Atiyah-Bott index on stratified manifolds

A tiy ah–Bott index on stratified manifolds ∗ V.E. Nazaik inskii, A.Y u. Savin, and B. Y u. Sterni n Abstract W e define A tiy ah–Bott index on stratified manifolds and express it in top ological terms. By w ay of example, we compute this in dex for geometric op erators on manifolds with edges. 1 In tro duction The pap er deals with elliptic theory on stratified manifolds. The sym b ol of a pseu- do differen tia l o p erator on a stratified manifold is a collection of sym b ols on the strata. The sym b o l on the stratum of maximal dimension, called the interior sym- b ol , is of particular imp ortance, since it is a scalar function on the cotangent bundle, while the sym b ols on the low er-dimensional strata are op erator- v alued. An op erat o r with elliptic in terior sym b ol defines an elemen t in the K -group of op erators with zero in terior sym b ol. 1 This elemen t, whic h we call the A tiyah–Bott index , has the following prop erties. • It is determine d by the interior sym b ol of the op erator and is a homotopy in v arian t of the interior sym b ol. • F or a smo oth manifo ld, this in v ariant coincides with the F redholm index. • The Atiy ah–Bott index is the obstruction to making a n op erator with inv ert- ible in terior sym b o l in vertible b y adding low er- order terms. • F or a manifold with b oundar y , t his index coincides with the A tiy ah–Bott ob- struction (see [4]) to the existence o f well-posed (F redholm) b oundary condi- tions for an elliptic o p erator in a b ounded domain. ∗ Research suppo rted in part by RFBR gr a nts Nos. 05-0 1-009 82 and 06 -01-0 0098 and DFG gr ant 436 RUS 113 /849/ 0-1 r “ K -theory and noncommut ative geometry of stratified manifolds.” 1 Similar inv ar iants w ere studied in other s ituations, e.g ., in [1, 2, 3]. 1 The main result of this pap er is a general formula expressing the A tiy ah–Bott index in top ological terms. W e also compute the range o f the index mapping, i.e., the K - group of op erato rs with zero interior sym b ol. This researc h was carried out during our stay at the Institute for Analysis, Han- no ve r Univ ersit y (German y). W e are gra teful to Professor E. Sch rohe and other mem b ers of the univ ersit y staff for their kind hospitalit y . 2 A tiy ah–B o tt index First, w e recall the k ey prop erties of pseudo differential op erators on stratified man- ifolds. F or detailed exp osition, e.g., see [5, 6, 7]. Stratified manifolds . Let M be a compact stratified manifold in the sense of [7]. Recall that M ha s a decreasing filtration M = M 0 ⊃ M 1 ⊃ M 2 . . . ⊃ M N ⊃ ∅ of length N by closed subsets M j suc h that the complemen t M j \ M j +1 (an o p en str atum ) is homeomorphic to the in terior M ◦ j of a compact manifold M j with corners (the blowup of M j ). Let us denote the blowup of M b y M . In addition, any x ∈ M j \ M j +1 has a neigh b orho o d homeomorphic to the pro duct U x × K Ω j , where U x is a neigh b o rho o d of x in M ◦ j and K Ω j = [0 , 1) × Ω j  { 0 } × Ω j is a cone whose base Ω j is a stratified manifold with filtr a tion of length < N . In particular, M N is a smo oth manifo ld. Pseudo differen tial op erators on stratified manifolds. Let Ψ( M ) b e the al- gebra of pseudo differen tial op erators of order zero on M a cting in the space L 2 ( M ) of complex-v alued functions. (F or the definition of t he alg ebra and of the measure defining the L 2 -space, w e refer t he reader to [8] or [5, 6].) The Calkin algebra (the algebr a o f symb ols ) is denoted b y Σ( M ). The sym b ol σ ( D ) of an op erator D on M is a collection σ ( D ) = ( σ 0 ( D ) , σ 1 ( D ) , ..., σ N ( D )) of sym b ols on the strata, where the sym b ol σ j ( D ) is defined o n the cosphere bundle S ∗ M j of the blo wup of the corresp onding stratum. The sym b ol σ 0 ( D ) on the stratum M \ M 1 of maximal dimension is a scalar function called the interior symb ol , a nd the remaining comp onents of the sym b ol are op erat or-v alued functions. 2 Definition of A t iy ah–Bott index. Let σ 0 : Ψ( M ) − → C ( S ∗ M ) b e the mapping taking eac h o p erator to it s interior sym b ol. This mapping is sur- jectiv e. Consider the short exact sequence 0 − → J − → Ψ( M ) σ 0 − → C ( S ∗ M ) − → 0 (1) of C ∗ -algebras, where J ⊂ Ψ( M ) is the ideal of op erators with zero in terior sym b ol. The b oundary mapping δ : K ∗ ( C ( S ∗ M )) − → K ∗ +1 ( J ) (2) induced in K -theory b y the exact sequence (1) is called the Atiyah–Bott index. Let us explain wh y the mapping (2) is called a n index. If M is a closed smo oth manifold, then M = M . (The blow up coincides with the original manifold.) More- o ve r, the k ernel of σ 0 coincides with the ideal K of compact op erators, and the non trivial b oundary mapping δ : K 1 ( C ( S ∗ M )) − → K 0 ( K ) ≃ Z is giv en b y the F redholm index. 2 It turns out that the Atiy a h–Bott index is the obstruction to making an o p erator with inv ertible inte rior sym b ol in v ertible b y adding op erators with zero inte rior sym b ol (cf. [3]). Theorem 2.1. L et A b e a matrix pseudo differ ential op er ator with invertible in terior symb o l σ 0 ( A ) o n a str atifie d manifold M . A ne c essary and suffici e n t c ondition that ther e exists an op er ator R with zer o interior symb ol such that A + R is invertible is that the A tiyah–B ott index δ [ σ 0 ( A )] ∈ K 0 ( J ) is zer o. The pro of is given in the App endix. 3 Main theo rem Analytic K -homology . Let us recall sev eral facts ab out analytic K -homology . (Detailed exp osition and further references can b e found in [9], Chapter 5.) Let M b e a compact stratified manifold. By D ( M ) w e denote the algebra o f lo c al op e r ators on L 2 ( M ), i.e., op erators compactly commuting with m ultiplications b y 2 Indeed, an element of K 1 ( C ( S ∗ M )) is determined by an inv ertible ma trix w ith entries in C ( S ∗ M ). Consider the matrix a s the symbol o f so me op era tor; then the ma pping takes (the equiv alence cla ss of ) the matrix to the F redholm index of that op er ator. 3 con tinuous functions on M . If M ′ is a closed subspace, then b y D ( M , M ′ ) ⊂ D ( M ) w e denote the ideal of lo c al ly c omp act op er ators . By definition, these are o p erators whose comp osition with m ultiplications b y con tin uous functions v anishing on M ′ is compact. The K -homo lo gy groups of M , M ′ , and M \ M ′ can b e defined as K ∗ ( M ) ≃ K ∗ +1  D ( M ) / K  , K ∗ ( M ′ ) ≃ K ∗ +1  D ( M , M ′ ) / K  , (3) K ∗ ( M \ M ′ ) ≃ K ∗ +1  D ( M ) / D ( M , M ′ )  . (4) Moreo ve r the K -homology sequence for the pair M ′ ⊂ M . . . → K ∗ ( M ′ ) → K ∗ ( M ) → K ∗ ( M \ M ′ ) ∂ → K ∗ +1 ( M ′ ) → . . . coincides with the K -theory sequence for the short exact sequence 0 → D ( M , M ′ ) / K − → D ( M ) / K − → D ( M )  D ( M , M ′ ) → 0 of C ∗ -algebras. Main theorem. Pseudodifferential op erators on stratified manifolds a r e lo cal (see [6]). On the other hand, op erators in J are lo cally compact with resp ect to the subspace M 1 ⊂ M ; i.e., their comp ositions with f unctions v anishing o n M 1 are compact. Th us, w e hav e the comm utative diagram 0 → J → Ψ( M ) → C ( S ∗ M ) → 0 i M 1 ↓ ↓ ↓ i M ◦ 0 → D ( M , M 1 ) → D ( M ) → D ( M )  D ( M , M 1 ) → 0 . (5) In view of the isomorphisms (3) and (4), the v ertical mappings in (5 ) induce homo- morphisms of K -groups in t o K -homology groups: i M 1 ∗ : K ∗ +1 ( J ) − → K ∗ +1  D ( M , M 1 ) / K  ≃ K ∗ ( M 1 ) , i M ◦ ∗ : K ∗ ( C ( S ∗ M )) − → K ∗  D ( M ) / D ( M , M 1 )  ≃ K ∗ +1 ( M \ M 1 ) . Theorem 3.1. I f M has no close d smo oth c omp onents, then the d iagr a m K ∗ ( C ( S ∗ M )) i M ◦ ∗   δ / / K ∗ +1 ( J ) i M 1 ∗   K ∗ +1 ( M \ M 1 ) ∂ / / e K ∗ ( M 1 ) c om mutes. Mor e over, i M 1 ∗ is a n isomorphism . Her e 4 • e K ∗ ( M 1 ) ≃ k er { K ∗ ( M 1 ) → K ∗ ( pt ) } is the r e duc e d K -h o molo gy gr oup. • ∂ is the b oundary mapping in exa c t K -homolo gy se quenc e of the p air M 1 ⊂ M . Pr o of. 1. Consider t he comm utative diagram 0 / / J j   / / Ψ   / / C ( S ∗ M ) / / 0 0 / / J / K / / Σ / / C ( S ∗ M ) / / 0 , where Ψ is the algebra of pseudo differen tial o p erators, Σ = Ψ / K is the algebra of sym b ols, and j is the natural pro jection. The b oundary mappings corresp ond- ing to the upp er and low er rows of t he diagram are compatible. Since j induces a monomorphism in K -theory (see the lemma b elo w), it suffices to compute the b oundary mapping correspo nding to t he lo w er exact sequence. Lemma 3.2. T h e n a tur al pr oje ction j : J → J / K induc es isomo rp h isms K 0 ( J ) ≃ K 0 ( J / K ) , K 1 ( J ) = k er  ind : K 1 ( J / K ) − → Z  . Pr o of. The b oundary mapping in the K -theory exact sequence of the pa ir J → J / K is the index mapping K 1 ( J / K ) − → K 0 ( K ) = Z . It is surjectiv e. The exact sequence o f pair K ⊂ J giv es the desired isomorphism. 2. Consider the commutativ e diagra m 0 → J / K → Σ → C ( S ∗ M ) → 0 ↓ ↓ ↓ 0 → D ( M , M 1 ) / K → D ( M ) / K → D ( M )  D ( M , M 1 ) → 0 . (6) As w e men tioned earlier, the seque nce of K -groups induced b y the lo wer r ow in (6) is isomorphic to the K -homolo g y exact sequence of the pair M 1 ⊂ M . Since M is assumed to hav e no closed smo oth comp o nents, there exists a no n- singular v ector field (a section M → S ∗ M of the cosphere bundle) o n the blowup M . It follow s that K ∗ (Σ) and K ∗ ( C ( S ∗ M )) ha ve direct summands K ∗ ( C ( M )), whic h are mapp ed isomorphically on to eac h other by σ 0 : Σ → C ( S ∗ M ). Hence (6) giv es the diagram K ∗ ( J / K ) → K ∗ (Σ) /K ∗ ( C ( M )) → K ∗ ( C ( S ∗ M )) /K ∗ ( C ( M )) → K ∗ + 1 ( J / K ) → i M 1 ∗ ↓ i M ∗ ↓ i M ◦ ∗ ↓ i M 1 ∗ ↓ K ∗ + 1 ( M 1 ) → K ∗ +1 ( M ) → K ∗ +1 ( M \ M 1 ) ∂ → K ∗ ( M 1 ) → (7) 5 of K - groups with exact rows . W e prov ed in [7] that i M ∗ and i M ◦ ∗ are isomorphisms. Hence i M 1 ∗ is an isomorphism as we ll b y the fiv e lemma. Now the commutativ e rightmos t square in (7), together with Lemma 3.2, giv es the desired comm utative diagram K ∗ ( C ( S ∗ M )) i M ◦ ∗   δ / / K ∗ +1 ( J ) i M 1 ∗   K ∗ +1 ( M \ M 1 ) ∂ / / e K ∗ ( M 1 ) where i M 1 ∗ is an isomorphism. The pro of of Theorem 3.1 is complete. Cohomological index formula . T o b e definite, w e write out a (co)homological form ula for an elemen t [ a ] ∈ K 1 ( C ( S ∗ M )) represen ted by an inv ertible sym b ol a on S ∗ M . Consider t he K - homology elemen t i M 1 ∗ δ [ a ] ∈ K 1 ( M 1 ) . (8) The Chern c hara cter of this elemen t is a rationa l homolog y class o n M 1 . Let us ev aluate the pairing of this class with an arbitrar y cohomolog y class on M 1 . T o this end, let Y ⊂ M b e a compact smo oth manifold of dimension dim M − 1, homeomorphic to the b oundary ∂ M = M \ M ◦ . (The smo oth structure is obtained b y smo othing the corners.) Next, let π : Y → M 1 b e the pro jection o n to the singularit y set M 1 . Corollary 3.3. F o r any [ a ] ∈ K 1 ( C ( S ∗ M )) and x ∈ H odd ( M 1 ) , one has D c h  i M 1 ∗ δ [ a ]  , x E = D c h [ a | Y ] · Td( T ∗ Y ⊗ C ) π ∗ x, [ T ∗ Y × R ] E , (9) wher e c h( i M 1 ∗ δ [ a ]) ∈ H odd ( M 1 , Q ) is the homolo gic al Chern char acter and [ a | Y ] ∈ K 0 c ( T ∗ Y × R ) is the r estriction of interio r symb ol a to Y . Pr o of. Since the Chern c hara cter is a rational isomorphism of K -theory and homol- ogy , it suffices to ev aluate the pairing with a n elemen t x of the form x = c h y , where y ∈ K 1 ( M 1 ). By Theorem 3.1, D c h  i M 1 ∗ δ [ a ]  , c h y E = D i M 1 ∗ δ [ a ] , y E = D ∂ i M ◦ ∗ [ a ] , y E . 6 It follows from the prop erties of the b oundary mapping ∂ in K -homology tha t D ∂ · i M ◦ ∗ [ a ] , y E = D π ∗ · ∂ 0 · i M ◦ ∗ [ a ] , y E , where ∂ 0 : K 0 ( M \ M 1 ) → K 1 ( Y ) is the b oundary mapping for the pair ∂ M ⊂ M . W e now tra nsfer the computation of the pairing to Y : D π ∗ ∂ 0 i M ◦ ∗ [ a ] , y E = D ∂ 0 i M ◦ ∗ [ a ] , π ∗ y E . By applying the A tiy a h–Singer f o rm ula on Y , w e obtain D ∂ 0 i M ◦ ∗ [ a ] , π ∗ y E = D c h [ a | Y ] · Td( T ∗ Y ⊗ C ) c h π ∗ y , [ T ∗ Y × R ] E . The righ t-hand side coincides with the desired cohomolog ical expres sion (9), since x = c h y . 4 Examples In this section, w e compute the A tiy ah–Bott index for geometric op erators on strat- ified manifolds with stratificatio n of length one. Suc h manifolds are called man- ifolds wi th e dges . Geometrically , a manifold with edges is o bta ined as follows. T ak e a smo oth manifold M whose b oundary ∂ M is fib ered ov er a smo oth base X , π : ∂ M → X . Then iden tify the p oin ts in eac h fib er of π . What is o btained is a manifold M with edge X . The blowup of M is just M , and the manifo ld Y in Corollary 3.3 is diffeomorphic to the b oundar y ∂ M . The Atiy ah–Bo tt index is zero for the Beltrami–Laplace a nd Euler op erators, since the r estrictions of their principal sym b o ls to Y giv e rise to zero elemen ts in K 0 c ( T ∗ Y × R ) . Let us (rationally) compute the A tiyah–Bott index of the D irac op erator. Supp ose t ha t M , X , and π a r e equipp ed with spin structures. Next, assume that the induced spin structure o n the total space ∂ M of the bundle π is compatible with the spin structure on ∂ M . Let D : S + ( M ◦ ) − → S − ( M ◦ ) b e the D ir a c op erator on M ◦ . (W e a ssume that the dimension o f M is ev en.) Prop osition 4.1. F or the Dir ac op er ator D on a manifold M with e dge X , the homolo gy c lass c h( i X ∗ δ [ σ 0 ( D )]) ∈ H ∗ ( X ) ⊗ Q is Poinc ar´ e dual to the c ohomolo gy c lass A ( X ) · π ∗ ( A (Ω)) , 7 wher e A ( X ) and A (Ω) ar e the A -classes of the b ase and of the fib ers of π , r esp e ctively, and π ∗ : H ∗ ( Y ) → H ∗− dim Ω ( X ) stands for inte gr ation over the fib ers. Pr o of. Cho osing a connection in π , w e o bt a in a decompo sition T ∗ M | Y ≃ π ∗ T ∗ X ⊕ ( T ∗ Ω ⊕ R ) of the restriction of the cota ngen t bundle to Y in to horizontal and v ertical comp o- nen ts. With regard to this decomp o sition, the sym b ol of the Dirac op erator ov er Y is the tensor pro duct of the pull-back of the sym b o l of t he Dirac op erato r on the base by the fa mily of Dirac op erat o rs in the fib ers. This giv es a factorization [ σ ( D ) | Y ] =  π ∗ [ σ ( D X )]  [ σ ( D Ω × R )] , of the corresp onding elemen t in K -theory , where [ σ 0 ( D X )] ∈ K dim X c ( T ∗ X ) and [ σ ( D Ω × R )] ∈ K dim Ω+1 c ( T ∗ Ω ⊕ R ). Let x ∈ H odd ( X ) b e a cohomology class. Let us compute t he pairing  c h( i X ∗ δ [ σ 0 ( D )]) , x  . By Corollary 3.3, this n umber is equal to  c h[ σ 0 ( D ) | Y ] Td( T ∗ Y ⊗ C ) π ∗ x, [ T ∗ Y × R ]  . Using the factorization of [ σ 0 ( D ) | Y ], w e see that this expression is equal to  c h[ σ ( D X )] c h[ σ ( D Ω × R )] Td( T ∗ X ⊗ C ) Td( T ∗ Ω ⊗ C ) π ∗ x, [ T ∗ Y × R ]  =  c h[ σ ( D X )] π ∗  c h[ σ ( D Ω × R )] Td( T ∗ Ω ⊗ C )  Td( T ∗ X ⊗ C ) x, [ T ∗ X ]  =  A ( X ) π ∗ ( A (Ω)) x, [ X ]  . (A t the last step, w e hav e used the standard transition from cohomology classes on the cotangen t bundle to cohomolog y classes on the manifolds themselv es by using the Thom isomorphism.) Th us,  c h( i X ∗ δ [ σ 0 ( D )]) , x  =  A ( X ) π ∗ ( A (Ω)) x, [ X ]  for all x ∈ H odd ( X ). It follo ws that the classes c h ( i X ∗ δ [ σ 0 ( D )]) and A ( X ) π ∗ ( A (Ω)) are P o incar´ e dua l. A similar computation o f t he Atiy a h–Bott index can b e carried out for the sig- nature op erato r. In this case, one should replace the A -classes b y the L - classes. 8 App endix. Pro of o f Theorem 2. 1 Auxiliary lemmas. The follo wing tw o lemmas a re standard. Lemma 4.2. L et A : H 1 → H 2 b e a c on tinuous family of F r e dholm op er a tors acting in sp ac es of se ctions of infinite-di m ensional Hilb ert bund les ov e r a lo c a l ly c omp act b as e X . Supp os e that A is invertible at infin ity. F or the existenc e of a c on tinuous family of finite r ank op er ators R ( x ) , x ∈ X , v a n ishing at infi n ity such that the family A + R is e v erywher e invertible, it is ne c essary and sufficient that ind A = 0 ∈ K 0 c ( X ) . Lemma 4.3. L et H b e an infinite-dimensional Hilb ert bund le over a lo c al ly c o m p ac t b as e X . T h en e ach element of K 1 ( C 0 ( X , K ( H ))) has a r epr esentative that is an invertible element of the unital algebr a C 0 ( X , K ( H )) + . Pro of of Theorem 2.1. If there exists an R with the desired prop erties, then A + R is in v ertible, [ A + R ] ∈ K 1 (Ψ( M )), a nd ( σ 0 ) ∗ [ A + R ] = [ σ 0 ( A )]. W e conclude that δ [ σ 0 ( A )] = δ (( σ 0 ) ∗ [ A + R ]) = 0, since the sequence . . . → K 1 (Ψ) ( σ 0 ) ∗ → K 1 ( C ( S ∗ M )) δ → K 0 ( J ) → . . . is exact. Let us prov e the con vers e. T o this end, let us recall sev eral prop erties of pseu- do differen tia l op erators on stratified manifolds (see [5 , 6]). The a lg ebra Ψ = Ψ( M ) is solv able of length (2 N + 1) ( e.g., see [5]), where N is the length of the stratificatio n of M , with comp osition series Ψ = Ψ 0 ⊃ Ψ 1 ⊃ Ψ 2 ⊃ Ψ 3 ⊃ . . . ⊃ Ψ 2 N ⊃ Ψ 2 N +1 = K ⊃ { 0 } . (10) T o describ e the ideals Ψ j in more detail, r ecall that the symbol o f a n op erator D is the collection σ ( D ) = ( σ 0 ( D ) , σ 1 ( D ) , ..., σ N ( D )) of sym b ols o n the strata. Here σ j ( D ) is a family of op erator s in L 2 ( K Ω j ), where K Ω j is cone with base Ω j , parametrized b y cosphere bundle S ∗ M j ; the conormal sym b ol (corresp onding to the cone tips) of this family do es not dep end on the co v ariables in S ∗ M j and is a family , pa r a metrized b y R × M j , of pseudo differen tia l op erators on the base Ω j of the cone. No w the ideals in (10) can b e described as follow s. The ideal Ψ 2 j + 1 ⊂ Ψ 2 j , j ≥ 0 , 9 consists of op erators with zero sym b ol σ j on the op en stratum M ◦ j , and the ideal Ψ 2 j ⊂ Ψ 2 j − 1 , j ≥ 1 , consists of op erators with zero conormal sym b ol σ c ( σ j ) of σ j on M ◦ j . Moreo ver, w e ha ve the isomorphisms Ψ 2 j / Ψ 2 j + 1 ≃ C ( S ∗ M j , K L 2 ( K Ω j )) (the mapping is defined b y the sym b ol σ j ) and Ψ 2 j − 1 / Ψ 2 j ≃ C 0 ( R × M j , K L 2 (Ω j )) (the mapping is defined b y the conormal sym b ol σ c ( σ j )). T o construct an in v ertible p erturbation of an op erator with trivial Atiy a h– Bo tt index, w e use Prop osition 4.4 b elow fo r j = 1 , 2 , . . . , 2 N + 1 . As a result, w e shall obta in an in v ertible o p erator with interior sym b ol equal to tha t of the original op erator. This will end the pro of of Theorem 2.1. Let A : L 2 ( M , C n ) → L 2 ( M , C n ) b e a matrix pseudo differen tial op erator. W e sa y that it is invertible mo dulo the ide al Ψ j if there exists a matrix op erator B suc h that the comp ositions AB and B A are equal to identit y mo dulo o p erators with matrix en tries in Ψ j . Prop osition 4.4. L et A b e a matrix pseudo diffe r ential op er ator on M such that for some j it is invertible mo dulo ide al Ψ j and δ j [ A ] = 0 , wher e δ j : K 1 (Ψ / Ψ j ) − → K 0 (Ψ j ) is the b oundary map i n K -the ory for the p air Ψ → Ψ / Ψ j . Then ther e exists a pseudo differ ential op er ator e A o n M that is invertible mo dulo the ide al Ψ j +1 , is e qual to A mo dulo Ψ j , and has zer o inde x δ j +1 [ e A ] = 0 in K 0 (Ψ j +1 ) . Pr o of. Let s 2 j = σ j and s 2 j − 1 = σ c ( σ j ). (In this notation, the ideal Ψ k +1 in Ψ k is determined by the condition s k = 0.) 0. Consider the dia gram K 1 (Ψ j / Ψ j +1 ) / / K 1 (Ψ / Ψ j +1 ) / / δ j +1   K 1 (Ψ / Ψ j ) δ ′′ / / δ j   K 0 (Ψ j / Ψ j +1 ) K 1 (Ψ j / Ψ j +1 ) δ ′ / / K 0 (Ψ j +1 ) γ / / K 0 (Ψ j ) / / K 0 (Ψ j / Ψ j +1 ) The middle columns a re parts of the exact sequence s for t he pairs Ψ → Ψ / Ψ j +1 and Ψ → Ψ / Ψ j . The ro ws are the exact sequences of the pa irs Ψ / Ψ j +1 → Ψ / Ψ j and 10 Ψ j → Ψ j / Ψ j +1 . Since the b oundary mapping is natural, it fo llo ws that the diagram comm utes. 1. Supp ose that δ j [ A ] = 0. Then δ ′′ [ A ] = 0. Hence the elemen t ind s j +1 = δ ′′ [ A ] ∈ K 0 (Ψ j / Ψ j +1 ) is zero. (Indeed, s j +1 is a F redholm family in v ertible at infinit y , since s j is in v ertible b y assumption.) Th us, b y Lemma 4.2, there exists a pseudodifferential op erator B = A mo d Ψ j on M inv ertible mo dulo Ψ j +1 . 2. Since γ δ j +1 [ B ] = δ j [ A ] = 0, w e ha v e δ j +1 [ B ] = δ ′ ( z ) f o r some z ∈ K 1 (Ψ j / Ψ j +1 ). 3. F or e A w e ta k e the comp osition e A = B Z − 1 , where Z ∈ Ψ + j is a pseudo differential op erat o r inv ertible mo dulo Ψ j +1 suc h tha t [ Z ] = z . (The existence of Z is guar an teed by Lemma 4.3.) It is clear that e A has the desired pro p erties: it is inv ertible mo dulo Ψ j +1 , is equal to A mo dulo Ψ j , and has zero index δ j +1 [ e A ] = 0, since δ j +1 [ e A ] = δ j +1 [ B ] − δ j +1 [ Z ] = δ j +1 [ B ] − δ ′ ( z ) = 0 b y construction. The pro of of Prop osition 4.4 is complete. References [1] H. Upmeier. T o eplitz op erators and index theory in sev eral complex v ariables. 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