Index formulas on stratified manifolds

Index form ulas on stratified manifolds ∗ A. Y u. Sa v in, B.Y u. Sternin No v em b er 3, 2018 Abstract Elliptic op erators on stratified manifolds with an y fin ite n um b er of str ata are considered. Un d er certain assump tions on the symbols of op erators, we obtain index form ulas, whic h express index as a su m of indices of elliptic op erators on the strata. 1 In tro d u ction This pap er deals with elliptic t heory on stratified manif o lds with stratification of arbitra r y length. Analytical asp ects of this theory (notio n of sym b ol, ellipticit y condition, finiteness theorem) are no w sufficien tly w ell w or ked out, at least for pseudodifferential opera t o rs of order zero (see [1–5]). Let us summarize the results o btained in the cited pap ers (b elow w e shall o nly deal with zero-order op erators). A stratified manifold is a union of a finite nu m b er o f op en strata. Eac h stratum is a smo oth manifold; and the strata are glued t o gether in some sp ecial w a y (see op. cit.). The principal sym b ol of a pseudo differen tial o p erator in this situation is a collection o f symbols on the strata. Eac h sym b o l is an op erato r-v alued function, defined on the cotangen t bundle (min us the zero section) of the corresp onding op en stratum. The s ym b o l ass igned to the stratum of maximal dimension play s a sp ecial role. This sym b ol, whic h is called the in terior symb o l of operator , ranges in o p erators in finite-dimensional v ector spaces. The ellipticity condition requires the in v ertibilit y of t he principal sym b ol of the o p erator, i.e., inv ertibilit y of sym b ols on all strata. Finiteness theorem asserts that an elliptic op erato r is F redholm in L 2 -spaces. An approac h to index formulas , i.e., form ulas, whic h express index in terms of the principal sym b o l of the op erator was prop osed in [6] for op era t ors on manifolds with isolated singularities. The idea of this appro ac h is to obtain f o rm ula for the index as a sum of homotopy in v arian ts o f sym bo ls on the stratum of maximal dimension and the singular p oints . Suc h index for mulas w ere obta ined in the cited pap er, and for manifolds with edges (simplest class of stratified manifolds with nonisolated singularities) in [7] and [8]. Note t ha t to obtain suc h form ulas, one assumes that the sym b ol of op erator is ∗ Suppo rted by RFBR grants 06-01 -0009 8 , 08-01-00 867, Pierr e Deligne Ba lzan prize in ma thema t- ics, a nd also by DFG grant DF G 436 R US 11 3/849 /0-1 r “ K -theory a nd nonco mm utative g eometry o f stratified manifolds” . 1 in v arian t with respect to some transformation of the cotangent bundle of the manifold ( symmetry c ondition ) . 1 In the presen t pap er, w e introduce a class of t r a nsformations of cotangent bundles o f stratified manifolds, and f or a n elliptic op erator in v a r ia n t under one o f the transformations w e giv e an explicit index form ula in terms of homotopy inv arian ts of ellip tic s ym b ols on the strata. Let us briefly describe the con ten ts o f the pap er. W e start with a brief summary of res ults on the geometry o f stratified manifolds and theory of pseudo differen tial op erato r s o n them, whic h are used in the follo wing sec tions (in our exp osition w e follow the terminology of the pap er [9]). In index t heory o f elliptic op erators on stratified manifolds an imp ortant role is pla y ed b y the surgery metho d (see [8, 10]). Th is metho d p ermits one to lo calize con tributions to the index of sym b ols on the strata . In this pap er, we in tro duce class of sur geries in phase s p ac e . Suc h surgeries are carried out on cotangen t bundles o f manifolds, r a ther than manifolds themselv es. This surgery tec hnique is treated in sections 3 and 4. T he class o f transformations of the cotangent bundle which w e consider in this pap er has the following imp orta nt prop erty: transformations from this class natura lly act on the algebra of principal sym b o ls of ψ DO. W e sa y that an op erator satisfies symmetry c ondition if its principal sym b ol is in v arian t under some mapping from this class (in what follo ws transformations a re denoted by G and op erators satisfying symmetry condition are called G -in v arian t). Then for a G -inv ar ian t op erator w e construct homotopy in v arian ts for eac h of the strata of the manifold. The index f orm ula express es the index of a G -inv aria nt elliptic op erator as a sum of these homoto py inv ariants of sym b ols o n the strata. A couple of words concerning the pro of o f index theorem. Using t he homotopy classi- fication of elliptic op erators on stra t ified manifolds [9 ], w e compute the con tributions to the index of the stra t a of lo w er dimensions. T o compute the contribution of the stratum of ma ximal dimension, we use surgery in phase space. Note, finally , that the a pplicatio n of surgery metho d in [7] w as substan tially hin- dered, b ecause the symmetry condition could o nly b e satisfied for op erato rs, for which the Atiy ah–Bott obstruction [11] is equal to zero. In the presen t pap er w e drop t his rather restrictiv e condition using a class of nonlo cal op erat ors. 2 Ψ D O on stratified manifolds Let us briefly recall some basic facts from the theory of op erator s on stratified manifolds, whic h are used in the presen t pap er. Detailed exp osition can b e found, e.g., in [3, 4, 9]. 1. Stratified manifolds. Let M b e a compact stratified manifold in the se nse of [9]. Recall that this means that M is a Hausdorff top o lo gical space with decreasing filtration of length r M = M 0 ⊃ M 1 ⊃ M 2 . . . ⊃ M r ⊃ ∅ 1 Note that in genera l one ca n not drop the symmetry condition: using metho ds of [7] it can b e shown that for an ar bitrary elliptic o per ator on a stratified manifold there is no decomp osition o f index a s a sum of homotopy inv ariants of symbols on the strata. 2 b y closed subsets M j (called str ata ), suc h that each complemen t M j \ M j +1 ≡ M ◦ j (called op en str atum ) is homeomorphic to the in terior M ◦ j of a compact manifold with corners 2 , denoted b y M j (whic h is called t he blow up of manifold M j ). In particular, manifold M r is smo oth. The blow up of manif o ld M is denoted b y M . There is a contin uous pro jec tion π : M − → M . Num b er r is called the length of str atific ation . In addition, stratified manifold has the followin g structure: eac h op en stratum M ◦ j has a neigh b orho o d U ⊂ M \ M j +1 homeomorphic to a lo cally-trivial bundle ov er M ◦ j K Ω j − → M ◦ j , (1) whose fib er ov er p oint x ∈ M ◦ j is the cone K Ω j ( x ) := [0 , 1) × Ω j ( x )  { 0 } × Ω j ( x ) with base Ω j ( x ); here w e suppo se tha t the base Ω j ( x ) of the cone is a strat ified manif o ld with stratification of length < r . Cotangen t bundles o f the strata are denoted by T ∗ M j ∈ V ect( M j ) , j ≥ 1 , T ∗ M ∈ V ect( M ) . Note that the bundle T ∗ M j is isomorphic to T ∗ M j (the isomorphism is not canonical). Let us call M ◦ = M ◦ 0 the smo oth str atum , while M 1 singular str atum . Example 2.1. Manifolds with stratification of length one are called mani f o lds with e dges . In this case the stratum M 1 is called e dge (it is a closed smo oth manifold). The comple- men t M \ M 1 is a smooth manifold, while some neighborho o d U of stratum M 1 fib ers o v er M 1 with fib er cone (1), where the base Ω 1 ( x ) of the cone is a smo oth manifold. The blo wup M is obta ined as follows: w e ta ke manifold M and in U replace bundle with fib er cone K Ω j ( x ) := [0 , 1) × Ω j ( x )  { 0 } × Ω j ( x ) b y bundle with fib er cylinder [0 , 1) × Ω j ( x ). 2. Pseudo differen t ial op erators. Let Ψ( M ) b e the algebra of pseudo differen tia l op erators ( ψ DO) of order zero on M , acting in the space L 2 ( M ) of complex v alued functions on M ( t he definition of this algebra ψ DO and the measure in the definition of the L 2 -space can be found, e.g., in [3, 4]). The algebra of principal symbols Ψ( M ) / K — quotien t b y the ideal of compact op erators — is denoted by Σ( M ). The principal sym b ol σ ( D ) of a n op erator D ∈ Ψ( M ) on a stratified manifo ld is a collection σ ( D ) = ( σ 0 ( D ) , σ 1 ( D ) , ..., σ r ( D )) (2) 2 Recall tha t a n n -dimensional manifold with co rners is a Hausdorff top ologica l space lo cally homeo- morphic to the pro duct R k + × R n − k , 0 ≤ k ≤ n with smo oth transitio n functions b etw een do mains of this t yp e. 3 of symbols o n the strata, w here symbol σ j ( D ), j ≥ 1 , is defined on the cotangen t bundle T ∗ M j min us the zero section of the cor r esp onding stratum. The sym b ol σ 0 ( D ), corre- sp onding to the smo oth stratum M ◦ , is called the interior symb ol and is a complex-v alued functions, while the remaining comp onents of the sym b ol are f unctions σ j ( D ) ∈ C ( T ∗ M j \ 0 , B ( L 2 ( K Ω j ))) with v alues in op erators, acting in L 2 -spaces on cones K Ω j . The measure on the cone K Ω j is described in [4]. The sym b ols on differen t strata are related by a certain compatibility condition, whic h is describ ed in the cited pap er. 3 Surgery in phase space 1. Endomorphisms of cotangen t bundle. The restriction of the cotangent bundle T ∗ M t o the subspace ∂ j M = π − 1 ( M ◦ j ) ⊂ M has direct sum decomp osition [9 ]: T ∗ M| ∂ j M ≃ π ∗ ( T ∗ M j ) ⊕ T ∗ Ω j ⊕ R , where the the second summand corresp onds to directions along the base of the bundle of cones (1), while the third summand correspo nds to the directions along the radial v ar iable on the cone. Definition 3.1. An endomorphism h ∈ End( T ∗ M ) of the cota ng en t bundle of M , whic h is defined in a neighborho o d of the b oundary ∂ M ⊂ M is called a dmissible , if fo r eac h j ≥ 1 one has h | ∂ j M = h j ⊕ I d ⊕ I d : π ∗ ( T ∗ M j ) ⊕ T ∗ Ω j ⊕ R − → π ∗ ( T ∗ M j ) ⊕ T ∗ Ω j ⊕ R , (3) where h j ∈ End( T ∗ M j ) , j ≥ 1 are some endomorphisms. W e also set h 0 = h . Remark 3.1. F or manifolds with edges (see Example 2.1), admissible endomorphisms are precisely those endomorphisms of T ∗ M , whic h a r e induced b y endomorphisms of the cotangen t bundle of the edge M 1 . Let us define the action h ∗ of an a dmissible endomorphism h on principal sym b ols on M : h ∗ ( σ 0 , σ 1 , ..., σ r ) = ( h ∗ 0 σ 0 , h ∗ 1 σ 1 , ..., h ∗ r σ r ) . (4) Here r is the length of stra tification and w e use the fact that for any j ≥ 1 the sym b o l σ j is an op erator- v alued function o n the space T ∗ M j , on whic h endomorphism h j acts fib erwise-linearly . W e note als o that the action on the in terior sym b ol σ 0 is defined only in a neighborho o d of the b oundary ∂ M . (The sym b ol (4) is we ll-defined, i.e., its comp onen ts h ∗ 0 σ 0 , h ∗ 1 σ 1 , ..., h ∗ r σ r satisfy the compatibility condition, whic h follows fr o m the admissibilit y of h .) 4 2. Statemen t of the problem. Let N b e a stratified manifold of the following for m. It has a neigh b orho o d U of the singular stratum N 1 , whic h is a disjoin t union U = U + ⊔ U − of tw o diffeomorphic op en submanifolds U + and U − . Let us fix diffeomorphism U + ≃ U − and consider U + and U − as t w o identical copies of U + . Let D : L 2 ( N , E ) − → L 2 ( N , F ) b e an elliptic op erator on N acting in sections of some bundles E , F ∈ V ect( M ) on the blo w up M . The restriction D | U = Π D Π (Π is the c haracteristic f unction of set U ) of op erator D to U can b e considered as a direct sum (mo dulo compact op erators) D | U = D + ⊕ D − : L 2 ( U + , E ⊕ E ) − → L 2 ( U + , F ⊕ F ) . (5) Hereinafter we supp ose that w e are g iven iden tifications E | U + ≃ E | U − and F | U + ≃ F | U − , whic h co v er diffeomorphism U + ≃ U − . Let us su pp ose that the principal sym bo ls of op erators D + and D − satisfy condition σ ( D − ) = h ∗ σ ( D + ) (6) o v er U + , where h ∈ End( T ∗ U + ) is an a dmissible endomorphism defined in U + . Lemma 3.1. Th e index ind D of op er ator D , which s atisfies c ondition (6) is determine d by t he interior symb o l of the op er ator. Pr o of. 1. Let D ′ b e an elliptic op erato r whic h satisfies condition (6) in U and the in terior sym b ol σ 0 ( D ′ ) is equ al to σ 0 ( D ). Then w e ha v e ind D − ind D ′ = ind D ( D ′ ) − 1 = ind D | U ( D ′ | U ) − 1 = ind D + ( D ′ + ) − 1 + ind D − ( D ′ − ) − 1 (7) (in the second equalit y w e to ok in to account the fa ct that D and D ′ differ by compact op erator in the interior, hence, w e can pass to their restrictions D | U , D ′ | U to U ). The in terior sym b ols of op erators D + ( D ′ + ) − 1 and D − ( D ′ − ) − 1 are equal to o ne, and the sym b ols on the singular strata for the second op erator are obtained from those fo r the first o p erator b y application of endomorphism h σ j ( D − ( D ′ − ) − 1 ) = h ∗ σ j ( D + ( D ′ + ) − 1 ) . 2. Since endomorphism h c hanges the sign of index (see Corollary 8.1), equation (7) giv es the desired eq ualit y: ind D − ind D ′ = 0 . Belo w, w e shall giv e an explicit form ula for the index ind D in terms of the principal sym b ol σ ( D ). 5 3. Surgery . Let T = T ∗ N / ∼ , (8) b e t he space obtained from T ∗ N b y identific ation of p oints in the closure of the sets T ∗ U + and T ∗ U − under the action of mapping h, whic h w e consider here as an isomorphism h : T ∗ U + → T ∗ U − . The space T is a v ector bundle with base, whic h is obtained from M b y iden tification of sets U + ∩ M ◦ and U − ∩ M ◦ under the action of diffeomorphism U + ≃ U − , whic h w as fixed ab ov e. Condition (6 ) implies that the inte rior sym b ols σ 0 ( D + ) a nd σ 0 ( D − ) are compatible with the iden tification (8) and define class [ σ 0 ( D ) , h ] ∈ K 0 c ( T ) (9) in top ological K -group with compact supp orts of the lo cally-compact space T . Let us supp ose that h rev erses o r ien tat io n of T ∗ U + . Then a c hain, whic h represen ts T ∗ N , defines cycle on T . D enote the homolog y class of this cycle by [ T ] ∈ H 2 n ( T ) , n = dim N . Let us defi ne the follo wing n umber ind t D = h c h[ σ 0 ( D ) , h ]Td( T ⊗ C ) , [ T ] i . (10) It is rational by construction (using results of [8], one can sho w, that this num b er is actually dy adic-rational). Remark 3.2. The inv ariant (10) can b e written as an in tegral ind t D = Z T ∗ N c h σ 0 ( D )Td( T ∗ N ⊗ C ) . (11) Here the in tegral is in terpreted a s iterated: we first in t egr a te ov er fib ers of the cota ngen t bundle and then in tegrate o v er base N . The in tegral o ve r base is w ell-defines, since the in tegrand is iden tically zero in a neighborho o d of the singular stratum. Indeed, when w e integrate o v er the fib ers T ∗ x U + , the contributions to the integral of t he comp onen ts σ 0 ( D + ) and σ 0 ( D − ) cancel, whic h follows from the fact the h is orien tation-rev ersing and condition (6). The next theorem b elongs to A.Y u. Sav in. Theorem 3.1 (surgery in phase space) . Supp os e that h ∈ End( T ∗ U + ) is an orientation- r eversing involution ( h 2 = I d ) and D is an el liptic ψ D O on N which satisfies c ond i tion (6) . The n one has ind D = ind t D . (12) 4 Pro o f of theorem 3.1 T o pro v e formula (12), let us introduce the following class of op erat o rs. 6 1. Admissible op erators. A b ounded op erator Q : L 2 ( N , E ) − → L 2 ( N , F ) , (13) is called admissible if the following three conditions are satisfied. 1. Q is ψ D O in a neigh b orho o d of N \ U and in a small neighbor ho o d of the singular stratum N 1 . 2. The r estriction Q | U =  Q ++ Q + − Q − + Q −−  : L 2 ( U + , E ⊕ E ) − → L 2 ( U + , F ⊕ F ) (14) of Q to U is a matrix of ψ D Os on U + (here w e use identifications U − ∼ U + , E | U − ≃ E | U + , F | U − ≃ F | U + ). 3. The o p erator Q in a small neigh b orho o d of N 1 satisfies condition (6), i.e., σ ( Q −− ) = h ∗ σ ( Q ++ ) . F or admissible op erators, ellipticit y and finiteness theorem a re formulated and pro v ed b y standard metho ds, and are left to the reader. 2. T op ological index. Cons ider the following inv ariant of a n admissible elliptic op er- ator Q ind t Q = Z T ∗ ( N \ U ) c h σ 0 ( Q )Td( T ∗ N ⊗ C ) + Z T ∗ U + c h σ 0 ( Q | U )Td( T ∗ U + ⊗ C ) (15) (recall that the restriction Q | U is considered as an op erator on U + , see (14)). Here the in tegrals ar e in terpreted a s iterated — first along the fib ers o f the cotangen t bundle, and then along the base. When w e integrate along the fib ers T ∗ x U + , x ∈ U + , in the second in tegral Z T ∗ U + c h σ 0 ( Q | U )Td( T ∗ U + ⊗ C ) , the contributions o f the components Q ++ and Q −− cancel eac h other in a neigh b orho o d of the singular stratum. This follows from the f act that h is or ien tat io n-rev ersing and condition (6). Num b er (15 ) is called top olo gic al index of op erator Q . When Q is lo cal, i.e., the off-diagonal comp onen ts in the decomp osition (14) are equal to zero, the inv aria n t (15 ), eviden tly , coincides with that define d in (10). F or this reason, we ke ep the old notation for this new in v a rian t. Lemma 4.1 (prop erties of top o lo gical index) . 1 . ind t Q i s a homotopy invarian t of the interior symb ol σ 0 ( Q ) ; 7 2 . When σ 0 ( Q ) do es not dep end on c ovariable s in a neighb orho o d of the singular str a- tum, o n e has ind Q = ind t Q ; 3 . When σ 0 ( Q ) do es not dep end on c ovaria b les in the c o mplement of U , the value of the functional ind t is determine d by the r estriction of the interior symb ol to U and one has ind t ( h ∗ Q ) = − ind t Q, (16) wher e h ∗ Q s tands for arbitr ary el liptic op er ator with princip a l symb ol h ∗ σ ( Q ) 4 . I f Q is a c omp o s i tion Q = Q 1 Q 2 of two admissible e l liptic op er a tors, wher e Q 1 and Q 2 b oth satisfy c ondition (6) , then ind t Q = ind t Q 1 + ind t Q 2 . Pr o of. The first claim is straightforw ard. The second w as pr ov ed in [1 2 , 13 ]. The third follo ws fro m the fact that h rev erses orien tation of the cotang ent bundle a nd hence, b y c hang e of v ariables f o rm ula in the in tegral, rev erses the sign of functional ind t . Finally , the fourth claim follows from the m ultiplicativity of the Chern character. 3. Homotop y to op erator of multiplication. Let D b e an elliptic op erato r with sym b ol satisfying condition (6). The left and r igh t ha nd sides of form ula (12) are homo- top y inv ariant in the set of admissible elliptic op erators. Thus , to prov e their equality , w e shall ch o ose a sp ecial represen tative in the homotopy class of op erator D . This represen- tativ e is constructed in the follow ing lemma. Consider a homeomorphism U + \ N + ≃ ∂ N + × (0 , 1 ). D enote by t the co ordinate along (0 , 1) and let U 1 / 2 b e the set ∂ N + × (1 / 2 − ε, 1 / 2 + ε ) ⊂ U + for some ε > 0. Lemma 4.2. Ther e exists l > 0 such that the op er ator D l = D ⊕ D ⊕ . . . ⊕ D | {z } N summa nds is hom otopic in the class of admissible op er ators to some op e r ator D ′ , whose symb o l do es not dep end on c ovariables in U 1 / 2 . Pr o of. 1. The mapping h ∗ : K 0 ( T ∗ M| ∂ M ) ⊗ C − → K 0 ( T ∗ M| ∂ M ) ⊗ C is equal to − I d (Lemma 8.1). Hence, the elemen t [ σ 0 ( D ) | t =1 / 2 ] = [ σ 0 ( D + ) | t =1 / 2 ⊕ h ∗ σ 0 ( D + ) | t =1 / 2 ] = ( I d + h ∗ )[ σ 0 ( D + ) | t =1 / 2 ] is equal to zero in K 0 ( T ∗ M| ∂ M ) ⊗ C . Therefore, there exists n um b er l suc h tha t the sym b ol σ 0 ( D l ) | t =1 / 2 is homotopic to a sym b ol, w hic h do es not depend on cov ariables. 2. The homotop y of the sym b ol σ 0 ( D l ) | t =1 / 2 can b e lifted to a homotop y o f op erator D l . 8 4. Surgery: cutting out the smo oth stratum. Cons ider the decomp osition of manifold N N = U ≥ 1 / 2 ∪ U < 1 / 2 (17) in to t w o subsets U ≥ 1 / 2 = ( N \ U ) ∪ ( ∂ N × [1 / 2 , 1)), U < 1 / 2 = N \ U ≥ 1 / 2 . Since the sym b ol of op erator D ′ in Lemma 4.2 do es not dep end on co v aria bles in the domain U 1 / 2 , this op erat o r is equal mo dulo compact op erato r s to the direct sum D ′ = A ⊕ B of its restrictions A = Π D ′ Π : L 2 ( U ≥ 1 / 2 , E ) − → L 2 ( U ≥ 1 / 2 , F ) B = (1 − Π) D ′ (1 − Π) : L 2 ( U < 1 / 2 , E ) − → L 2 ( U < 1 / 2 , F ) to the sets U ≥ 1 / 2 and U < 1 / 2 , resp ectiv ely . Thus , w e hav e ind D ′ = ind A + ind B . On the other hand, the top o lo gical index of D ′ is also equal to the sum ind t D ′ = ind t A + ind t B . By Lemma 4.1, Item. 2 w e ha ve ind A = ind t A . Let us pro v e that the top olog ical index of o p erator B is equal to its analytical index. 5. Index computation near singular stratum. Define o p erator of p erm utation T : L 2 ( U + , C ⊕ C ) − → L 2 ( U + , C ⊕ C ) T ( u 1 , u 2 ) = ( u 2 , u 1 ) ( T 2 = I d ) . Consider the admissible elliptic op erator B ( T ∗ h ∗ B ) − 1 . (18) Here T ∗ A = T AT − 1 — conj ug ation o f op erator A b y T . Its index is equal to ind( B ( T ∗ h ∗ B ) − 1 ) = ind B − ind T ∗ h ∗ B = ind B − ind h ∗ B = 2 ind B (19) (in the rig h tmo st equalit y w e used the fact that h ∗ rev erses the sign of index, see Corol- lary 8.1). By construction, for each j ≥ 1 the sym b ol σ j B ) is equal to σ j ( B + ) ⊕ h ∗ σ j ( B + ). Hence, the symbol σ j ( T ∗ h ∗ B ) is also equal to σ j ( B + ) ⊕ h ∗ σ j ( B + ), since h 2 = I d . The same formulas hold for the in terior symbol σ 0 ( B ) in the domain, where condition (6) is satisfied. No w, in a neigh bo rho o d of the singular stratum the sym b ols of the op erato r (18) are equal to one, i.e., this op erat o r is equal to the iden tity op erato r , mo dulo compact op erators. Th us, b y Lemma 4.1 (Items 2, 4, 3) w e ha v e the following c hain of equalities ind( B ( T ∗ h ∗ B ) − 1 ) Item 2 = ind t ( B ( T ∗ h ∗ B ) − 1 ) Item 4 = ind t B − ind t ( T ∗ h ∗ B ) Item 3 = = ind t B + ind t B = 2 ind t B . (20) Comparing (20) and (19), w e obtain the desired equalit y ind B = ind t B . Therefore, ind D ′ = ind t D ′ and hence ind D = ind t D . This completes the pro o f of Theorem 3.1. 9 5 Symmetries of sym b ol s Let M b e a stratified manifold and E , F ∈ V ect( M ) v ector bundles on the blowup M . Definition 5.1. A symmetry is a quadruple G = ( g , h, e, f ), where • g : M → M is a diffeomorphism of stratified manifold, whic h is defined in a neigh b orho o d U of the singular stratum M 1 ; • h ∈ End ( T ∗ M ) is a n admissible endomorphism defined in U ; • e, f are v ector bundle isomorphisms E | U e ≃ ( g ∗ E ) | U , F | U f ≃ ( g ∗ F ) | U . The differen tial o f diffeomorphism g defines a fib erwise-linear mapping dg : T ∗ M → T ∗ M , which cov ers g . D enote the induced maps on the strata b y g j : M j → M j , j ≥ 0 Consider no w a ψ DO D : L 2 ( M , E ) − → L 2 ( M , F ) . (21) Since algebras of ψ DO and their principal sym bo ls are diffeomorphism in v arian t, one has the follo wing action of symmetry G = ( g , h, e, f ) on the sym b ols: G ( σ j ) = f − 1  h ∗ j ( dg j ) ∗ σ j  e, j ≥ 0 . (22) This action is w ell defined (i.e., the sym b ol G ( σ ) = ( G ( σ 0 ) , G ( σ 1 ) , ..., G ( σ r )) enjoys com- patibilit y conditions). Note that the in terior sym b ol G ( σ 0 ) is defined only in neighborho o d U . Let ∂ T ∗ M j b e the restriction of the cotangen t bundle T ∗ M j to the b oundary ∂ M j ⊂ M j . Definition 5.2. Let G b e a symmetry . An elliptic op erator D is called G - invariant , if for all j ≥ 0 one has σ j ( D ) | ∂ T ∗ M j = G  σ j ( D ) | ∂ T ∗ M j  , (23) i.e., t he restrictions of the comp onen ts of the sym b ol t o the b oundaries of the corresp ond- ing strata are G -inv ariant. Without loss of generalit y w e shall assume throughout the follo wing that the in terior sym b ol enjo ys equality σ 0 ( D ) = G ( σ 0 ( D )) in the en tire domain U . Denote by G ( D ) an arbitr ary op erator with principal sym b ol G ( σ ( D )). 6 Homotop y inv arian ts of sym b ols Let D be a G -inv ariant elliptic operato r on M , where G = ( g , h, e, f ) is a sy mmetry . Supp ose that the symmetry satisfies the following additional condition: h : T ∗ M − → T ∗ M reve rses orien tation. It turns o ut that in this case one can construct nontrivial homotop y in v arian ts of the sym b ols on eac h of the strata M j , j ≥ 0. 10 1. In v arian t of t he interior sym b ol. Denote the disjoin t union of t wo copies of M b y N . Let us choose a neigh b orho o d of the singular stratum in N as a union U + ∪ U − of neigh b orho o d U + = U on the first cop y and U − = g ( U ) on the second copy . On b oth comp onen t s o f N w e consider op erator D , whic h is G -inv ariant, i.e. satisfies condition (23). There are isomorphisms E | U + g ∗ − 1 ◦ e − → E | U − and F | U + g ∗ − 1 ◦ f − → E | F − . Denote t he constructed op erator on N b y D ∪ D . A computation sho ws that the principal sym b ol o f this op erator satisfies condition (6) and hence, this op erator has top o logical index ind t ( D ∪ D ). Clearly , t his in v a rian t is determined b y the interior sym b ol σ 0 ( D ) and symmetry G . 2. Inv arian t s of symbols on the strata M j , j ≥ 1 . Consider the elliptic sy m b ol σ j ( D )[ Gσ j ( D )] − 1 (24) o v er the stratum M j . Denote by K the set of compact op erato rs. Lemma 6.1. The symb ol (24) has c o mp act fib er variation  σ j ( D )[ Gσ j ( D )] − 1  ( x, ξ ) −  σ j ( D )[ Gσ j ( D )] − 1  ( x, ξ ′ ) ∈ K , for al l nonzer o ξ , ξ ′ ∈ T ∗ x M j , as an op er ator-value d function on the bund le T ∗ M j − → M j . Pr o of. Indeed, b y [9], Prop osition 2.2 ) compact fib er v ariation prop erty is v alid, pro vided that the restriction of the sym b ol σ j − 1 ( D )[ Gσ j − 1 ( D )] − 1 to the b oundary ∂ T ∗ M j − 1 do es not dep end on cov aria bles. How ev er, b y the G -in v ariance (23) of sym b ol σ j − 1 ( D ), this expression is actually equal to the iden tit y sym b ol. On the other hand, the sym b ol ( 2 4) is constant on the b oundary ∂ T ∗ M j , where it consists of iden tit y opera t o rs (this time, b y G -in v aria nce of σ j ( D )). Th us, the elliptic sy m b ol (2 4) has compact fib er v a r ia tion a nd is equal to iden t ity on the b oundary of T ∗ M j ≃ T ∗ M j . This implies that this sym b ol on T ∗ M j can b e considered as an op erator-v alued sym b o l in the sense of Luk e [14] a nd w e can assign to it a F redholm op erator on M ◦ j , whic h is isomorphism at infinity . Denote this op erator b y Op  σ j ( D )[ Gσ j ( D )] − 1  : L 2 ( M ◦ j , L 2 ( K Ω j , F )) − → L 2 ( M ◦ j , L 2 ( K Ω j , F )) . (25) Remark 6.1. The index theorem for ψ D O with op era t o r-v alued sym b ols [1 4 ] giv es equal- it y ind Op  σ j ( D )( Gσ j ( D )) − 1  = p !  σ j ( D )( Gσ j ( D )) − 1  , where [ σ j ( D )( Gσ j ( D )) − 1 ] ∈ K 0 c ( T ∗ M ◦ j ) is the class of sym b ol in K -theory , and p ! : K 0 c ( T ∗ M ◦ j ) → K 0 ( pt ) = Z is the direct image mapping in K - theory induced b y the pro jection M j → { pt } to the one-p oint space. Index form ulas in cohomo lo gy can also b e obtained (see [15]). 11 7 Index the o rem The next theorem b elongs to B.Y u. Sternin. Theorem 7.1. Supp ose that an el liptic op e r ator D on a str atifie d manifold M is G - invariant, a n d the admissible endomorp h ism h : T ∗ M − → T ∗ M is an orientation- r eversing involution ( h 2 = I d ) . Then one has ind D = 1 2 ind t ( D ∪ D ) + 1 2 r X j =1 ind Op  σ j ( D )[ Gσ j ( D )] − 1  , (26) wher e the sum c on tain s indic es of el liptic o p er ators on the str ata M j with op er ator-value d symb ols in the sense of [14] e qual to σ j ( D )[ Gσ j ( D )] − 1 ( se e Se ction 6 ) . Remark 7.1. When r = 1, g = I d , e 2 = f 2 = I d , h 2 = I d , this theorem giv es index form ula on manifolds with edges, see [7, 8]. Pr o of. 1. Denote b y e D an op era t o r with sym b ol ( σ 0 ( D ) , Gσ 1 ( D ) , ..., Gσ r ( D )) . This op erator is w ell-defined, since the collection σ 0 ( D ) = Gσ 0 ( D ) , Gσ 1 ( D ) , ..., Gσ r ( D ) of sym b ols on the strat a is compatible (b ecause the action of symmetry G preserv es compatibilit y). By t he lo g arithmic prop erty of the index w e obtain ind D − ind e D = ind( D e D − 1 ) . (27) F urther, for all j ≥ 1 t he sym b ols σ j of o p erator D e D − 1 are equal to iden tit y on ∂ T ∗ M j , while the in terior sy m b ol is equal to the iden tit y on the en tire space T ∗ M . This implies that t he op erato r D e D − 1 can b e decomp osed (mo dulo compact op erato rs) as the pro duct D e D − 1 = r Y j =1 P j where P j is an op erator on M , whose sym b ols are equal to iden tit y , except the sym b ol σ j ( P j ), whic h is equal to σ j ( D )[ Gσ j ( D )] − 1 . W e get ind D e D − 1 = r X j =1 ind P j . Note now that it follo ws from the prop erties of the symbol of P j that outside arbitrarily small neighborho o d U of op en stratum M ◦ j op erator P j is equal to iden tity mo dulo com- pact op erators. Th us, P j is equal (mo dulo compact op erators) to the direct sum o f its 12 restriction Π P j Π to U (Π is the c haracteristic function o f U ) and the iden tity op erator, whic h acts o n functions o n the complemen t of U . Hence, we obta in ind P j = ind Π P j Π + ind(1 − Π) = ind Π P j Π . Let us no w ch o ose U suc h that it fib ers ov er the stratum M ◦ j with conical fib er. In this case the restriction Π P j Π of op erat o r P j to this neighborho o d can b e treated (see [9]) as an op erator o n M ◦ j with op erator-v alued sym b o l in t he sense of Luke equal to σ j (Π P j Π) = σ j ( D )[ Gσ j ( D )] − 1 . This gives us ind D − ind e D = r X j =1 ind Π P j Π = r X j =1 ind Op  σ j ( D )[ Gσ j ( D )] − 1  . (28) The righ t-hand side of this equalit y coincides with the sum in (26). 2. Consider no w t w o copies of manifo ld M . W e tak e op erato r D on the first copy , and e D on the second cop y and apply Theorem 3.1 to op erator D ∪ e D on the union of these manifolds. W e obtain ind D + ind e D = ind t ( D ∪ e D ) . (29) Since the in terior sym b ols of op erators D ∪ e D and D ∪ D are equal, w e get ind t ( D ∪ e D ) = ind t ( D ∪ D ). Th us, the sum of equations (28) and (29) giv es the desired formula (26). 8 App endix. Actions of symmetries in K -theory Consider the ideal J ⊂ Ψ( M ) of op erator s with zero interior sym b ol. Let h ∈ End( T ∗ M ) b e an admissible endomorphism. The action of this endomorphism on principal sym b ols obv iously restricts to t he a ction on the ideal J / K ⊂ Ψ ( M ) / K in the Calkin algebra. Belo w w e shall show that the induced action h ∗ : K ∗ ( J / K ) − → K ∗ ( J / K ) on the rational K -group is equal to ± I d . T o this end w e intro duce notation K ∗ ( J ) C = K ∗ ( J ) ⊗ C and define the sign of h by sgn h = h ∗ [ T ∗ M ◦ 1 ] [ T ∗ M ◦ 1 ] ∈ {± 1 } , where [ T ∗ M ◦ 1 ] ∈ H ev, c ( T ∗ M ◦ 1 ) is the fundamental class. Therefore, t he sign sgn h is equal to +1, if h preserv es orien tation of T ∗ M 1 and is − 1 otherwise. Prop osition 8.1. Supp ose that an admissi b le endomorphism h ∈ End( T ∗ M ) h as finite or der ( h N = 1 ) . Then one has h ∗ = (sgn h ) I d : K ∗ ( J / K ) C − → K ∗ ( J / K ) C . (30) 13 Pr o of. Consider a decreasing sequence of ideals J = J 0 ⊃ J 1 ⊃ J 2 ⊃ . . . ⊃ J r = K , where r is the length of stratification, a nd the ideal J j consists of op erato r s D with sym b ols σ k ( D ), whic h ar e equal to zero for k ≤ j. There are induced actions of h on the ideal J j / K and the quotien t J j /J j +1 . Let us pro v e by induction that the mapping h ∗ : K ∗ ( J j / K ) C − → K ∗ ( J j / K ) C is equal to (sgn h ) I d. 1. Ba se of induction j = r . In this case the prop osition is v alid, since J r = K and K ∗ ( J r / K ) = 0. 2. Inductiv e step. Let h ∗ = ( sgn h ) I d as an endomorphism o f the gro up K ∗ ( J j +1 / K ) C . Let us pro v e that the same equalit y is v alid for t he gro up K ∗ ( J j / K ) C . T o this end, consider the comm utat iv e diagram K ∗ ( J j +1 / K ) C − → K ∗ ( J j / K ) C − → K ∗ ( J j /J j +1 ) C (sgn h ) h ∗ ↓ ↓ (sgn h ) h ∗ ↓ ( sgn h ) h ∗ K ∗ ( J j +1 / K ) C − → K ∗ ( J j / K ) C − → K ∗ ( J j /J j +1 ) C , (31) where the row s are t w o iden tical copies of the K -theory exact sequence of the pair J j +1 / K ⊂ J j / K . Lemma 8.1. The mapping (sgn h ) h ∗ : K ∗ ( J j /J j +1 ) C − → K ∗ ( J j /J j +1 ) C is e qual to the id entity. Pr o of. 1. In [9] t he f o llo wing isomorphism w as obtained K ∗ ( J j /J j +1 ) ≃ K ∗ +1 c ( T ∗ M j +1 ) , (32) where the a ction h ∗ on the K -group of alg ebra in the left-hand side in (32) transforms to the a ctio n of endomorphism h j +1 ∈ End( T ∗ M j +1 ) on the top ological K -group in the righ t-hand side of the equalit y . 2. Since sgn h = sgn h j +1 (whic h is easy to see, b ecause h is admissible), to pro v e lemma, it suffices to sho w that the ma pping h ∗ j +1 : K ∗ c ( T ∗ M j +1 ) C − → K ∗ c ( T ∗ M j +1 ) C is equal to (sgn h ) I d . T o prov e this, we use isomorphisms K ∗ c ( T ∗ M j +1 ) C c h ≃ H ∗ c ( T ∗ M j +1 ) C ≃ Hom( H ∗ c ( M ◦ j +1 ) , C ) 14 (Chern character isomor phism and P oincare isomorphism — integration o v er fundamen- tal cycle [ T ∗ M j +1 ]), to transform the pro of that h ∗ j +1 = (sgn h ) I d from K -theory to cohomology . F or all x ∈ H ∗ c ( T ∗ M j +1 ) and y ∈ H ∗ c ( M ◦ j +1 ) w e hav e h h ∗ j +1 x, y i ≡ h ( h ∗ j +1 x ) y , [ T ∗ M ◦ j +1 ] i = h h ∗ j +1 ( xy ) , [ T ∗ M ◦ j +1 ] i = = h xy , ( h j +1 ) ∗ [ T ∗ M ◦ j +1 ] i = h xy , (sgn h ) [ T ∗ M ◦ j +1 ] i = (sgn h ) h x, y i (33) (here “ h· , ·i ” denotes pairing in cohomology , and w e used equalit y h ∗ j +1 ( y ) = y , whic h is v alid, b ecause y is a cohomology class on the base M ◦ j +1 ). Th us, w e obtain h h ∗ j +1 x, y i = (sgn h ) h x, y i (34) for all x ∈ H ∗ c ( T ∗ M j +1 ) and y ∈ H ∗ c ( M ◦ j +1 ). Since equalit y (34) is v alid fo r a ll y , w e obtain the desired equalit y h ∗ j +1 x = (sgn h ) x . Indeed, mappings in the leftmost a nd r ig h tmost columns o f the comm utativ e diagram (31) are iden tit y mappings (the left mapping is identit y by assumption, while t he rig h t mapping b y Lemma 8.1). Therefore, application of the follo wing algebraic Lemma 8.2 sho ws that the mapping (sgn h ) h ∗ in the middle column of the diagram (31) is also iden tit y . Lemma 8.2. Conside r c omm utative diagr am A − → B − → C I d ↓ ↓ h ↓ I d A − → B − → C of finite-dimensiona l ve c tor sp a c es and l i n e ar ma p pings w ith exact r ows. I f ma p ping h has finite or der ( h N = I d ) , then h = I d . Pr o of. Diag ram c hase giv es equalit y ( h − I d ) 2 = 0. This means that h = I d + e , where e 2 = 0 . Now w e use finite order condition and obt a in that h is diagonalized in some base. This implies that e = 0. Corollary 8.1. Under assumptions of Pr op osition 8.1 for arbitr ary el liptic op er ator D on M with interior symb ol, which do es not dep end on c ovariables, one has: ind( h ∗ D ) = (sgn h ) ind D , wher e h ∗ D stands for an arbitr ary el l i p tic op er a tor with prin c i p al symb ol h ∗ ( σ ( D )) . Pr o of. The index in this situation can b e considered as a functional K 1 ( J / K ) C − → C σ ∈ Mat N ( J / K ) 7− → ind Op( σ ) . By Prop osition 8.1 one has h ∗ [ σ ( D ) ] = (sgn h )[ σ ( D )]. Therefore, w e obtain the desired equalit y ind h ∗ D = (sgn h ) ind D . 15 References [1] B. A. Plamenevs ky and V. N. Senichk in. Repres en ta t ions of C ∗ -algebras of pseu- do differen tial op erators on piecewise-smooth manifolds. Algebr a i A naliz , 13 , No. 6, 2001, 124–17 4 . [2] V. Nistor. 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