aa r X i v : . [ m a t h . A P ] D ec Index formulas on stratified manifolds ∗ A. Yu. Savin, B.Yu. Sternin
November 3, 2018
Abstract
Elliptic operators on stratified manifolds with any finite number of strata areconsidered. Under certain assumptions on the symbols of operators, we obtain indexformulas, which express index as a sum of indices of elliptic operators on the strata.
This paper deals with elliptic theory on stratified manifolds with stratification of arbitrarylength. Analytical aspects of this theory (notion of symbol, ellipticity condition, finitenesstheorem) are now sufficiently well worked out, at least for pseudodifferential operators oforder zero (see [1–5]).Let us summarize the results obtained in the cited papers (below we shall only dealwith zero-order operators). A stratified manifold is a union of a finite number of openstrata. Each stratum is a smooth manifold; and the strata are glued together in somespecial way (see op. cit.). The principal symbol of a pseudodifferential operator in thissituation is a collection of symbols on the strata. Each symbol is an operator-valuedfunction, defined on the cotangent bundle (minus the zero section) of the correspondingopen stratum. The symbol assigned to the stratum of maximal dimension plays a specialrole. This symbol, which is called the interior symbol of operator, ranges in operators infinite-dimensional vector spaces. The ellipticity condition requires the invertibility of theprincipal symbol of the operator, i.e., invertibility of symbols on all strata. Finitenesstheorem asserts that an elliptic operator is Fredholm in L -spaces.An approach to index formulas, i.e., formulas, which express index in terms of theprincipal symbol of the operator was proposed in [6] for operators on manifolds withisolated singularities. The idea of this approach is to obtain formula for the index as asum of homotopy invariants of symbols on the stratum of maximal dimension and thesingular points. Such index formulas were obtained in the cited paper, and for manifoldswith edges (simplest class of stratified manifolds with nonisolated singularities) in [7]and [8]. Note that to obtain such formulas, one assumes that the symbol of operator is ∗ Supported by RFBR grants 06-01-00098, 08-01-00867, Pierre Deligne Balzan prize in mathemat-ics, and also by DFG grant DFG 436 RUS 113/849/0-1 r “ K -theory and noncommutative geometry ofstratified manifolds”. symmetry condition ). In the present paper, we introduce a class of transformations of cotangent bundles ofstratified manifolds, and for an elliptic operator invariant under one of the transformationswe give an explicit index formula in terms of homotopy invariants of elliptic symbols onthe strata.Let us briefly describe the contents of the paper.We start with a brief summary of results on the geometry of stratified manifolds andtheory of pseudodifferential operators on them, which are used in the following sections(in our exposition we follow the terminology of the paper [9]).In index theory of elliptic operators on stratified manifolds an important role is playedby the surgery method (see [8, 10]). This method permits one to localize contributionsto the index of symbols on the strata. In this paper, we introduce class of surgeries inphase space . Such surgeries are carried out on cotangent bundles of manifolds, ratherthan manifolds themselves. This surgery technique is treated in sections 3 and 4. Theclass of transformations of the cotangent bundle which we consider in this paper hasthe following important property: transformations from this class naturally act on thealgebra of principal symbols of ψ DO. We say that an operator satisfies symmetry condition if its principal symbol is invariant under some mapping from this class (in what followstransformations are denoted by G and operators satisfying symmetry condition are called G -invariant). Then for a G -invariant operator we construct homotopy invariants for eachof the strata of the manifold. The index formula expresses the index of a G -invariantelliptic operator as a sum of these homotopy invariants of symbols on the strata.A couple of words concerning the proof of index theorem. Using the homotopy classi-fication of elliptic operators on stratified manifolds [9], we compute the contributions tothe index of the strata of lower dimensions. To compute the contribution of the stratumof maximal dimension, we use surgery in phase space.Note, finally, that the application of surgery method in [7] was substantially hin-dered, because the symmetry condition could only be satisfied for operators, for whichthe Atiyah–Bott obstruction [11] is equal to zero. In the present paper we drop this ratherrestrictive condition using a class of nonlocal operators. Ψ DO on stratified manifolds Let us briefly recall some basic facts from the theory of operators on stratified manifolds,which are used in the present paper. Detailed exposition can be found, e.g., in [3, 4, 9].
1. Stratified manifolds.
Let M be a compact stratified manifold in the sense of [9].Recall that this means that M is a Hausdorff topological space with decreasing filtrationof length r M = M ⊃ M ⊃ M . . . ⊃ M r ⊃ ∅ Note that in general one can not drop the symmetry condition: using methods of [7] it can be shownthat for an arbitrary elliptic operator on a stratified manifold there is no decomposition of index as asum of homotopy invariants of symbols on the strata.
2y closed subsets M j (called strata ), such that each complement M j \M j +1 ≡ M ◦ j (called open stratum ) is homeomorphic to the interior M ◦ j of a compact manifold with corners ,denoted by M j (which is called the blow up of manifold M j ). In particular, manifold M r is smooth. The blowup of manifold M is denoted by M . There is a continuous projection π : M −→ M . Number r is called the length of stratification .In addition, stratified manifold has the following structure: each open stratum M ◦ j has a neighborhood U ⊂ M \ M j +1 homeomorphic to a locally-trivial bundle over M ◦ j K Ω j −→ M ◦ j , (1)whose fiber over point x ∈ M ◦ j is the cone K Ω j ( x ) := [0 , × Ω j ( x ) (cid:14) { } × Ω j ( x )with base Ω j ( x ); here we suppose that the base Ω j ( x ) of the cone is a stratified manifoldwith stratification of length < r .Cotangent bundles of the strata are denoted by T ∗ M j ∈ Vect( M j ) , j ≥ , T ∗ M ∈
Vect( M ) . Note that the bundle T ∗ M j is isomorphic to T ∗ M j (the isomorphism is not canonical).Let us call M ◦ = M ◦ the smooth stratum , while M singular stratum . Example 2.1.
Manifolds with stratification of length one are called manifolds with edges .In this case the stratum M is called edge (it is a closed smooth manifold). The comple-ment M \ M is a smooth manifold, while some neighborhood U of stratum M fibersover M with fiber cone (1), where the base Ω ( x ) of the cone is a smooth manifold. Theblowup M is obtained as follows: we take manifold M and in U replace bundle with fibercone K Ω j ( x ) := [0 , × Ω j ( x ) (cid:14) { } × Ω j ( x ) by bundle with fiber cylinder [0 , × Ω j ( x ).
2. Pseudodifferential operators.
Let Ψ( M ) be the algebra of pseudodifferentialoperators ( ψ DO) of order zero on M , acting in the space L ( M ) of complex valuedfunctions on M (the definition of this algebra ψ DO and the measure in the definition ofthe L -space can be found, e.g., in [3, 4]). The algebra of principal symbols Ψ( M ) / K —quotient by the ideal of compact operators — is denoted by Σ( M ).The principal symbol σ ( D ) of an operator D ∈ Ψ( M ) on a stratified manifold is acollection σ ( D ) = ( σ ( D ) , σ ( D ) , ..., σ r ( D )) (2) Recall that an n -dimensional manifold with corners is a Hausdorff topological space locally homeo-morphic to the product R k + × R n − k , 0 ≤ k ≤ n with smooth transition functions between domains of thistype.
3f symbols on the strata, where symbol σ j ( D ), j ≥ , is defined on the cotangent bundle T ∗ M j minus the zero section of the corresponding stratum. The symbol σ ( D ), corre-sponding to the smooth stratum M ◦ , is called the interior symbol and is a complex-valuedfunctions, while the remaining components of the symbol are functions σ j ( D ) ∈ C ( T ∗ M j \ , B ( L ( K Ω j )))with values in operators, acting in L -spaces on cones K Ω j . The measure on the cone K Ω j is described in [4]. The symbols on different strata are related by a certain compatibilitycondition, which is described in the cited paper.
1. Endomorphisms of cotangent bundle.
The restriction of the cotangent bundle T ∗ M to the subspace ∂ j M = π − ( M ◦ j ) ⊂ M has direct sum decomposition [9]: T ∗ M| ∂ j M ≃ π ∗ ( T ∗ M j ) ⊕ T ∗ Ω j ⊕ R , where the the second summand corresponds to directions along the base of the bundle ofcones (1), while the third summand corresponds to the directions along the radial variableon the cone. Definition 3.1.
An endomorphism h ∈ End( T ∗ M ) of the cotangent bundle of M , whichis defined in a neighborhood of the boundary ∂M ⊂ M is called admissible , if for each j ≥ h | ∂ j M = h j ⊕ Id ⊕ Id : 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 ≥ h = h . Remark 3.1.
For manifolds with edges (see Example 2.1), admissible endomorphismsare precisely those endomorphisms of T ∗ M , which are induced by endomorphisms of thecotangent bundle of the edge M .Let us define the action h ∗ of an admissible endomorphism h on principal symbols on M : h ∗ ( σ , σ , ..., σ r ) = ( h ∗ σ , h ∗ σ , ..., h ∗ r σ r ) . (4)Here r is the length of stratification and we use the fact that for any j ≥ σ j is an operator-valued function on the space T ∗ M j , on which endomorphism h j actsfiberwise-linearly. We note also that the action on the interior symbol σ is definedonly in a neighborhood of the boundary ∂M . (The symbol (4) is well-defined, i.e., itscomponents h ∗ σ , h ∗ σ , ..., h ∗ r σ r satisfy the compatibility condition, which follows from theadmissibility of h .) 4 . Statement of the problem. Let N be a stratified manifold of the following form.It has a neighborhood U of the singular stratum N , which is a disjoint union U = U + ⊔ U − of two diffeomorphic open submanifolds U + and U − . Let us fix diffeomorphism U + ≃ U − and consider U + and U − as two identical copies of U + .Let D : L ( N , E ) −→ L ( N , F )be an elliptic operator on N acting in sections of some bundles E, F ∈ Vect( M ) on theblow up M . The restriction D | U = Π D Π(Π is the characteristic function of set U ) of operator D to U can be considered as a directsum (modulo compact operators) D | U = D + ⊕ D − : L ( U + , E ⊕ E ) −→ L ( U + , F ⊕ F ) . (5)Hereinafter we suppose that we are given identifications E | U + ≃ E | U − and F | U + ≃ F | U − ,which cover diffeomorphism U + ≃ U − .Let us suppose that the principal symbols of operators D + and D − satisfy condition σ ( D − ) = h ∗ σ ( D + ) (6)over U + , where h ∈ End( T ∗ U + ) is an admissible endomorphism defined in U + . Lemma 3.1.
The index ind D of operator D , which satisfies condition (6) is determinedby the interior symbol of the operator.Proof.
1. Let D ′ be an elliptic operator which satisfies condition (6) in U and the interiorsymbol σ ( D ′ ) is equal to σ ( D ). Then we haveind D − ind D ′ = ind D ( D ′ ) − = ind D | U ( D ′ | U ) − = ind D + ( D ′ + ) − + ind D − ( D ′− ) − (7)(in the second equality we took into account the fact that D and D ′ differ by compactoperator in the interior, hence, we can pass to their restrictions D | U , D ′ | U to U ). Theinterior symbols of operators D + ( D ′ + ) − and D − ( D ′− ) − are equal to one, and the symbolson the singular strata for the second operator are obtained from those for the first operatorby application of endomorphism hσ j ( D − ( D ′− ) − ) = h ∗ σ j ( D + ( D ′ + ) − ) .
2. Since endomorphism h changes the sign of index (see Corollary 8.1), equation (7)gives the desired equality: ind D − ind D ′ = 0 . Below, we shall give an explicit formula for the index ind D in terms of the principalsymbol σ ( D ). 5 . Surgery. Let T = T ∗ N / ∼ , (8)be the space obtained from T ∗ N by identification of points in the closure of the sets T ∗ U + and T ∗ U − under the action of mapping h, which we consider here as an isomorphism h : T ∗ U + → T ∗ U − . The space T is a vector bundle with base, which is obtained from M by identification of sets U + ∩ M ◦ and U − ∩ M ◦ under the action of diffeomorphism U + ≃ U − , which was fixed above.Condition (6) implies that the interior symbols σ ( D + ) and σ ( D − ) are compatiblewith the identification (8) and define class[ σ ( D ) , h ] ∈ K c ( T ) (9)in topological K -group with compact supports of the locally-compact space T .Let us suppose that h reverses orientation of T ∗ U + . Then a chain, which represents T ∗ N , defines cycle on T . Denote the homology class of this cycle by[ T ] ∈ H n ( T ) , n = dim N . Let us define the following numberind t D = h ch[ σ ( D ) , h ]Td( T ⊗ C ) , [ T ] i . (10)It is rational by construction (using results of [8], one can show, that this number isactually dyadic-rational). Remark 3.2.
The invariant (10) can be written as an integralind t D = Z T ∗ N ch σ ( D )Td( T ∗ N ⊗ C ) . (11)Here the integral is interpreted as iterated: we first integrate over fibers of the cotangentbundle and then integrate over base N . The integral over base is well-defines, since theintegrand is identically zero in a neighborhood of the singular stratum. Indeed, whenwe integrate over the fibers T ∗ x U + , the contributions to the integral of the components σ ( D + ) and σ ( D − ) cancel, which follows from the fact the h is orientation-reversing andcondition (6).The next theorem belongs to A.Yu. Savin. Theorem 3.1 (surgery in phase space) . Suppose that h ∈ End( T ∗ U + ) is an orientation-reversing involution ( h = Id ) and D is an elliptic ψ DO on N which satisfies condition (6) . Then one has ind D = ind t D. (12) To prove formula (12), let us introduce the following class of operators.6 . Admissible operators.
A bounded operator Q : L ( N , E ) −→ L ( N , F ) , (13)is called admissible if the following three conditions are satisfied.1. Q is ψ DO in a neighborhood of
N \ U and in a small neighborhood of the singularstratum N .2. The restriction Q | U = (cid:18) Q ++ Q + − Q − + Q −− (cid:19) : L ( U + , E ⊕ E ) −→ L ( U + , F ⊕ F ) (14)of Q to U is a matrix of ψ DOs on U + (here we use identifications U − ∼ U + , E | U − ≃ E | U + , F | U − ≃ F | U + ).3. The operator Q in a small neighborhood of N satisfies condition (6), i.e., σ ( Q −− ) = h ∗ σ ( Q ++ ) . For admissible operators, ellipticity and finiteness theorem are formulated and provedby standard methods, and are left to the reader.
2. Topological index.
Consider the following invariant of an admissible elliptic oper-ator Q ind t Q = Z T ∗ ( N \ U ) ch σ ( Q )Td( T ∗ N ⊗ C ) + Z T ∗ U + ch σ ( Q | U )Td( T ∗ U + ⊗ C ) (15)(recall that the restriction Q | U is considered as an operator on U + , see (14)). Here theintegrals are interpreted as iterated — first along the fibers of the cotangent bundle, andthen along the base. When we integrate along the fibers T ∗ x U + , x ∈ U + , in the secondintegral Z T ∗ U + ch σ ( Q | U )Td( T ∗ U + ⊗ C ) , the contributions of the components Q ++ and Q −− cancel each other in a neighborhoodof the singular stratum. This follows from the fact that h is orientation-reversing andcondition (6).Number (15) is called topological index of operator Q . When Q is local, i.e., theoff-diagonal components in the decomposition (14) are equal to zero, the invariant (15),evidently, coincides with that defined in (10). For this reason, we keep the old notationfor this new invariant. Lemma 4.1 (properties of topological index) . . ind t Q is a homotopy invariant of the interior symbol σ ( Q ) ; . When σ ( Q ) does not depend on covariables in a neighborhood of the singular stra-tum, one has ind Q = ind t Q ;3 . When σ ( Q ) does not depend on covariables in the complement of U , the value ofthe functional ind t is determined by the restriction of the interior symbol to U andone has ind t ( h ∗ Q ) = − ind t Q, (16) where h ∗ Q stands for arbitrary elliptic operator with principal symbol h ∗ σ ( Q )4 . If Q is a composition Q = Q Q of two admissible elliptic operators, where Q and Q both satisfy condition (6) , then ind t Q = ind t Q + ind t Q . Proof.
The first claim is straightforward. The second was proved in [12, 13]. The thirdfollows from the fact that h reverses orientation of the cotangent bundle and hence, bychange of variables formula in the integral, reverses the sign of functional ind t . Finally,the fourth claim follows from the multiplicativity of the Chern character.
3. Homotopy to operator of multiplication.
Let D be an elliptic operator withsymbol satisfying condition (6). The left and right hand sides of formula (12) are homo-topy invariant in the set of admissible elliptic operators. Thus, to prove their equality, weshall choose a special representative in the homotopy class of operator D . This represen-tative is constructed in the following lemma.Consider a homeomorphism U + \ N + ≃ ∂N + × (0 , t the coordinate along(0 ,
1) and let U / be the set ∂N + × (1 / − ε, / ε ) ⊂ U + for some ε > Lemma 4.2.
There exists l > such that the operator D l = D ⊕ D ⊕ . . . ⊕ D | {z } N summands is homotopic in the class of admissible operators to some operator D ′ , whose symbol doesnot depend on covariables in U / .Proof.
1. The mapping h ∗ : K ( T ∗ M| ∂M ) ⊗ C −→ K ( T ∗ M| ∂M ) ⊗ C is equal to − Id (Lemma 8.1). Hence, the element[ σ ( D ) | t =1 / ] = [ σ ( D + ) | t =1 / ⊕ h ∗ σ ( D + ) | t =1 / ] = ( Id + h ∗ )[ σ ( D + ) | t =1 / ]is equal to zero in K ( T ∗ M| ∂M ) ⊗ C . Therefore, there exists number l such that thesymbol σ ( D l ) | t =1 / is homotopic to a symbol, which does not depend on covariables.2. The homotopy of the symbol σ ( D l ) | t =1 / can be lifted to a homotopy of operator D l . 8 . Surgery: cutting out the smooth stratum. Consider the decomposition ofmanifold
N N = U ≥ / ∪ U < / (17)into two subsets U ≥ / = ( N \ U ) ∪ ( ∂N × [1 / , U < / = N \ U ≥ / .Since the symbol of operator D ′ in Lemma 4.2 does not depend on covariables inthe domain U / , this operator is equal modulo compact operators to the direct sum D ′ = A ⊕ B of its restrictions A = Π D ′ Π : L ( U ≥ / , E ) −→ L ( U ≥ / , F ) B = (1 − Π) D ′ (1 − Π) : L ( U < / , E ) −→ L ( U < / , F )to the sets U ≥ / and U < / , respectively. Thus, we haveind D ′ = ind A + ind B. On the other hand, the topological index of D ′ is also equal to the sumind t D ′ = ind t A + ind t B. By Lemma 4.1, Item. 2 we have ind A = ind t A . Let us prove that the topological indexof operator B is equal to its analytical index.
5. Index computation near singular stratum.
Define operator of permutation T : L ( U + , C ⊕ C ) −→ L ( U + , C ⊕ C ) T ( u , u ) = ( u , u ) ( T = Id ).Consider the admissible elliptic operator B ( T ∗ h ∗ B ) − . (18)Here T ∗ A = T AT − — conjugation of operator A by T .Its index is equal toind( B ( T ∗ h ∗ B ) − ) = ind B − ind T ∗ h ∗ B = ind B − ind h ∗ B = 2 ind B (19)(in the rightmost equality we used the fact that h ∗ reverses the sign of index, see Corol-lary 8.1).By construction, for each j ≥ 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 = Id . Thesame formulas hold for the interior symbol σ ( B ) in the domain, where condition (6) issatisfied.Now, in a neighborhood of the singular stratum the symbols of the operator (18)are equal to one, i.e., this operator is equal to the identity operator, modulo compactoperators. Thus, by Lemma 4.1 (Items 2, 4, 3) we have the following chain of equalitiesind( B ( T ∗ h ∗ B ) − ) Item 2 = ind t ( B ( T ∗ h ∗ B ) − ) 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), we obtain the desired equality ind B = ind t B . Therefore,ind D ′ = ind t D ′ and hence ind D = ind t D .This completes the proof of Theorem 3.1.9 Symmetries of symbols
Let M be a stratified manifold and E, F ∈ Vect( M ) vector 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, which is defined in aneighborhood U of the singular stratum M ; • h ∈ End( T ∗ M ) is an admissible endomorphism defined in U ; • e, f are vector bundle isomorphisms E | U e ≃ ( g ∗ E ) | U , F | U f ≃ ( g ∗ F ) | U . The differential of diffeomorphism g defines a fiberwise-linear mapping dg : T ∗ M → T ∗ M , which covers g . Denote the induced maps on the strata by g j : M j → M j , j ≥ ψ DO D : L ( M , E ) −→ L ( M , F ) . (21)Since algebras of ψ DO and their principal symbols are diffeomorphism invariant, one hasthe following action of symmetry G = ( g, h, e, f ) on the symbols: G ( σ j ) = f − (cid:2) h ∗ j ( dg j ) ∗ σ j (cid:3) e, j ≥ . (22)This action is well defined (i.e., the symbol G ( σ ) = ( G ( σ ) , G ( σ ) , ..., G ( σ r )) enjoys com-patibility conditions). Note that the interior symbol G ( σ ) is defined only in neighborhood U . Let ∂T ∗ M j be the restriction of the cotangent bundle T ∗ M j to the boundary ∂M j ⊂ M j . Definition 5.2.
Let G be a symmetry. An elliptic operator D is called G - invariant , iffor all j ≥ σ j ( D ) | ∂T ∗ M j = G (cid:0) σ j ( D ) | ∂T ∗ M j (cid:1) , (23)i.e., the restrictions of the components of the symbol to the boundaries of the correspond-ing strata are G -invariant.Without loss of generality we shall assume throughout the following that the interiorsymbol enjoys equality σ ( D ) = G ( σ ( D )) in the entire domain U .Denote by G ( D ) an arbitrary operator with principal symbol G ( σ ( D )). Let D be a G -invariant elliptic operator on M , where G = ( g, h, e, f ) is a symmetry.Suppose that the symmetry satisfies the following additional condition: h : T ∗ M −→ T ∗ M reverses orientation. It turns out that in this case one can construct nontrivialhomotopy invariants of the symbols on each of the strata M j , j ≥ . Invariant of the interior symbol. Denote the disjoint union of two copies of M by N . Let us choose a neighborhood of the singular stratum in N as a union U + ∪ U − ofneighborhood U + = U on the first copy and U − = g ( U ) on the second copy.On both components of N we consider operator D , which is G -invariant, i.e. satisfiescondition (23). There are isomorphisms E | U + g ∗− ◦ e −→ E | U − and F | U + g ∗− ◦ f −→ E | F − . Denote the constructed operator on N by D ∪ D . A computation shows that the principalsymbol of this operator satisfies condition (6) and hence, this operator has topologicalindex ind t ( D ∪ D ). Clearly, this invariant is determined by the interior symbol σ ( D ) andsymmetry G .
2. Invariants of symbols on the strata M j , j ≥ . Consider the elliptic symbol σ j ( D )[ Gσ j ( D )] − (24)over the stratum M j . Denote by K the set of compact operators. Lemma 6.1.
The symbol (24) has compact fiber variation (cid:0) σ j ( D )[ Gσ j ( D )] − (cid:1) ( x, ξ ) − (cid:0) σ j ( D )[ Gσ j ( D )] − (cid:1) ( x, ξ ′ ) ∈ K , for all nonzero ξ, ξ ′ ∈ T ∗ x M j , as an operator-valued function on the bundle T ∗ M j −→ M j .Proof. Indeed, by [9], Proposition 2.2) compact fiber variation property is valid, providedthat the restriction of the symbol σ j − ( D )[ Gσ j − ( D )] − to the boundary ∂T ∗ M j − doesnot depend on covariables. However, by the G -invariance (23) of symbol σ j − ( D ), thisexpression is actually equal to the identity symbol.On the other hand, the symbol (24) is constant on the boundary ∂T ∗ M j , where itconsists of identity operators (this time, by G -invariance of σ j ( D )).Thus, the elliptic symbol (24) has compact fiber variation and is equal to identity onthe boundary of T ∗ M j ≃ T ∗ M j . This implies that this symbol on T ∗ M j can be consideredas an operator-valued symbol in the sense of Luke [14] and we can assign to it a Fredholmoperator on M ◦ j , which is isomorphism at infinity. Denote this operator byOp (cid:0) σ j ( D )[ Gσ j ( D )] − (cid:1) : L ( M ◦ j , L ( K Ω j , F )) −→ L ( M ◦ j , L ( K Ω j , F )) . (25) Remark 6.1.
The index theorem for ψ DO with operator-valued symbols [14] gives equal-ity ind Op (cid:0) σ j ( D )( Gσ j ( D )) − (cid:1) = p ! (cid:2) σ j ( D )( Gσ j ( D )) − (cid:3) , where [ σ j ( D )( Gσ j ( D )) − ] ∈ K c ( T ∗ M ◦ j )is the class of symbol in K -theory, and p ! : K c ( T ∗ M ◦ j ) → K ( pt ) = Z is the direct image mapping in K -theory induced by the projection M j → { pt } to theone-point space. Index formulas in cohomology can also be obtained (see [15]).11 Index theorem
The next theorem belongs to B.Yu. Sternin.
Theorem 7.1.
Suppose that an elliptic operator D on a stratified manifold M is G -invariant, and the admissible endomorphism h : T ∗ M −→ T ∗ M is an orientation-reversing involution ( h = Id ) . Then one has ind D = 12 ind t ( D ∪ D ) + 12 r X j =1 ind Op (cid:16) σ j ( D )[ Gσ j ( D )] − (cid:17) , (26) where the sum contains indices of elliptic operators on the strata M j with operator-valuedsymbols in the sense of [14] equal to σ j ( D )[ Gσ j ( D )] − ( see Section 6 ) . Remark 7.1.
When r = 1, g = Id , e = f = Id , h = Id , this theorem gives indexformula on manifolds with edges, see [7, 8]. Proof.
1. Denote by e D an operator with symbol( σ ( D ) , Gσ ( D ) , ..., Gσ r ( D )) . This operator is well-defined, since the collection σ ( D ) = Gσ ( D ) , Gσ ( D ) , ..., Gσ r ( D )of symbols on the strata is compatible (because the action of symmetry G preservescompatibility). By the logarithmic property of the index we obtainind D − ind e D = ind( D e D − ) . (27)Further, for all j ≥ σ j of operator D e D − are equal to identity on ∂T ∗ M j ,while the interior symbol is equal to the identity on the entire space T ∗ M . This impliesthat the operator D e D − can be decomposed (modulo compact operators) as the product D e D − = r Y j =1 P j where P j is an operator on M , whose symbols are equal to identity, except the symbol σ j ( P j ), which is equal to σ j ( D )[ Gσ j ( D )] − . We getind D e D − = r X j =1 ind P j . Note now that it follows from the properties of the symbol of P j that outside arbitrarilysmall neighborhood U of open stratum M ◦ j operator P j is equal to identity modulo com-pact operators. Thus, P j is equal (modulo compact operators) to the direct sum of its12estriction Π P j Π to U (Π is the characteristic function of U ) and the identity operator,which acts on functions on the complement of U . Hence, we obtainind P j = ind Π P j Π + ind(1 − Π) = ind Π P j Π . Let us now choose U such that it fibers over the stratum M ◦ j with conical fiber. Inthis case the restriction Π P j Π of operator P j to this neighborhood can be treated (see[9]) as an operator on M ◦ j with operator-valued symbol in the sense of Luke equal to σ j (Π P j Π) = σ j ( D )[ Gσ j ( D )] − . This gives usind D − ind e D = r X j =1 ind Π P j Π = r X j =1 ind Op (cid:16) σ j ( D )[ Gσ j ( D )] − (cid:17) . (28)The right-hand side of this equality coincides with the sum in (26).2. Consider now two copies of manifold M . We take operator D on the first copy, and e D on the second copy and apply Theorem 3.1 to operator D ∪ e D on the union of thesemanifolds. We obtain ind D + ind e D = ind t ( D ∪ e D ) . (29)Since the interior symbols of operators D ∪ e D and D ∪ D are equal, we get ind t ( D ∪ e D ) =ind t ( D ∪ D ). Thus, the sum of equations (28) and (29) gives the desired formula (26). K -theory Consider the ideal J ⊂ Ψ( M ) of operators with zero interior symbol.Let h ∈ End( T ∗ M ) be an admissible endomorphism. The action of this endomorphismon principal symbols obviously restricts to the action on the ideal J/ K ⊂ Ψ( M ) / K in theCalkin algebra. Below we shall show that the induced action h ∗ : K ∗ ( J/ K ) −→ K ∗ ( J/ K )on the rational K -group is equal to ± Id .To this end we introduce notation K ∗ ( J ) C = K ∗ ( J ) ⊗ C and define the sign of h bysgn h = h ∗ [ T ∗ M ◦ ][ T ∗ M ◦ ] ∈ {± } , where [ T ∗ M ◦ ] ∈ H ev,c ( T ∗ M ◦ ) is the fundamental class. Therefore, the sign sgn h is equalto +1, if h preserves orientation of T ∗ M and is − Proposition 8.1.
Suppose that an admissible endomorphism h ∈ End( T ∗ M ) has finiteorder ( h N = 1) . Then one has h ∗ = (sgn h ) Id : K ∗ ( J/ K ) C −→ K ∗ ( J/ K ) C . (30)13 roof. Consider a decreasing sequence of ideals J = J ⊃ J ⊃ J ⊃ . . . ⊃ J r = K , where r is the length of stratification, and the ideal J j consists of operators D withsymbols σ k ( D ), which are equal to zero for k ≤ j. There are induced actions of h on theideal J j / K and the quotient J j /J j +1 .Let us prove by induction that the mapping h ∗ : K ∗ ( J j / K ) C −→ K ∗ ( J j / K ) C is equal to (sgn h ) Id.
1. Base of induction j = r . In this case the proposition is valid, since J r = K and K ∗ ( J r / K ) = 0.2. Inductive step. Let h ∗ = (sgn h ) Id as an endomorphism of the group K ∗ ( J j +1 / K ) C .Let us prove that the same equality is valid for the group K ∗ ( J j / K ) C . To this end, considerthe commutative 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 rows are two identical 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 equal to the identity.Proof.
1. In [9] the following isomorphism was obtained K ∗ ( J j /J j +1 ) ≃ K ∗ +1 c ( T ∗ M j +1 ) , (32)where the action h ∗ on the K -group of algebra in the left-hand side in (32) transformsto the action of endomorphism h j +1 ∈ End( T ∗ M j +1 ) on the topological K -group in theright-hand side of the equality.2. Since sgn h = sgn h j +1 (which is easy to see, because h is admissible), to provelemma, it suffices to show that the mapping h ∗ j +1 : K ∗ c ( T ∗ M j +1 ) C −→ K ∗ c ( T ∗ M j +1 ) C is equal to (sgn h ) Id .To prove this, we use isomorphisms K ∗ c ( T ∗ M j +1 ) C ch ≃ H ∗ c ( T ∗ M j +1 ) C ≃ Hom( H ∗ c ( M ◦ j +1 ) , C )14Chern character isomorphism and Poincare isomorphism — integration over fundamen-tal cycle [ T ∗ M j +1 ]), to transform the proof that h ∗ j +1 = (sgn h ) Id from K -theory tocohomology. For all x ∈ H ∗ c ( T ∗ M j +1 ) and y ∈ H ∗ c ( M ◦ j +1 ) we have 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 we used equality h ∗ j +1 ( y ) = y , which isvalid, because y is a cohomology class on the base M ◦ j +1 ). Thus, we 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 equality (34) is valid for all y , weobtain the desired equality h ∗ j +1 x = (sgn h ) x .Indeed, mappings in the leftmost and rightmost columns of the commutative diagram(31) are identity mappings (the left mapping is identity by assumption, while the rightmapping by Lemma 8.1). Therefore, application of the following algebraic Lemma 8.2shows that the mapping (sgn h ) h ∗ in the middle column of the diagram (31) is also identity. Lemma 8.2.
Consider commutative diagram A −→ B −→ CId ↓ ↓ h ↓ IdA −→ B −→ C of finite-dimensional vector spaces and linear mappings with exact rows. If mapping h hasfinite order ( h N = Id ) , then h = Id .Proof. Diagram chase gives equality ( h − Id ) = 0. This means that h = Id + e , where e = 0. Now we use finite order condition and obtain that h is diagonalized in some base.This implies that e = 0. Corollary 8.1.
Under assumptions of Proposition for arbitrary elliptic operator D on M with interior symbol, which does not depend on covariables, one has: ind( h ∗ D ) = (sgn h ) ind D, where h ∗ D stands for an arbitrary elliptic operator with principal symbol h ∗ ( σ ( D )) .Proof. The index in this situation can be considered as a functional K ( J/ K ) C −→ C σ ∈ Mat N ( J/ K ) ind Op( σ ) . By Proposition 8.1 one has h ∗ [ σ ( D )] = (sgn h )[ σ ( D )]. Therefore, we obtain the desiredequality ind h ∗ D = (sgn h ) ind D. eferences [1] B. A. Plamenevsky and V. N. Senichkin. Representations of C ∗ -algebras of pseu-dodifferential operators on piecewise-smooth manifolds. Algebra i Analiz , , No. 6,2001, 124–174.[2] V. Nistor. Pseudodifferential operators on non-compact manifolds and analysis onpolyhedral domains. In Spectral geometry of manifolds with boundary and decom-position of manifolds , volume 366 of
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