with a ( p ossibly indeﬁnite) non- de gener ate Hermitian metr ic which is pr eserve d by the le afwise ﬂat st ructur e. A ssume that the pr oje ction P onto Ker(∆ E ) for the asso ciate d foliation F s of the homotopy gr oup oid of F is tr ansversely smo oth. Then, the (gr ade d) Chern-Connes char acter ch a ( P ) of the index bund le of the t wiste d le afwise signatur e op er ator e quals the twiste d higher harmonic signatur e of F , that is ch a ( P ) = σ ( F , E ) . Pr o of. As ch a is linear and 1 2 ( P ℓ ± τ P ℓ ) = π ± , w e need only show that ch a ( P j + τ P j ) = ch a ( P j − τ P j ) , for j = 0 , ..., ℓ − 1. Set P t = P j + tτ P j where − 1 ≤ t ≤ 1 . Then P t is a smo oth family of G inv ariant transversely smo oth idemp otents (since P j τ P j = 0 for j = 0 , ..., ℓ − 1) whic h connects P j + τ P j to P j − τ P j . It follows from Theorem 3.5 that ch a ( P j + τ P j ) = ch a ( P j − τ P j ).  Corollary 10 . 2. Under the hyp othesis of The or em 10.1, t he (gr ade d) Chern-Connes char acter ch a ( P ) of the index bund le of the le afwise signatur e op er ator with c o eﬃcients in E is a le afwise homotopy invariant. 50 M.-T. BENAMEUR AND J. L. HEITSCH The op erato r D E + is elliptic alo ng the lea ves of F s , and so pro duces, v ia a now classical construction due to Connes [C81], a K − theory inv ariant Ind a ( D E + ), the index o f the op erator D E + , which has a Chern-Connes character ch a (Ind a ( D E + )) ∈ H ∗ c ( M / F ), [BH04]. Conjecture 10.3 . Under the hyp othesis of The or em 10.1 , ch a (Ind a ( D E + )) = ch a ( P ) ∈ H ∗ c ( M / F ) . This conjecture has b een prov en when the sp ectrum of D E + is reasonably well b ehav ed, see [H95, HL99, BH08], where it is prov en for the holonomy gro up o id. The pr o ofs extend immediately to the homo to py group oid. It also holds for bo th group oids, w itho ut any extr a assumptions, whenever the pr o jection P belo ngs to Connes’ C ∗ -algebra of the foliation for the g roup oid in question. In particula r, it holds for the holonomy gro upo id ca se fo r an y folia tion whose leaves are the ﬁb ers o f a ﬁbratio n b e tw een closed manifolds, provided that P is transversely s mo oth. Recently , Azzali, Go ette and Sc hick have announced, [AGS], that they hav e prov en it for smo o th pro pe r submersions V → B with the ﬁbrewise action (freely and pr op erly disco nt inu ous) of a discrete gro up Γ such that the quotient V / Γ → B is a ﬁbration with compact ﬁb er, but only for bundles E which ar e g lobally ﬂat. Conjecture 10.3 should follow immediately for the homotop y gr oup oid pr ovided that their r esult extends to bundles which are only leafwise ﬂat. Recall, [BH04, GL03], that in Haeﬂiger cohomo logy , ch a (Ind a ( D E + )) = Z F L ( T F ) c h 2 ( E ) , where L ( T F ) is the c haracteris tic cla ss of T F asso ciated with the multiplicativ e s e quence Q j x j / tanh( x j ), and c h 2 ( E ) = P k 2 k ch k ( E ). Corollary 10.4. U n der the hyp othesis of The or em 10.1, and assuming Conje ctur e 10.3, Z F L ( T F ) c h 2 ( E ) is a le afwise homotopy invariant. Finally w e have the following. Deﬁnition 10. 5. Assume t he hyp othesis of The or em 10.1, but n ow F may have arbitr ary dimension. F or 0 ≤ j ≤ p = dim( F ) , deﬁne the j -t h twiste d high er Betti class β j ( F, E ) by β j ( F, E ) = ch a ( P j ) ∈ H ∗ c ( M / F ) . It is an interesting exer cise to sho w that, just as in the cas e of compa ct ﬁbra tio ns, the bundle deﬁned by the pro jection o nt o the leafwise ha rmonics (in the case E = M × C ) is a ﬂat bundle. That is, it admits a connection who s e curv ature is zero, so there are no higher terms in the β j ( F, M × C ). This is not the case in general. Theorem 10.6. (Comp ar e [HL91] ) Under the hyp othesis of The or em 10.1 with F al lowe d to have arbitr ary dimension, the twiste d higher Betti classes β j ( F, E ) , ar e le afwise homotopy invariants. Pr o of. W e o nly give a sketc h here of the pro of of the seco nd statement. Let f : ( M , F ) → ( M ′ , F ′ ) b e a smo oth leafwise homotopy equiv alence w ith smo oth ho motopy in verse g . The pull-back bundle f ∗ ( P ′ j ) is a smo oth bundle s ince it can b e r e a lized by the tra nsversely smo oth idemp otent P f j = f ∗ R ′∗ P ′ j g ∗ R ∗ P j . It can be endow ed with the pull-ba ck connection under f of the connection P ′ j ∇ ′ ν P ′ j , and hence the Cher n-Connes character of f ∗ ( P ′ j ) is given by ch a ( f ∗ ( P ′ j )) = f ∗ ch a ( P ′ j ) = f ∗ β j ( F ′ , E ′ ) . As in the proof o f our ma in theor em, one pr oves that P j : f ∗ (Ker(∆ E ′ j )) → Ke r(∆ E j ) is an iso morphism and that Q f j = P j P f j is a smo o th idemp otent with imag e Ker(∆ E j ), hence its Chern-Connes character coincides with the Betti class β j ( F, E ). As Q f j P f j = Q f j and P f j Q f j = P f j , the family Q t = tQ f j + (1 − t ) P f j ) is a smo oth homo topy by tr ansversely smo oth idempotents from Q f j to P f j . Therefore, P f j and Q f j hav e sa me Chern-Connes c haracter.  TWISTED HIGHER S IGNA TURES FOR F OLIA TIONS 51 11. Consequences of the Main Theorem In this s ection, we derive some imp orta nt consequences of Theorem 9 .1. In particular, we re- der ive some classic results for the No v iko v conjecture, a nd then give some general results for the Noviko v conjecture for groups and for foliations. Example 11.1 (Lusz tig , [Lu72]) . Let N b e a compact c onnected even dimensional Riemannian manifold. Set W = H 1 ( N ; R / Z ), and r ecall the na tur al (onto) map h 1 : W → Hom(H 1 ( N ; Z ); R / Z ). Cho ose a base p oint x o ∈ N . Then there is the natural (ont o) homomorphism h : W → Hom( π 1 ( N , x o ); R / Z ) g iven b y comp osing h 1 with the natural map π 1 ( N , x o ) → H 1 ( N , Z ). Thus for each elemen t w ∈ W , we hav e the homo morphism h ( w ) : π 1 ( N , x o ) → R / Z , which we may comp ose with the map x → exp(2 π ix ) to obtain the homo morphism h w : π 1 ( N , x o ) → S 1 ⊂ C . Denote by e N the universal covering of N . π 1 ( N , x o ) a c ts o n e N in the usual w ay , and on e N × W × C as follows. Let β ∈ π 1 ( N , x o ), and ( x, w, z ) ∈ e N × W × C , and deﬁne β · ( x, w , z ) = ( β x, w , h w ( β ) z ) . Set E = ( e N × W × C ) /π 1 ( N , x o ) , a complex bundle o ver ( e N × W ) /π 1 ( N , x o ) = N × W , whic h is leafwise ﬂat for the foliation F given by the ﬁbration M ≡ N × W → W . It is obvious that the usual metric on C deﬁnes a p os itive deﬁnite metric on E which is preser ved b y the leafwise ﬂat structure. As H 1 ( N ; R / Z ) is the ab e lianization of π 1 ( N , x o ), h is onto, and it is natural to call E the universal ﬂat C bundle for N . Then M , F , and E s atisfy the hyp o thesis of Theorem 9.1, s ince the preserved metr ic is positive deﬁnite. Note that if f : N → N ′ is a homotopy eq uiv alence, then ther e is a natur al ex tension of f to f : M , F → M ′ , F ′ which is a leafwise homo topy equiv alence, and f ∗ E ′ = E . Thus σ ( F, E ) is a homotopy in v ariant of the manifold N . By [BH04] (and assuming Conjecture 10.3 if necessar y), we have that σ ( F, E ) = Z N L ( T F ) c h 2 ( E ) ∈ H ∗ c ( M / F ) = H ∗ (H 1 ( N ; R / Z ); R ) . T o r e late this to Lusztig’s theorem on Noviko v conjecture, supp ose that π 1 ( N , x 0 ) = Z n . Deno te by g : N → B Z n = T n the map c lassifying the universal cov er e N → N (as a Z n bundle), and let α 1 , ..., α n be the natural basis of H 1 ( T n ; R ). Prop ositi o n 11 .2. ch 2 ( E ) = n Y i =1 (1 + 2 g ∗ ( α i ) ⊗ α i ) . Theorem 11.3 (Lusztig, [Lu72]) . The Novikov c onje ctur e is true for any c omp act manifold with fundamental gr oup Z n . Pr o of. σ ( F, E ) = Z N L ( T F ) c h 2 ( E ) = X i 1 < ···

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