The relative L^2 index theorem for Galois coverings
aa r X i v : . [ m a t h . OA ] S e p THE RELATIVE L INDEX THEOREM FOR GALOIS COVERINGS
MOULAY-TAHAR BENAMEUR
Abstract.
Gievn a Galois covering of complete spin manifolds where the base metric has PSC near infinity,we prove that for small enough ǫ >
0, the ǫ spectral projection of the Dirac operator has finite trace inthe Atiyah von Neumann algebra. This allows us to define the L index in the even case and we prove itscompatibility with the Xie-Yu higher index and deduce L versions of the classical Gromov-Lawson relativeindex theorems. Finally, we briefly discuss some Gromov-Lawson L invariants. Contents
1. Introduction 12. Dirac operators and Atiyah’s von Neumann algebra 43. Von Neumann trace of the spectral ǫ -projection 64. Proof of Theorem 3.1 85. Compatibility with the higher index 115.1. Review of the Xie-Yu higher index 115.2. Traces and numerical indices 136. Relative L index theory 166.1. The Φ-relative L index theorem 166.2. The torsion-free case 186.3. Some L Gromov-Lawson PSC-invariants 20Appendix A. A Rellich lemma and a second proof of Theorem 3.1 23References 261.
Introduction
In [GL:83], M. Gromov and B. Lawson introduced in their investigation of the topological space of metricsof positive scalar curvature on a given smooth manifold M , the notion of complete metrics having positivescalar curvature (PSC in the sequel) near infinity. They proved in particular that the Dirac operator D g associated with a spin structure which is compatible with such metric has finite dimensional kernel.This allowed them to introduce an integer invariant for such metrics on even dimensional spin manifolds,which is the Fredholm index of the Dirac operator D + g acting from the positive half-spinors to the negativeones, exactly as for closed spin manifolds, we denote this invariant by Spin( M, g ). Despite the closed case,the invariant Spin(
M, g ) depends on g and is not a topological invariant, although it does not dependon perturbations of the geometric data over compact subspaces. An already interesting class of exampleswhere such invariant plays a significant part corresponds to the “small” class of complete manifolds with ariemannian cylindrical end, where the invariant Spin( M, g ) can be shown to coincide with the Atiyah-Patodi-Singer index for a corresponding compact spin manifold with boundary [APS:75]. This observation allowsto relate Spin(
M, g ) with some standard spectral invariants but convinces as well that there will certainly beno simple formula for Spin(
M, g ) in the general case. The relative index theorem remedies for this by givinga formula for the difference of such indices when one has identifications near infinity. Gromov and Lawson
MSC (2010) 53C12, 57R30, 53C27, 32Q10.Key words: positive scalar curvature, relative index, L index theorem, spin structure, complete manifold. actually proved as well their so-called Φ-relative index theorem [GL:83] that we briefly review below, seealso [LM:89].Assume first that E is some hermitian bundle, over the complete spin riemannian manifold ( M, g ) with g having PSC near infinity, which has a connection ∇ E which is flat near infinity, then the twisted Diracoperator D g ⊗ E has again finite dimensional kernel and in the even case, the index of D + g ⊗ E , denotedSpin( M, g ; E ) is well defined. The simplest statement of the relative index theorem is the following formulawhere K is any compact subspace of M such that off K , g has PSC and E is flat:Spin( M, g ; E ) − dim( E ) Spin( M, g ) = Z K b A ( M, g ) ch > ( ∇ E ) , We have denoted as usual b A ( M, g ) the b A -polynomial in the Pontryagin forms of the Levi-Civita connectionassociated with g , and ch > ( ∇ E ) := ch( ∇ E ) − dim( E ) where ch( ∇ E ) denotes the Chern character polynomialin the curvature of the hermitian connection ∇ E . An obvious corollary is that the LHS is an obstructionto extending ∇ E to a flat connection on M . The general relative index theorem gives a similar formulafor the difference of the indices of two generalized Dirac operators on different smooth complete manifolds,such that the operators agree near infinity via some isometry. In this case the individual indices don’tmake sense in general, however the relative index (difference of the virtual two indices) still makes senseby replacing each manifold by some closed manifold obtained by chopping off the isometric open subspacesand attaching the same compact manifold with boundary to each of them, and then taking the differenceof the resulting indices, see [GL:83][page 119]. Then a similar relative index theorem holds by using wellchosen parametrices for the generalized Dirac operators. Finally, they also gave another version: the so-called Gromov-Lawson Φ-relative index theorem. Here, one assumes that both generalized Dirac operatorsare invertible near infinity so that the individual indices do make sense, but they don’t assume anymorethat the identification of the operators holds off some compact space but only on some union Φ of connectedcomponents of a neighborhood of infinity. The Φ-relative index is then defined similarly, but now one has tochop off the isometric parts and just glue the two remaining manifolds so that the resulting generalized Diracoperator is invertible near infinity, therefore has a well defined index. The index formula in this Φ-relativecase, identifies the index of this latter resulting generalized Dirac operator with the difference of the indicesof the generalized Dirac operators on the original manifolds.All these relative index theorems played a crucial role in their study of enlargeable manifolds and themoduli space of metrics of PSC. Gromov and Lawson could for instance deduce obstruction criteria in termsof (non-compact) enlargeability of closed manifolds, but also some similar criteria for complete non compactmanifolds, see the nice summary in [LM:89]. Already in the APS cylindrical ends case, it was for instancecombined with some topological results of Milnor, to recover in [GL:83] other non-trivial corollaries. Letus mention for instance their proof of the infiniteness of the number of connected components of the spaceof PSC metrics on the 7-sphere S . It is worthpointing out though that in the cylindrical ends case, allthe relative index theorems reviewed above become corollaries of the APS index theorem for manifolds withboundary, which provides a more precise answer since it gives an index formula for each individual manifold.The present paper is devoted to the L version of the Gromov-Lawson relative index theory together withsome applications. Our setting will be the category of riemannian Galois Γ-coverings π : f M → M over thecomplete riemannian manifold M , with Γ being a discrete countable group and we proceed now to describesome results that we obtained. Given a generalized Dirac operator over M , recall that D = ∇ ∗ ∇ + R withan explicit zero-th order operator R . We then consider its lift e D acting on the L -sections of the generalizedspinors e S over f M . The operator e D is affiliated with the Atiyah semi-finite von Neumann algebra M ofΓ-invariant bounded operators on L ( f M , e S ). The Atiyah trace on M is denoted τ and corresponds roughlyspeaking to integration over fundamental domains of the Galois covering. For any ǫ ≥
0, we may thenconsider the spectral projection P ǫ of e D corresponding to the intervalle [ −√ ǫ, + √ ǫ ], an element of M . Ourfirst result is the following ELATIVE L INDEX THEOREM 3
Theorem 1.1.
Assume that there exists κ > such that R ≥ κ Id off some compact subspace of M , thenthere exists ǫ > such that P ǫ has finite τ -trace. In particular, this proves in the even case that the L -index is well defined byInd (2) ( e D + ) = dim Γ (Ker e D + ) − dim Γ (Ker e D − ) . In the case where M is a spin manifold and D = D g is the spin-Dirac operator, this shows that if g hasPSC near infinity then the L -index is well defined. It will be denoted in this case Spin (2) ( f M , g ). When M = N × R for a closed spin odd dimensional manifold N , we deduce the L version of the Gromov-Lawsoninvariant i ( N ; g, g ′ ) that we denote by i (2) ( e N ; g, g ′ ) for any metrics g and g ′ with PSC.In [XY:14], Xie and Yu defined a higher relative index living in the K -theory of the group C ∗ -algebraand proved a higher version of the Φ-relative index theorem. We have thus privileged to use the Xie-Yuapproach to deduce our L versions of the relative index theorems. Inorder to achieve this program, we wereled to prove the compatibility of our L -index with the higher one through the usual regular trace on the C ∗ -algebra of Γ. Inorder to prove this compatibility result, we use results proved in [BR:15] together withthe crucial property of finite τ -trace of P ǫ , for some ǫ >
0. So our second result can be stated as follows
Theorem 1.2.
Assume that M is even dimensional and denote by D r the regular Michschenko-FomenkoDirac operator associated with the generalized Dirac operator D and by Ind( D + r ) its Xie-Yu higher index in K ( C ∗ r Γ) . If τ reg ∗ : K ( C ∗ r Γ) → R is the group morphism induced by the regular trace τ reg , then τ reg ∗ (cid:0) Ind( D + r ) (cid:1) = Ind (2) ( e D + ) . Applying this theorem together with a construction of the Φ-relative L -index, we could deduce thefollowing Φ-relative L index theorem: Theorem 1.3.
Assume that we have two Galois Γ -coverings f M → M and f M ′ → M ′ with generalized Diracoperators which both satisfy the invertibility condition near infinity R ≥ κ Id as in [GL:83] , and assumefurthermore that there exist unions Φ and Φ ′ of connected components of a neighborhood of infinity in M and M ′ respectively which are identified together with their Γ -coverings so that the generalized Dirac operators D and D ′ over Φ and Φ ′ are conjugate. Then the Φ -relative L -index Ind (2) ( e D, f D ′ ; Φ) is a well defined realnumber and we have the Φ -relative L index formula Ind (2) ( e D, f D ′ ; Φ) = Ind (2) ( e D + ) − Ind (2) ( f D ′ + ) . In the special case where the open subspaces Φ and Φ ′ have compact complements in M and M ′ , no needof any invertibility condition near infinity to define the L relative index Ind (2) ( e D, f D ′ ) for any generalizedDirac operators which are conjugate near infinity. In this case, we could deduce the following Atiyah relative L index formula which extends the classical result proved in [At:76]: Theorem 1.4.
Let f M → M and f M ′ → M ′ be two Galois Γ -coverings with generalized Dirac operators D and D ′ on M and M ′ respectively. Assume that the coverings are identified near infinity so that the lifts e D and f D ′ which are Γ -equivariantly conjugate near infinity. Then the relative L index Ind (2) ( e D, f D ′ ) is welldefined and agrees with the Gromov-Lawson relative index of the operators D and D ′ on M , i.e. Ind (2) ( e D, f D ′ ) = Ind( D, D ′ ) . When Γ is torsion free, we have the following partial but more precise result.
Theorem 1.5.
Assume that R is uniformly positive near infinity and that Γ is torsion free with rationallyonto maximal Baum-Connes assembly map K ( B Γ) → K ( C ∗ m Γ) . Then Ind (2 ( e D + ) = Ind( D + ) . In the spin case, we obtain the equality Spin (2) ( f M , g ) = Spin(
M, g ) , whenever g has uniform PSC nearinfinity and Γ is torsion free with rationally onto maximal Baum-Connes map. A corollary of this theorem M-T. BENAMEUR is the corresponding statement for the APS L index for coverings with boundaries. See Theorem 6.8.Our L index allows to consider for instance the generalized Cheeger-Gromov invariant κ Γ ( M, g ) := Spin (2) ( f M , g ) − Spin(
M, g ) , which is expected to be trivial for torsion free groups but provides an interesting invariant in general. In thecylindrical ends case considered by Atiyah-Patodi-Singer out of a compact spin manifold X with boundary Y having PSC, this invariant corresponds to the Cheeger-Gromov L rho invariant ρ Γ ( Y, g Y ), but in general wecan only show that it only depends on the geometry near infinity with, so far, no explicit spectral expressionfor instance. We recover for instance the well known fact that the Cheeger-Gromov ρ invariant induces a mapon the moduli space of connected components of the space of PSC metrics on any closed odd dimensionalmanifold modulo orientation preserving diffeomorphisms. This κ invariant for M = N × R as in [GL:83]with a Galois Γ-covering e N → N , yields the invariant κ Γ ( N ; g, g ′ ) := i (2) ( e N ; g, g ′ ) − i ( N ; g, g ′ ) . By the APS theorem this invariant is again just a difference of Cheeger-Gromov L rho invariants for thegiven metrics. When N has trivial real Pontryagin classes, one can extend similarly some results from[KS:93]. These and other applications will be carried out in a forthcoming paper. A short appendix givesa different proof of the finite τ -trace of the spectral projection P ǫ now using a von Neumann version of theRellich lemma for which we provide an independent proof. Acknowledgements.
The author would like to thank B. Ammann, P. Antonini, A. Carey, P. Carrillo-Rouse,J. Heitsch, V. Mathai and G. Yu for several discussions.2.
Dirac operators and Atiyah’s von Neumann algebra
Let Γ be a countable finitely generated discrete group and let π : f M → M be a Galois Γ-covering overthe smooth complete riemannian manifold ( M, g ). We endow f M with the lifted Γ-invariant metric e g , sothat it is also a complete riemannian manifold and π is an isometric covering. We fix a hermitian bundle S of generalized spinors over M , in the sense of [GL:83][Section I]. Its pull-back to f M is denoted e S . Wefix corresponding Lebesgue class measures on M and f M that we denote by dm or d e m respectively. Wethen consider the Hilbert spaces of L -sections of these spinor bundles S and e S that we denote by L ( M, S )and L ( f M , e S ) respectively. Notice that L ( f M , e S ) is a Hilbert space which is endowed with the unitaryrepresentation of Γ. Let F ⊂ f M be an open fundamental domain for the Galois cover. This means for usthat γF ∩ F = ∅ for any g = e , and that the collection ( gF ) γ ∈ Γ is a locally finite cover of f M . We mayassume as well that F equals the interior of its closure F if needed. It is then worth pointing out that thesubspace f M r ∪ γ ∈ Γ gF is d e m -negligible. This choice of fundamental domain F allows to identify the HilbertΓ-space L ( f M , e S ) with the Hilbert Γ-space ℓ Γ ⊗ L ( F, e S ) where Γ acts trivially on L ( F, e S ) ≃ L ( M, S )and by the left regular representation on ℓ Γ. See [At:76] for more details on this standard construction.The von Neumann algebra of bounded Γ-invariant operators on L ( f M , e S ) will be denote by M . Theelements of M are bounded operators T acting on the Hilbert space L ( f M , e S ) which commute with theunitary representation of Γ. This is a semi-finite von Neumann algebra which is isomorphic to B ( L ( F, e S )) ⊗N Γ where N Γ is the regular von Neumann algebra associated with Γ, say the weak closure of the leftregular representation in ℓ Γ. There is a faithful normal semi-finite positive trace τ on M correspondingthrough the isomorphism M ≃ B ( L ( F, e S )) ⊗ N Γ to the trace Tr ⊗ τ e where τ e is the finite trace on N Γgiven by θ < θ ( δ e ) , δ e > with δ e being the characteristic function at the neutral element e . Tr denotesas usual the trace of operators on the Hilbert space L ( F, e S ). It can also be defined directly as follows,see [At:76]. Denote by χ F the characteristic function of the fundamental domain F , i.e. the Borel functionwhich equals 1 on F and 0 on its complement. Then multiplication by the function χ F yields a projection M χ F : L ( f M , e S ) → L ( f M , e S ) whose image can be identified with L ( F, e S ). ELATIVE L INDEX THEOREM 5
Definition 2.1. [At:76]
For any non-negative operator T ∈ M , we set τ ( T ) := Tr( M χ F ◦ T ◦ M χ F ) ∈ [0 , + ∞ ] . That τ is a faithful normal semi-finite positive trace on M which does not depend on the choice offundamental domain F is straightforward, see again [At:76]. Any operator in M which corresponds to afinite sum of elementary tensors A i ⊗ θ i with trace class operators A i on the Hilbert space L ( F, e S ) andwith θ i ∈ N Γ, will obviously have finite τ -trace. Given a Hilbert Γ-subspace H of L ( f M , e S ), we denote by P H the Γ-invariant orthogonal projection onto H , a non-negative idempotent in M . The Γ-dimension of H ,denoted dim Γ ( H ), is defined as the trace of the projection P H ∈ M , i.e.dim Γ ( H ) := τ ( P H ) ∈ [0 , + ∞ ] . We can define in the same way the Γ-dimension of any Hilbert Γ-space which is unitarily equivalent to aHilbert Γ-subspace of L ( f M , e S ). In particular, given a Hilbert subspace H of L ( F, e S ), the Γ-dimension of ℓ Γ ⊗ H is well defined and coincides with the usual dimension of H . When H is finite dimensional, suchHilbert Γ-space what is called in [Lu:97] a finitely generated Hilbert Γ-space.When M is even dimensional, the generalized spin bundle S will be assumed to have a Z -grading and tosplit into S = S + ⊕ S − , we then have the corresponding Γ-equivariant splitting π ∗ ( S ) = e S = e S + ⊕ e S − . Let D be a generalized Dirac operator on M acting on the smooth sections of S , and denote by e D its Γ-invariantpull-back to the generalized Dirac operator on f M , acting on the sectiojns of e S . Recall that e D is a Γ-invariantfirst order differential operator which is odd for the grading, i.e. e D = e D − e D + ! . The orthogonal connection on S is denoted ∇ and its pull-back connection to e S is denoted e ∇ . Recall thatif ( e k ) k is a local orthonormal basis of tangent vectors then the operator ∇ ∗ ∇ is given by the formula ∇ ∗ ∇ = − X k (cid:16) ( ∇ e k ) − ∇ ∇ ek e k (cid:17) and similarly for e ∇ ∗ e ∇ . Recall as well that for smooth sections σ, σ ′ where at least one of them is compactly supported, one has h∇ ∗ ∇ σ, σ ′ i = h∇ σ, ∇ σ ′ i . For s ≥
0, the Sobolev space L s ( M, S ) (resp. L s ( f M , e S )) is defined as the completion of the space C ∞ c ( M, S )(resp. C ∞ c ( f M , e S )) of smooth compactly supported sections, for the Sobolev L -norm || σ || s := s X j =0 ||∇ j σ || , σ ∈ C ∞ c ( M, S ) , and similarly for f M using the Γ-invariant pull-back connection e ∇ . In particular, L ( M, S ) = L ( M, S ) and L ( f M , e S ) = L ( f M , e S ). We denote by Ω S the curvature tensor of the connection ∇ on S given byΩ S ( X , X ) = [ ∇ X , ∇ X ] − ∇ [ X ,X ] for any vector fields X , X . The pull-back f Ω S of Ω S to f M is nothing but the curvature tensor of the pull-back conneciton e ∇ on e S . Thefollowing theorem is classical, see also [GL:83][Propositions 2.4 & 2.5]: Theorem 2.2. [Li:63]
The operator ∇ ∗ ∇ is an essentially self-adjoint operator. Moreover, for u ∈ L ( f M , e S ) ,we have ∇ ∗ ∇ ( u ) = 0 ⇐⇒ ∇ u = 0 (i.e. u is parallel) . Moreover, the following generalized Lichnerowicz local formula holds D = ∇ ∗ ∇ + R with R := 12 X i,j e i · e j · Ω S ( e i , e j ) , M-T. BENAMEUR with ( e i ) i being any local orthonormal frame on f M . The corresponding relation holds obviously on f M since it satisfies the same riemannian conditions satisfiesby M . In the sequel, we shall sometimes also denote by R the pull-back operator e R which is defined by thesame formula. A standard calculation shows that when D is the spin-Dirac operator associated with a fixedspin structure associated with the SO bundle corresponding to the metric g on M , then R = κ Id S , where κ is the scalar curvature function of g . Proposition 2.3.
Under the previous assumptions, we have: (1) e D : C ∞ c ( f M , e S ) → C ∞ c ( f M , e S ) is essentially self-adjoint whose self-adjoint extension has domaincontained in L ( f M , e S ) ; (2) If the pointwise norm |R| of the zero-th order operator R is uniformly bounded on M , then thedomain of the self-adjoint extension of e D is exactly L ( f M , e S ) and e D gives a bounded operator from L ( f M , e S ) to L ( f M , e S ) ; (3) e D : C ∞ c ( f M , e S ) → C ∞ c ( f M , e S ) is essentially self-adjoint and its self-adjoint extension is a non-negative operator. (4) The kernels of the operators e D and e D on L ( f M , e S ) do coincide. Moreover, they are composed ofsmooth sections.Proof. Most of the items are standard results for generalized Dirac operators on complete manifolds, seefor instance [GL:83][Theorems 1.17, 1.23] and also [LM:89][Theorem II.5.7]. The coincidence of the domainswith the Sobolev spaces under the condition that R is bounded is a consequence of the Lichnerowicz formula,see [GL:83][Theorem 2.8]. For the second part of the last item, notice that if U is a relatively compact opensubspace of f M , then any L harmonic section restricts to a harmonic L section in L ( U, e S ). (cid:3) Given any open relatively compact subspace Ω of M and its inverse image e Ω in f M , the manifold e Ω hasbounded geometry and so is the restricted bundle e S | e Ω . Therefore, classical elliptic estimates apply and weobtain for instance the existence for any s ∈ Z of a constant C s > u ∈ L s ( e Ω , e S ): || u || L s ( e Ω , e S ) ≤ C s ( || u || L s − ( e Ω , e S ) + || e D ( u ) || L s − ( e Ω , e S ) )Here again the Sobolev spaces are defined as for f M completing the smooth compactly supported sections on e Ω. Using Proposition 2.3, we hence deduce that the norms || • || L s ( e Ω , e S ) and || • || L s − ( e Ω , e S ) + || e D ( • ) || L s − ( e Ω , e S ) are equivalent on L s ( e Ω , e S ).3. Von Neumann trace of the spectral ǫ -projection Let as before (
M, g ) be a complete riemannian manifold and assume now that there exists a compactsubspace K ⊂ M such that the zero-th order curvature operator R satisfies the relation R| M r K ≥ κ Id for some constant κ > . We shall refer to this condition as (uniform) invertibility near infinity . If D is a given generalized Diracoperator, then any twist of D by a hermitian connection on an extra bundle E → M yields a new gener-alized Dirac operator D E acting on the sections of the generalized spin bundle S ⊗ E . The operator R E corresponding to D E is then given by R E = R + 12 X i,j ( e i · e j ) ⊗ Ω E , where Ω E is the curvature tensor of E . In particular, if D is invertible near infinity and E is ǫ -almost flat fora sufficiently small ǫ , then D E will also be invertible near infinity. In the spin case, the spin-Dirac operator ELATIVE L INDEX THEOREM 7 is invertible near infinity if and only if the scalar curvature function κ satisfies κ ≥ κ off some compact subspace K for some constant κ > . We are now in position to state the main theorem of this section.
Theorem 3.1.
Assume that the generalized Dirac operator D acting on the sections of S over the completeriemannian manifold M is uniformly invertible near infinity. Then, for any Galois Γ -cover f M → M , thereexists ǫ > such that the spectral projection P ǫ of the Γ -invariant Laplacian e D , associated with the intervalle [0 , ǫ ] , has finite Γ -dimensional range, i.e. τ ( P ǫ ) < + ∞ . To emphasize the spin-Dirac operator associated with the metric g when there is an associated spinstructure, we shall sometimes denote it by D g and by e D g for the lift to f M . The spectral projection P ǫ isan element of Atiyah’s von Neumann algebra M = B ( L ( f M , e S )) Γ . When the Galois cover is the trivial onecorresponding to Γ = { e } , this theorem is due to Gromov and Lawson who prove moreover that in this casethere is a gap in the spectrum near 0 [GL:83]. They hence deduced that the Green operator associated with D is L -bounded [GL:83]. For e D this is not true for general infinite Γ as can be checked in simple examplesalready with M compact. Remark 3.2.
It is easy to see that the kernel projection P ǫ has a smooth Schwartz kernel k ǫ ( e m, e m ′ ) ∈ Hom( S m ′ , S m ) . Theorem 3.1 then implies that the integral R F tr( k ǫ ( e m, e m )) d e m , over any fundamental domain F of the Galois cover f M → M , is finite and coincides with τ ( P ǫ ) . It is worth pointing out that the finitenessof this integral can be proved directly but, even when Γ is trivial, it does not suffice to deduce Theorem 3.1. We shall give the proof of Theorem 3.1 in Section 4, and we also show in Appendix A.1 that a type IIRellich lemma also allows to deduce another proof of this theorem. We can deduce the following
Theorem 3.3.
Under the assumptions of Theorem 3.1 and when M is even dimensional, the Γ -invariantgeneralized Dirac operator e D + acting from L ( f M , e S + ) to L ( f M , e S − ) has a well defined L index Ind (2) ( e D + ) := dim Γ (Ker( e D + )) − dim Γ (Ker( e D − )) ∈ R . In the spin case with the metric g having PSC near infinity, this L -index is denoted Spin (2) ( f M , g ) and iscalled the L -genus of the riemannian Galois cover.Proof. It is clear from the very definition of the von Neumann trace τ that it can be restricted to operatorson e S ± and that for any diagonal non-negative operator T = (cid:18) T + T − (cid:19) , we have τ ( T ) = τ ( T + ) + τ ( T − ).We apply Proposition 2.3 to deduce thatKer( e D ) = Ker( e D ) = Ker( e D − e D + ) ⊕ Ker( e D + e D − ) . Therefore, + ∞ > dim Γ (Ker( e D )) = dim Γ (Ker( e D )) = dim Γ (Ker( e D − e D + )) + dim Γ (Ker( e D + e D − )) , and henceforth dim Γ (Ker( e D − e D + )) and dim Γ (Ker( e D + e D − )) are both finite non-negative real numbers whichsimilarly coincide respectively with dim Γ (Ker( e D + )) and dim Γ (Ker( e D − )). (cid:3) When M is a closed manifold, the invertibility near infinity condition disappears and the L -indexInd (2) ( e D ) is known to belong to the range of the additive K -theory map associated with the regular traceon the reduced C ∗ -algebra C ∗ r Γ. Moreover, by the covering Atiyah theorem [At:76], Ind (2) ( e D ) is then aninteger which coincides with the index Ind( D ) of the generalized Dirac operator on the base manifold M , atopological invariant. M-T. BENAMEUR Proof of Theorem 3.1
We denote for any Borel subspace A of f M by || • || A the L norm of the restriction to A . For a given φ ∈ L ( f M , e S ) we denote by β γ ( φ ) the non-negative real number || φ || γF , so that || φ || = X γ ∈ Γ β γ ( φ ) . We make the assumptions of Theorem 3.1 and our goal here is to give a direct proof inspired from theGromov-Lawson proof for trivial Γ. Recall that F denotes an open fundamental domain for the Galoiscovering π : f M → M of complete manifolds. We shall first concentrate on the kernel projection P = P and give a direct proof of its finite τ -trace which provides, after careful inspection, a quantitative estimatedepending on the covering and of the constant κ as well as the uniform norm of R on the compact space K . The proof for P ǫ with ǫ > ǫ = 0 without providing precise estimates. Since K is compact,we know that there exists a constant c > R ≥ − c Id in restriction to π − ( K ). Lemma 4.1.
For any φ ∈ Ker( e D ) and any γ ∈ Γ , the following estimate holds: || φ || γF ∩ π − ( K ) ≥ κ κ + c × q β γ ( φ ) . Proof.
Let φ ∈ Ker( e D ) be a given non trivial section and let us fix some γ ∈ Γ such that β γ ( φ ) >
0. Thenwe can assume that β γ ( φ ) = 1 and apply the generalized Lichnerowicz formula to φ to deduce0 = Z γF | ( e Dφ )( e m ) | S m d e m = Z γF | ( ∇ φ )( e m ) | S m ⊗ T ∗ m M d e m + Z γF hR u, u i S m d e m ≥ Z γF hR u, u i S m d e m. But Z γF hR u, u i S m d e m = Z γF ∩ π − ( K ) hR u, u i S m d e m + Z γF r π − ( K ) hR u, u i S m d e m. On the other hand for m ∈ M r K we have R m ≥ κ Id S m , therefore Z γF r π − ( K ) hR u, u i S m d e m ≥ κ (1 − || u || γF ∩ π − ( K ) ) , and we also have Z γF ∩ π − ( K ) hR u, u i S m d e m ≥ − c || u || γF ∩ π − ( K ) . Thus we deduce 0 ≥ κ (1 − || u || γF ∩ π − ( K ) ) − c || u || γF ∩ π − ( K ) = κ − ( κ + c ) || u || γF ∩ π − ( K ) , and hence the conclusion. (cid:3) We denote by || • || C ,A , for an open subspace A of f M , the uniform C -norm over A . Proposition 4.2.
Under the previous assumptions, and for any open relatively compact subspace L of M ,there exists a constant C = C ( L ) > such that for any φ ∈ Ker( e D ) and any γ ∈ Γ , we have || φ || C ,γF ∩ π − ( L ) ≤ C × q β γ ( φ ) . Proof.
Choose a compact subspace K ′ which is contained in π − L ∩ γF . Let Ω be an open neighborhood of K ′ in the open subspace γF which is relatively compact. Fix a smooth compactly supported function χ onΩ which equals 1 on K ′ . For any φ ∈ Ker( e D ), the uniform elliptic estimate recalled above insures that forany s ∈ N , there exists of a constant C ′ s such that || φ || L s ( K ′ , e S ) ≤ C ′ s (cid:16) || χφ || L s − (Ω , e S ) + || e D ( χφ ) || L s − (Ω , e S ) (cid:17) . ELATIVE L INDEX THEOREM 9
Since φ ∈ Ker( e D ), e D ( χφ ) = [ e D, M χ ]( φ ) where the operator [ e D, M χ ] is a zero-th order differential operatorwith compactly supported coefficients inside Ω, and hence there exists a constant C ′′ s such that || e D ( χφ ) || L s − (Ω , e S ) ≤ C ′′ s || φ || L s − (Ω , e S ) . Therefore, there exists of a constant C s > || φ || L s ( K ′ , e S ) ≤ C s || u || L s − (Ω , e S ) . Notice that theconstants C s , C ′ s and C ′′ s don’t depend neither on the group element γ nor on φ .Let now (Ω n ) ≤ n ≤ N be a finite collection of open relatively compacts subspaces of γF such that K ′ ⊂ Ω and Ω j ⊂ Ω j +1 for 1 ≤ j ≤ N − . Then we may apply the previous argument inductively to deduce the existence of a constant C s ( N ) suchthat for any γ ∈ Γ and any φ ∈ Ker( e D ): || φ || L s ( K ′ , e S ) ≤ C s ( N ) || u || L s − N ( γF, e S ) . Taking s = N we get a constant C ( N ) > γ ∈ Γ and any φ ∈ Ker( e D ): || φ || L N ( K ′ , e S ) ≤ C ( N ) q β γ ( φ ) . Taking the supremum over such subspaces K ′ of π − ( L ) ∩ γF , we deduce by the Beppo-Levi argument andsince our measure is regular (being a Lebesgue-class measure), || φ || L N ( π − ( L ) ∩ γF, e S ) ≤ C ( N ) q β γ ( φ ) . Since L is relatively compact, the Sobolev estimate implies that for N large enough ( N > dim( M ) + 1),there exists a constant C ′ such that || φ || C ,γF ∩ π − ( L ) ≤ C ′ || φ || L N ( π − ( L ) ∩ γF, e S ) The proof is now complete. (cid:3)
Proposition 4.3.
For any finite subset F := { e m j , ≤ j ≤ d } of the fundamental domain F , one defines abounded Γ -equivariant operator T F : Ker( e D ) −→ d M j =1 ℓ (Γ e m j ) ⊗ S m j by setting T F φ := ( T ( g e m j )) γ ∈ Γ , ≤ j ≤ d . Proof.
Choose any α > B ( e m j , α ) centered at e m i with radius α are disjoint from eachother and contained in the open fundamental domain F . Then since the metric is Γ-invariant, for any γ ∈ Γ,the same property is satisfied by the collection B ( γ e m j , α ) inside γF . Applying Proposition 4.2 with anyopen relatively compact subspace L of M which contains the projection of the union of the balls B ( e m j , α )for 1 ≤ j ≤ d , we deduce that for any e m ∈ B ( γ e m j , α ) and for any φ ∈ Ker( e D ), we have | φ ( γ e m j ) | ≤ (cid:0) | φ ( e m ) | + C α β γ ( φ ) (cid:1) , where C = C ( L ) is given by Proposition 4.2. Integrating this inequality over B ( γ e m j , α ) and suming over j ∈ { , · · · , d } and then over γ ∈ Γ, we deduce d X j =1 vol ( B ( e m j , α ))2 X γ ∈ Γ | φ ( γ e m j ) | ≤ X j,γ Z B ( γ e m j ,α ) | φ ( e m ) | d e m + C α d X j =1 vol ( B ( e m j , α )) X γ β γ ( φ ) . But X γ β γ ( φ ) = || φ || L ( f M, e S ) and X j,γ Z B ( γ e m j ,α ) | φ ( e m ) | d e m ≤ || φ || L ( f M, e S ) . Hence if m = inf ≤ j ≤ d vol ( B ( e m j , α )), and M = sup ≤ j ≤ d vol ( B ( e m j , α )), then we get X j,γ | φ ( γ e m j ) | ≤ dC α Mm × || φ || L ( f M, e S ) . Therefore, the sum P j,γ | φ ( γ e m j ) | converges so that( φ ( γ e m j )) j,γ ∈ d M i =1 ℓ (Γ e m j ) ⊗ S m j ≃ ℓ Γ ⊗ C d × dim( S ) . Moreover, since the constant dC α Mm does not depend on φ ∈ Ker( e D ), the map T F : Ker( e D ) −→ d M i =1 ℓ (Γ e m j ) ⊗ S m j given by T F ( φ ) := ( φ ( γ e m j )) j,γ is a bounded operator . It is finally obvious from its very definition that the operator T F is Γ-equivariant between the two Hilbert Γ-spaces Ker( e D ) and L di =1 ℓ (Γ e m j ) ⊗ S m j where this latter is induced from the regular representation throughthe isomorphism ℓ (Γ) ≃ ℓ (Γ e m j ). (cid:3) We are now in position to prove our theorem.
Proof. (of Theorem 3.1)Take any positive number ǫ ∈ ]0 ,
1] and fix a finite subset F ( ǫ ) = { e m j , ≤ j ≤ d } of F ∩ π − ( K ) such thatthe collection of open balls ( B ( e m j , ǫ )) ≤ j ≤ d provides an open cover of F ∩ π − ( K ). Applying Proposition4.3, we know that the operator T F ( ǫ ) from Ker( e D ) to the Hilbert Γ-space L dj =1 ℓ (Γ e m j ) ⊗ S m j given byevaluation at the points γ e m j for γ ∈ Γ and 1 ≤ j ≤ d , is a bounded Γ-equivariant operator. Assume thatdim Γ (Ker( e D )) = + ∞ , then we claim that the operator T F ( ǫ ) cannot be injective. Indeed, if it were injective,then denoting by H the Hilbert Γ-subspace of L dj =1 ℓ (Γ e m j ) ⊗ S m j which is the closure of the range of T F ( ǫ ) ,we would get an injective Γ-equivariant operator from Ker( e D ) to H which has dense range. But this showsthat dim Γ ( H ) = + ∞ , by a standard argument using the normality of the trace τ , see Lemma 4 in [BF:06]and its proof. Hence, we can find φ ∈ Ker( e D ) such that || φ || L ( f M, e S ) = 1 and φ ( γ e m j ) = 0 for any γ ∈ Γ and j ∈ { , · · · , d } . There then exists γ ∈ Γ such that β γ ( φ ) = R γF | φ ( e m ) | d e m >
0. Again replacing φ by φ √ β γ ( φ ) we can assumethat β γ ( φ ) = 1 and that φ vanishes on Γ F ( ǫ ). Denote now by L any open relatively compact subspace of M which contains K , one can take for instance the 1-neighborhood of K , say L = { m ∈ M, d ( m, K ) < } . Applying Proposition 4.2, we deduce the existence of
C >
0, independent of ǫ and φ , such that: | φ ( e m ) | ≤ C × ǫ, for any e m ∈ γF ∩ π − ( K ) . Hence || φ || γF ∩ π − ( K ) ≤ C ′ × ǫ for some constant C ′ > ǫ and φ . But Lemma 4.1then allows to conclude that there exists a constant C ′ > C ′ ǫ ≥ κ κ + c . If ǫ is small enough, we get a contradiction. Whence we conclude that dim Γ (Ker( e D )) < + ∞ .Inorder to complete the proof of Theorem 3.1, we now prove with a different direct method that P ǫ has τ -finite range. Notice that if σ is a non trivial L -section which belongs to the range of P ǫ , then || e Dσ || ≤ ǫ || σ || . ELATIVE L INDEX THEOREM 11
Therefore and using our assumption that
R ≥ κ Id off the compact subspace K , and that R ≥ − c Id overthe whole of M , we deduce that for ǫ small enough || σ || π − ( K ) ≥ κ − ǫκ + c || σ || . Said differently, if we denote by p K the Γ-invariant orthogonal projection from L ( f M , e S ) onto the Hilbertsubspace L ( π − ( K ) , e S ), then for ǫ small enough, the restriction of p K to the range of P ǫ is bounded below.We now choose a bounded Γ-invariant operator (a parametrix) e Q such that e S = Id − e Q e D is a smoothingoperator with finite propagation. This can be achieved for instance by lifting to f M a localized enough nearthe diagonal pseudodifferential parametrix for D modulo smoothing opertors on M , see for instance [At:76].Composing on the left with the projection p K we get p K e S = p K − ( p K e Q ) e D. Hence for σ in the range of P ǫ , we can write || p K e Sσ || ≥ || σ || π − ( K ) − √ ǫ || e Q || × || σ || ≥ (cid:18)r κ − ǫκ + c − √ ǫ || e Q || (cid:19) × || σ || . Hence for small enough ǫ , the operator p K e S is a Γ-equivariant isomorphism from the range of P ǫ to therange of p K e SP ǫ , the two being closed Γ-invariant subspaces of L ( f M , e S ). Since p K e S is a compact operatorrelative to the von Neumann algebra M , we conclude that P ǫ must be a compact operator relative to M andtherefore has a finite τ -trace. Indeed, one easily checks for instance that p K e S is a Hilbert-Schmidt operatorrelative to the trace τ , see again [At:76]. (cid:3) Compatibility with the higher index
Review of the Xie-Yu higher index.
Given two Hilbert modules E and E , the space of adjointableoperators from E to E will be denoted Mor( E , E ) and when E = E = E then we denote the resultingunital C ∗ -algebra by Mor( E ). Recall that the subspace K ( E r ) of Mor( E r ) composed of compact operators isa closed two-sided involutive ideal, see [K:80] for more details. In [XY:14], Xie and Yu proved that whenthe spin even dimensional manifold M has a complete metric g with PSC near infinity, then the operator D r obtained by twisting the spin-Dirac operator D g with the flat reduced Michschenko bundle, admits a welldefined higher index class Ind( D r ) ∈ K ( C ∗ r Γ) . Here C ∗ r Γ is the regular C ∗ -algebra associated with Γ and D r acts on the L sections of S ⊗ Ξ, withΞ = f M × Γ C ∗ r Γ being the flat Michschenko bundle with fibers C ∗ r Γ viewed as a module over itself. It is wellknown that D r is essentially self-adjoint and yields a regular self-adjoint operator, still denoted D r , withdomain contained (in general strictly) in the Hilbert module Sobolev space L ( M, S ⊗ Ξ) that one definesusing the spin connection ∇ as before twisted by the flat connection on the bundle of modules Ξ.Let us recall the construction of the higher index class from [XY:14], which as we shall see remains validfor all generalized Dirac operators which are invertible near infinity. We thus assume that the generalizedDirac operator D on M has square given by ∇ ∗ ∇ + R with a zero-th order operator R such that R ≥ κ Idoff some compact subspace K of M . There exists then a smooth compactly supported real valued function ρ ∈ C ∞ c ( M ) and a constant c > R ≥ ( c − ρ ) Id , holds over the whole of M . This discussions works as well for the Hilbert module operator D r by replacing R by its tensor product still denoted R by the fiberwise identity of Ξ. We shall denote as well by ρ the smoothfunction on f M which is the pull-back of ρ . We concentrate on the construction with the regular completions(the maximal completions are similar) and we add some details for the convenience of the reader. Let usconsider the operator F := D r ( D r + ρ ) − / acting on E r , where ( D r + ρ ) − / is by definition the square root of the self-adjoint non-negative bounded operator ( D r + ρ ) − , see [XY:14]. As we shall see this operatorcan also be defined equivalently as the inverse of the square root of the regular self-adjoint non-negativeoperator D r + ρ . Another equivalent and convenient definition is( D r + ρ ) − / := 1 π Z + ∞ ( D r + ρ ) − ( D r + ρ ) − + µ dµ √ µ , where the integral converges in the operator norm. The operator F is in fact an adjointable operator as weshall see. Notice first that we take a priori as initial domainDom( F ) = { σ ∈ E r | ( D r + ρ ) − / σ ∈ Dom( D r ) } , and F is then closed. In the same way the operator G := ( D r + ρ ) − / D r is closed when we take as domainthat of the operator D r . Moreover, for σ ∈ Dom( F ) and σ ′ ∈ Dom( D r ), we have hF σ, σ ′ i = h σ, G σ ′ i , and the domain of F can be seen to be exactly equal to the adjoint domain of G . Lemma 5.1.
The operator GF is a bounded (self-adjoint and non-negative) operator on the Hilbert module E r . In particular, so is F ∗ F and we have F ∗ F = GF .Proof. Notice that GF = ( D r + ρ ) − / D r ( D r + ρ ) − / has dense domain in E r . It is easy to check thatthe operator ( D r + ρ ) − / ( D r + ρ )( D r + ρ ) − / extends to the identity operator so that we can write for σ ∈ Dom( GF ): GF σ = σ − ( D r + ρ ) − / ρ ( D r + ρ ) − / σ. The operator ( D r + ρ ) − / ρ ( D r + ρ ) − / being bounded, we conclude that the operator GF extends to abounded operator on E r which coincides with the operator Id − ( D r + ρ ) − / ρ ( D r + ρ ) − / . Moreover, if σ ′ ∈ Dom( F ) is such that F σ ′ ∈ Dom( G ) = Dom( D r ) ⊂ Dom( F ∗ ) then F σ ′ ∈ Dom( F ∗ ) and we can writefor any σ ∈ E r hGF σ, σ ′ i = hF σ, F σ ′ i = h σ, F ∗ F σ ′ i = h σ, GF σ ′ i . We have used the relations
G ⊂ F ∗ and F = G ∗ . Notice finally that for any σ ∈ Dom( D r ) we have h ρ σ, σ i ≤ h ( D r + ρ ) σ, σ i . Therefore the operator GF is indeed non-negative. Recall that GF ⊂ F ∗ F , hence we conclude that we alsothe equality GF = F ∗ F of adjointable operators. (cid:3) Remark 5.2.
Using the above integral expression, it is easy to check that the densely defined commutator [ D r , ( D r + ρ ) − / ] extends to a compact adjointable operator. This is a consequence of the fact that ρ and [ D r , ρ ] have compact supports in M and ( D r + ρ ) − / is bounded from L to L , so that an easy applicationof a Rellich lemma in this context, see [XY:14] allows to conclude. Hence the operator F − G is a compactoperator on the Hilbert module E r . Proposition 5.3. [XY:14]
The closed operators F and G are adjointable. Moreover, the operators Id −F and Id −G are compact operators of the Hilbert module E r .Proof. Let us check first that the closed operator F is bounded. By the previous lemma, we know that if F ∗ is the adjoint of the closed operator F then the operator F ∗ F is a bounded non-negative operator onthe Hilbert module E r . Therefore the operator Id + F ∗ F is an adjointable self-adjoint invertible operator,and hence is surjective so that F is in fact a regular operator on E r . The square root of Id + F ∗ F is thenalso an adjointable self-adjoint invertible operator with the inverse given precisely by our previously definedoperator (Id + F ∗ F ) − / , by the classical properties of continuous functional calculus of regular self-adjointoperators. Now, F being regular, by the properties of the Woronowicz transform [Sk:90], we see that theoperator F (Id + F ∗ F ) − / is an adjointable operator. We conclude that the operator F = F (Id + F ∗ F ) − / (Id + F ∗ F ) / , ELATIVE L INDEX THEOREM 13 is adjointable with the adjoint given by F ∗ . As a corollary, we deduce that the densely defined operator G is also extendable to an adjointable operator which coincides with F ∗ .Computing on a dense subspace, one deduces the following equality of adjointable operatorsId −F = ( D r + ρ ) − / ρ ( D r + ρ ) − / + [( D r + ρ ) − , D r ] F . As we have already pointed out, the operator [( D r + ρ ) − , D r ] is a compact operator on the Hilbert module E r by a generalized version of the Rellich lemma [XY:14], and F is adjointable so that the second term inthe RHS is compact. On the other hand, and also by the compact support of ρ , the fact that ( D r + ρ ) − / is bounded with image in L , we deduce again by the Rellich lemma in the Mischenko-Fomenko calculus[XY:14] that the operator ( D r + ρ ) − / ρ ( D r + ρ ) − / is a compact operator on the Hilbert module E r .This ends the proof for F . Since G = F ∗ , the proof of the proposition is now complete. (cid:3) We deduce from the previous lemma that the operator G + := ( D + r D − r + ρ ) − / D + r : E + r −→ E − r , is a Fredholm operator between these Hilbert modules with quasi inverse given by G − := ( D − r D + r + ρ ) − / D − r : E − r −→ E + r . Therefore G + admits a well defined index class in K ( K ( E r )) where K ( E r ) is the C ∗ -algebra of compactoperators on the Hilbert module E r . We may as well use F + and we would end up with the same class asalready observed. This index class is given more precisely by the K -class [ e ] − [ f ] where e and f are the twoself-adjoint projections on E r which satisfy that e − f ∈ K ( E r ): e := (cid:18) ( S + ) S + G − G + S + (Id E + r + S + ) Id E − r − ( S − ) (cid:19) and f = (cid:18) E − r (cid:19) where S + = Id E + r −G − G + and S − = Id E − r −G + G − . That [ e ] − [ f ] defines a class in K ( K ( E r )) is standard. Definition 5.4. [XY:14]
The higher index class
Ind( D + r ) of D + r is by definition the image of the indexclass of G + in K ( C ∗ r Γ) under the isomorphism K ( K ( E r )) ≃ K ( C ∗ r Γ) induced by the Morita equivalence K ( E r ) ∼ C ∗ r Γ , so Ind( D + r ) ∈ K ( C ∗ r Γ) . Remark 5.5.
From the relations Id −F ∈ K ( E r ) and Id −G ∈ K ( E r ) , we deduce that the operator Q :=( D − r D + r + ρ ) − / D − r ( D + r D − r + ρ ) − / is a parametrix for D + r with remainders in K ( E ± r ) . We may as well give the same construction of the maximal index class Ind( D + m ) living in K ( C ∗ m Γ) byusing maximal completions everywhere. Notice that by using the natural C ∗ -homomorphism from C ∗ m Γ to C ∗ r Γ, one can as well recover the index class Ind( D r ) as the image of Ind( D m ), see for instance [BP:09].5.2. Traces and numerical indices.
The trivial 1-dimensional representation of Γ gives a C ∗ -algebrahomomorphism C ∗ m Γ → C , this is the so-called average trace τ av . In the same way the regular trace τ reg isa finite trace on C ∗ r Γ. These traces are defined on finitely supported functions f ∈ C Γ by τ av ( f ) := X γ ∈ Γ f ( g ) and τ reg ( f ) := f ( e ) . They induce the group homomorphisms τ av ∗ : K ( C ∗ m Γ) −→ R and τ reg ∗ : K ( C ∗ r Γ) −→ R . Theorem 5.6.
The following relations hold τ av ∗ (Ind( D m )) = Ind( D ) and τ reg ∗ (Ind( D r )) = Ind (2) ( e D ) , where Ind( D ) is the Gromov-Lawson index of the generalize Dirac operator D on M while Ind (2) ( e D ) is our L index of the lifted Γ -invariant generalized Dirac operator e D on f M . Remark 5.7.
We shall fully use for the proof of the second relation in Theorem 5.6 our Theorem 3.1, saythat there exists ǫ > such that P ǫ is τ -trace class in the von Neumann algebra M .Proof. Denote by π reg and π av the regular and average representations of C ∗ r Γ and C ∗ m Γ respectively thatwe can see as being both representations of C ∗ m Γ by using the natural morphism C ∗ m Γ → C ∗ r Γ. By[BR:15][Propositions 2.3 & 2.4], the composition Hilbert spaces E r ⊗ π reg ℓ Γ and E m ⊗ π av C are respec-tively isomorphic to L ( f M , e S ) and L ( M, S ) through the explicit isomorphisms denoted there Ψ reg and Ψ av .Let us concentrate on the second equality first. The regular self-adjoint operator D r gives after compositionwith the regular representation and through conjugation with Ψ reg the self-adjoint operator e D [BR:15], i.e.Ψ reg ( D r ⊗ π reg Id)Ψ − = e D. The mapping Φ reg : T Ψ reg ( T ⊗ π reg Id)Ψ − induces a well defined C ∗ -algebra morphism between theadjointable operators on E r and the bounded operators on L ( f M , e S ) which belong to the von Neumannalgebra M [BR:15]. The last property that we shall need from [BR:15] is that Φ reg sends the ideal K ( E r ) ofcompact operators to the ideal of τ -compact operators in M and intertwines the corresponding short Calkinexact sequences, i.e. 0 → K ( E r ) −−−−→ Mor( E r ) −−−−→ Mor( E r ) / K ( E r ) → Φreg y Φ reg y Φ reg y → K ( M , τ ) −−−−→ M −−−−→ M / K ( M , τ ) → K -theory are compatible, sayΦ reg , ∗ ◦ ∂ = ∂ ◦ Φ reg , ∗ : K (Mor( E r ) / K ( E r )) −→ K ( K ( M , τ )) . We then immediately deduce the following relationΦ reg , ∗ (cid:2) Ind( G + ) (cid:3) = Ind M (cid:0) Φ reg ( G + ) (cid:1) , where Ind M : K ( M / K ( M , τ )) → K ( K ( M , τ )) is the boundary map associated with the second exactsequence, and constructed as usual using any parametrix in M modulo K ( M , τ ). Now, recall that Ind( D + r ) =Ind( G + ) by definition, and by compatibility of functional calculi with respect to compositions of Hilbertmodules [Sk:90], we haveΦ reg ( G + ) = ( e D + e D − + ρ ) − / e D + =: e G + : L ( f M , e S + ) −→ L ( f M , e S − ) . The operator ( e D + e D − + ρ ) − / is thus defined by the continuous functional calculus for bounded Γ-invariantoperators on L ( f M , e S ) from the bounded operator ( e D + e D − + e ρ ) − .Now we can use any parametrix for e G + in M (not necessarily in the range of Φ reg ) inorder to representthe index class Ind( e G + ) ∈ K ( K ( M , τ )). Moreover, since e D is affiliated with M , we can also use directly anyparametrix for the operator e D . Now, by Theorem 3.1, there exists ǫ > P ǫ := 1 [0 ,ǫ ] ( e D ) is τ -trace class. We consider for such ǫ the odd bounded Borel function f ǫ : R → R given by f ǫ ( x ) := (cid:26) x if | x | > √ ǫ | x | ≤ √ ǫ ELATIVE L INDEX THEOREM 15
Then set e Q ǫ := f ǫ ( e D ), an element of M which is odd for the grading. Since 1 − xf ǫ ( x ) = 1 [0 ,ǫ ] ( x ), we haveId − e Q ǫ e D = Id − e D e Q ǫ = P ǫ and e Q ǫ = e Q − ǫ e Q + ǫ ! . Moreover, the operator e H := e Q ǫ ( e D + e ρ ) / is also a bounded operator which belongs to M and which isthen a parametrix for the operator e G . Indeed, one has for any C > e Q ǫ ( e D + C ) / = g ǫ ( e D ) with g ǫ ( x ) = (cid:26) x/ √ x + C if | x | > √ ǫ | x | ≤ √ ǫ Therefore, e Q ǫ ( e D + C ) / extends to a bounded operator which lives in M . Moreover, the operator( e D + C ) − / ( e D + ρ ) / also extends to a bounded operator which belongs to M . Hence, we deduce that e H belongs to M . On the other hand, we have e H e G = Id − P ǫ while e G e H = Id − ( e D + ρ ) − / P ǫ ρ ( e D + ρ ) − / . Therefore e H is a parametrix for e G modulo τ -trace class operators in K ( M , τ ). Therefore, the index class of e G can be represented by the two idempotents e and f in M such that e − f ∈ K ( M , τ ) given by (cid:18) P + ǫ e D − e D + + ρ ) − / e D + P + ǫ − ( e D + e D − + ρ ) − / P − ǫ ( e D + e D − + ρ ) / (cid:19) with f as before given by f = (cid:18) L ( f M, e S − ) (cid:19) . Notice that the operator P − ǫ ( e D + e D − + e ρ ) / is boundedand even τ -trace class since it equals( P − ǫ e D + e D − + P ǫ ρ )( e D + e D − + ρ ) − / , and the operator P − ǫ e D + e D − = P ǫ ( P − ǫ e D + e D − ) is bounded and even τ -trace class, and finally the operator( e D + e D − + ρ ) − / is bounded.If we now apply the trace τ to this K -theory class, then we get( τ ∗ ◦ Ind M )( e G + ) = τ ( P + ǫ ) − τ (( e D + e D − + ρ ) − / P − ǫ ( e D + e D − + ρ ) / )Since τ is a hypertrace and again since P − ǫ ( e D + e D − + e ρ ) / is τ -trace class while ( e D + e D − + e ρ ) − / is boundedand belongs to M , we can write τ (( e D + e D − + ρ ) − / P − ǫ ( e D + e D − + ρ ) / ) = τ ( P − ǫ ( e D + e D − + ρ ) / ( e D + e D − + ρ ) − / ) = τ ( P − ǫ ) . We conclude that ( τ ∗ ◦ Ind M )( e G + ) = τ ( P + ǫ ) − τ ( P − ǫ ) for any small enough ǫ > . Since the trace τ is normal, we can deduce from this equality that( τ ∗ ◦ Ind M )( e G + ) = τ ( P +0 ) − τ ( P − ) = Ind (2) ( e D + ) . We conclude from the whole previous discussion thatInd (2) ( e D + ) = [( τ ∗ ◦ Φ reg , ∗ ) ◦ Ind)] ( D + r ) = τ reg ∗ (Ind( D ∗ r ))The last equality is the consequence of the compatibility of the traces τ and τ reg with the Morita equivalence K ( E r ) ∼ C ∗ r Γ, say ( τ ∗ ◦ Φ reg , ∗ ) = τ reg ∗ ◦ M : K ( K ( M , τ ) −→ R , with M : K ( K ( E r )) → K ( C ∗ r Γ) being the isomorphism induced by the Morita equivalence. Details of theproof of this latter result can also be found in [BR:15].The proof for the trivial representation and the average trace is similar and actually simpler, since thevon Neumann algebra M has to be replaced by the bounded operators on the Hilbert space L ( M, S ) andthe spectrum of D admits a gap around zero, which allows to use a parametric for D which is the bounded Green operator. We leave these straightforward verifications for the interested reader, who can alternativelyalso use the needed compatibility results proved in [BR:15]. (cid:3) Relative L index theory The Φ -relative L index theorem. The relative index theorem of Gromov-Lawson [GL:83][Theorem4.18] can clearly be extended to Galois coverings with the expected statement about pairs of Galois coverings π : f M → M and π ′ : f M ′ → M ′ and Γ-operators e D and f D ′ which are invertible near infinity. For simplicity werestrict ourselves to the case of the spin Dirac operators for complete metrics which have PSC near infinity.More precisely, we assume that M and M ′ are endowed with complete metrics g and g ′ which both haveuniform PSC outside compact subspaces K and K ′ in M and M ′ respectively, and associated spin structures.Then the Γ-invariant Dirac operators e D = e D g and f D ′ = f D ′ g ′ acting on the sections of the hermitian spinorbundles e S and e S ′ both have well defined L indices Ind (2) ( e D ) and Ind (2) ( f D ′ ), thanks to Theorem 3.1.Moreover, we assume as in [GL:83] that there exists open submanifolds Φ ∈ M r K and Φ ′ ⊂ M ′ r K ′ which are unions of connected components of M r K and M ′ r K ′ respectively, such that all structures overΦ and Φ ′ are identified. So denoting by e Φ := π − (Φ) and e Φ ′ := π ′− ( e Φ ′ ), we assume that there exists aΓ-equivariant smooth spin-preserving isometry e Φ → e Φ ′ which is then covered by a bundle isometry : S | Φ → S ′ | Φ ′ , inducing a unitary U : L ( e Φ , e S ) → L ( e Φ ′ , e S ′ ) so that the Dirac operators are conjugated over e Φ and e Φ ′ , thatis as self-adjoint operators: f D ′ | Φ ′ = U ◦ e D | Φ ◦ U − . As in [GL:83][page 127], we consider a compact hypersurface H which is contained in Φ ≃ Φ ′ whichseparates off the infinite part of Φ ≃ Φ ′ . More precisely, we assume that there exists a compact subspace L of M containing H and K such that Φ r L and K are contained in different connected components of M r H . The similar assumption is thus insured as well in M ′ when we view H in M ′ . We may assume, upto modifying the metrics and spin structures near H , that the metrics are product metrics and the Diracoperators are of product type, in a collar neighborhood ( − ǫ, + ǫ ) × H of H in Φ ≃ Φ ′ . Like in [GL:83][page127], we chop off M and M ′ along H and attach the resulting riemannian manifolds with boundaries M and M ′ to build up the new complete riemannian spin manifold M ′′ := M ∐ H M ′ whose metric has aswell PSC near infinity. The spin structure (and orientation) on M ′′ agrees with that on M and with theopposite one on M ′ corresponding to the opposite Stieffel-Whitney 2-class, and the Dirac operator on M ′′ for the resulting spin structure is denoted D ′′ and agrees with D on M and with D op on M ′ .The Galois Γ-coverings f M → M and f M ′ → M ′ then yield a Galois Γ-covering f M ′′ → M ′′ in an obviousway. Indeed, H can be lifted to e H = π − ( H ) and all the constructions can be achieved upstairs in f M and f M ′ to build up the complete spin covering f M ′′ → M ′′ . Definition 6.1.
The Φ -relative L index of ( M, M ′ , Φ) is the L -index of the spin-Dirac operator f D ′′ + , i.e. Ind (2) ( e D, f D ′ ; Φ) := Ind (2) ( f D ′′ + ) ∈ R . We are now in position to state the main theorem of this paragraph.
Theorem 6.2 (The Φ-relative L -index theorem) . Under the previous notations, we have
Ind (2) ( e D, f D ′ ; Φ) = Ind (2) ( e D + ) − Ind (2) ( f D ′ + ) Proof.
Although this theorem can be proved directly by extending the Gromov-Lawson method, we shalldeduce it here from the main theorem of [XY:14][Theorem 4.2] by using our compatibility Theorem 5.6.Indeed, the construction of the higher Φ-relative index Ind( D , D ′ ; Φ) ∈ K ( C ∗ r Γ) carried out in [XY:14]shows that its image under the group homomorphism induced by the regular trace τ reg on C ∗ r Γ coincides
ELATIVE L INDEX THEOREM 17 with our Φ-relative L -index Ind (2) ( e D, f D ′ ; Φ) since this is a consequence of the compatibility result on Galoiscoverings again. The main theorem of [XY:14] gives the relationInd( D , D ′ ; Φ) = Ind( D + ) − Ind( D ′ + )Therefore, using Theorem 5.6, we conclude thatInd (2) ( e D, f D ′ ; Φ) = τ reg ∗ (Ind( D , D ′ ; Φ))= τ reg ∗ (cid:0) Ind( D + ) (cid:1) − τ reg ∗ (cid:16) Ind( D ′ + ) (cid:17) = Ind (2) ( e D + ) − Ind (2) ( f D ′ + ) . (cid:3) When the open submanifolds Φ and Φ ′ have compact complements in M and M ′ respectively, the Φ-relative L index can easily be sees to be independant of the choices of Φ and Φ ′ and it will be denotedInd (2) ( e D, f D ′ ). Moreover, in this case where the operators are identified near infinity, no need of any invert-ibility condition near infinity and the relative L index can be defined exactly using the process of choppingoff and pasting exactly as in [GL:83][page 119] that is now done with the Galois coverings. We can deducenow the following important consequence. Theorem 6.3 (The relative L -index theorem) . Under the assumptions of Theorem 6.2 with M r Φ and M ′ r Φ ′ compact subspaces of M and M ′ respectively, but without any condition on R now. The relative L -index Ind (2) ( e D, f D ′ ) is well defined, and we have Ind (2) ( e D, f D ′ ) = Ind( D, D ′ ) . In particular, in this case
Ind (2) ( e D, f D ′ ) ∈ Z . Here Ind(
D, D ′ ) is the Gromov-Lawson relative index for the Dirac operators downstairs in M and M ′ .Recall as well that Ind( D, D ′ ) = Z U b A ( T M ) − Z U ′ b A ( T M ′ ) , where U and U ′ are open relatively compact neighborhoods of L and L ′ respectively whose complementscorrespond under the isometry. This is indeed the content of the classical Gromov-Lawson relative indextheorem, see [GL:83][Theorem 4.18]. Proof.
From their very definitions, the indices Ind (2) ( e D, f D ′ ) and Ind( D, D ′ ) coincide respectively with the L -index of the Γ-invariant generalized Dirac operator on f M ′′ and with the classical index of the generalizedDirac operator on M ′′ . But here M ′′ is a closed manifold so that we can apply the classical Atiyah L -indextheorem for Galois coverings of closed manifolds and deduce the allowed equalityInd (2) ( e D, f D ′ ) = Ind( D, D ′ ) . (cid:3) Another direct consequence of the relative L -index Theorem 6.2 when we don’t assume anymore that M r Φ and M ′ r Φ ′ are compact, is the following convenient result. Theorem 6.4.
Under the assumptions of Theorem 6.2, assume that there exists a Galois Γ -covering π N : e N → N of spin manifolds with boundary such that N is compact, π − N ( ∂N ) = ∂ e N and the resulting Galois Γ -covering ∂ e N → ∂N is isometric to the Galois Γ -covering e H → H . We form as above the spin completemanifolds N := M ∐ H N and N ′ := M ′ ∐ H N with the resulting Dirac operators D and D ′ and their lifts e D and f D ′ to the covers. Then Ind (2) ( e D, f D ′ ; Φ) = Ind (2) ( e D +1 ) − Ind (2) ( f D ′ +1 ) . Proof.
Indeed, the relative L -index theorem applies to prove that the RHS coincides with the L -index ofthe Dirac operator on the smooth complete manifold obtained by chopping off N and N along H andattaching M to M ′ . But this latter complete spin manifold is exactly the manifold N used above to defineInd (2) ( e D, f D ′ ; Φ). Since all these construction can be done upstairs Γ-equivariantly, we immediately deducefrom Theorem 6.3 the allowed equality. (cid:3) Remark 6.5.
The Φ -relative L index Theorem 6.2, as well as all its corollaries listed above, can be proveddirectly without using the higher version obtained in [XY:14] , by adapting the proof given in [GL:83] to thetype II von Neumann setting. The torsion-free case.
In this second paragraph of applications, we shall show that the absolute
Gromov-Lawson index, in opposition to the relative one, already coincides with its L -version when thegroup Γ satisfies some extra conditions. Recall the famous Baum-Connes map and Baum-Connes conjecturefor the group Γ [BC:88]. In the torsion-free case, the conjecture concerns the bijectivity of the assembly map µ Γ : K ∗ ( B Γ) −→ K ∗ ( C ∗ r Γ) . This map µ Γ always factors through the K -theory of the maximal C ∗ -algebra. For K -amenable groups forinstance, K ∗ ( C ∗ r Γ) ≃ K ∗ ( C ∗ m Γ) and the Baum-Connes conjecture is equivalent to the so-called maximalBaum-Connes conjecture. We shall mainly be interested in the present paper in the maximal Baum-Connesmap and maximal Baum-Connes conjecture. This conjecture is false for some infinite property ( T ) groups,it is known to be true for all torsion free discrete subgroups of SO( N,
1) or SU( N,
1) and also for all torsionfree a-T-amenable groups which are K -amenable and for which the Baum-Connes conjecture was proved byHigson and Kasparov in [HK:01]. As an example, the maximal Baum-Connes map is thus an isomorphismfor all torsion free discrete amenable groups. In the sequel, only the even dimensional case will be neededand all the statements are relative to the even maximal Baum-Connes map for torsion free groups: µ Γ , max : K ( B Γ) −→ K ( C ∗ m Γ) . Theorem 6.6.
Let M be a complete even-dimensional riemannian manifold and D a generalized Diracoperator as before which is invertible near infinity. Assume that Γ is torsion free and the maximal Baum-Connes map is rationally onto. Then the following Atiyah-type formula holds for the Gromov-Lawson indices: Ind (2) ( e D + ) = Ind( D + ) . In particular,
Ind (2) ( e D + ) is an integer in this case.Proof. Since Γ is torsion free, the rational maximal Baum-Connes map can be described as µ Γ ⊗ Q : K geo0 ( B Γ) ⊗ Q → K ( C ∗ m Γ) ⊗ Q , where K geo0 ( B Γ) is the geometric Baum-Douglas K -homology group.The higher index class of the maximal completion of the generalized Dirac operator D m lives in K ( C ∗ m Γ)and hence there exist classes [ Z i , E i , f i ] in the geometric K -homology group K geo0 ( B Γ) and rational numbers q i such that P i µ Γ [ Z i , E i , f i ] ⊗ q i = Ind( D m ) ⊗ Q . Recall that each Z i is a smooth closed K -oriented evendimensional manifold with a complex vector bundle E i over Z i , and that f i : Z i → B Γ is a continuous mapwhich classifies some smooth Γ-covering e Z i → Z i , and [ Z i , E i , f i ] is the class of the triple ( Z i , E i , f i ) withrespect to some equivalence relations, see [BD:80]. Moreover, by definition, the map µ Γ assigns to any triple( Z, E, f ) as above the higher index class in K ( C ∗ m Γ) of the operator D m ( Z, E ) which is the maximal Diracoperator for the K -orientation twisted by some hermitian connection on E and by the flat connection on theMichscheko bundle associated with the Galois covering given by f . Applying the regular and average tracesto Ind( D m ) then gives τ reg ∗ Ind( D m ) = X i q i τ reg ∗ Ind( D m ( Z i , E i )) and τ av ∗ Ind( D m ) = X i q i τ av ∗ Ind( D m ( Z i , E i )) . ELATIVE L INDEX THEOREM 19
We may then apply the classical Atiyah covering index theorem for each operator D m ( Z i , E i ) on the closed K -oriented manifold Z i and deduce the equality τ reg ∗ Ind( D m ( Z i , E i )) = τ av ∗ Ind( D m ( Z i , E i ) , ∀ i. The conclusion follows. (cid:3)
It seems to us that the following is an interesting problem.
Problem 6.7.
Assume that Γ is torsion free. Then is is true that for any Galois Γ -cover of complete even-dimensional riemannian manifolds f M → M and any generalized Dirac operator D which is invertible nearinfinity, one has: Ind (2) ( e D + ) = Ind( D + )?An affirmative answer implies the weaker statement that for torsion free groups one would have Ind (2) ( e D ) ∈ Z . We end this paragraph with the corresponding statement for compact manifolds with boundary. Theorem 6.8.
Let Γ be a torsion free countable group such that the maximal Baum-Connes map for Γ is rationally onto. Let π X : e X → X be any Galois Γ -covering of compact manifolds with boundaries. Weassume that D X is a generalized Dirac operator on X which is a ”product operator” in a neighborhood ( − ǫ, × Y of the boundary Y given as in [APS:75] by σ ( ∂/∂t + D Y ) , and such that the generalized Diracoperator D Y on Y is L invertible. Then the L -index of the generalized Dirac operator e D with the globalAPS Γ -invariant boundary condition coincides with the integer index of the Dirac operator D downstairswith the classical global APS boundary condition. So we assume in particular that π X restricts to a Galois Γ-covering of the boundaries π Y : e Y → Y , where Y = ∂X and e Y = ∂ e X . Recall from [APS:75] that the generalized Dirac operator D with the global APSboundary condition has a well defined Fredholm index Ind AP S ( D ) ∈ Z . This result was generalized in [R:93]where the same result holds upstairs on e X for the Γ-invariant generalized Dirac operator e D subject to theΓ-invariant boundary condition, which then has a well defined L index Ind AP S (2) ( e D ) ∈ R . Therefore, underour assumptions on Γ, the statement of the theorem is thatInd AP S (2) ( e D ) = Ind AP S ( D ) . Proof.
As in [APS:75], we consider the smooth complete manifold M obtained by attaching the semi-cylinder Y × [0 , + ∞ ) to X along its boundary. Then the same construction upstairs gives a smooth complete manifold f M and a Galois Γ-covering π : f M → M which extends the Galois Γ-covering π X : e X → X . The operator D then ”extends” to a generalized Dirac operator D M which is invertible near infinity in M , namely outside X ⊂ M . By [APS:75][Proposition 3.11] and since the kernel of the Dirac operator on Y is trivial in ourcase, we deduce that the APS-index Ind AP S ( D ) coincides with the Gromov-Lawson index of the operator D M on M . The L index of the Γ-invariant operator g D M on f M is also well defined thanks to our Theorem3.1. Moreover, although more involved, it is still true that we have the equalityInd AP S (2) ( e D ) = Ind (2) ( g D M ) , this is the main result proved in [V:01], see also [An:13]. Therefore, applying our Theorem 6.6, we canconclude that Ind AP S (2) ( e D ) = Ind (2) ( g D M ) = Ind( D M ) = Ind AP S ( D ) . (cid:3) By using the APS index formula [APS:75] together with its L version as proved in [R:93], the previoustheorem implies that the Cheeger-Gromov rho invariant of the generalized Dirac operator e D on any Galoiscovering e Y → Y vanishes when Γ is torsion free with rationally onto full maximal Baum-Connes map. Thisresult was actually proved for general spin Γ coverings e Y → Y with PSC and torsion free Γ (not necessarilywith some multiple which bounds a compact spin manifold) but under the stronger assumption that Γ satisfies the maximal Baum-Connes conjecture, see [PS:07] and also [Ke:00]. According to our results, it iseasy to deduce that this full result (for torsion free groups) should hold for all generalized Dirac operators D such that R > Some L Gromov-Lawson PSC-invariants.
For any compact spin manifold X with boundary Y and for any g ∈ R + ( Y ) and any metric G on X which coincides with g + dt on a collar neighborhood of Y in X , we may extend G to a complete metric b G on the smooth manifold b X obtained out of X by attachinga semi-cylinder ( Y × [0 , + ∞ ) , g + dt ) at the boundary, see [APS:75]. Then set as in [GL:83] i ( X, g ) := Ind( d D X + ) , where d D X is the associated Dirac operator on b X . See [GL:83] where it was proved that Ind( d D X + ) only de-pends on X and g , this explains the notation. As explained in the end of the previous paragraph, it was provedin [APS:75][Proposition 3.11] that the invariant i ( X, g ) coincides with the APS-index Ind
AP S ( D X , B Y ) ofthe Dirac operator D + X on X acting on L -spinors with the global APS boundary condition B Y .In the same way, if e X → X is a Galois Γ-cover of manifolds with boundaries e Y and Y so that we alsohave the Galois cover e Y → Y . The Gromov-Lawson construction can then be done Γ-equivariantly and weobtain the invariant i (2) ( e X, g ) := Ind (2) (cid:16)d D e X + (cid:17) , where d D e X is the Γ-invariant Dirac operator defined similarly upstairs and Ind (2) (cid:16)d D e X + (cid:17) is its L indexobtained using our construction. Indeed, we have Lemma 6.9.
The index
Ind (2) ( d D e X ) does not depend on the choice of the metric G on X .Proof. Let us choose another metric G ′ which coincides as well with g + dt in a collar neighborhood of thebounday Y through the diffeomorphic identification with ( − ǫ, × Y . The two extended metrics and spinstructures on the augmented manifold b X , then coincide outside a compact subspace ( X itself here), so thatthe identity map allows to apply our L relative index Theorem 6.3 and deduce the formulaInd (2) (cid:16) \ D e X,G + (cid:17) − Ind (2) (cid:16) \ D e X,G ′ + (cid:17) = Z X b A ( G ) − Z X b A ( G ′ ) . The right hand side vanishes since it coincides with the index of the resulting generalized Dirac operatoron the double manifold 2 X . Notice that the metrics G and G ′ fit together to give a specific metric on thedouble, but the integral over 2 X of the Atiyah-Singer form does not depend on the metric. (cid:3) Following [GL:83] again, we can introduce an L -invariant associated with any two metrics of PSC ona smooth closed odd dimensional spin manifold N , whose spin cobordism class is not necessarily torsion.More precisely, assume that g and g are two metrics with PSC on N and consider the smooth manifold M = N × R endowed with any smooth metric G such that G = g + dt for t ≤ G = g + dt for t ≥ M, G ) is a spin manifold whose metric has PSC outside N × (0 , i ( N ; g , g ) := Ind( D + M,G ) does not depend on the choice of metric G for 0 < t <
1. It was used in [GL:83] to deduce the existence of infinitely many noncordant metrics of PSCon the sphere S .In the case where N is not simply connected, one can still define the Gromov-Lawson index i ( N ; g , g )but might use all normal covers e N → N associated with normal subgroups of the fundamental group π ( N )and the corresponding Galois covers f M = e N × R −→ M = N × R as we did before. More precisely, we introduce for any such Galois covering the invariant i (2) ( e N ; g , g ) := Ind (2) ( D + f M,G ) . ELATIVE L INDEX THEOREM 21
Notice that if we can find such metric G which has PSC on N × [0 ,
1] then for any such Galois cover, theΓ-invariant Dirac operator on f M will be L invertible, and hence will have trivial L index, i.e. in thiscase i (2) ( e N ; g , g ) = 0. This shows that i (2) ( e N ; g , g ) is trivial when g and g belong to the same PSCconcordance class, in particular when they are homotopic inside PSC metrics on N .Recall that the space of metrics of PSC on N is denoted by R + ( N ). Proposition 6.10. (1)
The invariant i (2) ( e N ; g , g ) does not depend on the metric G on N × (0 , . (2) One has i (2) ( e N ; g , g ) + i (2) ( e N ; g , g ) + i (2) ( e N ; g , g ) = 0 for any g i ∈ R + ( N ) . (3) For any compact spin Galois cover e X → X of manifolds with boundaries such that the boundary isthe Galois cover e Y → Y as previously defined, we have for any g , g ∈ R + ( Y ) : i (2) ( e X, g ) − i (2) ( e X, g ) = i (2) ( e Y , g , g ) . Hence, i (2) ( e X, g ) only depends on the concordance class of g in R + ( Y ) .Proof. Let G ′ be another smooth metric on M which coincides with g + dt for t ≤ g + dt for t ≥
1. Then the identity map M → M allows to apply our L relative index formula to deduce thatInd (2) (cid:16) D + f M,G (cid:17) − Ind (2) (cid:16) D + f M,G ′ (cid:17) = Z N × [0 , b A ( G ) − Z N × [0 , b A ( G ′ ) . But the latter difference of integrals coincides with an integral over N × S with S = [0 , / ∼ and with themetric G on N × [0 ,
1] and the leafwise metric f ∗ G ′ on N × [1 ,
2] with f : N × [1 , → N × [0 ,
1] given by f ( y, t ) := ( y, − t ). It is an obvious observation that we get in this way a well defined smooth metric b G onthe quotient N × S . Moreover, Z N × [0 , b A ( G ) − Z N × [0 , b A ( G ′ ) = Z N × S b A ( b G ) . The RHS being independent of the choice of metric, it obviously vanishes, and we conclude that Ind (2) ( D + f M,G ) =Ind (2) ( D + f M,G ′ ).Let us prove now the second item. We shall apply the L Φ-index theorem. More precisely, if we considertwo complete metrics on N × R defined as follows. The first one, denoted G is any metric which coincideswith g + dt on N × ( −∞ ,
0] and with g + dt on N × [1 , + ∞ ). The second, denoted G is any metric whichcoincides again with g + dt on N × ( −∞ ,
0] but with g + dt on N × [1 , + ∞ ). These corresponding twocomplete Γ-covers then are identified on e N × ( −∞ ,
0] by the identity map, and we may apply the Φ-relative L -index theorem since the metrics G and G both have PSC near infinity, we get i (2) ( e N ; g , g ) − i (2) ( e N ; g , g ) = i (2) ( e N ; g , g ) , and therefore since i (2) ( e N , g, g ′ ) + i (2) ( e N , g ′ , g ) = 0 by the sam argument:0 = i (2) ( e N ; g , g ) − i (2) ( e N ; g , g ) − i (2) ( e N ; g , g ) = i (2) ( e N ; g , g ) + i (2) ( e N ; g , g ) + i (2) ( e N ; g , g ) . It remains to prove the third item. Let G and G ′ be smooth metrics on X which coincide on a collarneighborhood U ǫ ≃ Y × [ − ǫ,
0] with g + dt and g + dt respectively. We are assuming again that e X → X is also a Galois Γ-cover over X which restricts to e Y → Y at the boundaries. Consider again the augmentedmanifold be X obtained by adding the semi-cylinder e Y × [0 , + ∞ ) with the corresponding metric lifted from Y × [0 , + ∞ ). We obtain our invariants i (2) ( e X, g ) and i (2) ( e X, g ) as before. We consider the new metric G ′′ on X which is defined as follows. On X r U ǫ/ ≃ Y × [ − ǫ/ ,
0] we take G ′′ = G ′ . On U ǫ/ we takea smooth leafwise metric which coincides with g + dt near − ǫ/ g + dt near 0. Using thesame construction as before, we get the invariant i (2) ( X, g ) by using this new leafwise metric. Recall that i (2) ( X, g ) does not depend on the choice of such metric. By applying the Φ-relative index theorem 6.2 to the Γ-covering manifold be X → b X with the metric associated with G ′ on the one hand, and the covering e Y × R → Y × R with the metric associated with G defining the invariant i (2) ( Y ; g , g ) on the other hand,we get i (2) ( e X, g ) − i (2) ( e Y , g , g ) = i (2) ( e X, g ) . Hence the conclusion.Hence we see from this relation that the difference i (2) ( e X, g ) − i (2) ( e X, g ) does not depend on the coveringbordant manifold e X → X . Moreover, if g ′ belongs to the concordance class of g then we know that theinvariant i (2) ( Y ; g ′ , g ) can be computed as the Γ-index of the Dirac operator on the Γ-cover e Y × R → Y × R for a metric which has PSC everywhere, and hence i (2) ( Y ; g ′ , g ) = 0. Therefore, we deduce that i (2) ( e X, g )only depends on the concordance class of g in R + ( Y, F Y ). (cid:3) We have the following corollary of Theorem 6.8.
Corollary 6.11.
Assume that the group Γ is torsion free and that the maximal Baum-Connes map for Γ isrationally onto, then the following Atiyah-type theorem holds: i (2) ( e N ; g , g ) = i ( N ; g , g ) , ∀ g , g ∈ R + ( N ) . Proof.
Just apply Theorem 6.8 to the Γ-covering e N × [0 ,
1] of the compact manifold with boundary N × [0 , (cid:3) When Γ has torsion, this corollary is false. Indeed, take the Lens space L = S / Z n where Z n actsdiagonally on S ⊂ C . Then if we fix the standard metric h on L which is induced by the standard metric g on S with constant scalar curvature equal to 1, then the map κ : π ( R + ( L )) −→ Z defined by κ ( g ) := i ( S ; g , g ) − n × i ( L ; h , g ) , has infinite range in Z . Notice that κ ( g ) = n × (cid:0) i (2) ( S ; h , g ) − i ( L ; h , g ) (cid:1) . To see this, just observe by theAPS formula that we have the following relation: κ ( g ) = n × ρ ( g ) − ρ ( g )2 , where ρ is the APS rho invariant, a difference of eta invariants [BP:09]. Then the main result proved in[BG:95] completes the argument. More generally, using the results of [BG:95], one can deduce that any4 k + 3-dimensional spin manifold, with k ≥ κ asdefined above. Remark 6.12.
By using Proposition 2.14 in [KS:93] , we see that g i ( L ; h , g ) already has infinite range. Given an orientation preserving diffeomorphism F ∈ Diff + ( N ) of N , we may transport any PSC metric g into the PSC metric F ∗ g . Notice also that the spin structure associated with g on M canonically determinesa spin structure associated with F ∗ g which is in fact the pull-back spin structure under F . By applyingagain the APS formula as well as its L version, we can deduce the following Theorem 6.13.
Let F ∈ Diff + ( N ) be an orientation preserving diffeomorphism. Then for any Galois cover e N → N with group Γ , we have: i (2) ( e N ; g , F ∗ g ) = i ( N ; g , F ∗ g ) . In particular, the invariant κ Γ g : g i (2) ( e N ; g , g ) − i ( N ; g , g ) induces a well defined map κ Γ g : π (cid:0) R + ( N ) / Diff + ( N ) (cid:1) −→ R . ELATIVE L INDEX THEOREM 23
Proof.
The first part of the theorem is an easy consequence of the APS theorem.Assume now that g is a given PSC metric on N , then by Proposition 6.10 i ( N ; g , F ∗ g ) − i ( N ; g , g ) = i ( N ; g, F ∗ g ) while i (2) ( e N ; g , F ∗ g ) − i (2) ( e N ; g , g ) = i (2) ( e N ; g, F ∗ g ) . Therefore, κ Γ g ( F ∗ g ) − κ Γ g ( g ) = i (2) ( e N ; g, F ∗ g ) − i ( N ; g, F ∗ g ) = 0 . (cid:3) Assume now that e N → N is the universal cover, so with Γ = π ( N ). A smooth orientation preservingdiffeomorphism F ∈ Diff + ( N ) of N as above then lifts to a smooth orientation preserving diffeomorphism e F of e N which is equivariant relative to an outer automorphism of the group Γ, and we can choose for thegiven F a representative automorphism that we denote by φ F ∈ Aut(Γ), which allows to define an action of Z on the group Γ. We denote by Γ ⋊ Z the semi-direct product, see for instance [KS:93, XY:14]. Notice thatwe could as well consider the transported metric e F ∗ e g of the pull-back metric e g on e N but it is an obviousobservation that the new metric e F ∗ e g is still Γ-invariant and induces the metric F ∗ g . Proposition 6.14.
Let e N → N be the universal cover of the closed riemannian manifold ( N, g ) withPSC. Denote by b N the closed manifold which is the quotient under the free proper action of Γ ⋊ Z , i.e. b N := ( e N × R ) / Γ ⋊ Z . Then, using the structures induced from those fixed on N , the index of the Diracoperator on b N coincides with the Gromov-Lawson index i ( N ; g, F ∗ g ) .Proof. We consider a metric G on N × [0 ,
1] which coincides with g near 0 and with F ∗ g near 1, then thismetric induces a smooth metric on the mapping torus b N , and we have Z N × (0 , b A ( G ) = Z N F b A ( T b N ) . Hence by [APS:75][Proposition 3.11], we may compute i ( N ; g, F ∗ g ) using the Atiyah-Patodi-Singer indexformula on N × [0 , i ( N ; g, F ∗ g ) = Z N × (0 , b A ( G ) − η ( D ( N,F ∗ g ) ) − η ( D ( N,g ) )2 . But we know that η ( D ( N,F ∗ g ) ) = η ( D ( N,g ) ), hence we get i ( N ; g, F ∗ g ) = Z N × (0 , b A ( G ) = Z b N b A ( T b N ) . The proof is complete since the latter integral is equal to the index of the Dirac operator by the Atiyah-Singerformula on b N . (cid:3) In [KS:93], the similar previous construction allowed to deduce the existence for any k ≥ N of dimension 4 k −
1, which moreover admits a metric of PSC, such that the image of themap Diff
Spin ( N ) −→ Z sending F to the index of the Dirac operator on the mapping torus b N , is Z for k even and 2 Z for k odd.Here Diff Spin ( N ) ⊂ Diff + ( N ) is the subgroup of spin-preserving diffeomorphisms. Appendix A. A Rellich lemma and a second proof of Theorem 3.1
We give in this appendix another proof of Theorem 3.1 based on the Rellich lemma and following thedirect method applied in [XY:14] for the higher index theory. It is worthpointing out that the present proofdoes not use Theorem 5.6 since this latter used in its proof Theorem 3.1. We have added a short proof ofthe Rellich lemma in this von Neumann context, which is ready for generalizations for instance to foliations.Given a compact subspace K of M , we denote by L s,K ( f M , e S ) the subspace of L s ( f M , e S ) composed of those sections which are supported within the Γ-compact subspace π − ( K ) of f M . Recall from Section 2 the semi-finite von Neumann algebra M = L ( f M , e S ) Γ with its trace τ . We have again chosen a fundamental domain F for the Galois cover π : f M → M with the usual properties. Lemma A.1. ( L Rellich lemma) Assume that for a fixed compact subspace K of M , an element T ∈ M factors through the inclusion L ,K ( f M , e S ) ֒ → L ( f M , e S ) via Γ -invariant operators. Then T belongs to the ideal K ( M , τ ) of τ -compact operators. Remark A.2.
When Γ is trivial, this is the exact statement of the classical Rellich lemma on M .Proof. Let ( f n ) n ≥ be a uniformly bounded sequence which belongs to L ,K ( f M , e S ) and let f ∈ L ,K ( f M , e S )be such that for any ϕ ∈ L ,K ( M, S ) ≃ L ,K ( F, e S ), we have X γ ∈ Γ h γ ∗ ( f n − f ) , ϕ i L ( F, e S ) −→ n → + ∞ . Then for any γ ∈ Γ, the restriction of γ ∗ ( f n − f ) to F defines a sequence in L ,K ( M, S ) which convergesweakly to 0 in that Hilbert space. Since K is compact in M , the classical Rellich lemma in M applies andwe can deduce that the sequence ( γ ∗ f n ) n ≥ converges to γ ∗ f in L ( M, S ) ≃ L ( F, e S ), say || f n − f || L ( γF, e S ) −→ n → + ∞ . Since X γ ∈ Γ || f n − f || L ( γF, e S ) = || f n − f || L ( f M, e S ) < + ∞ , an easy argument shows that the sequence ( f n ) n ≥ converges in L ( f M , e S ) to f .Suppose now that T ∈ M ⊂ B ( L ( f M , e S )) satisfies the assumptions of the lemma, i.e. T = ι ◦ A for someΓ-equivariant operator A : L ( f M , e S ) → L ,K ( f M , e S ). If a given uniformly bounded sequence ( σ n ) n ≥ from L ( f M , e S ) converges weakly relative to M in the sense of [Kf:82] to some section σ ∈ L ( f M , e S ), then for any ψ ∈ L ( M, S ) with || ψ || = 1, and using the τ rank one operator which is the orthogonal projection onto theline generated by ψ tensorized by the identity of ℓ Γ, we deduce that X γ ∈ Γ h g ∗ A ( σ n − σ ) , Aψ i L ,K ( F, e S ) = X γ ∈ Γ h g ∗ ( σ n − σ ) , A ∗ Aψ i L ( F, e S ) −→ n → + ∞ . We deduce from the previous discussion that (( ι ◦ A )( σ n )) n ≥ converges in L ( f M , e S ) to ( ι ◦ A )( σ ), and hencethat T is τ -compact, using again the compactness criterion of [Kf:82]. (cid:3) Recall that the generalized Dirac operator D satisfies the generalized Lichnerowicz formula D = ∇ ∗ ∇ + R , and that we assume that there exist a compact subspace K of M such that R| M r K is uniformly positive.Then we have fixed a compactly supported smooth real valued function ρ ∈ C ∞ c ( M ) and a constant c > R ≥ ( c − ρ ) Id over M. The operator ( e D + ρ ) / is then a well defined self-adjoint non-negative operator. Lemma A.3.
The operator ( e D + ρ ) / is injective with bounded inverse. Moreover, its inverse, denoted ( e D + ρ ) − / =: e T , is a bounded operator from L ( f M , e S ) to the Sobolev space L ( f M , e S ) . In particular, e T belongs to the von Neumann algebra M .Proof. Let σ ∈ Dom( e D ). Then h ( e D + ρ ) σ, σ i = h ( e ∇ ∗ e ∇ + R + ρ Id) σ, σ i ≥ || e ∇ σ || + c || σ || ≥ c || σ || . Hence the self-adjoint operator e D + ρ is injective and has a bounded inverse which obviously commuteswith Γ and belongs to the von Neumann algebra M . Indeed, its spectrum is contained in [ c, + ∞ ), so that ELATIVE L INDEX THEOREM 25 the spectrum of ( e D + ρ ) / is contained in [ √ c, + ∞ ). Therefore, the operator e T is a well defined boundednon-negative operator on L ( f M , e S ). Moreover, for σ ∈ L ( f M , e S ) we can write || ( e D + ρ ) − / σ || = h ( e D + ρ ) − σ, σ i + h ( e D + ρ ) − / e ∇ ∗ e ∇ ( e D + ρ ) − / σ, σ i≤ c || σ || + h ( e D + ρ ) − / [( e D + ρ ) − ( ρ + R )]( e D + ρ ) − / σ, σ i Notice that by the properties of the continuous functional calculus, we deduce that( e D + ρ ) − / ( e D + ρ )( e D + ρ ) − / = Id . Hence, we obtain since ρ Id +
R ≥ c Id: || ( e D + ρ ) − / σ || ≤ (1 + 1 c ) || σ || . Therefore, the operator ( e D + ρ ) − / is a bounded operator from L ( f M , e S ) to L ( f M , e S ) as announced. (cid:3) Remark A.4.
The same proof shows that for any λ ∈ R , the operator T λ := ( e D + ρ + λ ) − / extends toa bounded operator from L ( f M , e S ) to L ( f M , e S ) . The following proposition is a von Neumann version of results from [XY:14].
Proposition A.5.
For any λ ∈ R , consider the operator e F λ := e D ( e D + ρ + λ ) − / = e D e T λ , then: (1) e F λ extends to a bounded operator on L ( f M , e S ) which belongs to the von Neumann algebra M ; (2) The Γ -invariant operator e Q = ( e D + ρ ) − / e F is a self-adjoint parametrix for e D modulo τ -compactoperators with respect to the von Neumann algebra M .Proof. We notice again by the properties of the continuous functional calculus that e T λ ( e D + ρ + λ ) e T λ = Id , say is densely defined and extends to a bounded operator which is the identity operator. Therefore since ρ + λ is bounded on f M , we deduce that the operator e T λ e D e T λ also extends to a bounded operator on L ( f M , e S ). Hence we deduce by standard arguments that the operator e F λ also extends to a bounded operatoron L ( f M , e S ). On the other hand the Γ-invariance is obvious by construction since ρ is pulled-back from M ,hence e F λ ∈ M .We now prove that e F = e F is Fredholm with respect to M with quasi-inverse given by e F itself, this willfinish the proof. Notice that we can use the following integral expression where one easily checks that theintegral converges in the operator norm: e T = 2 π Z + ∞ ( e D + ρ + λ ) − dλ. This allows to see for instance that the commutator [ e D, e T ] extends to a bounded operator defined by theformula: [ e D, e T ] = 2 π Z + ∞ ( e D + ρ + λ ) − [ ρ , e D ]( e D + ρ + λ ) − dλ, and the operator [ ρ , e D ] being bounded while e D + ρ ≥ c Id, we can deduce the || ( e D + ρ + λ ) − [ ρ , e D ]( e D + ρ + λ ) − || ≤ || [ ρ , e D ] || c + λ ) . Moreover, e F = e D e T e D e T = e D e T − e D [ e D, e T ] e T = Id − ρ e T − e D [ e D, e T ] e T .
Similarly, we can write e D [ e D, e T ] = 2 π Z + ∞ e D [ e D, ( e D + ρ + λ ) − ] dλ = 2 π Z + ∞ e D ( e D + ρ + λ ) − [ ρ , e D ]( e D + ρ + λ ) − dλ Since ρ and [ e D, ρ ] are both Γ-compactly supported zero-th order Γ-invariant operators, the operator[ ρ , e D ]( e D + ρ + λ ) − is a compact operator with respect to M . Indeed, the operator ( D + ρ + λ ) − is bounded below from L ( f M , e S ) to L ( f M , e S ). On the other hand applying [ ρ , e D ] sends L ( M, S ; µ ) to the subspace L , Supp( ρ ) ( M, S ; µ )of L ( M, S ; µ ). Now the Rellich lemma A.1 allows to conclude that [ ρ , e D ]( e D + ρ + λ ) − is compact withrespect to the von Neumann algebra M . By the first item, the operator e D ( e D + ρ + λ ) − is bounded sothat for any λ ≥ e D ( e D + ρ + λ ) − [ ρ , e D ]( e D + ρ + λ ) − is compact with respect to the von Neumann algebra M . By the operator norm convergence of the integral,we deduce that e D [ e D, e T ] is compact with respect to the von Neumann algebra M . Finally, the same argumentagain allows to prove that the operator ρ e T is also compact with respect to M so that the proof is complete. (cid:3) Proof. (of Theorem 3.1)The self-adjoint Γ-invariant operator e Q = ( e D + ρ ) − / e D ( e D + ρ ) − / is an odd for the gradingparametrix for e D modulo compact operators with respect to M . Indeed, we have proved that e F − I is a compact operator with respect to the von Neumann algebra M , hence so is its adjoint, say:Id − e D e Q = e S ∈ K ( M , τ Γ ) and Id − e Q e D = e S ∗ ∈ K ( M , τ Γ ) . Hence, if we denote as before by P ǫ the orthogonal projection given by the functional calculus P ǫ := 1 [0 ,ǫ ] ( e D ),we can write P ǫ − P ǫ e Q ( e DP ǫ ) = P ǫ e S ∗ P ǫ . If σ ∈ Im( P ǫ ) then we thus get || P ǫ e S ∗ σ || ≥ (1 − √ ǫ || e Q || ) || σ || . Therefore, for ǫ > P ǫ e S ∗ P ǫ restricted to the image of P ǫ is injective with closedrange, and is therefore an isomorphism between Im( P ǫ ) and Im( P ǫ e S ∗ P ǫ ). Since P ǫ e S ∗ P ǫ is compact withrespect to the von Neumann algebra P ǫ M P ǫ due to the same property for e S ∗ with respect to M , we deducethat P ǫ must be compact with respect to P ǫ M P ǫ , that is that the projection P ǫ is compact with respect to M . We conclude again that P ǫ has finite τ -trace as allowed since any τ -compact projection is τ -finite. (cid:3) References [An:13] P. Antonini,
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Institut Montpellierain Alexander Grothendieck, UMR 5149 du CNRS, Universit´e de Montpellier
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