K-THEORY OF C -ALGEBRAS OF B-PSEUDODIFFERENTIAL OPERATORS

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1 K-THEORY OF C -ALGEBRAS OF B-PSEUDODIFFERENTIAL OPERATORS RICHARD MELROSE 1 AND VICTOR NISTOR 2 Astract. We compute K-theory invariants of algeras of pseudodifferential operators on manifolds with corners and prove an equivariant index theorem for operators invariant with respect to an action of R k. We riefly discuss the relation etween our results and the η-invariant. Introduction In this paper we analyze the K-groups of the norm closure of the algera Ψ 0 (M) of -pseudodifferential (or totally characteristic) operators acting on the compact manifold with corners M. In the case of a compact manifold with oundary this class of operators was introduced in [14], see also [15] and [11]. For the general case of a compact manifold with corners it was descried in [20]. There are closely related algeras which have the same completion, see [17]. The algera Ψ 0 (M) can e identified with a -closed sualgera of the ounded operators on L 2 (M) = L2 (M, Ω ) (corresponding to a logarithmically divergent measure), and its Fredholm elements can then e characterized y the invertiility of a joint symol consisting of the principal symol, in the ordinary sense, and an indicial operator (as for Fuchsian differential operators) at each oundary face, which arises y freezing the coefficients at the oundary face in question. In view of the invariance of the index with respect to small perturations [9], we consider (as in the case M = for the Atiyah-Singer index theorem, [3, 24]) the C -algera otained y norm closure, which we denote A(M). Its K-theory is easier to compute than that of the uncompleted algera. Just as in the case of a manifold without oundary, the principal symol map σ has a continuous extension to A(M) with values in C( S M), where S M S M as manifolds. The algera A(M) contains the algera of compact operators on L 2 (M), denoted K(L 2 (M)). Let Q(M) = A(M)/K(L2 (M)) e the quotient. If M =, then Q(M) is isomorphic to the algera C(S M) of symols. In the general case, we call Q(M) the algera of joint symols, since it involves oth the principal symol and extra morphisms giving the indicial operators. A model space N + H is associated to each oundary hypersurface H of M. As a manifold with corners N + H = [ 1, 1] H carries a natural action of R + = (0, ). This allows us to introduce the (completed) indicial algera at H, denoted A(H, M) and consisting of the R +-invariant elements of A(N + H). The indicial morphism at H localizes Date: Septemer 10, Partially supported y NSF grant DMS Partially supported y NSF Young Investigator Award DMS and a Sloan research fellowship. Preprints are availale y ftp from ftp-math-papers.mit.edu and from rm/rm-home.html or 1

2 2 RICHARD MELROSE AND VICTOR NISTOR A A(M) to In H,M (A) A(H, M). The joint symol map is the direct sum of the principal symol and the indicial operators at all oundary hypersurfaces. Its range is suject to compatiility conditions etween symol and indicial operators and etween indicial operators at the intersections of oundary hypersurfaces. Using these indicial maps, we construct a composition series for A(M) : (1) A(M) I 0 I 1... I n, n = dim M. The suquotients of this composition series are identified in Theorem 2: ( I l /I l+1 = C 0 R n l ; K(L 2 (F )) ), 0 l n, F F l (M) as the sum over the oundary faces of dimension l of the C algeras of continuous functions vanishing at infinity on R n l and taking values in the compact operators on an associated Hilert space (of dimension one when l = 0). The end cases are I n = K(L 2 (M)) and A(M)/I 0 = C( S M). The K-theory of each of these suquotients is readily computed, and this leads to a spectral sequence for the K-theory of A(M). To deduce the composition series (1), we first descrie joint symol maps at dimension l in the smooth (i.e. uncompleted) setting; the ideals I l are the completions of the null spaces of these morphisms. To show the appropriate exactness properties for the morphisms otained y continuous extension, we use lifting properties for the symol and indicial morphisms. In the particular case of a compact manifold with oundary, as already noted, the principal symol map induces an isomorphism of K 0 -groups, whereas each component of the oundary contriutes an extra copy of Z to K 1 ; this can e attriuted to spectral flow invariants [2]. More precisely, if M has q components, there is a short exact sequence 0 Z q K 1 (Q(M)) σ K1 (C( S M)) 0. If the oundary is connected, the index morphism Ind : K 1 (Q(M)) Z provides a splitting of this exact sequence. In a forthcoming note we will discuss the surjectivity of σ in relation to oundary conditions of Atiyah-Patodi-Singer type for elliptic operators, and use that discussion as a model for Fredholm oundary conditions on general manifolds with corners. We also compare the algeraic and topological K-theory of the uncompleted algera Ψ 0 (M), and therey interpret a result in [16] on the η-invariant in this setting. We conclude the paper with some results on the equivariant index of operators on manifolds equipped with a proper action of R k. In summary the contents of this paper are as follows. In the first section we recall ackground material and notation concerning manifolds with corners. In 2 the symol map and indicial morphisms for the algera of -pseudodifferential operators are discussed. In the next section the alternative description of the indicial morphism in terms of indicial families, otained y taking the Mellin transform, is descried. In 4 the continuous extension of the symol map to the closure of the algera of the -pseudodifferential operators in the ounded operators on L 2 is considered. We would like to thank Is Singer for a helpful discussion and Roert Lauter for alerting us to his related work in [13].

3 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 3 1. Manifolds with corners We shall work in the context of smooth manifolds with corners M. By definition, in such a space every point p M has coordinate neighorhoods diffeomorphic to [0, ) k R n k, where n is the dimension of M, k = k(p) is the codimension of the face containing p, and p corresponds to 0 under this isomorphism. The transitions etween such coordinate neighorhoods must e smooth up to the oundary; this is the same as eing extendile smoothly across the oundary. An open face is a path component of the set k M of all points p with a fixed k = k(p). The closure, in M, of an open face will e called a oundary face, or simply a face. A oundary face of codimension one may e called specifically a oundary hypersurface. In general, such a oundary face does not have a covering y coordinate neighorhoods of the type descried aove, ecause oundary points may e identified. To avoid this prolem, we demand, as part of the definition of a manifold with corners, that the oundary hypersurfaces e emedded. More precisely, this means that we assume that, for each oundary hypersurface H of M, there is a smooth function ρ H 0 on M, such that (2) H = {ρ H = 0}, where d(ρ H ) 0 at H. If p F, a face of codimension k, then exactly k of the functions ρ H vanish at p. Denoting them ρ 1,..., ρ k, the differentials dρ 1,..., dρ k must e linearly independent at p; it follows that the addition of some n k functions (with independent differentials on F at p) gives a coordinate system near p; in fact, this is what we shall mean y a coordinate system at p. We denote y F(M) the set of oundary faces of M, y F 1 (M) the set of oundary hypersurfaces H F(M) (i.e. faces of codimension 1) and, more generally, y F l (M), for 0 l n = dim M, the set of oundary faces of codimension l. It is also convenient to let F l (M) = F n l (M) denote the set of oundary faces of dimension l. In view of the assumed existence of oundary defining functions, (2), they are all manifolds with corners. Without loss of generality, it can e assumed that M is connected, and hence that there is a unique face of codimension 0, namely M. If F F l (M) then F k (F ) F k+l (M) consists of those G F k+l (M) which are contained in F. It is useful to make a choice of functions ρ H as in (2) and fix a metric h which locally at any point p has the form h = (dx 1 ) (dx k ) 2 + h 0 (y 1,..., y n k ), where x 1,..., x k, y 1,..., y n k are some local coordinates at p, and x 1 = ρ H1,..., x k = ρ Hk are the chosen defining functions. The existence of such a metric is shown in [10], for example. The choice of the functions ρ H for all H F 1 (M) estalishes a trivialization NF F R k of the normal undle to each oundary face. In fact, these undles are naturally decomposed as sums of trivial (ut not canonically so) line undles; namely the normal undles to the hypersurfaces containing F (3) NF = N F H. H F 1 (M), H F

4 4 RICHARD MELROSE AND VICTOR NISTOR We denote y N + F NF the closed set of normal vectors that point into the manifold M. They are exactly those vectors which have non-negative x-components. The group (0, ) k = R k + acts naturally on N + F y dilations. Consider the projective compactification of the closed half-line (as in (54) in the appendix) (4) [0, ) s s 1 [ 1, 1]. s + 1 The multiplicative action of (0, ) on [0, ) lifts to e smooth on [ 1, 1], so the k- fold application of this compactification emeds the inward pointing normal undle N + F = [0, ) k F to any oundary face of a manifold with corners into N + F = [ 1, 1] k F with the C structure on the compactification independent of the choice of oundary defining functions used to produce the trivialization; the action of R k + lifts to e smooth on N + F. These compactified inward-pointing normal undles to the oundary faces play an important rôle in the localization of operators at the oundary. In particular, the space N + F is a model for M near F. If G F is a pair of oundary faces then the closure in N + F of the union N + G F of the fiers over G forms a oundary face G NF N + F. Use of the oundary defining functions shows that there is a natural identification of the compactified inward-pointing normal undle of G NF, as a oundary face of N + F, with N + G : (5) N + G NF N + G. Now the action of (0, ) k y dilations lifts to an action on L 2 (N + F, Ω 1 2 ) : (6) λ ɛ (u)(x 1,..., x k, y 1,..., y n k ) = u(ɛ 1 1 x 1,..., ɛ 1 k x k, y 1,..., y n k ), ɛ = (ɛ 1,..., ɛ k ). This action is independent of the choice of defining functions; here Ω is the - density undle, with gloal section ν = ν/ H F 1 (M) ρ H. The exponential map associated to the Levi-Civita connection of a metric of product type, as descried aove, gives a diffeomorphism from a neighorhood V F of the zero section in N + F to an open neighorhood of F in M : Φ F = exp : V F M, V F N + F. Due to the particular choice of the metric h, Φ F is a diffeomorphism of manifolds with corners, which maps the zero section of NF onto F. Let ϕ F e a smooth function on M, 0 ϕ F 1, supported inside Φ F (V F ), and such that ϕ F = 1 in a neighorhood of F. Later we shall later use the maps (7) L F : L 2 (N + F, Ω 1 2 ) = L 2 (N + F, dx 1... dx k dy 1... dy n k ) L 2 (M, Ω 1 2 x 1... x ), k where L F (u) = ϕ F (u Φ 1 F ). The maps L F are well defined since supp ϕ F Φ F (V F ). The -pseudodifferential operators considered here are otained y a process of microlocalization of the Lie algera, V (M), of smooth vector fields which are tangent to all the oundary faces. As such, they are closely related to the -cotangent undle T M. This undle is naturally defined over any manifold with corners. Over the interior T M is canonically identified with T M, ut at a oundary point p

5 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 5 its fier is the space of equivalence class of differentials dρ H (8) a H + dφ, a H R, H F 1 (M), φ C (M), ρ H p H modulo the space of smooth differentials which, after eing pulled ack to F = H, vanish at p. It can e defined more naturally as the dual undle to the p H undle T M, with sections consisting precisely of the space V (M). 2. The algera of -pseudodifferential operators Two definitions of the algera of -pseudodifferential operators are recalled in the appendix. The most accessile of these starts from an explicit description of the algera Ψ (M) for the special, model, case of M = [ 1, 1]n. The general case is then otained y localization and Ψ (M) consists of operators from C c (M) to C (M). If M is a manifold without oundary this definition reduces to that of 1-step polyhomogeneous (i.e. classical) pseudodifferential operators in the usual sense. The second approach, readily shown to e equivalent to the first, is to define the appropriate class of kernels directly on a stretched version of M 2. This intrinsically gloal approach has the virtue of making many of the proofs elow transparent. We shall e concerned mainly here with the algera Ψ 0 (M) of (1-step polyhomogeneous) -pseudodifferential operators of non-positive integral order on M, a given compact manifold with corners. It is a -closed algera of ounded operators on L 2 (M) and is a Frechet space. As in the oundaryless case, the principal invariant of a pseudodifferential operator is its principal symol, it is a function on the -cotangent space. Let S M e the quotient of T M \ 0 y the fier action of (0, ) and let P m e the undle over S M with sections which are homogeneous functions of degree m on T M. Proposition 1. There is a natural short exact sequence (9) 0 Ψ m 1 (M) Ψ m (M) σ m C ( S M; P m ) 0, which is multiplicative if M is compact, where σ m (A) is determined y oscillatory testing in the sense that if ψ Cc (M), φ C (M) is real valued and a H R are such that the corresponding section α of T M given y (8) is non-vanishing over the support of ψ, then (10) σ m (A; α)ψ = lim λ λ m p H ρ iλa H H eiλφ A ( ρ iλa H H e iλφ ψ ). p H In case m = 0, we simplify the notation and write σ 0 = σ; the undle P 0 is canonically trivial, so the short exact sequence (9) ecomes (11) 0 Ψ 1 (M) Ψ 0 σ (M) C ( S M) 0. The algera Ψ 0 (M) acts as ounded operators on L2 (M), and (10) gives σ(a) A. However, when the oundary is non-trivial, the ideal Ψ 1 (M) does not map into the compact operators. To capture compactness, we need to consider the localization of the operators at oundary faces. To do so, we introduce a sualgera of Ψ 0 (N + F ),

6 6 RICHARD MELROSE AND VICTOR NISTOR where F F(M) and N + F is the compactified inward-pointing normal undle discussed aove. Definition 1. If M is a compact manifold with corners then, the indicial algera Ψ,I (N + F ) corresponding to a oundary face F M is the algera consisting of those -pseudodifferential operators on N + F which are invariant under the natural R k + action (6). The operators T Ψ,I (N + F ) of order at most m form a suspace denoted Ψ m,i (N + F ), so Ψ,I (N + F ) = m Ψm,I (N + F ). These spaces are delineated y the symol maps σ m defined aove, with M replaced y N + F. The R k + action on S N + F makes it a undle over SF M, the restriction of S M to F, with fier [ 1, 1] k. The symols of elements of Ψ 0,I (N + F ) are invariant under this action, so we define a reduced symol map (12) σ F : Ψ 0,I(N + F ) C ( S F M). It gives rise to a short exact sequence for the indicial operators (13) 0 Ψ 1,I (N + F ) Ψ 0,I(N + F ) σ F C ( S F M) 0. Every -pseudodifferential operator has an invariant indicial operator at each oundary face. To define it, let L F e as in (7). Theorem 1. For any oundary face F F(M) there is a surjective morphism In F,M : Ψ 0 (M) Ψ 0,I(N + F ) independent of any choices and uniquely determined y the property (14) In F,M (T )u = lim ɛi 0 (λ ɛ 1L F T L F λ ɛ )u for any u C c (N + F ), ɛ = (ɛ 1,..., ɛ k ), k eing the codimension of F. Although this is a asic result of the calculus, we outline a local proof and then descrie the gloal approach. Proof. Suppose that T Ψ 0 (M). As discussed in the Appendix, T is locally of the form (55). If x 1,..., x k are defining functions for the face to which p elongs and y 1,..., y n k are additional local coordinates then the defining formula (55) reduces to (15) T u(x, y) = 0 0 R n k T (x 1,..., x k, x 1,..., x k, y 1,..., y n k, y 1,..., y n k) u(x 1x 1,..., x kx k, y 1,..., y n k) dx 1 x 1... dx k x dy 1... dy n k, k where now T (s, x, y, y ) is conormal at x i = 1, y = y or smooth as the localizing functions are in the same or different coordinate patches; it is still rapidly decreasing as x i 0 or and now has compact support in y, y.

7 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 7 Since the computation is local, we can assume that T = L F T L F, and then (16) (λ ɛ 1T λ ɛ )u(x, y) = T (ɛ 1 x 1,..., ɛ k x k, x 1,..., x k, y 1,..., y n k, y 1,..., y n k) 0 0 R n k u(x 1x 1,..., x kx k, y 1,..., y n k) dx 1 x 1... dx k x dy 1... dy n k k (after a dilation in the x -variales). This shows immediately the existence of the limit as ɛ 0 in the statement, and that the the localized indicial operator is given y (17) In F,M (T )u = 0 0 R n k T (0,..., 0, x 1,..., x k, y 1,..., y n k, y 1,..., y n k) u(x 1x 1,..., x kx k, y 1,..., y n k) dx 1 x 1... dx k x dy 1... dy n k, k for any u C (N + F ), this is an element of Ψ 0,I (N + F ). To see the surjectivity of the indicial morphism for F, it is enough to work locally on F 2, since the invariance properties are preserved under such localization. Thus T can e assumed to have support in a product of coordinate patches, so takes the form (17). Inserting cut-off factors φ(x j ) and ψ(x j /x j ), for j = 1,..., k, where φ, ψ C (R) have supports near 0 and 1, respectively, and satisfy φ(0) = 1 and ψ(1) = 1, gives an element T Ψ 0 (M) with In F,M (T ) = T. In the gloal description of the kernels as distriutions on M 2, the stretched product of M with itself, the indicial morphism simply corresponds to the restriction of the kernel to a oundary face of M 2. Let H 1,..., H k e the oundary hypersurfaces containing F. Each of the oundary faces H i H i is lown up in the construction of M 2 from M 2 so corresponds to a oundary hypersurface ff(h i ) F 1 (M 2 ). Consider the component lying aove F of the intersection of these ff(h i ). It is canonically isomorphic to the corresponding face in the stretched product of the model space at F, (N + F ) 2, and In F,M (A) is the unique element of Ψ m,i (N + F ) with kernel having the same restriction as A to this face. Recall that if G F are oth oundary faces of M, then G determines a oundary face G F of N + F ; using the identification (5), the oundary maps can e iterated and identified directly from the formulæ in the proof aove. Corollary 1. If G F are oundary faces of M, then the indicial maps satisfy (18) In GF,N + F In F,M = In G,M. The null space of the indicial map for a oundary hypersurface is easily seen from the local coordinate discussion aove, or even more readily from the more direct gloal definition. Namely, for each H F 1 (M), there is a short exact sequence (19) 0 ρ H Ψ m (M) Ψ m (M) In H,M Ψ m,i(n + H, M) 0. This has a useful extension to several hypersurfaces.

8 8 RICHARD MELROSE AND VICTOR NISTOR Lemma 1. If H i F 1 (M), for i = 1,..., L, is a collection of oundary hypersurfaces, the joint null space of the indicial maps In Hi,M is ρ 1... ρ L Ψ 0 (M), where ρ i = ρ Hi are defining functions for the H i. Proof. Proceed y induction over L. By (19), if In H1,M (T ) = 0, then T = ρ 1 T 1. Now, from the fact that In Hi,M is a morphism: 0 = In Hi,M (T ) = In Hi,M (ρ 1 T 1 ) = ρ 1 Hi In Hi,M (T 1 ) = In Hi,M (T 1 ) = 0, i > 1. Applying the inductive hypothesis to T 1, for these L 1 hypersurfaces, gives the inductive hypothesis for L hypersurfaces. The -pseudodifferential operators of order m define ounded operators on the natural Soolev spaces A : H l (M) Hl m (M), for any l. As such, an operator A is compact if and only if its symol vanishes (hence it is in Ψ m 1 (M)) and all its indicial operators vanish, so it is in ρψ m (M), where ρ = H F 1 (M) ρ H, (and so, in fact, is in ρψ m 1 (M)). Let us note some examples of -pseudodifferential operators. If M is a manifold with corners and M M is emedded in a manifold without oundary, of the same dimension (say y douling across the oundary hypersurfaces), then the pseudodifferential operators of order m on M with kernels supported in M 2 are in Ψ m (M). Examples of the indicial operators can e otained in a similar way. Consider a pseudodifferential A operator of order m on R k F, where F =, which is invariant under all translations in R k and has its convolution kernel (on R k F 2 ) compactly supported. Then compactifying R k to [ 1, 1] k y first mapping each component x i R to t i = exp(x i ) (0, ) and then using the projective compactification, (4), gives an operator in Ψ m ([ 1, 1]k F ) which is (R +) k invariant. If F is realized as a oundary face of any manifold with corners M, this construction gives many elements of Ψ m,i (N + F ), enough to span the space modulo Ψ,I (N + F ). 3. Indicial family The indicial morphism is closely related to the fact that Ψ m (M) is invariant under conjugation y complex powers of each oundary defining function, i.e. (20) Ψ m (M) A ρ z H Aρz H Ψ m (M), z C is an isomorphism. Taking z = 1, it follows that Aρ H v = ρ H (ρ 1 H Aρ H)v vanishes on H F 1 (M), for any v C (M). Thus, if u C (H), then (21) A H u = (Aw) H, w C (M), w H = u defines an operator on C (H). This restriction map is a surjective morphism (22) Ψ m (M) H Ψ m (H). Using the Mellin transform, the relationship etween In H,M (A) and A H is easily seen to e In H,M (A)(dρ H ) z f = (dρ H ) z (ρ z H Aρz H) H f, where dρ H is a well-defined function on NH and hence a distriution on N + H. This follows directly from the limiting formulæ in the proof of Theorem 1. For a

9 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 9 oundary face of codimension k, the analogous result holds using the k defining functions for the oundary hypersurfaces containing F. It is therefore natural to define the indicial family of A Ψ m (M) at F F k(m) y In F,M (A; z 1..., z k ) = ( ρ z 1 1 ρ z 2 2 ρ z k k Aρ z 1 1 ρz 2 2 ) ρz k (23) k F Ψ m (F ), where A Ψ m,i (N + F ). Note that the definition does depend on the choice of defining functions for the oundary hypersurfaces containing F. Although it is straightforward to characterize the range of the map in (23), less precise information suffices for our purposes elow. Proposition 2. The indicial family In F,M (A; z) determines A Ψ m,i (N + F ) in the sense that if InF,M (A; z) vanishes for all z R k then A = 0. For any A (F ). If Ψ m,i (N + F ), InF,M (A; z) is an entire function of z C k with values in Ψ m m < 0, then, as operators on the Soolev spaces L 2 (F ) H m/2 (F ), (24) In F,M (A; z) 0, 1 2 m C(1 + z ) 1 2 m, z R k. InF,M The range of includes all entire functions of g(z) with values in the space C (F 2 ; πr Ω F ), of fully smoothing kernels, satisfying the estimates (25) sup (1 + z ) p f(z, ) <, Iz C for every C, p and seminorm on C (F 2 ; π R Ω F ) Proof. Consider the second result first. Fixing a positive gloal section of Ω, the elements of Ψ (M) correspond to smooth functions on M 2 vanishing to infinite order at all oundary hypersurfaces other than the ff(h), H F 1 (M). In the case of N + F, the elements of Ψ,I (N + F ) correspond exactly to those elements of C (F 2 [ 1, 1]k ) vanishing on all oundary hypersurfaces other than the ff(g) [ 1, 1] k, G F 1 (F ). In particular, (26) C (F 2 [ 1, 1] k ) = C (F 2 [ 1, 1] k ) Ψ,I (N + F ), since these are the smooth functions vanishing to infinite order at all oundary faces. Since the indicial family is otained y taking the Mellin transform in each of the variales in [1, 1] k, the Paley-Wiener theorem shows that entire smoothing operators satisfying (25) are in the range of In F,M. The first part of the statement follows from similar standard estimates for the Mellin transform (and hence the Fourier transform). 4. Joint symols By comining the definitions of the symol map in (10) and of the indicial operator in (14), the compatiility condition etween the two is immediately apparent σ F (In F,M (T )) = σ(t ) F, T Ψ 0 (M). These are the only compatiility conditions. This can e formalized y defining the joint symol j(t ) = σ(t ) In H,M (T ) C ( S M) Ψ 0,I(N + F ). H F 1 (M) H F 1 (M)

10 10 RICHARD MELROSE AND VICTOR NISTOR In view of (18), all the indicial operators for oundary faces of codimension greater than 1 can also e extracted from j(t ). Proposition 3. The joint symol map has as range the suspace { (a, SH ); a C ( S M), S H Ψ 0,I(N + H) such that σ H (S H ) = a H and In GF,N + H (S H) = In GF,N + H (S H ), F H H, F F(M) }. Proof. That these compatiility conditions on the range hold has already een shown. To prove the surjectivity of j it is convenient to prove the more general statement that for any collection of oundary hypersurfaces H i and and S i Ψ 0,I (N + H i ) there exists T Ψ 0 (M) with In H i,m (T ) = S i proved that the corresponding compatiility conditions are satisfied, that whenever F F(M) and F H i H j InGF,N (Si) = +Hi InGF,N +Hj (Sj). This is the desired result for the set of all oundary hypersurfaces and is already known from the exactness in (19) for one hypersurface. We proceed y induction over the numer L of hypersurfaces. By the surjectivity of the indicial map at H 1 we can choose T 1 Ψ 0 (M) so that In H 1,M (T 1 ) = S 1 ; set S i (1) = In H i,m (T 1 ), for i > 1. Consider the differences, S i S i Ψ0,I (N + H i ), for each i > 1. By Corollary 1 if F F(M) is a component of H H i and F i is the corresponding face of H i then In Fi,N + H i (S i S i ) = 0. Since ρ H Hi is a product of defining functions for these oundary faces, as oundary hypersurfaces of H i it follows that S i S i = ρ H Hi S i, with S i Ψ 0,I (N + H i ). Now the S i satisfy the compatiility conditions for the remaining L 1 hypersurfaces, therefore, y the inductive hypothesis, there exists T Ψ 0 (M) with In H i,m (T ) = S i, for i > 1. Then T = T 1 + ρ H T satisfies the requirements of the inductive hypothesis. More generally, if F is a oundary face of M, we can define a joint symol morphism for the indicial algera at F y (27) j F (T ) = σ F (T ) In H,F,M (T ) C ( SF M) Ψ 0,I(N + H). H F 1 (F ) H F 1 (F ) The same argument as in the proof of the proposition aove identifies the range of this morphism as the set of operators satisfying the ovious compatiility conditions: R F,M = { (f, T H ) C ( SF M) Ψ 0,I(N + H); H F 1 (F ) In H,F,M (T H ) = In H,F,M (T H ), H, H F 1 (F ) and σ H (T H ) = f H }. Proposition 4. For any oundary face F of M, the joint symol map at F gives a short exact sequence (28) 0 ρ F Ψ 1,I (N + F ) Ψ 0,I(N + F ) R F,M 0, where ρ F C (F ) is the product of oundary defining functions for the oundary hypersurfaces of F.

11 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 11 Comining the indicial operators at the oundary faces of a given dimension with the symol, consider (29) j l : Ψ m (M) C ( S M) Ψ m,i(n + F ), j l (A) = (σ m (A), In F,M ). F F l (M) The range of this map is the suspace satisfying the appropriate compatiility conditions on a and A F Ψ m,i (N + F ), F F l (M): (30) σ m (A F ) = a SF M, In G,F,M (A F ) = In G,F,M (A F ), F F l 1 (M), G F F. The null space is simply (31) with j F given in (27). ker(j l ) = {A Ψ 1 (M); In G,M = 0, G F l (M)}, 5. The norm closure, A(M) Next we discuss the properties of the algera, A(M), otained y taking the norm closure of Ψ 0 (M) as an algera of ounded operators on L2 (M). The compactified normal space of a oundary face is a special case and then we also denote y A(F, M) A(N + F ) the closure in norm of the invariant sualgera Ψ 0,I (N + F ) Ψ 0 (N + F ). The closure in norm of Ψ 1,I (N + F ) Ψ 0,I (N + M) will e denoted A (F, M) A(F, M). Thus A(M) = A(M, M) and A (M, M) will e similarly denoted A (M). Notice that A (F, M) is also the closure of Ψ,I (N + F ) Ψ 1,I (N + F ), since y standard properties of conormal distriutions, Ψ,I (N + F ) is dense in Ψ 1,I (N + F ) in the topology of ounded operators on L 2 (M). The same argument shows that A (F, M) is the closure of Ψ ɛ,i (N + F ) for any ɛ > 0. Each of these norm closed algeras of operators on a Hilert space is closed under conjugation, so y the theorem of Gelfand and Naimark they are all C -algeras. Below we will use the fact that any algeraic morphism of C -algera is continuous and has closed range [8]. In particular, as in the case of a compact manifold without oundary the symol map extends y continuity. For a locally compact space X, we shall denote y C 0 (X) the C -algera of those continuous functions on X that vanish at infinity. It is the norm closure in supremum norm of the algera C c (X) of continuous compactly supported functions on X. If X is a smooth manifold the set of compactly supported smooth functions will e denoted y Cc (X); it is also dense in C 0 (X). Proposition 5. The symol maps in (13) and (12) extend y continuity to surjective maps (32) σ : A(M) C( S M) and σ F : A(F, M) C( S F M). Proof. Consider first the full algera A(M). From the oscillatory testing property of the principal symol map, Proposition 1, it follows that σ F (T ) T, for all T Ψ 0 (M). Moreover, the principal symol morphism σ F is a -morphism, i.e. it satisfies σ F (T ) = σ F (T ).

12 12 RICHARD MELROSE AND VICTOR NISTOR Consequently its range is closed [8]. The same is true for the indicial algeras, just replacing M y N + F. Since the range of σ F contains C ( SF M), which is dense in C( SF M), the maps in (32) are surjective. Essentially the same proof shows that the indicial morphisms also extend to the norm closed algeras introduced aove. Proposition 6. For any oundary faces F of M, the indicial morphisms extend to surjective maps In F,M : A(M) A(F, M), and for any pair of oundary faces G F (33) In G,F,M : A(F, M) A(G, M) is defined y continuous extension of In GF, and hence satisfies,n + F In F,F,M = In F,M,M = In F,M for any triple of oundary faces F F F. and In F,F,M In F,F,M = In F,F,M Proof. It follows from the definition of the indicial morphisms In F,M given in Theorem 1 that they satisfy In F,M (T )u = lim ɛi 0 (λ ɛ 1L F T L F λ ɛ )u T u and hence In F,M (T ) T. This show that In F,M extends y continuity to the norm closure. The surjectivity follows from the corresponding surjectivity of the indicial maps in Theorem 1; the remainder of the proof now follows from Corollary Cross-sections In order to analyze the null spaces of the symol map, (32), and of the indicial morphism, (33), we construct a cross-section for In G,F,M. Proposition 7. For each F F(M) and each G F 1 (F ), there is a linear map λ F,G : A(G, M) A(F, M) with the following properties: (34) λ F,G (Ψ 0,I(N + G)) Ψ 0,I(N + F ), (35) (36) In G,F,M λ F,G (T ) = T, T A(G, M), λ F,G (T ) T, and, whenever G F(F ) is another face with G G, then (37) In G,F,M λ F,G (T ) = 0, if G G = In G,F,M λ F,G (T ) = λ G,K In K,G,M (T ), if K is a component of G G. Note that in (37), G G is either empty or else is a non-trivial union of oundary hypersurfaces of G. Proof. Initially, we define λ H,M, for every H F 1 (M), y the formula λ M,H (T ) = L H T L H, where L H is as in (7). For an aritrary pair (F, G), as in the statement, there exists a unique H F 1 (M) such that G is a component of F H; let the other components e G i,

13 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 13 i = 1,..., L. Using the local coordinate representations aove we see that the indicial operator In F,M λ M,H (T ) depends only on In G,H (T ) and the In Gi,H(T ). Hence we can define the linear section λ F,G y the requirement (38) λ F,G In G,H (T ) = In F,M λ M,H (T ), T A(H, M) with In Gi,H,M (T ) = 0, i = 1,..., L. Here we use the disjointness of these oundary hypersurfaces to conclude that In G,H.M is still surjective onto A(G, M) when the domain is restricted as in (38). Consider now three faces F, G and G, G F 1 (F ), G F, as in the statement of the proposition. Let H F 1 (M) e such that G is a component of F H, with the G i as aove. Then, for all T A(H, M) with In Gi,H(T ) = 0 Thus In G,F λ F,G In G,H (T ) = In G,F In F,M λ M,H (T ) = In G,F λ M,H. In G,F λ F,G In G,H (T ) = 0, if G H =. This is enough to conclude the proof in the case G G =. On the other hand, if G H and K is one of its components, then it is necessarily a oundary hypersurface of G, so λ G,K is defined and In G,F λ F,G In G,H (T ) = λ G,K In K,G In F,F 0 (T ), where we have used the definition (38), the properties of the indicial morphisms proved in the previous proposition. This completes the proof of the Proposition. By placing extra conditions on the functions φ F it is actually possile to define λ G,F satisfying In G,F λ G,F = Id and λ G,G λ G,F = λ G,F. Corollary 2. Let T F A(F, M), respectively T F Ψ 0,I (N + F ), F F 1 (F ), satisfy the compatiility condition In G,F (T F ) = In G,F (T F ) for all pairs F, F and any connected component G of F F. Then we can find T A(F, M), respectively T Ψ 0,I (N + F ), such that T F = In F,F (T ) and T C max T F, where the constant C > 0 depends only on the face F. Proof. Let F 1 (F ) = {F 1, F 2,..., F m } and define T 1 = λ F,F1 (T F1 ) A(F, M) (respectively T 1 Ψ 0,I (N + F )) and T l+1 = T l + λ F,Fl+1 (T Fl+1 In Fl+1,F (T l )). We will prove y induction on l that In Fj,F (T l ) = T Fj for all indices j l. Indeed, for l = 1, this is the asic property of the sections λ. We first prove the inductive statement for l + 1 and j l. If F j F l+1 =, then In Fj,F λ F,Fl+1 = 0. If F j F l+1, we have In Fj,F λ F,Fl+1 = λ Fj,F j F l+1 In Fj F l+1,f l+1.

14 14 RICHARD MELROSE AND VICTOR NISTOR Using the inductive hypothesis for l and the compatiility relation from the assumptions of the Corollary, we have In Fj F l+1,f l+1 (T Fl+1 In Fl+1,F (T l )) = In Fj F l+1,f l+1 (T Fl+1 ) In Fj F l+1,f (T l ) = In Fj F l+1,f l+1 (T Fl+1 ) In Fj F l+1,f j In Fj,F (T l ) = In Fj F l+1,f l+1 (T Fl+1 ) In Fj F l+1,f j (T Fj ) = 0. Thus in oth cases In Fj,F (T l+1 ) = In Fj,F (T l ) = T Fj. Finally, for j = l + 1, In Fj,F λ F,Fl+1 = Id and In Fl+1,F (T l+1 ) = In Fl+1,F (T l ) T Fl+1 In Fl+1,F (T l ) = T Fl+1. From this construction we may take C = 3 m 1. It is also possile to construct a cross-section for the symol map; however, we content ourselves with the existence of suitale liftings. Proposition 8. For any compact manifold with corners M there is a constant C such that for any a C ( S M) there exists A Ψ 0 (M) with (39) σ(a) = a and A L 2 (M) C a L ( S M). In the case of a normal space N + F, if a C ( SF M), then A can e chosen in Ψ 0,I (N + F ). Proof. It suffices to assume that a is real-valued. We will prove the general statement following (39) using induction over the maximal codimension of oundary faces for F (not M, the manifold of which it is a oundary face.) Note that we already know the symol map to e surjective, it is the norm estimate on an element in the preimage of a that we need. The asic case where M is compact without oundary is well known. Indeed, for any A Ψ 0 (M) with symol a, the spectrum of A outside the disk of radius a L (S M) is discrete and consists of finite rank smooth eigenspaces. Since A can e replaced y its self-adjoint part, it splits as a sum of the orthogonal actions on the eigenspaces corresponding to eigenvalues in z 2 a L and those outside this disk. The latter part is a smoothing operator, so sutracting it gives (39) with C = 2. To complete the initial step in the induction we need the more general, indicial, case with F a manifold without oundary which is a oundary face of M. Thus, given a C ( SF M), we need to find A Ψ0,I (N + F ) with symol a and satisfying (39). We can replace M y N + F = [ 1, 1] k F and then a can e interpreted as a smooth function on the sphere undle of R k T F. Consider the compact manifold F = T k F, T k = R k /Z k eing the standard torus. Now, T F = T k (R k T F ) under the standard R k action. Thus a can e interpreted as an R k invariant function on S F. As such the discussion for a compact manifold applies, and gives A 1 Ψ 0 ( F ) with symol a satisfying (39). In fact, A 1 can e taken to have kernel supported in any preassigned neighorhood of the diagonal in F 2 ; it is only necessary to take a sufficiently fine partition of unity, φ i on F and discard all terms φ i A 1 φ j, where the supports of φ i and φ j are disjoint. Furthermore, if this neighorhood is invariant under the diagonal R k action, it can e assumed that A is invariant, y averaging (over the dual torus). Now such a sufficiently small

15 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 15 neighorhood of the diagonal in F 2 can e identified unamiguously as the image under projection on oth factors of a neighorhood of the diagonal in (R k F ) 2 which is invariant under the diagonal R k action. The kernel A 1 lifts to a unique R k invariant kernel A with support in this neighorhood of the diagonal. As noted in 2 operators of this type are in Ψ 0 ([ 1, 1]k F ). Certainly, A has symol a, and the estimate (39) holds, for a larger C. Proceeding y induction suppose that the result is known for all faces F themselves having oundary faces only up to codimension k 1 in any compact manifold with corners. Suppose F has oundary faces up to dimension k and a C ( SF M) is given. Order the oundary hypersurfaces of F as H 1, H 2,..., H L, and let H i e the corresponding oundary faces of M. The inductive hypothesis applies to a H 1 giving A 1 Ψ 0,I (N + H i ). Using the section for indicial operators, choose A 1 Ψ 0,I (N + F ) with In Hi,F,M (A 1) = A 1. Sutracting the symol of A 1 from the given a, we can now assume that a H1 = 0. Proceeding successively with the oundary faces, we can assume that a Hi = 0, for i < j, provided we can then construct A j Ψ0,I (N + F ) with symol a j such that a j = a on H i, i j (so vanishes for i < j). Simply choose A j Ψ 0,I (N + F ) as aove, for j = 1, y extension of A j Ψ0,I (N + H j ) with symol A Hj. By construction and the properties of the section for indicial operators, the symols of all the In Hi,F,M (A j ), for i < j, vanish. Proceeding in this way, and then summing the A i over the oundary hypersurfaces of F gives an element A Ψ 0,I (N + F ) which satisfies the norm estimate (39) and has all In Hi,H,M (A ) with the correct symols. Thus we are reduced to the case that a C ( SF M) vanishes when restricted to each of the SH M with H a oundary hypersurface of F, i.e. vanishes at the oundary of F. Let ρ e the product of defining functions for the oundary hypersurfaces of F. Thus we can choose B Ψ 0,I (N + F ) with σ F (Bρ) = a. Select φ C (R) with 0 φ(r) 1, φ(0) = 1 and ρ(r) = 0, r > 1 2. Then the function φ(ρ/δ) C (F ) is 1 on the oundary ut with support in ρ < 1. In this case we can cut off close to the oundary of F. Thus Bρφ(ρ/δ) δ B and σ F (Bρφ(ρ/δ) = aφ(ρ/δ). Choosing δ small we are finally reduced to the case that the symol takes the form a = (1 φ(ρ/δ))a, so vanishes identically near the oundary of F. Returning to the eginning of the induction, we can simply doule F across all its oundary hypersurfaces to a manifold without oundary and choose an appropriately ounded A with symol a. Again cutting off the kernel near the oundary of F, in oth factors, does not change the symol and gives an element of Ψ 0,I (N + F ). This completes the inductive step. 7. Symol sequences Using these cross-sections, we can now analyze the short exact sequences for the symol maps on the completed algeras. Proposition 9. The symol map (12) gives a short exact sequence (40) 0 A (F, M) A(F, M) σ F C( SF M) 0, where A (F, M) is the norm closure of Ψ 1,I (N + F ).

16 16 RICHARD MELROSE AND VICTOR NISTOR Proof. Only the exactness at A(F, M) remains to e shown. By continuity of the symol map, the algera A (F, M) is contained in the null space of σ F, so suppose A A(F, M) and σ F (A) = 0. By definition, there is a sequence B n Ψ 0,I (N + F ) with B n A in norm. Continuity of the symol map shows that a n = σ F (B n ) 0 in L. Using Proposition 8 we can choose A n Ψ 0,I (N + F ) with σ(a n ) = a n and A n C a n 0. Then B n A n A in norm and σ F (B n A n ) = 0, so A A (F, M). For a given face F of M we are particularly interested in the joint symol morphism j F and the replacement for (28) for the completed algeras. Denote y K(H) the algera of compact operators on a Hilert space H, Proposition 10. For any oundary face F of codimension k in M the (continuous extension of) the joint symol map at F gives a short exact sequence (41) 0 K F,M A(F, M) R F,M 0, where there is an isomorphism of C algeras (42) K F,M C 0 (R k ; K(L 2 (F )) and R F,M = { (f, T H ) C( S F M) H F 1 (F ) A(H, M); In G,H,M (T H ) = In G,H,M (T H ), G, H, H F 1 (F ), G H H and σ H (T H ) = f H }. Proof. Use of the sections for the indicial morphisms and the lifting property for the symol map, as aove, shows that the norm completion of R F,M in (28) is precisely R F,M as defined aove. Similarly, as in the proof aove, the sequence (41) is exact if K F,M is interpreted as the norm completion of the null space, ρ F Ψ 1,I (N + F ) in (28). Thus the significant part of the proposition is the identification of the null space, equation (42). This identification follows from Proposition 2. Similar considerations apply to the maps j l in (29). Proposition 11. For each l, the map j l extends y continuity to a morphism defining a short exact sequence F F l (M) (43) 0 I l A(M) B l,m 0, where B l,m C( S M) Ψ 0,I (N + F ) is the sualgera fixed y the compatiility conditions in (30), and where the null space I l if given y (44) I l = {A A (M); In F,M (A) = 0, F F l (M)}, just the closure of the space in (31). 8. Composition series Using these results on the joint symols we can now see that the null spaces of the morphisms j l give a composition series for the completed algera.

17 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 17 Theorem 2. The norm closure A(M) of the algera of -pseudodifferential operators of order zero on the compact connected manifold with corners M has a composition series A(M) I 0 I 1... I n, n = dim(m), consisting of the closed ideals in (44); the partial quotients are and (45) I l /I l+1 σ 0 : A(M)/I 0 F F l (M) C0 (S M), C 0 (R n l, K(L 2 (F )), 0 l n. The composition series and the isomorphisms are natural with respect to maps of manifolds with corners which are local diffeomorphisms. The last isomorphism reduces to I n = K(L 2 (M)). Also I 0 /I 1 = C 0 (N F ) C 0 (R n ), F F 0 (M) since K(L 2 (F )) = C if F has dimension 0. F F 0 (M) Proof. The fact that the principal symol induces an isomorphism σ : A(M)/J 0 C( S M) was proved in Proposition 9. That the ideals I l form a composition series for A(M) follows directly from their definition in (43). To examine the partial quotients consider j l+1 acting on I l. Essentially y definition its null space is I l+1. Since the symol and the indicial operators on faces of dimension less than l already vanish on I l the map j l can e replaced y the direct sum of the symol maps at faces of dimension l + 1. In fact, this gives a short exact sequence j l+1 0 I l+1 I l F F l+1 (M) K F,M 0, j l = F F l+1 (M) In F,M. The surjectivity here follows from Proposition 10. The identification in (42) of K F,M now leads immediately to the isomorphisms in (45). The naturality of the composition series follows from the naturality of the principal symol and of the indicial maps. The indicial algeras A(F, M) have similar composition series which are compatile with the indicial morphisms. Theorem 3. The algera A(F, M) has a composition series A(F, M) J 0 J 1... J n, n = dim(f ), where J 0 = Ψ 1,I (N + F ) = ker σ F and J l is the closure of the ideal of order 1, pseudodifferential operators whose indicial parts vanish on all faces F F of dimension less than l. The partial quotients are determined y the natural isomorphisms σ : A(F, M)/J 0 C0 ( SF M), and J l /J l+1 C 0 (R n l, K(L 2 (F ))), 0 l n. F F l (F ) Proof. The proof consists of a repetition of the arguments in the proof of the preceding theorem, replacing M y F.

18 18 RICHARD MELROSE AND VICTOR NISTOR We have the following generalization of Proposition 10. Corollary 3. If F is a face of codimension k in M, then passage to indicial families gives an isomorphism A (F, M) C 0 (R k, A (F )). Proof. The map Ψ ( n 1),I (N + F ) C (R k, Ψ ( n 1),I (N + F 0 )). is compatile with the composition series of A(F, M) and A(F 0, M 0 ) of the aove Theorem and induces an isomorphism on the partial quotients (after completing in norm). The density property in the aove corollary completes the proof. Corollary 4. If M 1 and M 2 are two manifolds with corners then A (F 1 F 2, M 1 M 2 ) A (F 1, M 1 ) min A (F 2, M 2 ). The tensor product min is the minimal tensor product of two C -algeras and is defined as the completion in norm of A (M 1 ) A (M 2 ) acting on L 2 (M 1 M 2 ). (The space L 2 (M 1 M 2 ) is the Hilert space tensor product L 2 (M 1 ) ˆ L 2 (M 2 ), it is the completion of the algeraic tensor product L 2 (M 1 ) L 2 (M 2 ) in the natural Hilert space norm.) Proof. We will assume that F i = M i, the general case eing proved similarly. We have Ψ (M 1 ) Ψ (M 2 ) Ψ (M 1 M 2 ). From the density of Ψ,I (N + F ) in A (F, M), discussed at the eginning of 5, we conclude the existence of a morphism χ : A (M 1 ) min A (M 2 ) A (M 1 M 2 ) which preserves the composition series. Moreover, y direct inspection the morphisms induced y χ on the suquotients are isomorphisms. If follows that χ is an isomorphism as well. For a compact manifold with oundary the results can e made even more explicit. The theorem elow was also otained y Lauter [13]. Theorem 4. If M is a compact manifold with oundary then I 0 = I n 1, I n 1 /I n C 0 (R, K M ) and A(M)/I 0 = C 0 ( S M). The algera Q(M) = A(M)/I n has the following fiered product structure Q(M) Q C 0 ( S M) A( M), Q = {(f, T ), f M = σ M (T )}. The indicial algera of the oundary, A( M), fits into an exact sequence 0 G C 0 (R, K(L 2 (G)) A( M) σ M C 0 ( S M M) 0, where G ranges through the connected components of M. 9. Computation of the K-groups Our starting point for the computation of the K-groups of the algeras discussed in the previous section is the short exact sequence, of C -algeras, 0 K(L 2 (M)) A(M) Q(M) 0.

19 K-THEORY OF B-PSEUDODIFFERENTIAL OPERATORS 19 This exact sequence gives rise to the fundamental six-term exact sequence in K- theory (see [5]) (46) K 0 (K(L 2 (M))) K 0(A(M)) K 1 (Q(M)) K 0 (Q(M)) 0 K 1 (A(M)) K 1 (K(L 2 (M))), where we are particularly interested in the K-groups of Q(M). Now K 0 (K(L 2 (M))) Z, and K 1 (K(L 2 (M))) 0, so the right vertical map is zero. The left vertical arrow represents the index map. Consider an m m matrix P with values in the -pseudodifferential operators on M. If P is fully elliptic, in the sense that its image j(p ) in M m (Q(M)) is invertile, and hence defines an element [j(p )] K 1 (Q(M)), then (47) [j(p )] = Ind(P ) = dim ker P dim ker P Z K 0 (K(L 2 (M))), see [5, 6, 12]. We proceed to study the exact sequences 0 I l /I l+1 I l 1 /I l+1 I l 1 /I l 0 corresponding to the composition series descried in Theorem 2. We know that { K i (C 0 (R j Z if i + j is even, (48), K)) 0 otherwise. We shall fix these isomorphisms uniquely as follows. For j = 0, K 0 (K(L 2 (M))) Z will e the dimension function; it is induced y the trace. For j > 0 we define the isomorphisms in (48) y induction to e compatile with the isomorphisms Z K 2l j+1 (C 0 (R j 1, K)) K 2l j (C 0 ((0, ) R j 1, K)) K 2l j (C 0 (R j, K)), where the oundary map corresponds to the exact sequence of C -algeras 0 C 0 ((0, ) R j 1, K) C 0 ([0, ) R j 1, K) C 0 (R j 1, K) 0. For any C -algera A, set SA = C 0 (R, A) = C 0 (R) min A, S k A = C 0 (R k, A). Define F 0 = {(0,..., 0, 0)} R l 1, F 0 = {(0,..., 0)} [0, 1) R l 1, and M 0 = [0, 1) n l+1 R l 1, F 0 F 0 M 0. Also, let H = [0, 1) and 0 L 1 L 0 A(H) e the canonical composition series of A(H), such that L 0 = A (H), L 0 /L 1 C 0 (R), L 1 K, see Theorem 2. Lemma 2. With K 1 = K(L 2 (R l 1 )) there is a commutative diagram 0 ker(in F 0,F 0 ) A (F 0, M 0 ) A (F 0, M 0 ) 0 0 S n l L 1 K 1 S n l L 0 K 1 S n l+1 K 1 0

20 20 RICHARD MELROSE AND VICTOR NISTOR in which all vertical arrows are isomorphisms, the ottom exact sequence is otained from 0 L 1 L 0 C 0 (R) 0 y tensoring with C 0 (R n l, K 1 ), and the oundary map : K n l+1 (C 0 (R n l+1, K 1 )) K n l+1 (A (F 0, M 0 )) K n l (ker(in F 0,F 0 )) K n l (C 0 (R n l, K 1 )) is (the inverse of) the canonical isomorphism. Proof. Let H l 1 = [0, 1) l 1 and F 1 = {(0,..., 0)} H, F 1 H l 1. It follows from the corollary 4 that the algera A (F 0, M 0 ) is isomorphic to A (F 1, H l 1 ) A (R l 1 ). Moreover, A (R l 1 ) = K(L 2 (R l 1 )). The corollary 3 further gives A (F 1, H l 1 ) C 0 (R n l, A (F 1, F 1 )) = C 0 (R n l, A (H)). The first commutative diagram then is just an expression of the composition series of A (H), Theorem 2. Then an easy argument reduces the computation of the connecting morphism to that of the connecting morphism of the Wiener-Hopf exact sequence (i.e. the Wiener-Hopf extension). This is a well known and easy computation. It amounts to the fact that the multiplication y z has index 1 on the Hardy space H 2 (S 1 ) of the unit circle S 1. See [5] for more details. From Theorem 2 we then know that Z if n l + i is even K i (I l /I l+1 ) F F l (M) 0 otherwise. Here n = dim M. Fix from now on an orientation of the normal undle NF to each face F of M, including M itself. No compatiilities are required. This uniquely determines the aove isomorphisms. This choice of orientations fixes an incidence relation [F : G] etween oundary faces. If F / F 1 (G) and G / F 1 (F ), then we set [F : G] = 0. If F F 1 (G) then an orientation of NF induces canonically an orientation of NG. If this orientation of G coincides with the given one, then [F : G] = 1, if it is the opposite orientation, then [F : G] = 0. Finally, G F 1 (F ) then [F : G] = [G : F ]. Theorem 5. Suppose n l + i is even. Then the matrix of the oundary map : K i 1 (I l 1 /I l ) Z Z K i (I l /I l+1 ) F F l 1 (M) F F l (M) is given y the incidence matrix. If n l + i is odd, then = 0. Proof. Let e F K i 1 (I l 1 /I l ) and e F K i (I l /I l+1 ) e the canonical generators of these groups. We need to show that (e F ) = [F : F ]e F. F F l (M) The idea of the proof is to reduce the computation to the case M = M 0, F = F 0 and F = F 0 considered in the preceding lemma: M 0 = H n l+1 R l 1, F 0 = {(0,..., 0)} H R l 1, F 0 = {(0,..., 0, 0)} R l 1, and F = F 0 the face of minimal dimension. Choose a point p F. There exists a diffeomorphism ϕ : M 0 M of manifolds with corners onto an open neighorhood of p such that p ϕ(f 0). Since we