DIRAC INDEX CLASSES AND THE NONCOMMUTATIVE SPECTRAL FLOW

ERIC LEICHTNAM AND PAOLO PIAZZA

June 9 2001

Abstract. We present a detailed proof of the existence-theorem for noncommutative spectral sections, see [32]. We apply this result to various indextheoretic situations, extending to the noncommutative context results of Booss-Wojciechowski, Melrose-Piazza and Dai-Zhang. In particular, we prove a variational formula, in K_{∗}(C_{r}^{∗}(Γ)), for the index classes associated to 1-parameter family of Dirac operators on a Γ-covering with boundary; this formula involves a noncommutative spectral flow for the boundary family. Next, we establish an additivity result, in K_{∗}(C_{r}^{∗}(Γ)), for the index class defined by a Dirac-type operator associated to a closed manifold M and a map r : M → BΓ when we assume that M is the union along a hypersurface F of two manifolds with boundary: M = M_{+} ∪_{F} M_{−}. Finally, we prove a defect formula for the signatureindex classes of two cut-and-paste equivalent pairs (M_{1}, r_{1} : M_{1} → BΓ) and (M_{2}, r_{2} : M_{2} → BΓ), where

M_{1} = M_{+} ∪_{(F,ϕ_{1})} M_{−} , M_{2} = M_{+} ∪_{(F,ϕ_{2})} M_{−}

and ϕ_{j} ∈ Diff(F). The formula involves the noncommutative spectral flow of a suitable 1-parameter family of twisted signature operators on F. We give applications to the problem of cut-and-paste invariance of Novikov’s higher signatures on closed oriented manifolds.

Contents

1. Introduction and main results 2 2. Proof of the existence of noncommutative spectral sections 6 2.1. Very full projections and a lifting theorem 6 2.2. Spectral cuts and very full projections 9 2.3. Proof of the technical Lemma. 10 2.4. Existence of spectral sections in the odd dimensional case 13 2.5. Difference classes and the noncommutative spectral flow 16 2.6. Existence of spectral sections in the even dimensional case 17 3. Manifolds with boundary, spectral sections and index classes 18 3.1. APS and b-index classes in the even dimensional case 18 3.2. APS and b-index classes in the odd dimensional case 20 3.3. Proof of Theorem 7. 23 4. A splitting formula for index classes 25 4.1. The even dimensional case. 25 4.2. The odd dimensional case. 27 5. Cut and paste invariance for higher signatures 28 5.1. A defect formula. 28 5.2. Remarks 31 5.3. Vanishing spectral flow in the even-dimensional case 32 5.4. Vanishing spectral flow in the odd dimensional case 34 References 38

1. Introduction and main results

The Connes-Moscovici higher index theorem on Galois coverings [7] can be seen as a family-index theorem but with a noncommutative parameter space given by the reduced C^{∗}-algebra of the covering group [6]. It is natural to ask to what extent results established for families of Dirac operators parametrized by a topological space X can be carried over to the noncommutative context. The past 20 years of noncommutative geometry show that such generalizations have, besides their own beauty, deep and interesting geometric applications. See Alain Connes’ book [6] for a wide and fascinating treatment of many of these generalizations.

We consider the following results in family-index theory:

• on a closed manifold N, the existence of spectral sections for a family of Dirac operators with trivial index class ([27] [28]);

• on a manifold with boundary M, the definition of the Atiyah-Patodi-Singer index class and of the b-index class associated to the choice of a spectral section for the boundary family ([27] [28]);

• the equality of these two index classes ([27] [28])

• the relative index theorem, giving a formula for the difference of two index classes associated to two different choices of spectral sections ([27] [28])

• on a closed manifold N = M_{+} ∪_{F} M_{−}, union along a hypersurface F of two manifolds with boundary, the splitting formula for the index class associated to a family of Dirac operators ([8]).

• the notion of higher spectral flow and its relationships with family index

theory ([9]) For an informal presentation of some of these results we refer the reader to the surveys [30] [25]. We shall also consider the following result in ordinary (i.e. numerical) index theory:

• Let M_{1} = M_{+} ∪_{(F,ϕ_{1})} M_{−} and M_{2} = M_{+} ∪_{(F,ϕ_{2})} M_{−}, ϕ_{j} ∈ Diff(F), ∂M_{+} = F = −∂M_{−} be two even-dimensional manifolds that are cut-and-paste equivalent; let D /_{1}, D /_{2} be two cut-and-paste equivalent operators of Dirac-type. Then the difference indD /_{2}^{+} − indD /_{1}^{+} is equal to the spectral flow of a natural family of operators on F ([4]).

The main goal of this paper is to establish noncommutative generalizations of the above results and give geometric applications to the problem of cut-and-paste invariance of Novikov’s higher signatures on a closed oriented manifold.

The first result, i.e. the existence of noncommutative spectral sections, was claimed in the odd-dimensional case by F. Wu in the preprint [32]. Unfortunately Wu’s preprint was never published; moreover, as pointed out to us by Michel Hilsum, the proof given by Wu contained one unjustified step. For these reasons we felt it was necessary to present a full proof of the existence of noncommutative spectral sections; the missing step turned out to be non-trivial and one section of this paper is entirely devoted to a rigorous proof of this particular result. Notice that there was much more in Wu’s preprint than just the existence-theorem for noncommutative spectral sections. Of course we cannot report here on Wu’s further results and we strongly feel that it is a pity that the interesting Mathematics established in [32] will most likely never be published.

Some of our papers on higher index theory freely use the existence-theorem stated in Wu’s unpublished preprint as well as the consequent definition, given there, of noncommutative spectral flow (this definition is an immediate generalization of the family case treated in [9]). In these articles of ours, [18], [19], [21], some of the above generalizations are presented. More precisely:

- the definition, in the even-dimensional case, of the b-index class associated to a Dirac operator on a Galois covering with boundary once a spectral section for the boundary operator is chosen, is given in [18]. (Notice that our main interest in [18] was in proving an index formula, in noncommutative topological de Rham homology, for the Chern character of such an index class.)

- a formula for the Chern character of the noncommutative spectral flow in terms of higher eta invariants is also proved in [18].

- the gluing formula for index classes is already stated in [21].

- the existence-theorem for spectral sections on even-dimensional closed manifolds is proved in [19] assuming the theorem in the odd dimensional case.

- finally, always in [19], we prove the relative index theorem for even dimensional Galois coverings with boundary.

In writing this article we were also motivated by the wish of completing the picture that has emerged so far from these articles. We shall be more specific in each particular section.

The paper is organized as follows. In Section 2 we recall the notion of noncommutative spectral section and prove the fundamental result that a noncommutative spectral section for a Dirac-type operator exists if and only if the index class defined by this operator is trivial in K-Theory. We also recall the notion of difference class and of noncommutative spectral flow. In Section 3 we recall from [18] and [33] the definition of b-index class and APS-index class associated to a Dirac-type operator on an even-dimensional Galois covering with boundary; these classes depend on the choice of a noncommutative spectral section for the boundary operator. We then prove the equality of this two index classes. We end Section 3 by extending these results to odd dimensions, using suspension as in [28]. In Section 4, following ideas of Bunke and Dai-Zhang, we prove a gluing formula for index classes. In Section 5 we discuss applications to the problem of cut-and-paste invariance for higher signatures. Let us briefly explain what is the problem and what are the techniques that we shall be using.

Recall from [13] that two manifolds M_{1}, M_{2} are SK-equivalent if

M_{1} = M_{+} ∪_{(F,ϕ_{1})} M_{−} , M_{2} = M_{+} ∪_{(F,ϕ_{2})} M_{−} (1)

with ∂M_{+} = F = −∂M_{−} and ϕ_{j} ∈ Diff(F). In words, M_{1} and M_{2} are obtained by gluing two manifolds with boundary but the gluing diffeomorhisms are different. Incidentally, SK stands for Schneiden=Cutting and Kleben=Pasting. The signature of a manifold is a SK-invariant: σ(M_{1}) = σ(M_{2}) for M_{1}, M_{2} as above. An analytic proof of this fact is given in [4]. The argument given there is a consequence of a more general formula concerning the numerical indeces of two Dirac-type operators obtained one from the other by a cut-and-paste construction; the formula expresses the difference of the numerical indeces in terms of the spectral flow of a suitable 1-parameter family of operators on F. For the particular case of the signature operator this spectral flow turns out to be zero, as it is simply the spectral flow of a 1-parameter family {D_{F} (θ)}_{θ∈S^{1}}, of odd signature operators on F parametrized by a path of metrics. In this last step the cohomological significance of the zero-eigenvalue has been used. Since the signature of a manifold is equal to the index of the signature-operator, we obtain finally

(2) σ(M_{2}) − σ(M_{1}) = indD_{2} − indD_{1} = sf({D_{F} (θ)}_{θ∈S^{1}}) = 0

In Section 5 we shall be concerned with the higher analogue of this argument. We thus consider a pair (M, r : M → BΓ), with M a closed oriented manifold and r a continuous map into the classifying space of a dicrete group Γ. To such a pair one can attach a set of numbers, the Novikov’s higher signatures σ(M, r; [c]) defined by

σ(M, r; [c]) =< L(M) ∪ r^{∗}[c], [M] > ,

with [c] ∈ H^{∗}(BΓ, ℚ) = H^{∗}(Γ, ℚ). Two pairs

(M_{1}, r_{1} : M_{1} → BΓ) and (M_{2}, r_{2} : M_{2} → BΓ)

are said to be SK-equivalent if M_{1} and M_{2} satisfy (1) and (r_{1})_{|M_{+}} ≃ (r_{2})_{|M_{+}}, (r_{1})_{|M_{−}} ≃ (r_{2})_{|M_{−}}, where ≃ means homotopy equivalence. One can ask whether two SK-equivalent pairs (M_{1}, r_{1}), (M_{2}, r_{2}) have the same higher signatures. It was observed by John Lott in [24] (Remark 4.1), that results in [13], [29] can be reinterpreted as giving examples where the answer to this question is negative; one can then try to find conditions on Γ and F so as to ensure that the higher signatures are indeed SK-invariants. The first result in this direction is given by Leichtnam-Lott-Piazza in [14], as a Corollary of the higher APS-index formula proved there and the higher index formula proved in [22] (see Corollary 0.4 in [14]). The condition on the group Γ is that it be Gromov hyperbolic or virtually nilpotent. The condition on F is recalled in detail in Sect. 5.2/5.3 below: it is an homotopy-invariant condition of the pair (F, r_{|F}) and can be briefly formulated as a gap condition, in middle degree, for the spectrum of the differential-form Laplacian on the Γ-covering associated to (F, r_{|F}). This condition was first introduced by Lott in [23], where a regularization of the apriori divergent integral defining the higher eta invariant for the signature operator was proposed. These ideas were further pursued in the articles [19], [24], [14].

A different treatment of the cut-and-paste result proved in [14] was subsequently given in [15]. In that paper techniques from algebraic surgery are used; the condition on F remains the same but the group Γ must only satisfy the property that the Baum-Connes map be rationally injective.

In Section 5 we shall reobtain these results using a higher version of the result of Booss-Wojciechowski, i.e. a higher analogue of (2). We briefly describe the result. Let us denote by D_{M_{j}}^{sign,r_{j}}, j = 1, 2, the Mishchenko-Fomenko signature operators associated to the two SK-equivalent pairs (M_{j}, r_{j}) and a choice of metric g_{j} on M_{j}; then for the corresponding index classes, in K_{∗}(C_{r}^{∗}(Γ)), the following defect formula holds:

(3) Ind D_{M_{1}}^{sign,r_{2}} − Ind D_{M_{1}}^{sign,r_{1}} = sf({D_{F} (θ)}_{θ∈S^{1}}) .

On the right hand side of the above formula the noncommutative spectral flow of a suitable 1-parameter family of twisted signature operators on F appears. Formula (3) is a consequence of two fundamental results:

- the gluing formula;

- a variational formula for the index classes, in K_{∗}(C_{r}^{∗}(Γ)), associated to a 1parameter family of Dirac operators on a Γ-covering with boundary.

It is the latter formula that involves a noncommutative spectral flow. Notice that the noncommutative spectral flow is a class in K_{∗}(C_{r}^{∗}(Γ)); as such it should be more precisely called a higher noncommutative spectral flow. Numerical (i.e. lower) spectral flows in the noncommutative context have already been considered, for example, in the work of V. Perera, J. Kaminker and J. Phillips; see [11] and references therein.

It is important to notice that, in contrast with the numerical case treated in (2), the noncommutative spectral flow appearing in (3) will in general be different from 0, as it is associated to a family of twisted signature operators, with the twisting bundle depending on the parameter θ. A similar formula holds for any Dirac-type operator once the notion of SK-equivalent operators is given. By employing symmetric spectral sections, as in [19], we shall show that under the cited assumption on F this noncommutative spectral flow is automatically zero. This fact, together with the assumption that the Baum-Connes assembly map is rationally injective, allows one to reobtain the results of [15].

Notice that in the even dimensional case a recent preprint [10] of Hilsum sharpens further these results by slightly weakening the assumptions on F. We refer the reader to his paper for the detailed statement of this weaker assumption and also for a somewhat different treatment of boundary value problems in the noncommutative context, more in the spirit of [4]. It would be interesting to know whether the higher APS index formula proved in [18] can be established for these more general boundary value problems (one advantage of spectral sections being that they are specific enough for local index-theory arguments to be carried over).

Notice also that the cut and paste results in the odd-dimensional case are somewhat more involved than in the even-dimensional case and cannot be obtained by simply crossing with S^{1}. This phenomenon is already present in [14], [15] and will be further clarified in the present paper. Acknowledgements. We heartily thank Ralf Meyer for pointing out the relevance of the notion of very full projection and for suggesting the statement of Theorem 1 as well as the structure of its proof. We thank Michel Hilsum for alerting us of the unjustified step in Wu’s preprint and Etienne Blanchard for a helpful comment. The second author wishes to thank John Lott for interesting discussions concerning the cut and paste invariance of higher signatures. Finally, we thank again Michel Hilsum and Ralf Meyer for some very useful comments on an earlier version of the paper.

Both authors are memebers of the RTN “Geometric Analysis” of the European Community. The second author has received financial support from the “Shortterm mobility program” of the Consiglio Nazionale delle Ricerche, from Ministero dell’Università e della Ricerca Scientifica e Tecnologica (Cofinanziamento MURST ’99) and by the Istituto Nazionale di Alta Matematica through the GNSAGA. The writing of this paper was completed in April 2001 while the second author was visiting the MSRI, Berkeley, as a member of the Spectral Invariants program (from March 1st to May 12th 2001); he thanks the institute for the invitation and partial financial support.

2. Proof of the existence of noncommutative spectral sections

We begin this section with two functional analytic results that will be needed in the proof of the existence-theorem. The definition of noncommutative spectral section and the statement of the existence-theorem are given in Section 2.4 below.

2.1. Very full projections and a lifting theorem.

We consider in this section a unital C^{∗}−algebra A and denote by H_{A} = l^{2}(A) = l^{2}(ℕ)⊗_{ℂ}A the standard A−Hilbert module. We consider B_{A} the algebra of A−linear continuous adjointable operators from H_{A} to itself and denote by K_{A} the subalgebra of B_{A} of all the operators T such that both T and T^{∗} are A−compact. We then have the following exact sequence:

0 → K_{A} → B_{A} → C_{A} = B_{A}/K_{A} → 0

where C_{A} denotes the (generalized) Calkin algebra. We recall that (see [2]) the K−theory groups K_{∗}(B_{A}) vanish and that K_{∗}(K_{A}) ≃ K_{∗}(A). The standard six terms exact sequence in K−Theory gives the following isomorphism δ called the index map:

δ : K_{0}(C_{A}) → K_{1}(K_{A}) ≃ K_{1}(A) (4)

We shall now recall what a very full projection is, see [2] (Sect. 6.11). The relevance of this notion in our problem has been pointed out to us by Ralf Meyer.

Definition 1. A projection p (= p^{∗}) of a unital C^{∗}−algebra A is a said to be very full if there exists an isometry u ∈ A (u^{∗}u = 1 = 1_{A}) such that uu^{∗} ≤ p. In other words, p contains a projection which is equivalent to 1.

The following Theorem and the precise structure of its proof has been suggested to us by Ralf Meyer.

Theorem 1. (Ralf Meyer) Let p be a very full projection of C_{A} such that the K-Theory class [p] of p in K_{0}(C_{A}) is zero (i.e. the index δ(p) is zero) and 1 − p is very full. Then there exists a projection P ∈ B_{A} such that π(P) = p where π : B_{A} → C_{A} = B_{A}/K_{A} is the canonical projection.

Proof. The proof of the Theorem relies on the next two Propositions.

Proposition 1. Let p be a very full projection of C_{A}. Then there exists an isometry W ∈ B_{A} such that π(W)π(W^{∗}) ≤ p.

Proof of the Proposition. By assumption there exists W_{1} ∈ B_{A} such that π(W_{1}) is an isometry of C_{A} satisfying π(W_{1})π(W_{1})^{∗} ≤ p and W_{1}^{∗}W_{1} − Id ∈ K_{A}. Recall that l^{2}(A) = ⊕_{n≥0}^{⊥}Ae_{n}; for any N ∈ ℕ, we denote by P_{N} ∈ B_{A} the projection from l^{2}(A) onto Im P_{N} = ⊕_{n≥N}^{⊥}Ae_{n}. Then there exists N ∈ ℕ such that: ||P_{N}(W_{1}^{∗}W_{1} − Id)P_{N}|| ≤ \frac{1}{4} where || · || denotes the C^{∗}− norm of B_{A}.

Now we consider and fix a unitary isomorphism χ : l^{2}(A) → Im P_{N} ⊂ l^{2}(A) We have:

||χ^{−1}P_{N}(W_{1}^{∗}W_{1} − Id)P_{N}χ|| ≤ \frac{1}{4}; (5)

we identify P_{N}χ with an element of B_{A} so that (P_{N}χ)^{∗} = χ^{−1}P_{N}. We then set W_{2} = W_{1}P_{N}χ ∈ B_{A} so that W_{2}^{∗} = χ^{−1}P_{N}W_{1}^{∗}. From (5) we deduce that:

||W_{2}^{∗}W_{2} − Id || = ||χ^{−1}P_{N}(W_{1}^{∗}W_{1} − Id)P_{N}χ|| ≤ \frac{1}{4}

So W_{2}^{∗}W_{2} ∈ B_{A} is invertible and Im W_{2}^{∗} = l^{2}(A). Therefore W_{2}^{∗} admits a polar decomposition W_{2}^{∗} = V \sqrt{W_{2}W_{2}^{∗}}, Im W_{2} = Im W_{2}W_{2}^{∗} is closed in l^{2}(A), ker W_{2}^{∗} = ker W_{2}W_{2}^{∗} and

l^{2}(A) = Im W_{2}W_{2}^{∗} ⊕^{⊥} ker W_{2}W_{2}^{∗} (6)

(See Theorem 15.3.8 of [31] for details). Moreover V^{∗}V (resp. V V^{∗}) is the orthogonal projection onto Im W_{2}W_{2}^{∗} (resp. Im W_{2}^{∗}W_{2} = l^{2}(A)), so V V^{∗} = Id. From the definition of W_{2} we get W_{2}W_{2}^{∗} = W_{1}W_{1}^{∗} + W_{1}(P_{N} − Id)W_{1}^{∗} and thus π(W_{2}W_{2}^{∗}) = π(W_{1}W_{1}^{∗}). Since by hypothesis π(W_{2}W_{2}^{∗}) = π(W_{1}W_{1}^{∗}) ≤ p, we obtain the Proposition as a consequence of the following Lemma by taking W = V^{∗}.

Lemma 1. We have π(V^{∗}V) = π(W_{2}W_{2}^{∗}).

Proof. Using the decomposition (6) and the fact that Im W_{2}W_{2}^{∗} is closed and defines a A−Hilbert module, one sees that W_{2}W_{2}^{∗} induces a (bi-)continuous bijection:

W_{2}W_{2}^{∗} : Im W_{2}W_{2}^{∗} → Im W_{2}W_{2}^{∗}

still denoted W_{2}W_{2}^{∗}. We denote by S its inverse; by extending S by 0 on ker W_{2}W_{2}^{∗}, we get an element of B_{A} still denoted S. Thus, by definition of the polar decomposition W_{2}^{∗} = V \sqrt{W_{2}W_{2}^{∗}}, we have W_{2}W_{2}^{∗}S = V^{∗}V. Now we consider the universal representation of the Calkin algebra j : C_{A} → B(H_{C_{A}}). By applying j ∘ π to the previous identity we get: j(π(W_{2}W_{2}^{∗}))j(π(S)) = j(π(V^{∗}V)); so the range of the projection j(π(V^{∗}V)) is contained in the range of the projection j(π(W_{2}W_{2}^{∗})). Moreover from the identity W_{2}W_{2}^{∗} = V^{∗}V W_{2}W_{2}^{∗} = W_{2}W_{2}^{∗}V^{∗}V, we get that the range of j(π(W_{2}W_{2}^{∗})) is contained in the one of j(π(V^{∗}V)). Since j is injective, we see that π(V^{∗}V) = π(W_{2}W_{2}^{∗}) which proves the Lemma.

Proposition 2. Let p be a very full projection of C_{A} such that 1 − p is also very full. Then there exists a projection p′′ of C_{A} such that for any k, l ∈ ℕ^{∗}, one can find an A−linear bicontinuous isomorphism:

χ_{k,l} : l^{2}(A) → ( Π_{i=1}^{k}l^{2}(A) ) ⊕ l^{2}(A) ⊕ ( Π_{j=1}^{l}l^{2}(A) ) = H_{k,l}

such that

p = χ_{k,l}^{−1}(1_{k} ⊕ p′′ ⊕ 0_{l})χ_{k,l} (7)

where 1_{k} ⊕ p′′ ⊕ 0_{l} denotes the obvious diagonal-by-blocks element of the Calkin algebra C_{A}(k, 1, l) associated with H_{k,l} and u → χ_{k,l}^{−1}uχ_{k,l} induces an isomorphism between C_{A} and C_{A}(k, 1, l).

Proof. We shall treat (only) the case k = l = 1 from which the general case follows immediately.

By considering a decomposition l^{2}(A) ≃ l^{2}(A) ⊕ l^{2}(A), we choose and fix a projection Π_{1} ∈ B_{A} such that ker Π_{1} and Im Π_{1} are both isomorphic to l^{2}(A). According to Proposition 1, we fix an isometry W ∈ B_{A} such that π(W)π(W^{∗}) ≤ p. We set Ŵ = W Π_{1}, so Ŵ^{∗} = Π_{1}W^{∗}. Since W^{∗}W = Id we get:

Ŵ^{∗}Ŵ = Π_{1}, ŴŴ^{∗} = W Π_{1}W^{∗}, W W^{∗}ŴŴ^{∗} = ŴŴ^{∗}W W^{∗} = ŴŴ^{∗}

From this, we deduce that ŴŴ^{∗} ≤ W W^{∗}. So W W^{∗} − ŴŴ^{∗} = W (Id −Π_{1})W^{∗} is a projection,

Im W W^{∗} = Im ŴŴ^{∗} ⊕^{⊥} Im(W W^{∗} − ŴŴ^{∗})

and we get:

l^{2}(A) = Im ŴŴ^{∗} ⊕^{⊥} Im(W W^{∗} − ŴŴ^{∗}) ⊕^{⊥} ker W W^{∗} (8)

Since the map:

Im Ŵ^{∗}Ŵ = Im Π_{1} ≃ l^{2}(A) → Im ŴŴ^{∗}
y → Ŵ(y)

induces an isomorphism, one sees that

Im ŴŴ^{∗} ≃ l^{2}(A). (9)

Moreover, since W W^{∗} −Ŵ^{∗}Ŵ = W (Id −Π_{1})(W (Id −Π_{1}))^{∗}, one proves in the same way that Im(W W^{∗}−Ŵ^{∗}Ŵ) ≃ l^{2}(A). Thanks to Kasparov’s Stabilization Theorem, we then have:

Im(W W^{∗} − Ŵ^{∗}Ŵ) ⊕^{⊥} ker W W^{∗} ≃ l^{2}(A). (10)

Now we go back to the proof of the Proposition. Since we have pπ(W W^{∗}) = π(W W^{∗})p = π(W W^{∗}) (ie p ≥ π(W W^{∗})) and π(W W^{∗}) ≥ π(ŴŴ^{∗}), we see that p ≥ π(ŴŴ^{∗}) and p ≥ π((W W^{∗} − ŴŴ^{∗}). Using the decomposition (8), we can identify B_{A} as a set of (3, 3)− matrices; accordingly, we then see that p(∈ C_{A}) is of the following diagonal form by blocks:

p = 1 ⊕ 1 ⊕ q_{1} (11)

We set 1 ⊕ q_{1} = p′, p′ is a projection of the Calkin algebra associated with the left hand-side of (10). Since 1 − p is very full, 1 − p′ is also very full. Now we apply to 1 − p′ and to Im(W W^{∗} − Ŵ^{∗}Ŵ) ⊕^{⊥} ker W W^{∗} the analogue of decomposition (8) Repeating for 1 − p′ the arguments that have been used for p, i.e. (9), (10), (11), we obtain the following decomposition:

Im(W W^{∗} − Ŵ^{∗}Ŵ) ⊕^{⊥} ker W W^{∗} ≃ l^{2}(A) ⊕ l^{2}(A) (12)

With respect to this decomposition we can write 1−p′ = q_{2} ⊕1 (diagonal by blocks) where q_{2} is a projection in C_{A}. So, from (8) and (12), we get a decomposition l^{2}(A) ≃ l^{2}(A) ⊕ l^{2}(A) ⊕ l^{2}(A) such that:

p = 1 ⊕ (1 − q_{2}) ⊕ 0.

One then gets the Proposition by setting p′′ = 1 − q_{2}.

End of the proof of Theorem 1.

We use the projection p′′ ∈ C_{A} associated to p as in Proposition 2.

Lemma 2. The K−theory class [1] of 1 in K_{0}(C_{A}) is zero.

Proof of the Lemma. We recall that the index map (4) δ is an isomorphism and is also the connecting map associated with the six terms exact sequence in K−Theory. Since the projection 1 in the Calkin algebra C_{A} is the image of the projection defined by Id in B_{A}, we have δ([1]) = 0. Since δ is an isomorphism, the Lemma is proved.

Since [p] is (by assumption) zero, the previous Lemma then shows that [p′′] = 0. Thus there exists k, l′ ∈ ℕ^{∗} and a unitary element u ∈ U(M_{k+1+l′}(C_{A})) such that:

u(1_{k} ⊕ 0_{1} ⊕ 0_{l′})u^{−1} = 1_{k} ⊕ p′′ ⊕ 0_{l′}.

We set l = k + 2l′ + 1 and z = diag(u, u^{−1}). We then have:

z(1_{k} ⊕ 0_{1} ⊕ 0_{l})z^{−1} = 1_{k} ⊕ p′′ ⊕ 0_{l}.

Since z belongs to the connected component of 1 in U(M_{k+1+l}(C_{A})) (see [2]) , we can find Z in U(M_{k+1+l}(B_{A})) such that π(Z) = z. Now we apply Proposition 2 with k and l = k + 2l′ + 1 and we set:

P = χ_{k,l}^{−1}Z(Id_{k} ⊕0_{1} ⊕ 0_{l})Z^{−1}χ_{k,l} ∈ B_{A}. (13)

We thus get a projection P of B_{A}; using the two equalities (7), (13) and the fact that π(Z) = z, one sees that that π(P) = p which proves Theorem 1.

2.2. Spectral cuts and very full projections.

We now consider a finitely generated group Γ ; we shall apply the above results in a particular case to be described below, where the C^{∗}-algebra A will be equal to C_{r}^{∗}(Γ), the reduced C^{∗}−algebra associated to the group Γ.

It should be remarked, however, that all arguments leading to the main result in this section and to the existence-theorem in the next section, can be easily generalized to any Dirac-type operator acting on the sections of a A-vector bundle in the sense of Mishchenko-Fomenko, with A any unital C^{∗}-algebra.

We shall briefly denote by Λ the reduced C^{∗}-algebra C_{r}^{∗}(Γ).

We next consider a smooth compact riemannian manifold M of dimension 2m+1, a continuous map f : M → BΓ and a complex hermitian clifford module E → M endowed with a unitary clifford connection. Let M̃ → M be the Γ−normal cover of M associated with f : M → BΓ. Let V_{f} = Λ ×_{Γ} M̃ → M be the Λ−flat bundle associated to these data. Then we denote by D the Λ−Dirac type operator acting on C^{∞}(M, E ⊗_{ℂ} V_{f}), the set of smooth sections of the bundle E ⊗_{ℂ} V_{f} → M. In fact D defines a self-adjoint unbounded regular operator acting on the Λ−Hilbert module L_{Λ}^{2}(M, E⊗_{ℂ}V_{f}), the completion of C^{∞}(M, E⊗_{ℂ}V_{f}) with respect to the Λ−hermitian scalar product < ; > associated to the above data. Of course, L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}) is isomorphic to l^{2}(Λ), and the algebra of continuous Λ−linear adjointable operators BL_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}) is isomorphic to B_{Λ}. We recall that for any F ∈ C^{0}([−∞, +∞], ℂ), F (D) is well defined as an element of BL_{Λ}^{2}(M, E⊗_{ℂ}V_{f}). Moreover if KL_{Λ}^{2}(M, E⊗_{ℂ}V_{f}) denotes the ideal of adjointable Λ−compact operators then the following Calkin algebra:

C_{Λ}(E) = \frac{BL_{Λ}^{2}(M, E ⊗_{ℂ} V_{f})}{KL_{Λ}^{2}(M, E ⊗_{ℂ} V_{f})}

is isomorphic to C_{Λ}.

Now we recall the following:

Definition 2. A smooth map χ ∈ C^{∞}(ℝ, [0, 1]) is said to be a spectral cut if there exists two reals a < b such that for any t < a, χ(t) = 0 and for any t > b, χ(t) = 1.

Theorem 2. Let χ be any spectral cut,then χ(D) and Id −χ(D) define two very full projections of the Calkin algebra C_{Λ}(E).

Proof. We will treat only the case of χ(D), the case of Id −χ(D) being similar. We shall need the following

Technical Lemma. Let χ be a spectral cut. There exists an orthonormal sequence (e_{k})_{k∈ℕ} of elements of L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}) such that ∀ k, l ∈ ℕ, < e_{k}; e_{l} >= δ_{k,l}1_{Γ} and χ(D)(e_{k}) = e_{k}.

Granting this result, we shall now prove Theorem 2. We denote by F ⊂ L_{Λ}^{2}(M, E⊗_{ℂ} V_{f}) the closure of ⊕_{k∈ℕ}^{⊥}Λe_{k}. Thus, F is Λ−Hilbert module contained in the range of χ(D). We denote by p_{F} = ∑_{k∈ℕ} < .; e_{k} > e_{k} the orthogonal projection onto F, p_{F} belongs to B_{Λ}L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}). We fix a Λ−unitary isomorphism θ : L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}) → F and denote by j the canonical injection j : F → L_{Λ}^{2}(M, E⊗_{ℂ}V_{f}). Then j∘θ defines an element of B_{Λ}L_{Λ}^{2}(M, E ⊗_{ℂ}V_{f}) whose adjoint is given by (j ∘θ)^{∗} = θ^{∗}∘p_{F}. Then one checks easily that:

θ^{∗} ∘ p_{F} ∘ j ∘ θ = Id_{L_{Λ}^{2}(M,E⊗_{ℂ}V_{f})}, j ∘ θ ∘ θ^{∗} ∘ p_{F} = p_{F} .

But, according to the technical Lemma we have

χ(D) ∘ p_{F} = p_{F} = (χ(D) ∘ p_{F})^{∗} = p_{F} ∘ χ(D),

therefore we deduce that, in the Calkin algebra, the projection χ(D) contains the very full projection p_{F}. Theorem 2 is proved.

2.3. Proof of the technical Lemma.

This section can be skipped at first reading. Recall that the principal symbol of D (acting on sections of E) is σ(D)(x; ξ) = \sqrt{−1}c(ξ) where c(ξ) denotes the Clifford multiplication by ξ ∈ T_{x}^{∗}M; thus the principal symbol of D is σ(D)(x; ξ) = \sqrt{−1}c(ξ) ⊗_{ℂ} Id_{V_{f}}. Let x ∈ M and ξ ∈ S_{x}^{∗}M (the unit cotangent sphere) then (\sqrt{−1}c(ξ))^{2} = Id and:

(14) E_{x} = ker(\sqrt{−1}c(ξ) − Id) ⊕^{⊥} ker(\sqrt{−1}c(ξ) + Id).

In other words, the two bundles ker(\sqrt{−1}c(ξ) ± Id) → S_{x}^{∗}M define a direct orthogonal sum decomposition of the trivial pulled back bundle π^{∗}(E_{x}) → S_{x}^{∗}M where π : S_{x}^{∗}M → {x} denotes the canonical projection. We start with an elementary result.

Lemma 3. Let ξ ∈ S_{x}^{∗}M, we then have: 1]. For any spectral cut χ, the principal symbol σ(χ(D))(x; ξ) is the orthogonal projection of π^{∗}(E_{x}) onto ker(\sqrt{−1}c(ξ) − Id). 2].

dim ker(\sqrt{−1}c(ξ) − Id) = dim ker(\sqrt{−1}c(ξ) + Id) =

\frac{1}{2}dim E_{x} ≥ 2^{m−1}.

Proof. 1]. We denote by |D|^{−1} the operator defined as the inverse of D on ImD ∈ L^{2}(M, E) and by zero on ker D. The identity (14) shows that the principal symbol σ(Id +\frac{D}{|D|})(x, ξ) is the orthogonal projection of π^{∗}(E_{x}) onto ker(\sqrt{−1}c(ξ) − Id). Since χ(D)−(Id +\frac{D}{|D|}) is a smoothing operator, we see that χ(D) and Id +\frac{D}{|D|} have the same principal symbol, 1] is thus proved.

2]. This result is implicitly proved in the literature; we give the proof for the sake of completness. By writing the clifford module E_{x} as a direct sum of irreducible clifford modules, one can reduce oneself to the case where E_{x} itself is irreducible. Recall then that the Clifford algebra Cl(ℂ^{2m+1}) is isomorphic to the algebra M_{2^{m}}(ℂ) ⊕ M_{2^{m}}(ℂ) so that Cl(ℂ^{2m+1}) admits to inequivalent complex irreducible modules S^{+}, S^{−} of the same dimension 2^{m}, thus E_{x} is one of the S^{±} and dim E_{x} = 2^{m}. Moreover, there exists an algebra isomorphism θ:

θ : Cl(ℂ^{2m+2}) → End_{ℂ}(S^{+} ⊕ S^{−}).

As usual, the Cl(ℂ^{2m+1}) is identified to the even part Cl^{0}(ℂ^{2m+2}) of the clifford algebra Cl(ℂ^{2m+2}). by the isomorphism induced by the map ξ ∈ ℂ^{2m+1} → e_{2m+2}·ξ where e_{2m+2} is the last vector of the canonical basis of ℝ^{2m+2}. For any ξ ∈ ℂ^{2m+1}, we set c(ξ) = θ(e_{2m+2} · ξ), the operator c(ξ) sends S^{±} into itself and its restriction to E_{x} (identified to one of the S^{±}) coincides with the clifford multiplication ( also denoted c(ξ) above ) induced by the principal symbol σ(D)(x; ξ) of D. So we have to prove that for any ξ ∈ S^{2m} = {η ∈ ℝ^{2m+1}/ ||η|| = 1}:

dim ker(\sqrt{−1}c(ξ) − Id)_{|S^{±}} = dim ker(\sqrt{−1}c(ξ) + Id)_{|S^{±}}.

Since m ≥ 1 ( and thus 2m + 2 > 2 ), Definition 3.20 and Proposition 3.21 of [3] show that:

Tr\sqrt{−1}c(ξ)_{|S^{+}} = Tr\sqrt{−1}c(ξ)_{|S^{−}} (15)

Now we consider the Chirality operator β = (\sqrt{−1}^{m+1})e_{1} … e_{2m+1} ; in this formula (e_{1}, …, e_{2m+1}) is the canonical oriented basis of ℝ^{2m+1}. Let ξ ∈ S^{2m}, recall (see [3] page 109-110) that β · e_{2m+2} · ξ = −e_{2m+2} · ξ · β and θ(β)c(ξ) = −c(ξ)θ(β).

Since θ(β) sends S^{±} onto S^{∓} we then get:

(16) ker(\sqrt{−1}c(ξ) ± Id)_{|S^{±}} ≃ ker(\sqrt{−1}c(ξ) ∓ Id)_{|S^{∓}}.

Moreover, since (\sqrt{−1}c(ξ))^{2} = Id we have

(17) Tr\sqrt{−1}c(ξ)_{|S^{±}} = dim ker(\sqrt{−1}c(ξ) − Id)_{|S^{±}} − dim ker(\sqrt{−1}c(ξ) + Id)_{|S^{±}}

Using the three equations (15), (16) and (17) we get

dim ker(\sqrt{−1}c(ξ) − Id)_{|S^{±}} = dim ker(\sqrt{−1}c(ξ) + Id)_{|S^{±}}.

which proves the Lemma.

Lemma 4. Let χ be a spectral cut. There exists a strictly increasing sequence of real numbers (a_{k})_{k∈ℕ} such that χ is equal to 1 on ]a_{0} − 1, +∞[, lim a_{k} = +∞ and for any k ∈ ℕ, one can find ϕ_{k} ∈ C_{comp}^{∞}(]a_{k}, a_{k+1}[, ℝ) and u_{k} ∈ L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}) such that < ϕ_{k}(D)(u_{k}); ϕ_{k}(D)(u_{k}) > is invertible in Λ.

Proof. We fix a spectral cut χ_{1} whose support is contained in [a_{0}+1, +∞[, thus χ is equal to 1 on an open neighbourhood of the support of χ_{1}. We are going to show the existence of v ∈ L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}) such that < χ_{1}(D)(v); χ_{1}(D)(v) > is invertible in Λ. Then one can find ψ ∈ C_{comp}^{∞}(ℝ, [0, 1]) (with ψ ≡ 1 on a big enough interval) such that < ψ(D)χ_{1}(D)(v); ψ(D)χ_{1}(D)(v) > is invertible in Λ. This will imply the Lemma in the following way: we fix a_{1} > a_{0} such that the support of ϕ_{0} = ψχ_{1} is contained in ]a_{0}, a_{1}[, then, to construct ϕ_{1}, we proceed as above, replacing a_{0} by a_{1}. Thus inductively, we construct the a_{k} and the ϕ_{k} satisfying the requirements of the Lemma.

Let U be a (small) non empty open subset of M over which the bundles E and V_{f} are trivial and for which there exists a diffeomorphism Ψ : U → B(0, r_{0}) onto an open ball B(0, r_{0}) of ℝ^{n} where n = 2m + 1 is the dimension of M. For each r ∈]0, r_{0}], we shall set U_{r} = Ψ^{−1}(B(0, r)) ⊂ U. Until the end of the proof of Lemma 4 we identify each U_{r} (r ∈]0, r_{0}]) with the open ball B(0, r) of ℝ^{n} and E_{|U_{r}} with U_{r} × ℂ^{N}. Let D denote the Dirac operator acting on the sections of E. Let a ∈ C^{∞}(M, [0, 1]) with support contained in a small chart and such that a ≡ 1 on a neighbourhood of \overline{U}. Since the principal symbol σ(aχ_{1}^{2}(D))(x; ξ) of aχ_{1}^{2}(D) is equal to σ(aχ_{1}^{2}(D))(x; ξ) ⊗ Id, we can find a Λ−pseudo-differential operator R of order −1 acting on the sections of E ⊗_{ℂ} V_{f} such that for any u ∈ C_{c}^{∞}(U; E) one has:

aχ_{1}^{2}(D)(u1_{Γ}) = aχ_{1}^{2}(D)(u)1_{Γ} + R(u1_{Γ}).

We then observe that: < aχ_{1}^{2}(D)(u1_{Γ}); u1_{Γ} > is equal to

< χ_{1}(D)(u1_{Γ}); χ_{1}(D)(u1_{Γ}) >= ||χ_{1}(D)(u)||_{L^{2}}^{2}1_{Γ}+ < R(u1_{Γ}); u1_{Γ} > (18)

where 1_{Γ} denotes the neutral element of Γ, so that u1_{Γ} is indeed a section of E⊗_{ℂ}V_{f}. The idea of the rest of the proof of Lemma 4 consists in finding a u ∈ C_{c}^{∞}(U; E) such that ||χ_{1}(D)(u)||_{L^{2}}^{2} = 1 and || < R(u1_{Γ}); u1_{Γ} > ||_{Λ} < 1, where || · ||_{Λ} is the C^{∗}-norm of Λ.

For any (x; ξ) ∈ T^{∗}U \ 0 we shall denote by p(x; ξ) the projection of E_{x} onto ker(\sqrt{−1}c(\frac{ξ}{|ξ|_{x}})−Id) along ker(\sqrt{−1}c(\frac{ξ}{|ξ|_{x}})+Id). Let r ∈]0, r_{0}/4] and α ∈ C_{c}^{∞}(U; [0, 1]) which is equal to 1 on U_{2r}; the choice of r will be made small enough according to the next sublemma. Then there exists a pseudo-differential operator T of order −1 such that for any u ∈ C_{c}^{∞}(U_{r}; E) one has:

χ_{1}(D)(u)(x) = α(x)\frac{1}{(2π)^{n}} ∫_{|ξ|≥1} exp(\sqrt{−1}x.ξ) p(x; ξ)û(ξ)dξ + T (u)(x). (19)

We fix a point (0; ξ_{0}) ∈ S^{∗}U_{r} (where 0 is the origin of ℝ^{n}) and a small contractible open neighbourhood C(ξ_{0}) of ξ_{0} in S_{0}^{∗}U_{r}. We still denote by C(ξ_{0}) the open conic neighbourhood of ξ_{0} in T_{0}^{∗}U_{r} \ 0 induced from C(ξ_{0}) by ℝ^{∗+}−homogeneity. Since the bundle defined by the range of p(0; ξ) is trivial over C(ξ_{0}), we may find v ∈ S(ℝ^{n}; ℂ^{N}) such that ∫_{ℝ^{n}} |v̂(ξ)|^{2}dξ = (2π)^{n} and

supp v̂ ⊂ C(ξ_{0}) ∩ {ξ ∈ ℝ^{n}/ |ξ| ≥ 1/2}
∀ξ ∈ C(ξ_{0}), p(0; ξ)(v̂(ξ)) = v̂(ξ).

Let β ∈ C_{c}^{∞}(U_{r}; [0, 1]) be such that β ≡ 1 on an open neigborhood of 0 ∈ ℝ^{n}. We then set for any real R ≥ 2, u_{R}(x) = β(x)R^{\frac{n}{2}} v(Rx). Lemma 4 is an easy consequence of the following:

Sublemma 1. If r > 0 (and thus also supp α) is small enough then one has:

1) lim_{R→+∞} ||u_{R}||_{L^{2}} = 1 and lim_{R→+∞} ||u_{R}||_{H^{−1}} = 0.

2) \underline{lim}_{R→+∞}||χ_{1}(D)(u_{R})||_{L^{2}} > 2/3.

Proof of Lemma 4 admitting Sublemma 1.

With the notations of Equation (18), let C be the operator norm of R ∈ B_{Λ}(H_{Λ}^{−1}(M, E ⊗ V_{f}); L_{Λ}^{2}(M, E ⊗ V_{f})). According to Lemma 1 we choose r > 0 and R > 2 such that, setting u = \frac{u_{R}}{||χ_{1}(D)(u_{R})||_{L^{2}}} , we have ||u||_{L^{2}} ≤ 3 and ||u||_{H^{−1}} ≤ \frac{1}{10C+10}. Then, using the Cauchy-Schwartz inequality one has immediately:

|| < R(u1_{Γ}); u1_{Γ} > ||_{Λ} < 1

where || ||_{Λ} denotes the C^{∗}−norm of Λ. Next, using Equation (18), one checks easily that < χ_{1}(D)(u1_{Γ}); χ_{1}(D)(u1_{Γ}) > is invertible in Λ which proves Lemma 4.

Proof of Sublemma 1. 1) is easy and left to the reader. Let us sketch the proof of 2). According to 1) and Equation (19) we have to consider the following term:

α(x)\frac{1}{(2π)^{n}} ∫_{|ξ|≥1} exp(\sqrt{−1}x.ξ) p(x; ξ)û_{R}(ξ)dξ. (20)

Then, by writing p(x; ξ) − p(0; ξ) = ∑_{j=1}^{n} x_{j}g_{j}(x; ξ), we see that at the expense of shrinking r (and thus supp α), the L^{2}−operator norm of the pseudo-differential operator ∑_{j=1}^{n} x_{j}g_{j}(x; D_{x}) will be small so that we may replace the term (20) by

α(x)\frac{1}{(2π)^{n}} ∫_{|ξ|≥1} exp(\sqrt{−1}x.ξ) p(0; ξ)û_{R}(ξ)dξ. (21)

Now recall that 1 − α(x) is identically zero on a neighbourhood of supp β. Then, since the operator with the following Schwartz kernel:

(1 − α(x))\frac{1}{(2π)^{n}} ∫_{|ξ|≥1} ∫_{ℝ^{n}} exp(\sqrt{−1}x.ξ) p(0; ξ)β(y)dξdy

is smoothing we may replace (thanks to 1)) the term (20) by

v_{R}(x) = \frac{1}{(2π)^{n}} ∫_{|ξ|≥1} exp(\sqrt{−1}x.ξ) p(0; ξ)û_{R}(ξ)dξ.

But, using standard analysis and all the properties of the function v, one checks that lim_{R→+∞} ||v_{R}||_{L^{2}(ℝ^{n})} = 1, which proves Sublemma 1.

Recall that our goal in this section is to provide a proof of the following Technical Lemma: Let χ be a spectral cut. There exists an orthonormal sequence (e_{k})_{k∈ℕ} of elements of L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}) such that ∀ k, l ∈ ℕ, < e_{k}; e_{l} >= δ_{k,l}1_{Γ} and χ(D)(e_{k}) = e_{k}. However, by setting for each k ∈ ℕ (and with the notation of Lemma 4):

e_{k} = (< ϕ_{k}(D)(u_{k}), ϕ_{k}(D)(u_{k}) >)^{−1/2} ϕ_{k}(D)(u_{k})

one deduces immediately this result from Lemma 4.

2.4. Existence of spectral sections in the odd dimensional case.

We begin by recalling the definition of noncommutative spectral section. This definition is motivated by the family (i.e. commutative) case treated in Melrose-Piazza [27]; the definition of noncommutative spectral section was given in the unpublished paper [32] and recalled in [33] (which is a published paper).

Definition 3. (F. Wu) A projection P ∈ BL_{Λ}^{2}(M; E ⊗_{ℂ} V_{f}) is said to be a noncommutative spectral section for D if there exist two spectral cuts χ_{1}, χ_{2} such that χ_{2} ≡ 1 on a neighbourhood of the support of χ_{1}, and Im χ_{1}(D) ⊂ Im P ⊂ Im χ_{2}(D).

We recall that D defines an index class Ind(D) ∈ K_{1}(Λ); one way to define this class is to apply the isomorphism (4) to the projection, in the Calkin algebra C_{Λ}(E), defined by \frac{1}{2}(F + 1), with F = (1 + D^{2})^{− \frac{1}{2}} D. The following Theorem is stated by Wu in [32] [33].

Theorem 3. There exists a spectral section for D if and only if the index class Ind(D) vanishes in K_{1}(Λ).

Remark. Although we state the theorem for the particular C^{∗}-algebra Λ = C_{r}^{∗}(Γ), it should be clear that the proof easily extends to any Dirac-type operator acting on the sections of a A-bundle, as in the work of Mishchenko-Fomenko, with A any unital C^{∗}-algebra. Proof. First we remark that for any smooth spectral cut χ, we have Ind(D) = δ([χ(D)]); indeed \frac{1}{2}(F + 1) = χ(D) in the Calkin algebra.

We start by proving that if a spectral section P exists, then δ([χ(D)] = 0. Let us fix a spectral cut χ. By the very definition of spectral section, P and χ(D) induce the same element in the Calkin algebra C_{Λ}(E). So the K−theory class [χ(D)] ∈ K_{0}(C_{Λ}(E)) is the image of [P] ∈ K_{0}(B_{Λ}L_{Λ}^{2}(M; E⊗_{ℂ}V_{f})) by the map induced by the projection π: B_{Λ}L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f}) → C_{Λ}(E). Since K_{0}(B_{Λ}L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f})) = {0}, we see immediately that [χ(D)] = 0 and thus that δ([χ(D)] = 0.

We shall now prove the converse. First of all we state the most important consequence of our results in the preceeding 3 sections. Let us fix a spectral cut χ. From Theorems 1 and 2 we infer that there exists a projection Q in B_{Λ}L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f}) such that Q − χ(D) is a Λ−compact operator. As pointed out to us by Michel Hilsum, the existence of such a projection Q was claimed without explanation in [32]. Equipped with this fundamental result, we can now finish the proof. All the arguments we give from here until the end of the this section are due to F. Wu. First we have the the following Lemma, which is a noncommutative generalization of Lemma 1 of [27].

Lemma 5. (F. Wu) Let E be a Λ−Hilbert module isomorphic to l^{2}(Λ). Let Q = Q^{2} = Q^{∗} ∈ B_{Λ}(E) be a projection and J ∈ B_{Λ}(E) such that ||Q − J || ≤ \frac{1}{2}. Then there is a function f, holomorphic in a neighborhood of the spectrum of |QJ^{∗}|, such that P = f(J QJ^{∗}) ∈ B_{Λ}(E) is the orthogonal projection onto J Q(E). Furthermore, if Q − J ∈ K_{Λ}(E), then P − Q ∈ K_{Λ}(E) too.

Proof. For any x ∈ Q(E) one has x = Q(x) = J (x) + (Q − J)(x). Hence we have:

∀x ∈ Q(E), ||J (x)|| ≥ \frac{1}{2}||x||.

From this inequality, one deduces easily that J Q has closed range. Hence (J Q)^{∗} = QJ^{∗} has also closed range and Theorem 15.3.8 of [31] shows that:

J Q(E) = ker(QJ^{∗})^{⊥} = ker(|QJ^{∗}|)^{⊥} = |QJ^{∗}|(E)

E = ker(|QJ^{∗}|) ⊕ J Q(E) = ker(|QJ^{∗}|^{2}) ⊕ J Q(E) = ker(J QJ^{∗}) ⊕ J Q(E).

Therefore 0 is an isolated point in the spectrum of |QJ^{∗}| and hence of |QJ^{∗}|^{2} = J QJ^{∗}. Let f(λ) = 1 for λ in a neighborhood of Spec(|QJ^{∗}|^{2}) \ {0} and f(λ) = 0 for λ near 0. Then f is holomorphic in a neighborhood of Spec(|QJ^{∗}|^{2}) and:

P = f(|QJ^{∗}|^{2}) = f(J QJ^{∗})

is the orthogonal projection onto J Q(E).

Now we show that if in addition Q − J ∈ K_{Λ}(E) then P − Q ∈ K_{Λ}(E). To this end, we first observe that, since P(E) = J Q(E), we have P(J Q) = J Q. Since π(J Q) = π(Q) in the Calkin algebra C_{Λ}(E), this implies that:

π(P)π(Q) = π(P)π(J Q) = π(PJ Q) = π(J Q) = π(Q) ∈ C_{Λ}(E).

Taking adjoints on both sides, noticing that both π(P) and π(Q) are self-adjoint elements in C_{Λ}(E), we get:

π(Q) = π(Q)π(P). (22)

On the otherhand, since J : Q(E) → P(E) is an isomorphism, J^{−1}P ∈ B_{Λ}(E) is well-defined, satisfying J (J^{−1}P) = P. Passing to C_{Λ}(E) and noticing that π(Q)π(J) = π(J)π(Q), we obtain π(Q)π(P) = π(P). From equation (22) we get π(P) = π(Q) in C_{Λ}(E) which proves the Lemma.

Let φ ∈ C_{c}^{∞}(ℝ; [0, 1]) such that φ(t) = 1 for t ∈ [−1, 1] and φ(t) = 0 for |t| ≥ 2. We set φ_{N}(t) = φ(t/N) for N ∈ ℕ^{∗}. One checks easily that as N → +∞:

φ_{N}(D)(χ(D) − Q) → χ(D) − Q

in the norm of K_{Λ}(E). Thus there is N_{0} such that:

||(Id −Q) − J || ≤ \frac{1}{2},

where

J = Id −χ(D) − φ_{N_{0}}(D)(χ(D) − Q).

Applying Lemma 5 to Id −Q instead of Q we see that there is a projection P_{1} = P_{1}^{∗} = P_{1}^{2} in B_{Λ}(E) such that (Id −Q) − (Id −P_{1}) =∈ K_{Λ}(E) and

(Id −P_{1})(E) = J (Id −Q)(E).

Choose another spectral cut χ_{0} such that

χ_{0}(t) = 0 on Supp(1 − χ) ∪ [−2N_{0}, 2N_{0}].

Then we have (1 − χ_{0})(1 − χ) = (1 − χ) and (1 − χ_{0})φ_{N_{0}} = φ_{N_{0}}. Thus we have:

(Id −χ_{0}(D))(Id −χ(D)) = (Id −χ(D)) , (Id −χ_{0}(D))φ_{N_{0}}(D) = φ_{N_{0}}(D).

This implies that (Id −χ_{0}(D))J = J. Hence we have:

(Id −P_{1})(L_{Λ}^{2}(M; E⊗_{ℂ}V_{f})) = Im(Id −χ_{0}(D))J(Id −Q) ⊂ (Id −χ_{0}(D))(L_{Λ}^{2}(M; E⊗_{ℂ}V_{f})).

Let χ_{1} be a spectral cut such that χ_{0}(t) = 1 on a neighborhood of Supp(χ_{1}). Then we get, for any x ∈ L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f}):

χ_{1}(D)(P_{1}(x)) = χ_{1}(D)(x) − χ_{1}(D)((Id −P_{1})(x)).

Since Im(Id −P_{1}) ⊂ (Id −χ_{0}(D))(L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f})), there is y ∈ L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f}) such that (Id −P_{1})(x) = (Id −χ_{0}(D))(y). Hence:

χ_{1}(D)((Id −P_{1})(x)) = χ_{1}(D)(Id −χ_{0}(D))(y) = 0.

Therefore χ_{1}(D)P_{1} = χ_{1}(D). Taking the adjoint shows also P_{1}χ_{1}(D) = χ_{1}(D). In particular, we have found a spectral cut χ_{1} and a projection P_{1} such that Imχ_{1}(D) ⊂ Im(P_{1}).

We now proceed to modify the projection P_{1} to a spectral section of D. Let ψ be a spectral cut such that ψ(t) = 1 on [−1, +∞[, we set ψ_{N}(t) = ψ(t/N). Since χ_{1}(D) − P_{1} is a Λ−compact operator, one checks easily that

ψ_{N}(D)(χ_{1}(D) − P_{1}) → χ_{1}(D) − P_{1}.

Thus there exists N_{0} > 0 such that ψ_{N_{0}}(t)χ_{1}(t) = χ_{1}(t) and:

||P_{1} − χ_{1}(D) + ψ_{N_{0}}(D)(χ_{1}(D) − P_{1})|| = ||P_{1} − ψ_{N_{0}}(D)P_{1}|| < \frac{1}{2}.

Applying Lemma 5, we obtain a projection P = P^{2} = P^{∗} in B_{Λ}L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f}) such that

Im(P) = ψ_{N_{0}}(D)P_{1}(L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f})).

Let χ_{2} be a smooth spectral cut such that χ_{2}(t)ψ_{N_{0}}(t) = ψ_{N_{0}}(t). Then χ_{2}(D)ψ_{N_{0}}(D)P_{1} = ψ_{N_{0}}(D)P_{1} and we have:

P(L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f})) = Im(χ_{2}(D)ψ_{N_{0}}(D)P_{1}) ⊂ Imχ_{2}(D).

On the other hand, since P_{1}χ_{1}(D) = χ_{1}(D), we have:

χ_{1}(D) = ψ_{N_{0}}(D)χ_{1}(D) = ψ_{N_{0}}(D)P_{1}χ_{1}(D).

This implies that:

Imχ_{1}(D) = Im(ψ_{N_{0}}(D)P_{1}χ_{1}(D)) ⊂ Im(ψ_{N_{0}}(D)P_{1}) = ImP.

Therefore, Imχ_{1}(D) ⊂ ImP ⊂ Imχ_{2}(D) and P is a spectral section of D. Theorem 3 is proved.

2.5. Difference classes and the noncommutative spectral flow.

We recall the notion of difference class associated to two spectral sections. The definition is already given in [32] [33] and it is an easy and natural extension to the noncommutative setting of the definition given in ([27]). We shall need the following proposition of Wu whose proof (left to the reader) uses the same technique as in the proof of Theorem 3.

Proposition 3. If there exists a spectral section for D, then for any spectral cut χ_{1}, there is a spectral cut χ_{2} such that χ_{1}χ_{2} = χ_{1}, and a spectral section R satisfying Im χ_{1}(D) ⊂ Im R ⊂ Im χ_{2}(D). Similarly, for any spectral cut χ_{2}, there is a spectral cut χ_{1} such that χ_{1}χ_{2} = χ_{1}, and a spectral section Q satisfying Im χ_{1}(D) ⊂ Im Q ⊂ Im χ_{2}(D).

This Proposition implies immediately the following:

Corollary 1. Let P_{1}, P_{2} be two spectral sections of D. Then there exist spectral sections R, Q such that for i = 1, 2

P_{i}R = RP_{i} = P_{i} and P_{i}Q = QP_{i} = Q.

Definition 4. Let P_{1} and P_{2} be two spectral sections of D. Then there exists a class [P_{1} − P_{2}] ∈ K_{0}(Λ), called the difference class, which is defined as follows: choose a spectral section Q of D satisfying P_{i}Q = Q = QP_{i} for i ∈ {1, 2}, such a spectral section Q exists by Corollary 1. Then P_{1} − Q and P_{2} − Q are projections in K_{Λ}(L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f})) and [P_{1} − P_{2}] := [P_{1} − Q] − [P_{2} − Q] is well defined as a class in K_{0}(K_{Λ}(L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f}))) = K_{0}(Λ).

Remark. The definition is well posed, independent of the choice of Q. The proof of this fact proceeds, mutatis mutandis, as in [27], page 127.

Now we recall Wu’s definition of noncommutative spectral flow; this is an extension to the noncommutative context of the notion of higher spectral flow given in [9].

Let (f_{u})_{u∈[0,1]} be a continuous family of maps f_{u} : M → BΓ. Let (g_{u})_{u∈[0,1]}, (h_{u})_{u∈[0,1]} and (∇^{E,u})_{u∈[0,1]} be continuous families of respectively riemannian metrics on M, hermitian metrics on E and Clifford unitary connections on (E, h_{u}) with respect to g_{u} for each u ∈ [0, 1]. Let (D_{u})_{u∈[0,1]} be the associated continuous family of Dirac type operators, where each D_{u} acts on L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f_{u}}). Recall that there exists an isomorphism

U : K_{1}(C^{0}([0, 1]; ℂ) ⊗ Λ ) ≃ K_{1}(Λ)

which is implemented by the evaluation map f(·) ⊗ λ → f(0)λ. Now, assume that the index δ([χ(D_{0})]) vanishes in K_{1}(Λ), then using the above isomorphism U, one gets that δ([χ((D_{u})_{u∈[0,1]})]) vanishes in K_{1}(C^{0}([0, 1]) ⊗ Λ). Summarizing: the family (D_{u})_{u∈[0,1]} admits a (total) spectral section P = (P_{u})_{u∈[0,1]}.

Definition 5. If Q_{0} (resp. Q_{1}) is a spectral section associated with D_{0} (resp. D_{1}) then the noncommutative spectral flow sf((D_{u})_{u∈[0,1]}; Q_{0}, Q_{1}) from (D_{0}, Q_{0}) to (D_{1}, Q_{1}) through (D_{u})_{u∈[0,1]} is the K_{0}(Λ)−class:

sf((D_{u})_{u∈[0,1]}; Q_{0}, Q_{1}) = [Q_{1} − P_{1}] − [Q_{0} − P_{0}] ∈ K_{0}(Λ).

This definition does not depend on the particular choice of the total spectral section P = (P)_{u∈[0,1]}. If morever the family is periodic (ie D_{1} = D_{0}) and if we take Q_{1} = Q_{0} then sf((D_{u})_{u∈[0,1]}; Q_{0}, Q_{0}) does not depend on the choice of Q_{1} = Q_{0}.

Remarks.

1. For a motivation behind Definition 5 we refer the reader to the original work of Dai-Zhang [9].

2. The definition of noncommutative spectral flow, together with some noncommutative extensions of results of Dai-Zhang, have been already published in [18]. In particular, a formula for the Chern character of the noncommutative spectral flow was proved there (Theorem 5.3); the formula involves the higher eta invariant.

3. The definition of noncommutative spectral flow can be given for any 1-parameter family of Dirac-type operators, not necessarily associated to a variation of classifying map, of the metrics and of the connection. In fact, the definition can be easily extended to the more general situation of Dirac-type operators on A-bundles, as in the work of Mishchenko-Fomenko.

2.6. Existence of spectral sections in the even dimensional case.

Now we consider a smooth closed riemannian manifold M of dimension 2m, a continuous map f : M → BΓ and a complex hermitian ℤ_{2}−graded Clifford module E = E^{+} ⊕ E^{−} → M endowed with a unitary Clifford connection. Let M̃ → M be the Γ−normal cover of M associated with f : M → BΓ. Let V_{f} = Λ ×_{Γ} M̃ → M be the Λ−flat bundle associated to these data. Then we denote by D the Λ−Dirac type operator acting on C^{∞}(M, E ⊗_{ℂ} V_{f}) associated to these data. In fact,

D = (\begin{matrix}0 & D^{−} \\ D^{+} & 0 \end{matrix})

defines an odd self-adjoint unbounded regular operator acting on the Λ−Hilbert module L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f}).

Let τ denotes the grading of L_{Λ}^{2}(M, E⊗_{ℂ}V_{f}) induced by the grading-automorphism τ_{E} defining the splitting E = E^{+} ⊕ E^{−} We clearly have Dτ = −τD.

Definition 6. A Cl(1)-spectral section for D is a spectral section P with the additional property that

Pτ + τP = τ .

Lemma 4.3 of [19] shows the following:

Theorem 4. D admits a Cl(1)-spectral section P ∈ B_{Λ}(L_{Λ}^{2}(M, E ⊗_{ℂ} V_{f})) if and only if the index of D^{+} is zero in K_{0}(Λ).

3. Manifolds with boundary, spectral sections and index classes

3.1. APS and b-index classes in the even dimensional case.

Let (M, g) be a smooth riemannian manifold of dimension n = 2m with boundary, E = E^{+} ⊕E^{−} a ℤ_{2}−graded hermitian Clifford module endowed with a unitary Clifford connection ∇. We assume that on a collar neighborhood (≃ [0, 1] × ∂M) of ∂M, g and ∇ have a product structure. We consider also a finitely generated group Γ, a classifying map f : M → BΓ and denote by Λ the reduced C^{∗}−algebra of Γ. Let M̃ → M be the associated Γ−normal cover. We consider V_{f} = Λ ×_{Γ} M̃, a flat bundle over M. These data define a ℤ_{2}−graded Λ−linear Dirac type operator D acting on C^{∞}(M, V_{f} ⊗_{ℂ} E) which, in a collar neightborhood (≃ [0, 1] × ∂M), may be written as:

D = (\begin{matrix}0 & D^{−} \\ D^{+} & 0 \end{matrix}) ,

where D^{+} = σ_{D^{+}}(dx)(\frac{∂}{∂x} + D_{0}), D_{0} is the induced boundary operator acting on the sections of (V_{f} ⊗_{ℂ} E^{+})_{|∂M} and D^{−} = σ^{−}(\frac{∂}{∂x} + σ^{+}D_{0}σ^{−}) where we have set σ^{±} = σ_{D^{±}}(dx). We now consider the Λ−Hilbert modules L_{Λ}^{2}(M, V_{f} ⊗ E) and L_{Λ}^{2}(∂M, (V_{f} ⊗ E^{±})_{|∂M}) defined by the above data. Then the operator D_{0} defines a self-adjoint regular unbounded Λ−linear operator acting on L_{Λ}^{2}(∂M, (V_{f} ⊗ E)_{|∂M}). By the cobordism invariance of the index class (see [18], Proposition 2.3) we know that Ind D_{0}^{+} is zero in K_{1}(Λ); thus Theorem 3 shows that D_{0} admits spectral section P ∈ BL_{Λ}^{2}(∂M, (V_{f} ⊗E^{+})_{|∂M}), moreover (see [18], Theorem 2.6 1]), one may assume that P is a Λ−pseudodifferential operator of order zero. Then we define an odd operator acting on L_{Λ}^{2}(∂M, (V_{f} ⊗ E)_{|∂M}) by:

B_{P} = (\begin{matrix}0 & (Id −P)σ^{−} \\ σ^{−}P & 0 \end{matrix})

Next we introduce the domain dom(D_{P}) of D associated with the global APSboundary condition defined by P :

dom(D_{P}) = {ξ ∈ H_{Λ}^{1}(M, V_{f} ⊗ E)/B_{P}(ξ_{|∂M}) = 0}

and will denote by D_{P} the restriction of D to dom(D_{P}). In a similar and obvious way one defines D_{P}^{±}. We also set:

dom(D_{P}^{2}) = {ξ ∈ dom(D_{P})/D(ξ) ∈ dom(D_{P})}.

Then Wu has shown in [33], page 174, that the heat operator H(s) = e^{−sD_{P}^{+}D_{P}^{−}} is well defined for s real > 0 as a Λ−compact operator sending L_{Λ}^{2}(M, V_{f} ⊗ E) into dom(D_{P}^{2}) and that Id +D_{P}^{2} is invertible with an inverse being Λ−compact. Thus D_{P}^{−} ∘∫_{0}^{+∞} H(s)ds is a parametrix for D_{P}^{+} and D_{P}^{+} is indeed a Λ−Fredholm operator from dom(D_{P}^{+}) to L_{Λ}^{2}(M, V_{f} ⊗ E^{−}); we will denote by Ind^{AP S}(D^{+}, P) ∈ K_{0}(Λ) the associated higher (APS) index class.

Now we are going to recall the definition of the b−higher index class (see [18]) associated to D^{+}. According to [33], [18] there exists a Λ−compact self-adjoint operator A_{P}^{0} ∈ KL_{Λ}^{2}(∂M, (V_{f} ⊗ E)_{|∂M}) such that D_{0}^{+} + A_{P}^{0} is invertible and

P = \frac{1}{2}(Id +\frac{D_{0}^{+} + A_{P}^{0}}{|D_{0}^{+} + A_{P}^{0}|}).

It is proved in [18] (Proposition 2.10) that the operator A_{P}^{0} can be chosen to be a Λ-smoothing operator.

We add to M a cylinder [0, 1]×∂M and still denote by M the extended manifold. Then we extend the metric g to a b−metric having a product structure (\frac{dx}{x})^{2} + g_{∂M}(y) on [0, 1/2] × ∂M (this amounts to add an infinite cylinder). Then we denote by ^{b}D the b − Λ-operator associated to D acting on L_{b,Λ}^{2}(M, V_{f} ⊗ E) (see Section 11 of [16]). Let ρ ∈ C_{c}^{∞}(ℝ, ℝ^{+}) be a nonnegative even smooth test function such that ∫_{ℝ} ρ(x)dx = 1. We set ρ_{ϵ}(x) = \frac{1}{ϵ}ρ(\frac{x}{ϵ}) and then consider the Fourier transform of ρ_{ϵ}:

ρ̂_{ϵ}(z) = ∫_{ℝ} e^{−itz}ρ(\frac{t}{ϵ})ϵ^{−1}dt.

Then there exists an operator A_{P} ∈ Ψ_{b,Λ}^{−∞}(M, V_{f} ⊗ E^{+}, V_{f} ⊗ E^{−}) such that the indicial family of ^{b}D^{+} + A_{P} is given by:

I(^{b}D^{+} + A_{P}, z) = D_{0} + iz + ρ̂_{ϵ}(z)A_{P}^{0}

and is invertible for any z ∈ ℝ (see [27] and Lemma 6.1 of [18]). So^{b}D^{+}+A_{P} belongs to the b − Λ−calculus and is a Λ−Fredholm operator from H_{b,Λ}^{1}(M, V_{f} ⊗ E^{+}) to L_{b,Λ}^{2}(M, V_{f} ⊗E^{−}). We shall denote by Ind_{b}(^{b}D^{+}, P) ∈ K_{0}(Λ) the associated higher index class.

Theorem 5. With the above notations, the following equality holds:

Ind^{AP S}(D^{+}, P) = Ind_{b}(^{b}D^{+}, P).

Proof. Let φ ∈ C^{∞}([0, 1]; [0, 1]), such that φ(x) = 0 for \frac{3}{4} ≤ x ≤ 1 and φ(x) = 1 for 0 ≤ x ≤ \frac{1}{2}. For each t ∈ [0, 1], one defines two perturbed Dirac type Λ−linear operators by setting: D^{+}(t) = σ^{+}(\frac{∂}{∂x} + D_{0} + tφ(x)A_{P}^{0}) and: D^{−}(t) = σ^{−}( \frac{∂}{∂x} + σ^{+}(D_{0} + tφ(x)A_{P}^{0})σ^{−}). We then set:

D(t) = (\begin{matrix}0 & D^{−}(t) \\ D^{+}(t) & 0 \end{matrix}) .

Wu has constructed (see [33]) for each s > 0 a heat operator e^{−sD^{2}(1)} which sends L_{Λ}^{2}(M, V_{f} ⊗ E) into dom(D_{P}^{2}). Observe that for each t ∈ [0, 1], D^{2}(1) − D^{2}(t) is a smoothing operator. Thus, using Duhamel expansion’s formula one then defines for each s > 0 the heat operator e^{−sD^{2}(t)} which sends L_{Λ}^{2}(M, V_{f} ⊗ E) into dom(D_{P}^{2}). Then,

D^{−}(t) ∫_{0}^{+∞} e^{−sD^{+}(t)D^{−}(t)}ds

defines a parametrix for D^{+}(t). So the (D^{+}(t))_{t∈[0,1]} define a continuously family of Λ−Fredholm operators from H_{Λ}^{1}(M, V_{f} ⊗ E^{+}) to L_{Λ}^{2}(M, V_{f} ⊗ E^{−}). Thus for any t ∈ [0, 1], Ind^{AP S}(D(t)^{+}, P) = Ind^{AP S}(D(1)^{+}, P). Now, following closely the arguments of Section 10 of [14], one easily checks that D(1)^{+} induces a Λ−Fredholm operator, denoted ^{cyl}D(1)^{+} from H_{b,Λ}^{1}(M, V_{f} ⊗ E^{+}) to L_{b,Λ}^{2}(M, V_{f} ⊗ E^{−}) and that:

Ind(^{cyl}D(1)^{+}) = Ind_{b}(^{b}D^{+}, P) ∈ K_{0}(Λ)
Ind(^{cyl}D(1)^{+}) = Ind^{AP S}(D(1)^{+}, P).

Since we have seen that Ind^{AP S}(D(1)^{+}, P) = Ind^{AP S}(D(0)^{+}, P) the Theorem is proved.

By combining the previous Theorem and Proposition 6.1 of [19], i.e. the relative index theorem for b-index classes, one gets at once the following relative index Theorem for AP S-index classes:

Theorem 6. With the above notations, let P, P′ be two spectral sections for D_{0}. Then one has:

Ind^{AP S}(D^{+}, P) − Ind^{AP S}(D^{+}, P′) = [P′ − P] ∈ K_{0}(Λ).

3.2. APS and b-index classes in the odd dimensional case.

We shall now address the odd-dimensional case. Let (M, g) be a smooth riemannian manifold with boundary of dimension n = 2m + 1, E a hermitian Clifford module endowed with a unitary Clifford connection ∇. We assume that in a collar neighborhood of ∂M, (≃ [0, 1] × ∂M = {(x, y)}), g and ∇ have a product structure. We fix a Clifford action cl_{∂} of T^{∗}∂M on E_{0} = E_{|∂M} by setting for any η ∈ T^{∗}∂M, cl_{∂}(η) = cl(dx)cl(η) where cl denotes the Clifford action of T^{∗}M on E. We set τ = \sqrt{−1}cl(dx); since τ^{2} = Id, E_{0} = E_{0}^{+} ⊕ E_{0}^{−} is ℤ_{2}−graded with E_{0}^{±} = ker(τ − ± Id).

We consider a finitely generated group Γ, and a classifying map f : M → BΓ. These data define as before a Λ−linear Dirac type operator D acting on C^{∞}(M, V_{f} ⊗_{ℂ} E) which, in a collar neighborhood (≃ [0, 1] × ∂M), may be written as:

D = σ_{D}(dx)(\frac{∂}{∂x} + D_{0})

where D_{0} is the induced boundary ℤ_{2}−graded operator acting on the sections of (V_{f} ⊗_{ℂ} E)_{|∂M}. The operator D_{0} is a self-adjoint regular unbounded Λ−linear operator acting on L_{Λ}^{2}(∂M, (V_{f} ⊗E)_{|∂M}); we know that IndD_{0} is zero in K_{0}(Λ), so Theorem 4 shows that D_{0} admits a Cl(1)−spectral section P ∈ BL_{Λ}^{2}(∂M, (V_{f} ⊗E)_{|∂M}); by definition we have τP +Pτ = τ. One may assume as in the even case that P is a Λ−pseudodifferential operator of order zero. We fix such a Cl(1)−spectral section P in the rest of this section. We can find (proceeding as in the even-dimensional case) a self-adjoint odd Λ−compact operator A_{P}^{0} acting on L_{Λ}^{2}(∂M, (V_{f} ⊗ E)_{|∂M}) such that D_{0} + A_{P}^{0} is invertible and

P = \frac{1}{2}(Id +\frac{D_{0} + A_{P}^{0}}{|D_{0} + A_{P}^{0}|}). (23)

The fact that A_{P}^{0} be odd is implied by the condition τP + Pτ = τ, as in [28].

We denote by ^{b}D the b−operator induced by D. Proceeding as in [27] (Lemma 9) and in [18], one checks the existence of a self-adjoint operator A_{P} ∈ Ψ_{b,Λ}^{−∞}(M, V_{f} ⊗ E, V_{f} ⊗ E) with indicial family given by:

I(^{b}D + A, z) = D_{0} + iz + ρ̂_{ϵ}(z)A_{P}^{0} (24)

This indicial family is invertible for any z ∈ ℝ.

Proposition 4. The operator ^{b}D + A_{P} is self-adjoint regular on L_{b,Λ}^{2}(M, V_{f} ⊗ E).

Proof. We follow [33] page 174. Proceeding as in [20] (page 328) and as in [27] (Proposition 8) one constructs for any s > 0 a heat operator H(s) = e^{−s(^{b}D+A_{P})^{2}} such that R = ∫_{0}^{+∞} e^{−s} H(s)ds sends L_{b,Λ}^{2}(M, V_{f} ⊗ E) into H_{b,Λ}^{2}(M, V_{f} ⊗ E) and (Id +(^{b}D + A_{P})^{2}) ∘ R = Id. Thus (Id +(^{b}D + A_{P})^{2}) is surjective. Moreover, for any ξ, ξ′ ∈ H_{b,Λ}^{1}(M, V_{f} ⊗ E) one has:

< (^{b}D + A_{P})(ξ_{1}); ξ_{2} >=< ξ_{1}; (^{b}D + A_{P})(ξ_{2}) > ∈ Λ

which completes the proof of the Proposition.

Proceeding along the same lines, one proves the following:

Proposition 5. Let D_{P}^{AP S} be the operator induced by D defined on the domain dom(D_{P}^{AP S}) = {ξ ∈ H_{b,Λ}^{1}(M, V_{f} ⊗ E)/ P(ξ_{|∂M}) = 0}. Then D_{P}^{AP S} is a self-adjoint unbounded regular operator on L_{Λ}^{2}(M, V_{f} ⊗ E).

Definition 7. One defines two suspended families indexed by t ∈ [0, π] by the following two formulae: 1] (Σ(^{b}D + A_{P}))_{t} := \sqrt{Id +(^{b}D + A_{P})^{2}} cos t + \sqrt{−1}(^{b}D + A_{P}) sin t. The family Σ(^{b}D + A_{P}) = (Σ(^{b}D + A_{P}))_{t∈[0,π]} defines a (C_{c}^{0}(]0, π[) ⊗ Λ)−linear operator acting on C_{c}^{0}(]0, π[) ⊗ H_{b,Λ}^{1}(M, V_{f} ⊗ E).

2] (Σ(D_{P}^{AP S}))_{t} := \sqrt{Id +(D_{P}^{AP S})^{2}} cos t + \sqrt{−1}D_{P}^{AP S} sin t.

The easy proof of the next Proposition is left to the reader.

Proposition 6. 1] The index class Ind(Σ(^{b}D + A_{P})) is well defined in K_{0}(C_{c}^{0}(]0, π[) ⊗ Λ) ≃ K_{1}(Λ). 2] The index class Ind(Σ(D_{P}^{AP S})) is well defined in K_{0}(C_{c}^{0}(]0, π[) ⊗ Λ) ≃ K_{1}(Λ).

We can use suspension in order to extend to the odd dimensional case Definition 4. We follow Proposition 4 of [28].

Definition 8. Consider two Cl(1)−spectral sections P_{1}, P_{2} of D_{0}, let A_{P_{j}}^{0} ( j ∈ {1, 2}) be a ℤ_{2}−graded self-adjoint Λ−compact operator acting on L_{Λ}^{2}(∂M; (V_{f} ⊗ E)_{|∂M}) such that for each j ∈∈ {1, 2}, D_{0} + A_{P_{j}}^{0} is invertible and:

P_{j} = \frac{1}{2}(Id +\frac{D_{0} + A_{P_{j}}^{0}}{|D_{0} + A_{P_{j}}^{0}|}).

Then for any t ∈ [0, π],

D_{0}^{j}(t) = τ( Id +(D_{0} + A_{P_{j}}^{0})^{2} )^{1/2} cos t + (D_{0} + A_{P_{j}}^{0}) sin t

is invertible, where τ = \sqrt{−1}cl(dx) is the involution of E_{∂M}. Let P_{j}(t) denote the projection onto the positive part of D_{0}^{j}(t). According to Corollary 1, we consider a continuous family (Q(t))_{t∈[0,π]} of projections in B_{Λ}L_{Λ}^{2}(∂M; (V_{f} ⊗ E)_{|∂M}) such that for any t ∈ [0, π] and j ∈ {1, 2}, P_{j}(t)Q(t) = Q(t)P_{j}(t) = Q(t). Then

[P_{1} − P_{2}] = [ (P_{1}(t) − P_{2}(t))_{t}] = [ (P_{1}(t) − Q(t))_{t}] − [ (P_{2}(t) − Q(t))_{t}]

is a well defined class in K_{0}(C_{c}^{0}(]0, π[) ⊗ Λ) ≃ K_{1}(Λ) which does not depend on the choice of A_{P_{j}}^{0} ( j ∈ {1, 2}) and (Q(t))_{t∈[0,π]}.

Remark. In fact the previous definition of difference class [P_{1}−P_{2}] holds equally well when one replaces ∂M (resp. f : ∂M → BΓ) by a closed manifold F (resp. a map f′ : F → BΓ) which is not necessarily the boundary of a manifold.

There is a third index class that one can consider. Following [28], we are going to associate a Dirac suspension to the Λ−operator ^{b}D + A_{P}. For any θ ∈ ℝ/2πℤ, we denote by L_{θ} the line bundle over ℝ/πℤ obtained by identifying (0, v) with (π, exp(−\sqrt{−1}θ)v), v ∈ ℂ. Notice that the first circle has length 2π whereas the second circle has length π. We then denote by L the bundle of C^{0}(ℝ/πℤ)−modules over ℝ/2πℤ whose fiber over θ ∈ ℝ/2πℤ is given by L_{θ} = C^{0}(ℝ/πℤ; L_{θ}) (recall that the L_{θ} are topologically trivial). We set:

SΛ = C^{0}(ℝ/πℤ) ⊗ Λ.

Proposition 7. One can define a Dirac suspension of^{b}D+A_{P} which is a ℤ_{2}−graded SΛ−Fredholm operator, ^{b}D_{P}^{Σ}:

H_{b,SΛ}^{1}(ℝ/2πℤ × M; L ⊗_{ℂ} V_{f} ⊗ E ⊗ ℂ^{2}) → L_{b,SΛ}^{2}(ℝ/2πℤ × M; L ⊗_{ℂ} V_{f} ⊗ E ⊗ ℂ^{2})

which thus has a well defined index class Ind(^{b}D_{P}^{Σ}) ∈ K_{0}(SΛ).

Proof. We follow Section 5 of [28] and explain the modifications that are needed in the present noncommutative context. We introduce the following operator acting on C^{∞}(ℝ/2πℤ, L):

∂^{L} = −\sqrt{−1}\frac{∂}{∂θ} + \sqrt{−1}\frac{t − \frac{1}{2}}{π} .

Then, for each θ ∈ ℝ/2πℤ and n ∈ ℤ, we consider the section e_{n}(t, θ) of L_{θ} defined by e_{n}(t, θ) = e^{\sqrt{−1} nθ} if t ∈ [0, π[ and e_{n}(t, θ) = e^{\sqrt{−1}(n−1)θ} if t = π. Next, we fix ϵ > 0 and δ > 0 two small (enough) reals and a function f_{ϵ,δ} ∈ C_{c}^{∞}([0, π]×ℝ) which is equal to zero on \{0\} × ℝ and \{1\} × ℝ and equal to one on [δ, π − δ] × [\frac{−1}{ϵ}, \frac{1}{ϵ}]. Then we introduce the C^{0}(ℝ/πℤ)−linear operator R_{ϵ,δ} acting on C^{∞}(ℝ/2πℤ, L) and defined by the following Schwartz kernel:

K(R_{ϵ,δ})(t, θ, t′, θ′) = ∑_{n∈ℤ} f_{ϵ,δ}(t, n + \frac{(t − 1/2)}{π})δ_{t=t′} ⊗ (e_{n}(t, θ) ⊗ \overline{e_{n}(t, θ′)}).

Then the required Dirac suspension ^{b}D_{P}^{Σ} is defined to be:

(\begin{matrix}0 & ∂^{L} ⊗ Id + Id ⊗\sqrt{−1}D + R_{ϵ,δ} ⊗ \sqrt{−1}A \\ ∂^{L} ⊗ Id − Id ⊗\sqrt{−1}D − R_{ϵ,δ} ⊗ \sqrt{−1}A & 0 \end{matrix}) .

This operator is the sum of a Dirac b-SΛ-operator and a b-SΛ operator of order −∞; moreover its indicial family is invertible for z ∈ ℝ; it follows from [18] that the operator is SΛ-Fredholm. The Proposition is proved.

In the next Theorem we relate the three index classes introduced above; we also state the relative index theorem in the odd-dimensional case. We devote the next subsection to its proof.

Theorem 7. With the notation introduced above we have: 1] Ind(Σ(^{b}D + A_{P})) = Ind(Σ(D_{P}^{AP S})) 2] Let j : K_{0}(C_{c}^{0}(]0, π[)⊗Λ) → K_{0}(SΛ) be the map induced by the inclusion ]0, π[→ ℝ/πℤ. Then

j(Ind(Σ(^{b}D + A_{P}))) = Ind(^{b}D_{P}^{Σ}) ∈ K_{0}(SΛ) .

3] Let P_{1}, P_{2} be two Cl(1)−spectral sections for D_{0}, then:

Ind(Σ(D_{P_{1}}^{AP S})) − Ind(Σ(D_{P_{2}}^{AP S})) = [P_{2} − P_{1}] ∈ K_{1}(Λ).

We can also give the odd dimensional version of definition 5 thus introducing the noncommutative spectral flow for an odd dimensional manifold F endowed with a ℤ_{2}−graded hermitian Clifford module E.

Let (f_{u})_{u∈[0,1]} be a continuous family of maps f_{u} : F → BΓ. Let (g_{u})_{u∈[0,1]}, (h_{u})_{u∈[0,1]}, and (∇^{E,u})_{u∈[0,1]} be continuous families of respectively riemannian metrics on F, hermitian metrics on E, and Clifford unitary connections on (E, h_{u}) with respect to g_{u} for each u ∈ [0, 1]. Let (D_{u})_{u∈[0,1]} be the associated continuous family of odd Dirac type operators (as above) where each D_{u} acts on L_{Λ}^{2}(M; E ⊗_{ℂ} V_{f_{u}}). Assume that the index Ind D_{0}^{+} vanishes in K_{0}(Λ). Then Ind(D_{u}^{+})_{u∈[0,1]} vanishes in K_{0}(C^{0}([0, 1]) ⊗ Λ) and thus (D_{u})_{u∈[0,1]} admits a Cl(1)−spectral section (P_{u})_{u∈[0,1]}.

Definition 9. If Q_{0} (resp. Q_{1}) is a Cl(1)−spectral section associated with D_{0} (resp. D_{1}) then the noncommutative spectral flow sf((D_{u})_{u∈[0,1]}; Q_{0}, Q_{1}) from (D_{0}, Q_{0}) to (D_{1}, Q_{1}) through (D_{u})_{u∈[0,1]} is the K_{1}(Λ)−class:

sf((D_{u})_{u∈[0,1]}; Q_{0}, Q_{1}) = [Q_{1} − P_{1}] − [Q_{0} − P_{0}] ∈ K_{1}(Λ).

This definition does not depend on the particular choice of the spectral section (P_{u})_{u∈[0,1]}. If moreover the family is periodic (ie D_{0} = D_{1}) and if we take Q_{0} = Q_{1} then sf((D_{u})_{u∈[0,1]}; Q_{0}, Q_{0}) does not depend on the choice of Q_{0} = Q_{1}. Remark. As in the even case, the definition of noncommutative spectral flow can be given for any 1-parameter family of Dirac-type operators.

3.3. Proof of Theorem 7.

This Section can be skipped at first reading. We shall adapt to the noncommutative context the arguments in [28]. This proceeds in a rather straightforward way; in fact, the suspension-arguments do not distinguish very much between the commutative and noncommutative context.

1] Our goal is to show that Ind(Σ(^{b}D + A_{P})) = Ind(Σ(D_{P}^{AP S})). We adapt to the noncommutative setting the proof of Proposition 5 of [28]; we shall also employ arguments used in a similar Proposition proved in [14] (Theorem 10.1) Let φ ∈ C^{∞}([0, 1]; [0, 1]) such that φ(x) = 1 for 0 ≤ x ≤ 1/2 and φ(x) = 0 for 3/4 ≤ x. By considering the family of operators:

D(t) = σ_{D}(\frac{∂}{∂x} + D_{0} + tφ(x)A_{P}^{0})

and using an homotopy argument one can replace, in the definition of D_{P}^{AP S}, D by D(1). We shall denote by D_{P}^{AP S}(1) the associated self-adjoint Λ−Fredholm operator. Then let μ > 0 such that [−μ, μ] ∩ spec(D_{0} + A_{P}^{0}) = ∅. We observe that we do not change the index class if, in Definition 7, we take instead of ((Σ(D_{P}^{AP S}))_{t}) the family: (D_{P}^{AP S}(1) + s\sqrt{−1} Id)_{s∈[−μ/2,μ/2]}. Now, for any s ∈ [−μ/2, μ/2] we set:

P_{s\sqrt{−1}} = \frac{1}{2}(Id +\frac{D_{0} + A_{P}^{0} + s\sqrt{−1}τ}{|D_{0} + A_{P}^{0} + s\sqrt{−1}τ|}).

Proceeding as in [28] page 303, using the family (P_{ts\sqrt{−1}})_{t∈[0,1]} and an homotopy argument, one checks easily the following:

Lemma 6. Ind(Σ(D_{P}^{AP S})) coincides with the index class defined by the suspended family (parametrized by s ∈ [−μ/2, μ/2]):

D_{P}^{AP S}(1) + s\sqrt{−1} Id : Dom_{s} → L_{Λ}^{2}(M, V_{f} ⊗ E)

Dom_{s} = \{u ∈ H_{Λ}^{1}(M, V_{f} ⊗ E)/P_{s\sqrt{−1}}(ξ_{|∂M}) = 0\}.

Now we extend M to a manifold M_{e}, with cylindrical end and the boundary defining functions related by x = exp(x̃). Let us fix χ ∈ C^{∞}(ℝ, ℝ) such that χ(x) = 1 for x < 1 and χ(x) ∈ C^{∞}(M_{e}) has support in the collar neighborhood of the boundary. Following closely [28] page 303, we embedd Dom_{s} into H_{b,Λ}^{1}(M_{e}, V_{f} ⊗E) and obtain the decomposition:

Dom_{s} ⊕ G_{s} = H_{b,Λ}^{1}(M_{e}, V_{f} ⊗ E),

where

G_{s} = \{u ∈ H_{b,Λ}^{1}(M_{e}, V_{f} ⊗ E)/ P_{s\sqrt{−1}}(u_{|x=1}) = u_{|x=1} and ∀x > 1 u = χ(x)u_{|x=1}\}.

Moreover, D(1) induces an operator, denoted ^{cyl}D_{P} from H_{b,Λ}^{1}(M_{e}, V_{f} ⊗ E) to L_{b,Λ}^{2}(M_{e}, V_{f} ⊗ E). Then, proceeding as in [28] page 303-304, one checks easily that for s ∈] − μ, μ[, ^{cyl}D_{P} − \sqrt{−1}s Id induces an invertible operator from G_{s} onto its range denoted H_{s} and that

L_{b,Λ}^{2}(M_{e}, V_{f} ⊗ E) = L^{2}(M, V_{f} ⊗ E) ⊕ H_{s}.

From this and Lemma 6, we conclude that Ind(Σ(D_{P}^{AP S})) = Ind(Σ(^{cyl}D_{P})). Lastly, an homotopy argument (see Section 10 of [14]) shows that: Ind(Σ(^{b}D + A_{P})) = Ind(Σ(^{cyl}D_{P})) which completes the proof of 1].

2] Let j : K_{0}(C_{c}^{0}(]0, π[) ⊗ Λ) → K_{0}(SΛ) be the map induced by the inclusion ]0, π[→ ℝ/πℤ. Our goal is to prove that j(Ind(Σ(^{b}D+A_{P}))) = Ind(^{b}D_{P}^{Σ}) ∈ K_{0}(SΛ). We observe that L_{b,SΛ}^{2}(ℝ/2πℤ × M; L ⊗_{ℂ} V_{f} ⊗ E ⊗ ℂ^{2}) is isomorphic to the field of continuous Λ−Hilbert modules over ℝ/πℤ given by: ⊕_{n∈ℤ}ℂe_{n}(t, .) ⊗ L_{b,Λ}^{2}(M; V_{f} ⊗ E ⊗ ℂ^{2}). Now an easy translation in the noncommutative setting of the proof of the Proposition 6 of [28] (pages 308-309) allows to finish the proof of 2]

3] We shall now prove the relative index theorem in the odd dimensional case: Ind(Σ(D_{P_{1}}^{AP S})) − Ind(Σ(D_{P_{2}}^{AP S})) = [P_{2} − P_{1}] ∈ K_{1}(Λ). We follow the proof of Proposition 11 of [28] and explain the modifications allowing to extend it to our noncommutative setting. We fix three real numbers, a < 0, b > 1 and ϵ > 0. We consider a function ψ_{ϵ}(t, λ) ∈ C^{∞}([0, π) × ℝ) which vanishes on the complement of the open rectangle ]0, π[×]a, a + bπ[ and is equal to one on the rectangle [ϵ, π − ϵ] × [a + ϵ, a + bπ − ϵ]. Next we introduce the C^{0}(ℝ/πℤ)−linear operator R_{ϵ} acting on C^{∞}(ℝ/2πℤ, L) and defined by the following Schwartz kernel:

K(R_{ϵ}(t, θ, t′, θ′)) = ∑_{n∈ℤ} ψ_{ϵ}(t, n + \frac{(t − 1/2)}{π})δ_{t=t′} ⊗ (e_{n}(t, θ) ⊗ \overline{e_{n}(t, θ′)}).

We fix two operators A_{P_{j}}^{0} (j ∈ {1, 2}) as in Definition 8 and we associate to them two operators A_{P_{j}}^{0} as in (24). Then, we consider the Dirac suspension ^{b}D_{P_{j}}^{Σ} defined by

(\begin{matrix}0 & ∂^{L} ⊗ Id + Id ⊗\sqrt{−1}D + R_{ϵ} ⊗ \sqrt{−1}A_{P_{j}} \\ ∂^{L} ⊗ Id − Id ⊗\sqrt{−1}D − R_{ϵ} ⊗ \sqrt{−1}A_{P_{j}} & 0 \end{matrix}) .

Next we observe that L_{b,SΛ}^{2}(ℝ/2πℤ × ∂M; L ⊗_{ℂ} V_{f} ⊗ E_{|∂M}) is isomorphic to the field of continuous Λ−Hilbert modules over ℝ/πℤ given by V_{0}(t) ⊕ V (t) where:

V_{0}(t) = ℂe_{0}(t, .) ⊗ L_{Λ}^{2}(∂M; V_{f} ⊗ E_{|∂M})

V (t) = ⊕_{n∈ℤ\{0}}ℂe_{n}(t, .) ⊗ L_{Λ}^{2}(∂M; V_{f} ⊗ E_{|∂M})

where we have written V_{f} instead of (V_{f})_{|∂M}.

Now we consider the two following families of operators:

(25) (Id +(D_{0} + A_{P_{j}}^{0})^{2})^{1/2}τ cos t + (D_{0} + A_{P_{j}}^{0}) sin t

(Id +(D_{0} + ψ_{ϵ}(t, \frac{(t − 1/2)}{π})A_{P_{j}}^{0})^{2})^{1/2}τ cos t + (D_{0} + ψ_{ϵ}(t, \frac{(t − 1/2)}{π})A_{P_{j}}^{0}) sin t, (26)

where j ∈ {1, 2}. Observe that for each t ∈ (0, π], the operators of (25) and (26 ) are invertible, then by considering the projections onto their positive part, one defines the class [P_{2}^{ϵ} − P_{1}^{ϵ}] ∈ K_{1}(Λ) exactly as in Definition 8. Proceeding as in [28] (pages 316-317), one checks that:

[P_{2}^{ϵ} − P_{1}^{ϵ}] = Ind(Σ(D_{P_{1}}^{AP S})) − Ind(Σ(D_{P_{2}}^{AP S})).

Since the operators in (25) and (26 ) are invertible and coincide for at least one real t ∈]1/2, π], we have [P_{2}^{ϵ} − P_{1}^{ϵ}] = [P_{2} − P_{1}]. The Theorem is now completely proved.

4. A splitting formula for index classes

4.1. The even dimensional case.

In this section we shall establish a splitting (or gluing) formula for index classes; this result is stated without proof in [21]. Our first task consists in computing higher APS-index classes for the cylinder in the even-dimensional case.

Let N be a closed manifold of dimension 2m + 1, f : N → BΓ a continuous map and E → N be a Clifford hermitian module over N endowed with a unitary Clifford connection. Let D_{N} be the associated Λ−Dirac type operator acting on L_{Λ}^{2}(N, V_{f} ⊗ E). We fix a constant map c : [−1, 1] → BΓ. Let D be the associated ℤ_{2}−graded Dirac type operator acting on L_{Λ}^{2}([−1, 1] × N, V_{f×c} ⊗ E ⊗ ℂ^{2}), with boundary operator equal to D_{N}. We fix a spectral section P at {1} × N for the boundary operator D_{N} (of D) and consider the spectral section Id −P at {−1}×N.

Lemma 7.

Ind^{AP S}(D, Id −P, P) = 0 ∈ K_{0}(Λ).

Proof. Let A_{P}^{0} be a self-adjoint Λ−compact operator such that D_{N} + A_{P}^{0} is invertible and P is the projection onto to the positive part of D_{N} + A_{P}^{0}. We consider the operator ^{cyl}D_{P} defined by:

(\begin{matrix}0 & ^{cyl}D_{P}^{−} \\ ^{cyl}D_{P}^{+} & 0 \end{matrix})

where ^{cyl}D_{P}^{+} = cl(\frac{dx}{x−1})(\frac{∂}{∂x} + D_{N} + A_{P}^{0}) and ^{cyl}D_{P}^{−} is the adjoint of ^{cyl}D_{P}^{+}. Then, proceeding as in Section 10 of [14], one checks that ^{cyl}D_{P} is Λ−Fredholm from H_{b,Λ}^{1}([−1, 1] × N, V_{f×c} ⊗ E ⊗ ℂ^{2}) to L_{b,Λ}^{2}([−1, 1] × N, V_{f×c} ⊗ E ⊗ ℂ^{2}) and that Ind^{AP S}(D, Id −P, P) = Ind(^{cyl}D_{P}^{+}). But using Mellin transform on the cylinder one sees that ^{cyl}D_{P}^{2} is invertible, because ξ^{2} + (D_{N} + A_{P}^{0})^{2} is invertible on L^{2}(ℝ, dξ) ⊗ L_{Λ}^{2}(N, V_{f} ⊗ E). So the index class is zero, which proves the Lemma.

Lemma 8. We use the notations of the previous Lemma. Let P and Q be two spectral sections for D_{N} at {1} × N. Then:

Ind^{AP S}(D, Id −Q, P) = [Q − P] ∈ K_{0}(Λ).

Proof. Theorem 7 implies that:

Ind^{AP S}(D, Id −Q, Q) − Ind^{AP S}(D, Id −Q, P) = [Q − P].

Since Lemma 7 implies that Ind^{AP S}(D, Id −Q, Q) = 0 one gets immediately the Lemma.

Now we are going to prove the gluing formula for index classes; before stating it, we introduce suitable notations. Let (M, g) be a smooth closed riemannian manifold dimension n = 2m, E = E^{+} ⊕E^{−} a ℤ_{2}−graded hermitian Clifford module endowed with a unitary Clifford connection ∇. We consider also a finitely generated group Γ, a classifying map f : M → BΓ and denote by Λ the reduced C^{∗}−algebra of Γ. Let M̃ → M be the associated Γ−normal cover. We consider the following flat bundle over M:

V_{f} = Λ ×_{Γ} M̃.

These data define a ℤ_{2}−graded Λ−linear Dirac type operator D acting on L_{Λ}^{2}(M, V_{f}⊗ E). This operator has a well defined index class Ind(D^{+}) ∈ K_{0}(Λ). Let F be a closed cutting hypersurface of M such that M = M_{+} ∪ M_{−} where M_{±} are two manifolds whose (common) boundary is F. We assume that all these data have a product structure near F. Let P and Q be two spectral sections for the boundary operator of the operator induced by the restriction D_{|M_{±}} of D to M_{±}.

Theorem 8. Ind (D^{+}) = Ind^{AP S} (D_{|M_{+}}^{+}, P) + Ind^{AP S}(D_{|M_{−}}^{+}, Id −Q) + [P − Q].

Proof. One just has to adapt to the noncommutative setting (and the even dimensional case) the proof of Theorem 3.1 of [8]. Thus following an idea of Bunke [5], one considers a unitary map W = a + b + c − d from

L_{Λ}^{2}(M, V_{f} ⊗ E) ⊕ L_{Λ}^{2}([−1, 1] × F, (V_{f} ⊗ E)_{[−1,1]×F} )

to

L_{Λ}^{2}(M_{+}∪[0, 1]×F, (V_{f} ⊗E)_{M_{+}∪[0,1]×F} )⊕L_{Λ}^{2}(M_{−}∪[−1, 0]×F, (V_{f} ⊗E)_{M_{−}∪[−1,0]×F} )

where a, b, c, and d are defined as in [8] page 315. Then one finishes the proof as in [8] by using Lemma 8 instead of Theorem 3.3 of [8]. We omit the (easy) details.

Remark. In this even-dimensional case, but for more general boundary value problems, M. Hilsum has also established the above gluing result, with a different proof. See [10], Theorem 9.2. Notice that our formula

Ind (D^{+}) = Ind^{AP S} (D_{|M_{+}}^{+}, P) + Ind^{AP S}(D_{|M_{−}}^{+}, Id −P)

is a direct consequence of the APS-theory on Galois coverings developed in [18], [33] and of the ideas of Bunke (and Dai-Zhang). This much has been known to us since the appearance of [8] and of [17], but the lack of geometric applications of such a formula has held us from publishing the result until now; the applications we shall give, in Section 5, to the cut-and-paste invariance of higher signatures seem to be a good enough motivation to finally write down the (easy) proof.

4.2. The odd dimensional case.

Our next task consists in computing higher APS-index classes for the cylinder in the odd dimensional case. Let N be a closed manifold of dimension 2m, f : N → BΓ a continuous map and E → N be a ℤ_{2}−graded Clifford hermitian module over N endowed with a unitary Clifford connection. Let D_{N} be the associated Λ−Dirac type operator acting on L_{Λ}^{2}(N, V_{f} ⊗E. We fix a constant map c : [−1, 1] → BΓ. Let D be the associated ℤ_{2}−graded Dirac type operator acting on L_{Λ}^{2}([−1, 1]×N, V_{f×c}⊗ E ⊗ ℂ^{2}), with boundary operator equal to D_{N}. We fix a Cl(1)−spectral section P at {1} × N for the boundary operator D_{N} (of D) and consider the Cl(1)−spectral section Id −P at {−1} × N.

The proof of the following Lemma is basically the same as the one of Lemma 7 and will be omitted.

Notation. In order to simplify the notation we denote by Ind^{AP S}(D, P) the K_{1}- index class appearing in the statement of Theorem 7.

Lemma 9.

Ind^{AP S}(D, Id −P, P) = 0 ∈ K_{0}(C_{c}^{0}(]0, π[) ⊗ Λ) ≃ K_{1}(Λ).

Lemma 10. We use the notations of the previous Lemma. Let P and Q be two Cl(1)−spectral sections for D_{N} at {1} × N. Then:

Ind^{AP S}(D, Id −Q, P) = [Q − P] ∈ K_{1}(Λ).

Proof. Theorem 7.3] implies that:

Ind^{AP S}(D, Id −Q, Q) − Ind^{AP S}(D, Id −Q, P) = [Q − P].

Since Lemma 9 implies that Ind^{AP S}(D, Id −Q, Q) = 0 one gets immediately the Lemma.

Let (M, g) be a smooth closed riemannian manifold dimension n = 2m + 1, E a hermitian Clifford module endowed with a unitary Clifford connection ∇. As in the even case we consider a finitely generated group Γ, a classifying map f : M → BΓ and denote by Λ the reduced C^{∗}−algebra of Γ and by

V_{f} = Λ ×_{Γ} M̃.

the associated flat bundle. These data define a Λ−linear unbounded self-adjoint Dirac type operator D acting on L_{Λ}^{2}(M, (V_{f} ⊗ E)). This operator has a well defined index class Ind(D) ∈ K_{1}(Λ) which is defined by suspension. Let F be a closed cutting hypersurface of M such that M = M_{+} ∪ M_{−} where M_{±} are two manifolds whose F has (common) boundary is F. We assume that all these data have a product structure near F. Let P and Q be two Cl(1)−spectral sections for the boundary operator of the operator induced by the restriction D_{|M_{±}} of D to M_{±}. Then we have:

Theorem 9. Ind (D) = Ind^{AP S} (D_{|M_{+}}, P) + Ind^{AP S}(D_{|M_{−}}, Id −Q) + [P − Q].

Proof. Once again the proof of Theorem 3.1 of [8] (which follows an idea of Bunke [5]) extends without problems; we simply have to use Lemma 10 instead of Theorem 3.3 of [8]. We omit the details.

5. Cut and paste invariance for higher signatures

5.1. A defect formula.

Let (M_{i}, r_{i}) (1 ≤ i ≤ 2) be a closed oriented smooth manifold endowed with a continuous map r_{i} : M_{i} → BΓ.

Definition 10. [13] We say that (M_{1}, r_{1}) and (M_{2}, r_{2}) are SK-equivalent if there exist decompositions

M_{1} = M_{+} ∪_{(F,ϕ_{1})} M_{−} M_{2} = M_{+} ∪_{(F,ϕ_{2})} M_{−}

with

∂M_{+} = F = −∂M_{−} , ϕ_{i} ∈ Diff(F)

and such that (r_{1})_{|M_{±}} ≃ (r_{2})_{|M_{±}} (≃ meaning homotopy equivalence). F is, by assumption, an embedded hypersurface in M_{1} and in M_{2} which is endowed with the orientation induced by the one of M_{+} and the inward normal vector to ∂M_{+}. For simplicity we assume that F is two-sided (i.e. the normal bundle to F is trivial).

Let (M_{i}, r_{i}) (1 ≤ i ≤ 2) be as in Definition 10. We denote by M̃_{i} the coverings defined by r_{i}. Thus M̃_{i} := (r_{i})^{∗}EΓ. We also let F̃_{i} → F be the covering associated with (r_{i})_{|F} : F → BΓ. We consider the flat bundles V_{i} = M̃_{i} ×_{Γ} Λ over M. We now introduce riemannian metrics g_{i} on M_{i} and endow the coverings M̃_{i} with the lifted metrics. We denote by D_{M_{i}}^{sign,r_{i}} the associated signature operator on M_{i} with coefficients in V_{i}. The restriction of this operator to M_{±} will be denoted by D_{M_{±}}^{sign,r_{i}}; we use the precise notation D_{∂M_{±}}^{sign,r_{i}} for the associated boundary operators.

Our goal in this Section is to express the difference

Ind D_{M_{2}}^{sign,r_{2}} − Ind D_{M_{1}}^{sign,r_{1}} ∈ K_{∗}(C_{r}^{∗}(Γ))

in terms of a noncommutative spectral flow, on F, naturally associated to the above geometric data.

First we need a general result (see [9] for the family case). Let X be an evendimensional riemannian manifold with boundary and let E be a ℤ_{2}-graded complex vector bundle over X. We assume that there exists a 1-parameter family of riemannian metrics {g_{u}}_{u∈[1,2]} on X and a 1-parameter family of hermitian metrics {h_{u}^{E}}_{u∈[1,2]} on E so that (E, h_{u}) is a unitary ℤ_{2}-graded Clifford module for (X, g_{u}). We also assume that there is a 1-parameter family of connections ∇^{E,u} that are unitary with respect to h_{u}^{E} and Clifford with respect to the Levi-Civita connection associated to g_{u}. We assume that these data depend continuously on u ∈ [1, 2]. Let D /_{u} be the Dirac operator associated to (g_{u}, h_{u}^{E}, ∇^{E,u}). Let f_{u} : X → BΓ , u ∈ [1, 2] be a 1-parameter family of continuous maps. Let {D /_{u}}_{u∈[1,2]}, D /_{u} : C^{∞}(X, E ⊗ V_{u}) → C^{∞}(X, E ⊗ V_{u}), be the resulting family of C_{r}^{∗}(Γ)-linear operators.

Proposition 8. Let us denote by {(D /_{u})_{0}} the family of boundary-operators associated to {D /_{u}, u ∈ [1, 2]}. We fix noncommutative spectral sections P_{1}, P_{2} for (D /_{1})_{0} and (D /_{2})_{0} respectively. Then

(27) Ind^{AP S}(D /_{2}^{+}, P_{2}) − Ind^{AP S}(D /_{1}^{+}, P_{1}) = sf({(D /_{u})_{0}}; P_{2}, P_{1}) in K_{0}(C_{r}^{∗}(Γ))

Proof. We are going to extend to the noncommutative setting the proof of Theorem 5.2 of [9]. Let (Q(u))_{u∈[1,2]} be a total spectral section for the family {(D /_{u})_{0}, u ∈ [1, 2]}. Then we get a continuous family of elliptic boundary value problems (D /_{u}, Q(u) ), 1 ≤ u ≤ 2. By the homotopy invariance of the K-theoretic index class we get:

Ind(D /_{1}^{+}, Q(1)) = Ind((D /_{2}^{+}, Q(2)).

By the relative index Theorem 6 we get:

Ind(D /_{1}^{+}, Q(1)) = Ind(D /_{1}^{+}, P_{1}) + [P_{1} − Q(1)]

and

Ind(D /_{2}^{+}, Q(2)) = Ind(D /_{2}^{+}, P_{2}) + [P_{2} − Q(2)].

The Proposition is then an immediate consequence of the definition of the higher spectral flow and of the three previous equalities. Remark. One can immediately extend the previous Proposition to the odd dimensional case; in such a case we need to fix Cl(1)-spectral sections. Notation. In order to treat simultaneously the odd and even dimensional case, from now on we shall not write the superscript + in the even dimensional case (see formula (27)); thus both in the even and in the odd-dimensional case we shall simply write Ind^{AP S}(D /, P) for the index class associated to D / and the choice of a spectral section for the boundary operator. We shall implicitly choose Cl(1)- spectral sections in the odd-dimensional case.

Now, for two SK-equivalent pairs (M_{1}, r_{1}), (M_{2}, r_{2}) as in Definition 10, we can prove a defect formula for the difference Ind D_{M_{2}}^{sign,r_{2}} − Ind D_{M_{1}}^{sign,r_{1}} ∈ K_{∗}(C_{r}^{∗}(Γ)). First we perturb the metrics g_{1} and g_{2} so as to be product-like near F. Thanks to the additivity formula proved in Section 4 we have

(28) Ind(D_{M_{1}}^{sign,r_{1}}) = Ind^{AP S}(D_{M_{+}}^{sign,r_{1}}, P_{1}) + Ind^{AP S}(D_{M_{−}}^{sign,r_{1}}, 1 − P_{1})

with P_{1} a noncommutative spectral section for the boundary operator D_{∂M_{+}}^{sign,r_{1}}. Similarly,

(29) Ind(D_{M_{2}}^{sign,r_{2}}) = Ind^{AP S}(D_{M_{+}}^{sign,r_{2}}, P_{2}) + Ind^{AP S}(D_{M_{−}}^{sign,r_{2}}, 1 − P_{2})

with P_{2} a noncommutative spectral section for D_{∂M_{+}}^{sign,r_{2}}. Consider (g_{1})_{|M_{+}} and (g_{2})_{|M_{+}} and let g_{+,u}, with u ∈ [1, 2] a path of riemannian metrics on M_{+} connecting them. We can choose these metrics to be product-like near the boundary of M_{+}. By assumption there is a continuous family of classifying maps r_{+,u} : M_{+} → BΓ, u ∈ [1, 2], connecting (r_{1})_{|M_{+}} and (r_{2})_{|M_{+}}. Thus, there is a 1-parameter family of Dirac-type operators on M_{+}, {D_{M_{+}}^{sign,u}}_{u∈[1,2]} with boundary family {D_{∂M_{+}}^{sign,u}}_{u∈[1,2]}. We can thus apply the previous Proposition. We obtain

Ind^{AP S}(D_{M_{+}}^{sign,r_{2}}, P_{2}) − Ind^{AP S}(D_{M_{+}}^{sign,r_{1}}, P_{1}) = sf({D_{∂M_{+}}^{sign,u}}; P_{2}, P_{1}) in K_{∗}(C_{r}^{∗}(Γ)) (30)

Notice that it does not make sense to take a path joining g_{1} and g_{2} (indeed, M_{1} and M_{2} are in general different manifolds).

We apply the same reasoning to M_{−}. Thus by assumption there is a continuous family of classifying maps r_{−,u} : M_{−} → BΓ, u ∈ [1, 2], connecting (r_{1})_{|M_{−}} and (r_{2})_{|M_{−}} and, we choose a path of metrics g_{−,u}, u ∈ [1, 2], connecting (g_{1})_{|M_{−}} and (g_{2})_{|M_{−}}. We obtain a family of operators D_{M_{−}}^{sign,u} connecting D_{M_{−}}^{sign,r_{2}} and D_{M_{−}}^{sign,r_{1}} together with the formula

Ind^{AP S}(D_{M_{−}}^{sign,r_{2}}, 1 − P_{2}) − Ind^{AP S}(D_{M_{−}}^{sign,r_{1}}, 1 − P_{1}) = sf({D_{∂M_{−}}^{sign,u}}; 1 − P_{2}, 1 − P_{1}) (31)

Summarizing, in K_{∗}(C_{r}^{∗}(Γ)) :

Ind D_{M_{2}}^{sign,r_{2}} − Ind D_{M_{1}}^{sign,r_{1}} = sf({D_{∂M_{+}}^{sign,u}}; P_{2}, P_{1}) + sf({D_{∂M_{−}}^{sign,u}}; 1 − P_{2}, 1 − P_{1}) (32)

Notice that

D_{∂M_{+}}^{sign,1} = −D_{∂M_{−}}^{sign,1} , D_{∂M_{+}}^{sign,2} = −D_{∂M_{−}}^{sign,2}

but that, in general, D_{∂M_{+}}^{sign,u} ≠ −D_{∂M_{−}}^{sign,u} for u ∈ (1, 2).

Let us consider the S^{1} (i.e. periodic) family {D_{F} (θ)}_{θ∈S^{1}} obtained taking the first family from 1 to 2 and the opposite of the second family from 2 to 1. Thus, with an obvious and harmless abuse of notation,

(33) {D_{F} (θ)}_{θ∈S^{1}} := {D_{∂M_{+}}^{sign,u}}_{u∈[1,2]} ∪ {−D_{∂M_{−}}^{sign,u}}_{u∈[2,1]}.

We shall use the fact that, in general, the spectral flow is additive with respect to the composition of path; this means that if {A(t)}_{t∈[a,b]} is a 1-parameter family of Dirac-type operators as in Definitions 5 or 9 and if {B(t)}_{t∈[b,c]} is a second such family such that B(b) = A(b), then, for the family {D(t)}_{t∈[a,c]} := {A(t)} ∪ {B(t)} the following additivity-formula holds

(34) sf({D(t)}; P_{a}, P_{c}) = sf({A(t)}; P_{a}, P_{b}) + sf({B(t)}; P_{b}, P_{c}) in K_{∗}(C_{r}^{∗}(Γ))

This formula follows immediately from the definition. Since

sf({D_{∂M_{−}}^{sign,u}}_{u∈[1,2]}; 1 − P_{2}, 1 − P_{1}) = sf({−D_{∂M_{−}}^{sign,u}}_{u∈[2,1]}; P_{1}, P_{2})

we can then write

(35) sf({D_{∂M_{+}}^{sign,u}}; P_{2}, P_{1}) + sf({D_{∂M_{−}}^{sign,u}}; 1 − P_{2}, 1 − P_{1}) = sf({D_{F} (θ)}_{θ∈S^{1}})

obtaining, finally, a proof of the following

Theorem 10. The following formula holds in K_{∗}(C_{r}^{∗}(Γ)):

(36) Ind D_{M_{2}}^{sign,r_{2}} − Ind D_{M_{1}}^{sign,r_{1}} = sf({D_{F} (θ)}_{θ∈S^{1}})

where {D_{F} (θ)}_{θ∈S^{1}} is the periodic family of twisted signatures operators on F defined by (33).

One important consequence of our discussion so far is the following

Corollary 2. Assume that Γ is such that the assembly map of the Baum-Connes conjecture is rationally injective. Let (M_{1}, r_{1}), (M_{2}, r_{2}) be two SK-equivalent pairs as in Definition 10. If, with the above notations,

sf({D_{F} (θ)}_{θ∈S^{1}}) = 0 in K_{∗}(C_{r}^{∗}(Γ)) ⊗_{ℤ} ℚ

then the higher signatures of (M_{1}, r_{1}) and (M_{2}, r_{2}) are the same.

Proof. It is proven in [15] (just after Theorem 0.5 ) that, under the stated assumption on Γ, the equality of the two index classes imply the equality of the higher signatures. The Proposition is then a consequence of Theorem 10.

5.2. Remarks.

1. Formula (36) is a higher version of formula (25.22) in [4]. In the numerical case treated there, the numerical spectral flow is also equal to the numerical index of the mapping torus defined by the diffeomorphism ψ = (ϕ_{2})^{−1} ∘ ϕ_{1}. We shall now discuss the extension of the latter result to the higher context. We consider two SK-equivalent pairs (M_{1}, r_{1}), (M_{2}, r_{2}) as in Definition 10. We consider the mapping torus M(F, ψ), which is obtained from F × [1, 2] ⊔ F × [1, 2] by identifying (ϕ_{1}(x), 1) (in the first F ×[1, 2]) with (x, 1) ( in the second F ×[1, 2] ) and similarly (ϕ_{2}(x), 2) with (x, 2), for any x ∈ F. As explained in [13] (page 16), see also [15], the mapping torus M(F, ψ) comes with a map r̂: M(F, ψ) → BΓ such that

(37) (M_{2}, r_{2}) − (M_{1}, r_{1}) is bordant to (M(F, ψ), r̂)

Let us recall the argument: one considers the space Y obtained from M_{1} × [0, 3] ⊔ M_{2} × [0, 3] and the following identifications: for x ∈ F identify (x, t) ∈ ∂M_{+} × [0, 1] with (ϕ_{1}(x), t) ∈ ∂M_{×}[0, 1] and (x, t) ∈ ∂M_{+} × [2, 3] with (ϕ_{2}(x), t) ∈ ∂M_{−} × [2, 3]. Then, after smoothing, ∂Y = M_{1} − M_{2} − M(F, ψ). Moreover the two homotopies (r_{j})_{|M_{+}} ≃ (r_{j})_{|M_{−}} can be used to define a map R : Y → BΓ, with R_{|M_{j}} = r_{j}; one then sets r̂ := R_{|M(F,ψ)} which proves (37). Using the metrics considered in the discussion preceeding the statement of Theorem 10 we can endow the mapping torus M(F, ψ) with a riemannian metric; we shall then denote by D_{M(F,ψ)}^{sign,r̂} the signature operator with coefficients in the flat bundle associated with r̂.

Proposition 9. One has:

(38) Ind D_{M_{2}}^{sign,r_{2}} − Ind D_{M_{1}}^{sign,r_{1}} = Ind D_{M(F,ψ)}^{sign,r̂}.

Proof. According to (37) the difference of pairs [M_{1}, r_{1}] − [M_{2}, r_{2}] is bordant to [M(F, ψ), r̂]. By the cobordism invariance of the index class (see [18], Proposition 2.3), one gets immediately the Proposition. From the previous Proposition and Theorem 10 we immediately obtain the following higher analogue of Proposition 25.1 in [4]

Corollary 3. With the notation introduced above the following formula holds in K_{∗}(C_{r}^{∗}(Γ))

Ind D_{M(F,ψ)}^{sign,r̂} = sf({D_{F} (θ)}_{θ∈S^{1}}) (39)

2. We have obtained formula (39) by exploiting two different ways for computing the difference Ind D_{M_{2}}^{sign,r_{2}} − Ind D_{M_{1}}^{sign,r_{1}}. In fact, a direct proof of this fact is essentially given in [32] using the intersection product in KK-theory. The results of Wu apply to any periodic family giving, at the K-theoretic level, the usual relationship between the spectral flow and the index.

3. Let B_{Γ}^{∞} be the Connes-Moscovici algebra associated to Γ. Then, as a consequence of (39) we have that

(40) Ch (Ind D_{M(F,ψ)}^{sign,r̂}) = Ch (sf({D_{F} (θ)}_{θ∈S^{1}}))

holds in the topological non commutative de Rham homology \overline{H}_{∗}(B_{Γ}^{∞}) (see [12]) This formula, a weaker version of (39), can be proved directly without using the unpublished paper of Wu and using instead results of [9] (Theorem 3.2) together with their noncommutative generalizations in [18] (Theorem 5.3). We omit the details.

4. One can also establish formula (38) using the gluing formula for index classes. In the even-dimensional case, this is the route that is taken in [10]. See Theorem 14.1 there. The cut and paste results for higher signatures are then obtained in [10], always in the even-dimensional case, by giving sufficient conditions for the index class of the mapping torus to be zero. Corollary 3 shows why the approach we follow here, directly inspired by [14], and the approach in [10] are compatible.

5. For future applications, we briefly point out that the defect formula (36) is still valid if one replaces the signature operators by Dirac type operators. In this generality one needs to give the appropriate definitions. Thus, we assume that we are given metrics g_{1} on M_{1} and g_{2} on M_{2} and Clifford bundles E_{1} on M_{1} and E_{2} on M_{2}. We let D_{1} and D_{2} be two Dirac type operators on these bundles. We assume that the two bundles are obtained by a clutching construction involving two Clifford bundles E_{+} → M_{+}, E_{−} → M_{−} and two bundles isomorphisms (E_{+})_{|∂M_{+}} ≃ (E_{−})_{|∂M_{−}} covering respectively ϕ_{1} and ϕ_{2}. Given two classifying maps r_{j} : M_{j} → BΓ (1 ≤ j ≤ 2) as above one defines two Λ−linear Dirac type operators D_{M_{j}}^{r_{j}}. Moreover, assuming that the two paths of metrics {g_{+,u}, u ∈ [1, 2] } and {g_{−,u}, u ∈ [1, 2] } considered in Section 5.1 induce (now) a Clifford action on respectively E_{+} and E_{−} for each u ∈ [1, 2], one may define an associated S^{1}−family ({D_{F} (θ)}_{θ∈S^{1}}) so that the defect formula (36) holds verbatim. A typical example of this situation is given when M_{j} are spin manifolds and the operators are Dirac operators. We leave the details to the interested reader.

5.3. Vanishing spectral flow in the even-dimensional case.

In this subsection we consider two SK-equivalent pairs (M_{1}, r_{1}), (M_{2}, r_{2}) as in Definition 10 such that dimM_{1} = dimM_{2} = 2m. We shall give sufficient conditions for the noncommutative spectral flow appearing in the defect formula of Theorem 10 to be zero. Let F be as in the previous sub-section and consider r = (r_{1})_{|F} : F → BΓ. Following [23] and then [14] we shall make the following: Assumption A. Let Ω_{(2)}^{ℓ}(F, V_{r}) denote the L_{Λ}^{2}-completion of Ω^{ℓ}(F, V_{r}). We assume that the operator

d : Ω_{(2)}^{m−1}(F, V_{r}) → Ω_{(2)}^{m}(F, V_{r}),

with domain equal to Λ-Sobolev space H_{Λ}^{1}, has closed image. This assumption appeared for the first time in [23] (see Section 7); there it was assumed that the L^{2}−spectrum of the differential form laplacian of F̃ (the covering defined by r : F → BΓ), acting on the vector space of differential forms of degree m, has a full gap at zero. Assumption A (which is a homotopy-invariant assumption) is equivalent to the slightly weaker assumption that the differential form laplacian of F̃ has a strictly positive spectrum on Ω^{m−1}(F̃)/ ker(d). We refer the reader to [14] for a proof of this fact and for examples where Assumption A is fulfilled.

We first recall (and slightly extend) the definition of symmetric spectral section [19]. Under Assumption A we have

Ω_{(2)}^{m}(F, V_{r}) = dΩ_{(2)}^{m−1}(F, V_{r}) ⊕ (dΩ_{(2)}^{m−1}(F, V_{r}))^{⊥}
Ω_{(2)}^{m−1}(F, V_{r}) = d^{∗}Ω_{(2)}^{m}(F, V_{r}) ⊕ (d^{∗}Ω_{(2)}^{m}(F, V_{r}))^{⊥}.

Then, we can write Ω^{∗}(F, V_{r}) = V ⊕ W where (we abbreviate Ω_{V}^{∗} = Ω^{∗}(F, V_{r})):

V = d^{∗}Ω_{V}^{m} + dΩ_{V}^{m−1}, W = Ω_{V}^{<} ⊕ Ω_{V}^{>}
Ω_{V}^{<} = Ω_{V}^{0} ⊕ … ⊕ Ω_{V}^{m−2} ⊕ (d^{∗}Ω_{V}^{m})^{⊥},
Ω_{V}^{>} = (dΩ_{V}^{m−1})^{⊥} ⊕ Ω_{V}^{m+1} ⊕ … ⊕ Ω_{V}^{2m−1}.

It is clear that D_{F}^{sign,r} sends V ( resp. W) into itself. Using Assumption A and proceeding as in the proof of Lemma 2.1 of [14], one checks easily that D_{F}^{sign,r} induces an invertible operator on the L_{Λ}^{2}−completion of V (with domain H^{1}) and we denote by Π_{>} the projection onto the positive part. Moreover, the proof of Theorem 2 shows immediately that the restriction of D_{F}^{sign,r} to the L_{Λ}^{2}−completion of W admits a spectral section. Notice that although W is not the space of sections of a Λ−bundle over F the proof of Theorem 2 still applies in this case. Then, Lemma 4.3 of [19] shows that D_{F}^{sign,r} admits a symmetric spectral section P in the sense that P is diagonal with respect to the decomposition Ω^{∗}(F, V_{r}) = V ⊕W and:

P_{|V} = Π_{>}, P_{|W} ∘ α + α ∘ P_{|W} = α

where α is the involution of W equal to identity on Ω_{V}^{<} and to minus the identity on Ω_{V}^{>}. Recall from [19] Prop. 4.4 that if Q is another symmetric spectral section then [P − Q] = 0 in K_{0}(Λ) ⊗_{ℤ} ℚ. We immediately obtain the following

Proposition 10. Let (M_{1}, r_{1}), (M_{2}, r_{2}) be two SK-equivalent pairs as in Definition 10. We set r := (r_{1})_{|F}. If Assumption A holds for (F, r) then

sf({D_{F} (θ)}_{θ∈S^{1}}) = 0 in K_{∗}(C_{r}^{∗}(Γ)) ⊗_{ℤ} ℚ.

Proof. In the definition of sf({D_{F} (θ)}_{θ∈S^{1}}), we can assume that all the spectral sections involved are symmetric. Since, by definition, the noncommutative spectral flow is given in terms of difference classes associated to these symmetric spectral sections, it follows by Proposition 4.4 in [19] that sf({D_{F} (θ)}_{θ∈S^{1}}) is zero in K_{0}(C_{r}^{∗}(Γ)) ⊗_{ℤ} ℚ. The Proposition is proved.

Remark. Notice that because of the homotopy-invariance of Assumption A we could equivalently set r := (r_{2})_{|F}.

We have now obtained an analytic proof of the main result in [15]

Theorem 11. Assume that Γ is such that the assembly map of the Baum-Connes conjecture is rationally injective. Let (M_{1}, r_{1}) and (M_{2}, r_{2}) be two SK-equivalent pairs as in Definition 10. We assume that Assumption A holds for (F, r) with r := (r_{1})_{|F}. Then the higher signatures of (M_{1}, r_{1}) and (M_{2}, r_{2}) are the same.

The proof is a direct consequence of Proposition 10 and of Corollary 2.

Remark. The previous Theorem was first proved by Leichtnam-Lott-Piazza in [14] in the case of virtually nilpotent or Gromov Hyperbolic groups. It was then extended by Leichtnam-Lueck ([15]), using techniques from algebraic surgery, to the case where Γ satisfies the assumption of Theorem 11.

5.4. Vanishing spectral flow in the odd dimensional case.

In this subsection we consider two SK-equivalent pairs (M_{1}, r_{1}), (M_{2}, r_{2}) as in Definition 10 such that dimM_{1} = dimM_{2} = 2m + 1. Let F be as in the previous section and consider r = (r_{1})_{|F} : F → BΓ.

As in the even case we shall make the following assumption on (F, r): Assumption B. The operator

d : Ω_{(2)}^{m−1}(F, V_{r}) → Ω_{(2)}^{m}(F, V_{r})

with domain equal to Λ-Sobolev space H_{Λ}^{1}, has closed range.

It is proved in [14] that Assumption B is equivalent to the hypothesis that L^{2}−spectrum of the differential form laplacian of F̃ acting on the vector space of differential forms of degree m admits a gap at zero. Assumption B (in the slightly stronger form of requiring L^{2}-invertibility in middle degree) appears for the first time in Section 7 of [23]. We refer the reader to [14] for examples where Assumption B is satisfied.

Now we recall results and notations allowing to introduce the concept of symmetric Cl(1)−spectral section. We define the rescaled flat exterior derivative d̃ : Ω^{∗}(F, V_{r}) → Ω^{∗+1}(F, V_{r}) by setting:

∀ω ∈ Ω^{|ω|}(F, V_{r}), d̃(ω) = i^{|ω|}dω.

Then using the Hodge duality operator ∗ one defines the involution τ:

∀ω ∈ Ω^{|ω|}(F, V_{r}), τ(ω) = i^{−|ω|(2m−|ω|)} ∗ ω.

We refer to the Section 1 of [14] for the definition of the quadratic form Q and the Λ−hermitian scalar product <; > on Ω^{∗}(F, V_{r}). The signature operator D_{F}^{sign,r} is then equal to d̃− τd̃τ.

Let H^{m}(F, V_{r}) denote the m−th cohomology group of the locally constant sheaf V_{r}. Using Hodge duality and Assumption B one proves easily that the two following orthogonal decomposition holds:

Ω_{(2)}^{m}(F, V_{r}) = H^{m}(F, V_{r}) ⊕ \overline{τd̃τ(Ω^{m+1}(F, V_{r}))} ⊕ \overline{d̃(Ω^{m−1}(F, V_{r}))}.
Ω_{(2)}^{∗}(F, V_{r}) = V ⊕ W with
V = Ω_{(2)}^{m}(F, V_{r}) ⊕ D_{F}^{sign,r}(H_{Λ}^{1}(F; ∧^{m}T^{∗}F ⊗ V_{r}) ) , W = V^{⊥}

V being a closed complementable Λ−Hilbert submodule of Ω_{(2)}^{∗}(F, V_{r}). We observe that both D_{F}^{sign,r} and the grading τ send V (resp. W) into itself.

We shall consider the involution α of W such that α induces Id (resp. − Id) on the differential forms of degree < m (resp. > m).

Proposition 11. At the expense of replacing F by two disjoint copies of F, one can find a finitely generated projective Λ-submodule N of V such that:

0) D_{F}^{sign,r}(N) ⊂ N, N^{+} ≃ N^{−}.

1) τ(N) ⊂ N and H^{m}(F, V_{r}) ⊂ N as a Λ-submodule.

2) N admits a lagrangian Λ-submodule L with respect to the quadratic form Q and the orthogonal Λ−projection P_{L} from N onto L is well defined.

3) V = N ⊕ N^{⊥}. We will say that such an L is a stable lagrangian of H^{m}(F, V_{r}) (associated with N).

Proof. The index class of the signature operator on F with values in V_{r} is zero in K_{0}(Λ), by cobordism invariance. Notice that this index class is the sum of the index classes associated to the restrictions D_{V}, D_{W} of the signature operator D_{F}^{sign,r} to V and W respectively. Under Assumption B, one checks easily that

α(D_{W}^{+} )^{∗}α = D_{W}^{+}

and so 2 Ind D_{W}^{+} = 0 in K_{0}(Λ). Replacing F by the disjoint union of two copies of F, we may assume that Ind D_{W}^{+} = 0 in K_{0}(Λ). Therefore, we may conclude that the index class associated to D_{F}^{sign,r} restricted to V is also zero. Extending the proof of Proposition 2 page 295 of [28] to the case of (D_{F}^{sign,r})_{|V} along the lines of [19] (see Lemma 4.3), we infer that there exists a real number R > 0 such that for any spectral section P′ as in the statement of the Theorem, N = ker(P′ + τP′τ) is a finitely generated projective submodule of V containing H^{m}(F, V_{r}) and such that:

a) N is the range of a projection of V and thus N ⊕ N^{⊥} = V.

b) N = N^{+} ⊕ N^{−}, with N^{±} = {ω ; τω = ±ω} and N^{+} isomorphic to N^{−}. Let then j : N^{+} → N^{−} be a unitary isomorphism; our lagrangian L is given by

{ω + j(ω) , ω ∈ N^{+}} .

The Proposition is proved.

Remarks.

1. Notice that, as in [28], page 295, we have in fact proved the following more precise statement: one can find a real number R > 0 such that for any spectral section P′ of (D_{F}^{sign,r})_{|V} satisfying P′ ∘ ξ((D_{F}^{sign,r})_{|V}) = 0 for any ξ ∈ C^{∞}(ℝ, ℝ), ξ ≡ 0 on [R, +∞[, the module N := ker(P′ + τP′τ) is finitely generated projective and satisfies the 4 properties of the above Proposition.

2. This notion of stable lagrangian is essentially the same as the one used in Section 3 of [14].

3. In the previous Proposition one may assume that N is included in the range of ψ(D_{F}^{sign,r}_{|V}) for a suitable function ψ ∈ C_{comp}^{∞}(ℝ; ℝ).

Now, let L be a stable lagrangian of H^{m}(F, V_{r}) (associated with N) as in Proposition 11. We observe that D_{F}^{sign,r} is diagonal with respect to the decomposition

Ω_{(2)}^{∗}(F, V_{r}) = (N ⊕ N^{⊥}) ⊕ W

and that the restriction of D_{F}^{sign,r} to N^{⊥} is invertible. We shall denote by Π_{>}(N^{⊥}) the APS projection onto the positive part of the restriction of D_{F}^{sign,r} to N^{⊥}. We then may give the following:

Definition 11. Let R be a spectral section ∈ B_{Λ}(W) for D_{F}^{sign,r}_{|W} which is Cl(1) for both α and τ. The self-adjoint projection

P(L, R) = P_{L} ⊕ Π_{>}(N^{⊥}) ⊕ R

of B_{Λ}(Ω_{(2)}^{∗}(F, V_{r})) is called symmetric Cl(1)−spectral section for D_{F}^{sign,r}.

One checks the existence of a spectral section R as in the above Proposition by proceeding as in the proof of Lemma 4.3 of [19]. Therefore, there exists such symmetric Cl(1)−spectral sections.

Let L and L′ be two stable lagrangians for H^{m}(F, V_{r}), as in Proposition 11. The proof of the Proposition and the arguments of [28] (page 295) shows that if in the statement of Proposition 11 one replaces the real number R by a bigger one then N^{±} gets replaced by N^{±} ⊕ M^{±} where (D_{F}^{sign,r})^{+} induces an isomorphism from the finitely generated projective C_{r}^{∗}(Γ)−module M^{+} onto the C_{r}^{∗}(Γ)−module M^{−}. Thus, at the expense of replacing L by

{ ω ⊕ β + j(ω) ⊕ (D_{F}^{sign,r})^{+}(β), (ω, β) ∈ N^{+} × M^{+} }

(and similarly for L′), one may assume that both L and L′ are associated with the same N.

Definition 12. Let L, L′ be two lagrangians as above. Let P_{L}, P_{L′} be the Λendomorphisms of N given by the orthogonal projections onto L, L′ respectively. According to Definition 8 these two projections define a difference class [P_{L}−P_{L′}] ∈ K_{0}(C_{c}^{0}(]0, π[) ⊗ Λ) = K_{1}(Λ); we set [L − L′] := [P_{L} − P_{L′}]

Remarks.

1. The class [L − L′] ∈ K_{1}(Λ) depends only on L and L′.

2. One can give a different definition of the class [L − L′] ∈ K_{1}(Λ) (see [14] Section 3): if j, j′ are two unitary isomorphims N^{+} → N^{−} such that L = {ω + j(ω)/ω ∈ N^{+}}, L′ = {ω + j′(ω)/ω ∈ N^{+}}, one defines the class [L − L′] to be [j ∘ (j′)^{−1}] ∈ K_{1}(Λ). The two definitions should be compatible through the suspension isomorphism, but we have not looked into the details.

We have the following relative result (where we are always under Assumption B):

Proposition 12. Let L and L′ be two stable lagrangians as in Definition 12. Let P(L, R) and P(L′, R′) be two symmetric Cl(1)−spectral sections for D_{F}^{sign,r} as in Definition 11, then one has

[P(L, R) − P(L′, R′)] = [L − L′] in K_{1}(Λ) ⊗_{ℤ} ℚ.

Proof. We set P_{1} = P(L, R) and P_{2} = P(L′, R′) and assume that L and L′ are associated with (the same) N. We then consider the odd self-adjoint Λ−compact operator A_{P_{j}}^{0} associated with P_{j} (j ∈ {1, 2}) as in Definition 8. Thus D_{F}^{sign,r} + A_{P_{j}}^{0} is invertible and we can moreover assume that A_{P_{j}}^{0} is diagonal with respect to the decomposition

Ω_{(2)}^{∗}(F, V_{r}) = (N ⊕ N^{⊥}) ⊕ W

and that its restriction to W, denoted A_{j}^{0}(W), is odd with respect to the grading α. Of course we can assume that the restrictions of A_{P_{j}}^{0} to N^{⊥} coincide.

Now, for j ∈ {1, 2} and t ∈ [0, π], we denote by P_{j}(t) the projection onto the positive part of

τ( Id +[D_{F}^{sign,r}_{|W} + A_{j}^{0}(W)]^{2} )^{\frac{1}{2}} cos t + (D_{F}^{sign,r}_{|W} + A_{j}^{0}(W)) sin t.

Since τα = −ατ and both D_{F}^{sign,r}_{|W} and the A_{j}^{0}(W) are odd with respect to the grading α, one checks immediately that α(P_{1}(t)−P_{2}(t)α = −(P_{1}(t)−P_{2}(t) for any t ∈ [0, π]. Therefore

[(P_{1}(t) − P_{2}(t))_{t∈[0,π]}] = −[(P_{2}(t) − P_{1}(t))_{t∈[0,π]}] in K_{0}(C_{c}^{0}(]0, π[) ⊗ Λ) (41)

and this K−theory class is rationnally zero.

Next, we denote by A_{j}^{0}(N) the restriction of A_{P_{j}}^{0} to N for j ∈ {1, 2}. For t ∈ [0, π] we denote by Q_{j}(t) the projection onto the positive part of

τ( Id +[D_{F}^{sign,r}_{|N} + A_{j}^{0}(N)]^{2} )^{\frac{1}{2}} cos t + (D_{F}^{sign,r}_{|N} + A_{j}^{0}(N)) sin t.

Then by Definition 12 of the difference class one has:

[(Q_{1}(t) − Q_{2}(t))_{t∈[0,π]}] = [L − L′] in K_{0}(C_{c}^{0}(]0, π[) ⊗ Λ) ≃ K_{1}(Λ) (42)

By combining the two equations (41) and (42) and recalling Definition 8, one gets immediately the result of the Proposition.

We recall that (M_{1}, r_{1}) and (M_{2}, r_{2}) are two SK-equivalent pairs of dimension 2m+1 as in Definition 10. We consider the periodic family (D_{F} (θ))_{θ∈[0,2π]} (defined in (33)). By following the proof of Proposition 11 we may construct two continuous families of spaces N_{θ} and V_{θ}, each of them being associated with D_{F} (θ), so that (N_{θ})_{θ∈[0,2π]} defines a bundle over [0, 2π] of finitely generated projective C_{r}^{∗}(Γ)− module with N_{0} = N_{2π}

Let (L_{θ})_{θ∈[0,2π]} be a continuous family of stable lagrangians of (D_{F} (θ))_{θ∈[0,2π]}; there exists a total spectral section (P_{θ})_{θ∈[0,2π]} for the periodic family (D_{F} (θ))_{θ∈[0,2π]} where, for each θ ∈ [0, 2π], P_{θ} can be chosen to be a Cl(1)−symmetric spectral section of the form:

P_{θ} = P(L_{θ}, R_{θ})

with the spectral sections R_{θ} varying continuously with respect to θ ∈ [0, 2π]. By the definition of noncommutative spectral flow of a periodic family and by Proposition 12 we obtain:

sf({D_{F} (θ)}_{θ∈S^{1}}) := [P(L_{2π}, R_{2π}) − P(L_{0}, R_{0})] = [L_{2π} − L_{0}]

Summarizing for a 1-parameter family of stable lagrangians as above:

(43) Ind D_{M_{2}}^{sign,r_{2}} − Ind D_{M_{1}}^{sign,r_{1}} = [L_{2π} − L_{0}]

Definition 13. We shall say that the above family of stable lagrangians is K_{1}(C_{r}^{∗}(Γ))- periodic if [L_{2π} − L_{0}] = 0 in K_{1}(C_{r}^{∗}(Γ)).

We can state, finally, the main result of this subsection:

Proposition 13. Let (M_{1}, r_{1}) and (M_{2}, r_{2}) be two SK-equivalent pairs of dimension 2m + 1 as in Definition 10. If Assumption B holds for (F, r = (r_{1})_{|F}) and if the family (D_{F} (θ))_{θ∈[0,2π]} admits a K_{1}(C_{r}^{∗}(Γ))-periodic family of stable lagrangians then

sf({D_{F} (θ)}_{θ∈S^{1}}) = 0 in K_{1}(Λ) ⊗_{ℤ} ℚ.

and so

Ind D_{M_{2}}^{sign,r_{2}} = Ind D_{M_{1}}^{sign,r_{1}} in K_{1}(Λ) ⊗_{ℤ} ℚ.

The previous Proposition implies the main result of [15]:

Theorem 12. Assume that Γ is such that the assembly map of the Baum-Connes conjecture is rationally injective. Let (M_{1}, r_{1}) and (M_{2}, r_{2}) be two SK-equivalent pairs of dimension 2m + 1 as in Definition 10. If Assumption B holds for (F, r) and if the family (D_{F} (θ))_{θ∈[0,2π]} admits a K_{1}(C_{r}^{∗}(Γ))-periodic family of stable lagrangians then (M_{1}, r_{1}) and (M_{2}, r_{2}) have the same higher signatures.

Remarks.

1. The previous Theorem was first proved by Leichtnam-Lott-Piazza in [14] in the case of virtually nilpotent of Gromov Hyperbolic groups. It was then extended by Leichtnam-Lueck ([15]), using techniques from algebraic surgery, to the case where Γ satisfies the condition of Theorem 12. (Notice that in [14] [15] the result is stated under the slightly stronger assumption that it should exists a ψ-invariant stable lagrangian, with ψ = ϕ_{1} ∘ ϕ_{2}^{−1}. However, Proposition 3.7 in [14] shows that this stronger assumption can be relaxed as in Theorem 12 above; the same is true for [15], using results from algebraic surgery about stable formations.)

2. As a final remark we point out that our K-theoretic approach to the cut-and-paste invariance of higher signatures can be extended to foliations: in that case the higher signatures are defined in [1] and one can ask whether these higher signatures are invariant under an appropriate notion of cut and paste. One of the advantages of the K-theoretic approach presented here is that, at least formally, can be reproposed in the foliation-case. This is not the case either with the original approach in [14] (which uses local index formulae) or with the algebraic-surgery approach of [15]. In fact, finding a proof of the cut-and-paste results in these papers that could be generalized to the foliations was one of the motivations for proving Theorem 10. We plan to deal with the technicalities of this program in a future publication.

References

[1] P. Baum, A. Connes. Leafwise homotopy equivalence and rational Pontrjagin classes. Foliations, Advances Studies in Pure Math., 5, North-Holland, Amsterdam, 1985, p. 1-14.

[2] B. Blackadar. K-Theory of operator algebras, Math. Sciences Research Institute Publications, 1998.

[3] N. Berline, E. Getzler, and M. Vergne. Heat kernels and Dirac operators, Springer Verlag 298, 1992.

[4] B. Booss-Bavnek, K. Wojciechowski Elliptic boundary problems for Dirac operators, Mathematics: Theory and Applications, Birkhauser 1993.

[5] U. Bunke. On the gluing problem for the η−invariant, Journal of Differential Geometry, 41, 1995, p. 397-448.

[6] A. Connes. Noncommutative Geometry. Academic Press. San Diego.

[7] A. Connes, H. Moscovici. Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology. 29, 1990, p. 345-388.

[8] X.Dai, W. Zhang. Splitting of the family index, Commun. Math. Phys. 182, 1996, p. 303-318.

[9] X.Dai, W. Zhang. Higher Spectral Flow, Journal of Funct. Analysis, 157, 1998, p. 432-469.

[10] M. Hilsum. Index classes of Hilbert modules with boundary, Preprint Paris 6, March 2001.

[11] J. Kaminker and V. Perera. Type II spectral flow and the eta invariant, Canad. Math. Bull., 43, 2000, p. 68-73

[12] M. Karoubi. Homologie cyclique et K-Théorie, Astérisque 149, 1987.

[13] U. Karras, M. Kreck, W. Neumann and E. Ossa. Cutting and pasting of manifolds. SK-groups, Publish or Perish, Boston 1973.

[14] E. Leichtnam, J.Lott and P.Piazza. On the homotopy invariance of higher signatures for manifolds with boundary, Journal of Differential Geometry, 54, 2000, p. 561-633.

[15] E. Leichtnam, W.Lueck. On the cut and paste property of higher signatures of a closed oriented manifold, Preprint 2000, to appear in Topology.

[16] E. Leichtnam, P. Piazza. The b−pseudo-differential calculus on Galois coverings and a higher Atiyah-Patodi-Singer index theorem, Mémoires de la Société Mathématiques de France 68, 1997.

[17] E. Leichtnam, P. Piazza. Higher Atiyah-Patodi-Singer index theory C. R. Acad. Sci. Paris, 325, Ser I (1997), p. 399-404.

[18] E. Leichtnam, P. Piazza. Spectral sections and higher Atiyah-Patodi-Singer index theory on Galois coverings, GAFA, Vol. 8, 1998, p. 17-58.

[19] E. Leichtnam, P. Piazza. A Higher Atiyah-Patodi-Singer Index theorem for the signature operator on Galois Coverings, Annals of Global Analysis and Geometry, 18, 2000, p. 171-189.

[20] E. Leichtnam, P. Piazza. Homotopy invariance of twisted higher signatures on manifolds with boundary, Bull. Soc. math. France, 127, 1999, p. 307-331.

[21] E. Leichtnam, P. Piazza. On higher eta invariants and metrics of positive scalar curvature. Preprint June 2000.

[22] J. Lott. Superconnections and higher index theory GAFA, 2, p. 421-454.

[23] J. Lott. Higher eta invariants, K-Theory, 6, 1992, p. 191-233.

[24] J. Lott. Signatures and higher signatures on S^{1}-quotients. Math. Annalen 316, 2000, p. 617-657.

[25] R. Mazzeo, P. Piazza. Dirac operators, heat kernels and microlocal analysis. Part II: Analytic surgery. Rendiconti di Matematica e delle sue applicazioni (Roma),18 1998.

[26] R. Melrose. The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, 4, 1993, A K Peters.

[27] R. Melrose, P. Piazza. Families of Dirac operators, boundaries and the b-calculus , Journal of Differential Geometry, 46, 1997, p. 99-180.

[28] R. Melrose, P. Piazza. An index theorem for families of Dirac operators on odd-dimensional manifolds with boundary , Journal of Differential Geometry, 46, 1997, p. 287-334.

[29] W. Neumann. Manifold cutting and pasting groups. Topology, 14, 1975, p. 237-244.

[30] P. Piazza. Dirac operators, heat kernels and microlocal analysis. Part I: Family index theory. Rendiconti del Circolo Matematico di Palermo, Supplemento, Serie II, 49 1997.

[31] N. Wegge-Olsen. K−Theory and C^{∗}−algebras, Oxford University Press, 1993.

[32] F. Wu. The noncommutative spectral flow, unpublished preprint, 1997.

[33] F. Wu. The Higher Γ−index for coverings of manifolds with boundaries, Fields Institute Communications, Volume 17, 1997, p. 169-183.

Institut de Jussieu et CNRS, Etage 7E, 175 rue du Chevaleret, 75013, Paris, France

   E-mail address: leicht@math.jussieu.fr

Dipartimento di Matematica G. Castelnuovo, Università di Roma “La Sapienza”, P.le Aldo Moro 2, 00185 Rome, Italy

   E-mail address: piazza@mat.uniroma1.it