RELATIVE CHERN CHARACTER, BOUNDARIES AND INDEX FORMULÆ

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1 RELATIVE CHERN CHARACTER, BOUNDARIES AND INDEX FORMULÆ PIERRE ALBIN AND RICHARD MELROSE.9A; Revise: ; Run: August 1, 28 Abstract. For three classes of elliptic pseuoifferential operators on a compact manifol with bounary which have geometric K-theory, namely the transmission algebra introuce by Boutet e Monvel [5], the zero algebra introuce by Mazzeo in [9, 1] an the scattering algebra from [16] we give explicit formulæ for the Chern character of the inex bunle in terms of the symbols incluing normal operators at the bounary of a Freholm family of fibre operators. This involves appropriate escriptions, in each case, of the cohomology with compact supports in the interior of the total space of a vector bunle over a manifol with bounary in which the Chern character, mapping from the corresponing realization of K-theory, naturally takes values. Contents Introuction 1 1. Relative cohomology Homotopy invariance an moule structure Manifol with bounary Sphere bunle of a real vector bunle Bunle with line subbunle Bunle over manifol with bounary Bunle with line subbunle over the bounary Pseuoifferential operators an relative K-theory Chern character an the families inex theorem Scattering families inex theorem Zero families inex theorem Transmission families inex theorem 29 Appenix A. Normal operator 31 Appenix B. Resiue traces 33 Appenix C. Operator value forms 36 References 37 Introuction Among the ifferent algebras of pseuoifferential operators on a compact manifol with bounary, those for which the stable homotopy classes of the Freholm elements reuce to the relative K-theory of the cotangent bunle can be expecte to have the simplest, most local, inex formulæ. These cases inclue the algebra 1

2 2 PIERRE ALBIN AND RICHARD MELROSE of transmission operators introuce by Boutet e Monvel, for which this is shown in essence in [5], the algebra of zero pseuoifferential operators of [11] an the algebra of scattering pseuoifferential operators of [16]. For a Freholm family of such operators efine on the fibers of a fibration, 1 Z M the families inex, compose with the Chern character, therefore gives a map the same in all three cases 2 K T M /B in KB Ch H even B, M = M \ M. This map is always given by the general Atiyah-Singer formula incluing here the extension by Atiyah an Bott just as in the bounaryless case in [2, 3] ψ B, 3 ChInA = ψ! Ch[A] TZ where Ch[A] is the Chern character, 4 Ch : K T M /B H even c T M /B. In fact, the ientification, in the three cases, of the stable homotopy classes of the Freholm elements with the relative K-theory involves non-trivial homotopies. As a result 3 is not really a formula for the families inex. Here we give much more explicit formulæ for the Chern character in terms of the symbolic ata etermining the Freholm conition for the family. In each case this correspons to the ellipticity of the interior symbol, uniformly up to the bounary in an appropriate sense, together with the invertibility of a bounary family. To represent the Chern character we construct relative erham chain complexes, all with cohomology Hc T M /B, tailore to each setting an then construct Chern-Weil forms epening on the leaing symbolic ata. Then ψ! is the generalize integration map on cohomology from T M/B to B. To explain the strategy behin these explicit versions of the formulæ 3, consier the familiar case of an elliptic family of classical pseuoifferential operators A Ψ m M/B; E where E = E +, E is a superbunle i.e. a Z 2 -grae bunle over M an the fibration 1 has compact bounaryless fibres. This is the families setting of Atiyah an Singer an the analytic inex is a map as in 2, in this case the Chern character gives ChInA : KT M/B H even B. The K-class is fixe by the invertible symbol of A, a = σ m A C S M/B; home N m where N m is a trivial real line bunle capturing the homogeneity. Feosov in [6] gives an explicit formula for the Chern character of the compactly supporte K- class etermine by E an a as a erham class with compact support on T M/B. Moifying his approach slightly we consier the representation of Hc W, for any real vector bunle over M, as the hypercohomology of the relative complex 5 C M; Λ C SW ; Λ, D = π

3 RCCBIF 3 where π : W M is the bunle projection an SW = W \O M /R + is the sphere bunle of M. The Chern-Weil formula given by Feosov in terms of connections an curvatures, naturally fits into this representation as the class 6 Ch[A] = ChE, a = ChE Cha C M; Λ even C S M/B; Λ o Cha = 1 2πi ChE = tr e ω+ tr e ω, tr a 1 ae wt t where wt = 1 tω + + ta 1 ω a + 1 2πi t1 ta 1 a 2. The push-forwar map on cohomology, ψ!, becomes integration of the secon form in 5 an the Atiyah-Singer formula, 3, becomes an explict fibre integral 7 ChInA = Cha Tψ. S M/B We procee to iscuss formulæ essentially as explicit as 6 an 7 for the Chern character of the inex bunle for families of pseuoifferential operators on a manifol with bounary, where the uniformity conitions up to the bounary correspon to one of the three cases mentione above. The simplest of the three cases, really because it is the most commutative, correspons to the scattering calculus of [16]. This arises geometrically in the context of asymptotically flat manifols an in particular inclues constant-coefficient ifferential operators on a vector space, thought of as acting on the raial compactification as a compact manifol with bounary. Thus, the fibration 1 now, an from now on, has fibres which are compact manifols with bounary an we consier a fully elliptic family A Ψ m scm/b; E. As well as a uniform version of the symbol, a = σa C sc S M/B; home N m acting on a bounary-rescale version of the cosphere bunle there is an invariantly-efine bounary symbol b = βa C sc T M M/B; home N m. Together these two symbols form a smooth section of the bunle over the bounary of the raial compactification sc T M/B which is continuous at the corner. Full ellipticity of the family A reuces to invertibility of this joint symbol an this is equivalent to the requirement that A be a family of Freholm operators on the natural geometric Sobolev spaces. Since sc T M/B is a topological manifol with bounary, these full symbols provie a chain space for the K-theory with compact supports in the interior an the analytic inex becomes a well-efine map as in 2. To give an explicit formula for the Chern character in this scattering setting we use a similar relative chain complex to 5, now associate to a vector bunle π : W M over a manifol with bounary. Namely, C M; Λ { u, v; u C SW ; Λ, v C W M ; Λ an ι u = ι v }, 8 π D = φ, φ = π ι. The cohomology of this chain complex is Hc W, the compactly supporte cohomology of W restricte to the interior of M, an the Chern character is represente by the explicit forms 9 Ch[A] = ChE, a, b = ChE Cha, Chb

4 4 PIERRE ALBIN AND RICHARD MELROSE where Cha an Chb are given by Feosov s formula 6. The Atiyah-Singer formula 3 then follows an in this case it becomes quite clear that the bounary symbol is a rather complete analogue of the usual symbol an enters into the inex formula in the same way. The other two contexts in which we give such an explicit inex formula are similar, but more complicate an less symmetric between the bounary an the cosphere bunle the high-momentum limit because there is more resiual noncommutativity at the bounary. Note that we are only consiering those settings in which the analytic inex gives a map as in 2 always the same map of course! This restriction correspons to the fact, as we shall see, that the only non-commutativity which remains at the bounary is in the normal variables tangential non-commutativity, as occurs in the cusp or b-calculi, inuces an analytic inex map from a ifferent K-theory in place of the relative K-theory in 2 an this leas to more subtlety in the inex formula. Consier next the zero calculus which correspons geometrically to asymptotically hyperbolic or conformally compact manifols an is so name because it quantizates the Lie algebra of vector fiels which vanish at the bounary of any compact manifol with bounary as oppose to the scattering calculus which quantizes the smaller Lie algebra in which the normal part also vanishes to secon orer. As note this Lie algebra is not commutative at the bounary, rather it is solvable with tangential part forming an Abelian subalgebra on which the normal part acts by homotheity. As a result the final formula has a truly regularize Chern character, an eta form, coming from the normal part. Again we consier a family of pseuoifferential operators, now A Ψ m M/B; E for a fibration 1 with fibres compact manifols with bounary moelle on Z. The fully ellipticity of such a family, corresponing to its being Freholm on the natural zero Sobolev spaces, reuces to the invertibility of the uniform symbol, a = σ m A C S M/B; home N m together with the invertibility of the normal operator, or equivalently the reuce normal family. The former takes values in the bunle, over the bounary, of invariant pseuoifferential operators on the Lie group associate to the tangent solvable Lie algebra mentione above. The conition here is invertibility on the appropriate Sobolev spaces since this inverse, because of the appearance of non-trivial asymptotic terms, lies in a larger space of pseuofferential operators on the group. The reuce normal operator, RNA, correspons to ecomposition of the normal operator in terms of the representations of the solvable group. In the case of the zero calculus it is less obvious, but shown in [1], that this symbolic ata, the invertibility of which fixes the Freholm property of A, also gives a chain space for the relative K-theory [A] = [a, RNA] KT M/B. Similarly, the chain complex leaing again to cohomology with compact supports is more involve an epens on more of the structure of the zero cotangent bunle. More abstractly, consier a real vector bunle W over M with restriction to the bounary having a trivial real line subbunle L W M, set U = W M /L an 1 C M; Λ k C SW L; Λ k 1 C SU; Λ k 3, C SW L; Λ k 1 = { a, γ C SW ; Λ k 1 C L; Λ k 1 ; i ±a = i ±γ } D = φ 1 π, φ 1 = SW π, φ 2 = ν SW φ 2 L ι, πsu νl.

5 RCCBIF 5 Here i ± are the restriction to the two bounary components of L which are also submanifols of SW, the maps π are the various bunle projections an ν SW, ν L are well-efine push-forwar maps along the fibres of L, the first from SW to SU an the secon from L to M. As alreay anticipate the cohomology of this complex is canonically isomorphic to the compactly supporte cohomology of W over M. The reuce normal operator for the zero calculus is a family of pseuoiferential operators on an interval, although with ifferent uniformity behaviour at the two ens. It is fixe by the choice of a splitting of the zero cotangent bunle over the bounary an the choice of a metric an is then naturally parametrize by S M/B which is SU in the preceeing paragraph. In terms of the representation of the cohomology with compact supports of T M /B given by 1, with W = T M/B, the Chern character is given by the explicit forms 11 Ch[A] = Cha, RNA = ChE Cha, ChIb RNA ηrna, where I b RNA is the moel operator of the reuce normal operator at one en of the interval, known as the inicial family, an ηn is given by an expression similar to that efining Cha but taking into account that the operators involve are not trace-class an o not commute, namely 1 2πi 1 t R Tr b,sc N 1 N e ω N t + t R Tr b,sc N e ω N t N 1 t. The renormalize trace appearing in this formula is efine in Appenix B following previous work of the secon author an Victor Nistor [18]. As in the scattering case we obtain an explicit representative of the Chern character. Finally, we consier the calculus of Boutet e Monvel, the transmission calculus, which contains classical elliptic bounary value problems an their parametrices. Geometrically this calculus correspons to Riemannian manifols with bounary an so the connection to relative K-theory is immeiate an alreay present in [5]. An element of the transmission calculus is represente by a matrix of operators A = γ+ A + B T K : Q C X; E + C X; E C X; F + C X; F acting on the superbunles E over X an F over X. Whether or not a family A Ψ m tmm/b; E, F is Freholm is again etermine by invertibility of two moel operators, the interior symbol an the bounary symbol. An operator is elliptic if the former is invertible an fully elliptic if they are both invertible. Similarly to the zero calculus, the moel operator at the bounary is a family of pseuoifferential operators parametrize by the cosphere bunle of the bounary, but here these operators are of Wiener-Hopf type. As before there is a convenient escription of the relative cohomology of the cotangent bunle over the interior that is compatible with the represention of a K-class by a fully elliptic family of transmission operators. For this consier W, L, an U as escribe in 1 above, together with a restricte space of sections of SW, C ± SW ; Λ = {α C SW ; Λ : i L +α = i L α, i L +α = i L α},

6 6 PIERRE ALBIN AND RICHARD MELROSE an form the complex 12 C M; Λ k C± SW ; Λ k 1 C M; Λ k 2 C SU; Λ k 3, D = φ 1 π, φ 1 = SW i φ 2, φ 2 = ν SW, πsu. M with the notation as in the previous paragraph. The cohomology of this complex is again canonically isomorphic to the compactly supporte cohomology of W over M. If a an N are respectively the interior symbol an bounary symbol of a fully elliptic family of transmission operators, then the Chern character of the associate K-theory class is represente by ChE, F, a, N = ChE, Cha, ChE F, ηn in the complex 12 with W, L, an U again equal to T M/B, the normal bunle to the bounary, an T M/B respectively. The form ηn is formally the same as in the zero calculus, but the renormalization of the trace is one in a ifferent fashion, ue to Feosov see 3.25 below. As alreay mentione, central to this iscussion is the fact that the K-theory escribe by fully elliptic families in these cacluli is the topological K-theory of the cotangent bunle in the interior. A well-known consequence is that quantization of an elliptic symbol to a Freholm operator is only possible if the Atiyah-Bott obstruction of the symbol vanish. One way aroun this is to quantize into other pseuoifferential calculi. For instance, the b-calculus, which is well-aapte to manifols with asymptotically cylinrical ens, is universal in the sense that any elliptic symbol can be quantize to a Freholm operator. Whereas the calculi escribe above are asymptotically non-commutative in at most the normal irection, the b calculus is in the same sense asymptotically non-commutative in all irections. This is relate to the fact that the eta invariants escribe above are local in the bounary an global in the normal irection to the bounary while the eta invariant in the Atiyah-Patoi-Singer inex theorem is global in the bounary. An analysis of the Chern character for the b-calculus an relate calculi is the subject of an ongoing project of the secon author with Frééric Rochon [19]. This manuscript is ivie into three parts. Section 1 is evote to a iscussion of cohomology. We escribe the approach to relative cohomology that we will follow together with some stanar properties an work out various ways of representing Hc T M /B. In section 2 the represention of a class in K c T M /B by a family of fully elliptic operators in either the scattering, zero, or transmission calculus is recalle. Finally, in section 3, we put these iscussions together an obtain explicit formulæfor the Chern character Ch : K c T M /B Hc even T M /B as escribe above. In each case an explicit Atiyah-Singer formula for the Chern character of the inex bunle is given using only the appropriate moel operators. C k i 1. Relative cohomology Suppose that Ci,, i = 1,..., N, are Z-grae ifferential complexes an φ i : C k+1 fi i+1 for 1 i < N, is a complex of chain maps between them, so

7 RCCBIF 7 φ i = φ i+1. Then the complexes can be rolle up into one complex. In fact only the cases N = 2 an N = 3 arise here, so consier first N = 2 : 1.1 C k 1 1 C k 1 C k+1 1 φ C k f 2 φ C k+1 f 2 φ C k+2 f 2. Then φ inuces a map of the corresponing hypercohomologies which we can also enote φ : H1 k H k+1 f 2. The relative chain complex 1.2 C k, D, C k = C1 k C k f 2, D = φ is such that inclusion an projection give an exact sequence 1.3 C k f 2 ι C k p C k 1. The hypercohomology of 1.2, enote H C, φ, may be compute from a spectral sequence, in this case a rather simple one corresponing to the fact that the long exact sequence associate to 1.3 is 1.4 H k f 2 ι H k C, φ p H k 1 For N = 3 the φ i give a commutative iagram φ H k f C k 1 1 C k 1 C k+1 1 φ 1 C k f1 2 φ 1 C k+1 f1 2 φ 1 C k+2 f1 2 φ 2 C k+1 f1 f2 3 The rolle up ouble complex is 1.6 C k, D, C k = φ 2 C k+2 f1 f2 3 φ 2 C k+3 f1 f2 3 3 Ci k, D = φ 1. φ 2 i=1 Of course, the secon two rows in 1.5 are an example of 1.1, so give the complex 1.7 C 2,, Ck 2 = C2 k C k f2 3, = such that 1.8 Ck f 1 2 The hypercohomologies therefore give long exact se- is a short exact sequence. quences as in H k f 2 H k f 3 ι C k p C k 1 φ 2 ι H k C, φ ι H k p H k 1 p H k 2 φ 1 Hk+1 f 2 φ 2 H k+1 f 3. Thus the case N > 2 reuces to an iteration of N = 2 cases.

8 8 PIERRE ALBIN AND RICHARD MELROSE The most obvious case of relative cohomology in a erham setting arises from a smooth map between compact manifols possibly with corners 1.1 ψ : M X, with C 2 = C M; Λ, C 1 = C X, Λ an D = ψ. In this case we may enote the relative cohomology by H M, ψ. Note that this is precisely how relative cohomology is efine in [4, 6]. Several variants of relative cohomology, leaing to stanar cohomology groups, are of interest here, only the last an most funamental for the zero inex formula correspons to N = Homotopy invariance an moule structure. In all of the cases we will consier below the complexes Ci will be built up from ifferential forms an will inherit more structure. We point out some of these properties for later use Moule structure. First, suppose that each of the C i has a grae prouct i : C j i Ck i C j+k i α i β = α i β + 1 α α i β, φ i α i β = φ i α i+1 φ i β. Lemma 1.1. If N = 3 an each C i has a grae prouct as above, then H C, φ is a moule over H C 1 through H C, φ H C 1 α i, γ s.t. Proof. Notice that if α i C k D α i γ = α 1 1 γ φ 1 α 1 1 γ α 2 2 φ 1 γ φ 2 α 2 2 φ 1 γ + α 3 3 φ 2 φ 1 γ α 1 1 γ, α 2 2 φ 1 γ, α 3 3 φ 2 φ 1 γ H C, φ α 1 = D α i γ + 1 k 1 f1 α 2 γ. 1 f1 f2 α 3 Thus if f 1 an f 2 are o D satisfies a Leibnitz rule with respect to. In any case it is true that if γ is close an α i is D-close then α i γ is D-close an represents a class in H M, φ epening only on the class of γ in H C 1 an α i in H M, φ Homotopy invariance. Recall the usual proof of homotopy invariance of the Chern character on a close manifol. A one-parameter family of connections on a fixe bunle over a space X, can be interprete as a single connection on the same bunle pulle-back to X [, 1]. Since the Chern character of this connection is close, the cohomology class of the Chern character of the family of connections is constant in the parameter. What is important here is that the space of forms on X [, 1] r is really two copies of the space of forms on X, Ω X [, 1] r = Ω X r Ω 1 X an that the ifferential becomes with respect to this splitting. r The spaces C i below will generally be irect sums of spaces of ifferential forms with perhaps some compatibility conitions. In orer to carry out the classical

9 RCCBIF 9 argument for the homotopy invariance of the Chern character, assume that a oneparameter family of elements α r of C efine an element of C with 1.11 Ci = C i C 1 i = C i + r C 1 i an that the action of an φ i is extene to C so that with respect to this splitting φi = an φ r i =. φi Then D acts on C = C C 1 by φ 1 D = r φ 2 r φ 1 r φ 2 Lemma 1.2. If the spaces C i have the properties of ifferential forms as escribe above an α r is a D close element of C then α an α 1 are cohomologous in C. Proof. Write α i = αt, i αn i = α i t + r αn i with respect to the splitting Then the fact that α is D-close implies that i α i 1 α = r α t r = α 1 n, φ 1 αn 1 αn, 2 φ 2 αn 2 + αn 3 r = D α 1 n, αn, 2 αn 3 r, hence i α an i 1 α represent the same class in the cohomology of C, D Push-forwar. Notice that, if M is oriente an im Y < im M, there is a well-efine integral 1.12 : HM, ψ C. M Inee, two representatives of the same class in H M, ψ iffer an element in the image of D, i.e. of the form u, ψ u v, but v = by Stoke s theorem an M M ψ u = for imensional reasons, so the value of the integral is not affecte. More generally, if we have a pair C1 C2, C 1 C 2 of Z-grae complexes with chain maps φ, φ as in 1.2, an maps between them Ci ψ i C l i satisfying ψ i = ψ i an fitting into the commutative iagram 1.13 C k 1 φ C k+1 f 2 ψ 1 C k l 1 eφ ψ 2 Ck l+1 f 2

10 1 PIERRE ALBIN AND RICHARD MELROSE we get a map 1.14 H C, φ ψ H l C, φ. This situation occurs for instance for pull-back iagrams: Assume X f Z is a fibration of oriente manifols an E π Z is a vector bunle, then we have f SE eπ X Ω f SE eπ Ω X ef SE π Z f = ef Ω l SE π f Ω l Z where l = im X/Z an f enotes the push-forwar of ifferential forms by integration along the fibers. Then 1.13 commutes essentially because the fibers of f an f coincie an we get a map H f SE, π H SE, π inuce by f via 1.14, which we can call push-forwar by f. Alternately, consier a smooth fibration N h Γ with im N > im Γ an the vertical cosphere bunle S N/Γ π N. We can use 1.14 to get a map 1.15 H S N/Γ, π h H Y by taking l = 2 im N/Γ 1, C 1 =, C 2 = Ω Y an mapping between Ω S N/Γ an Ω Y by the push-forwar of forms along the map h π. In this case 1.13 commutes because forms pulle back from N push-fowar to zero along h π. This push-forwar map will be use in the formula for the families inex theorem below. Note that 1.12 is a particular case with Γ = {pt} Manifol with bounary. A stanar case of 1.1 arises when X is a compact manifol with bounary an ψ is the inclusion map for the bounary. Thus C 1 = C X; Λ an ψ = ι : X X, C 2 = C X; Λ. Then Lemma 1.3. For ι : X X the inclusion of the bounary of a compact manifol with bounary an 1.16 C k = C X; Λ k C X; Λ k 1, H k C, φ = H k X, ι = H k X; X = H k c X \ X is relative cohomology in the usual sense. Proof. This is completely stanar but a brief proof is inclue for the sake of completeness an for later generalization. If u, v C k satisfies D u, v = then there is T u, v C k 1 such that DT u, v u, v = ω, with 1.17 ω C c X ; Λ k. Taking a prouct neighbourhoo of the bounary, U = [, 1 x X, an enoting Y = X a smooth form ecomposes on U as 1.18 u = u t x + x u n x, u t C [, 1 Y ; Λ k Y, u n C [, 1 Y ; Λ k 1 Y with ifferential 1.19 u = Y u t + x x u tx Y u n x.

11 RCCBIF 11 So if u, v satisfies Du, v =, 1.2 Y u t =, x u tx Y u n x = in x < 1, v = u t. Choosing a cutoff function ρ C [, 1] with ρx = 1 in x < 1 2, ρx = in x > 3 4, consier 1.21 T u, v = ρx x u n s s + ρxv,. This satisfies DT u, v = u + ω, v where ω C X; Λ k has support away from the bounary as require in Thus the complex retracts to the subcomplex of erham forms with compact support in the interior an the cohomology is therefore the cohomology of X relative to its bounary. Corollary 1.4. A particular case of 1.1 arises with X = U, the raial compactification of a real vector bunle U over a compact manifol Y an X = SU = U \ Y /R + the sphere bunle, so 1.22 H X, ι = H c U Sphere bunle of a real vector bunle. Although Corollary 1.4 is a compact representation of the compactly-supporte cohomology of a real vector bunle over a manifol it is not the most natural one for inex theory. In view of the contractibility of the fibres, instea of the inclusion of the sphere bunle as the bounary of the raial compactification of W we may consier instea simply the projection 1.23 π : SW X. Denote by H k SW, π the cohomology of the complex 1.24 C X; Λ C SW ; Λ 1, D = π We get the same cohomology with π instea of π, but the latter leas to better signs in the expressions for the Chern characters below. Lemma 1.5. For any real vector bunle over a compact manifol without bounary, H k SW, π = H k c W. Proof. Recall that the cohomology with compact supports of W may be represente by the erham cohomology of smooth forms with compact support on W. Let i : X W be the inclusion of the zero section an choose a metric on W so as to have a prouct ecomposition 1.25 W \ i X = SW R + an enote the projection onto the left factor by R. Given a close k-form u on W with compact support consier the map 1.26 Φ u = i u, 1 k 1 R u C X; Λ k C SW ; Λ k 1.

12 12 PIERRE ALBIN AND RICHARD MELROSE We can think of introucing polar coorinates as pulling-back to the space SW R +, say via a map β, in terms of which we have β u = u t + u n r, with i r u t = so Φ u = u t, 1 k 1 u n r where we enote the interior prouct with r by i r. Notice that Φ v = X i v, 1 k R [ SW v n + 1 k r v t r ] = X i v, π i v SW [ 1 k 1 R v] = DΦ v, so Φ efines a map on cohomology which is easily seen to be an isomorphism for instance by using the commutative iagram... H k 1 SW H k SW, π H k X i Φ i π H k SW... i... H k 1 SW Hc k W an the Five Lemma. H k W π H k SW Bunle with line subbunle. A variant of the setting of Lemma 1.5 arises in the inex formula for perturbations of the ientity in the zero algebra. There a real vector bunle, W Y, has a trivial line subbunle L W. Setting U = W/L 1.27 H SU, π = H c U = H +1 c W since W U L with L by assumption trivial. For our purposes there is another more useful complex giving the same cohomology. Consier the raial compactification of L, L, obtaine by attaching to each fiber of L the points at ±, L +, L, an the forms C ± L; Λ = {α C L; Λ : i L +α = i L α, i L +α = i L α}. We use the erham complexes C k 1 = C ± L; Λ k an C k 2 = C SU; Λ k with the chain map π ν L : C k 1 C k 1 2, where ν L : C± L; Λ C Y ; Λ 1 is push-forwar uner π SW restricte to L to form the complex C k = C1 k C2 k 2 note f = 2 with ifferential π ν L. The cohomology of this complex will be enote H SU, L. Notice that for α C± L; Λ we have ν L α = ν L α cf. Lemma 1.8 below. Lemma 1.6. There are natural isomorphisms in cohomology 1.28 H k C, π ν L = H k c W = H k 1 SU, π.

13 RCCBIF 13 Proof. We have the following commutative iagram relating the long exact sequences for H SU, L an H SU, π escribe in 1.4,... H k 2 SU H k SU, L H k L π ν L H k 1 SU... i ν L... H k 2 SU H k 1 SU, π H k 1 Y π H k 1 SU... Since the cohomologies of L an Y are isomorphic via ν L the Five Lemma shows that as require. H k SU, L ν L = H k 1 SU, π = H k 1 c U = H k c W For future reference we point out that from the proof of this lemma an that of Lemma 1.5 the map 1.29 Φ L : Cc W ; Λ k C± L; Λ k C SU; Λ k 2 efine by Φ L ω = i ω, L 1k 1 ν L R W ω, where i L is the inclusion of L into W an ν L is the push-forwar from SW to SU, inuces an isomorphism in cohomology between H k c W an H k 1 SU, L Bunle over manifol with bounary. A more general case, which arises in the inex formula for the scattering algebra, correspons to a vector bunle W over a compact manifol with bounary X. Let W be the raial compactification of W. Its bounary consists of two hypersurfaces, SW, the part at infinity an W X, the part over the bounary. Set 1.3 C 1 = C X; Λ an C 2 = { α, β; α C SW ; Λ, β C W X ; Λ an ι α = ι β }, the pairs of forms on the two hypersurfaces with common restriction to the corner SW X. This gives a complex as in 1.3 with C k = C1 k C2 k 1 an ifferential π 1.31 D =, where φ = φ π i Lemma 1.7. The cohomology of the complex 1.3 with ifferential 1.31 reuces to the cohomology with compact support in W X\ X. Proof. The cohomology of W X is canonically isomorphic to the cohomology of X uner the pull-back map. Thus a close form on W X is the sum of a form on X pulle-back to W X an an exact form on W X. We point out that the same is true for a form β on W X if β is a form pulle-back from X this follows from the Hoge ecomposition of forms an the previous statement or from the proof of [4, Cor ]. Thus from D a, α, β = it follows that β = π β + β an hence 1.32 a =, π a = α, i a = β. ν L i

14 14 PIERRE ALBIN AND RICHARD MELROSE Choose a prouct neighborhoo of the bounary U = [, 1] x X an a corresponing ecomposition of SW U = [, 1]x V with V = SW X. In this neighborhoo an 1.32 becomes a = a t x + a n x x, a t C X; Λ k, a n C X; Λ k 1, α = α t x + α n x x, α t C V ; Λ k 1, α n C V ; Λ k 2, X a t =, 1 k x a t + X a n =, V α t = π a t, 1 k x α t + V α n = π a n, a t = β. Choose a smooth function ρ C X that is ientically equal to one if x < 1/2 an ientically equal to zero if x > 3/4 an efine T C k 1 by x x T = ρ x 1 k+1 a n s s β, 1 k 1 α n s s + ι β, β. Then, for some ω C k, DT = ρ x a, α, β + ρ x ω so that a, α, β DT = ã, α, with ã an α forms supporte in X. Thus the complex retracts to the subcomplex of forms supporte in X, an from Lemma 1.5 this complex computes the cohomology with compact support of W over X Bunle with line subbunle over the bounary. The representations of relative cohomology that will be use for the inex of zero operators an for operators in Boutet e Monvel s transmission calculus are closely relate. Consier a compact manifol with bounary, X, a real vector bunle W over X which over the bounary has a trivial line subbunle L, an enote the quotient bunle over the bounary by U = W X /L. The compactly supporte cohomology of W will be represente as in 1.3, that of its restriction to the bounary either as in 1.4 for the zero calculus or 1.3 for the transmission calculus. An appropriate version of the inclusion map then gives the compactly supporte cohomology of the interior much as in 1.2. Notice that SW X \ {L +, L } fibers over SU an can be ientifie with SU R since L is trivial, thus there is a push-forwar map ν SW : C SW, Λ k C SU, Λ k 1 which however oes not commute with. Lemma 1.8. If α C SW, Λ k then 1.35 ν SW SW α = SU ν SW α + 1 k π i L +α π i L α. Thus if i L +α = i L α then ν L α = ν L α. Proof. Introuce polar coorinates aroun L ± in SW X i.e., blow them up to get a map 1.36 SU L β SW X. The pre-image of L ± will still be enote L ±. The push-forwar is given by a ν L β a an there are no integrability issues since SW X is compact. In local coorinates, for a a form of egree k, β a = a s + a s s, with a, a C SU, Λ = β a = SU a s + SU a s + 1 k s a s s.

15 RCCBIF 15 Hence ν L β a = R a s s an ν L β SW a = SU ν L β a + 1 k s a s s giving R Introucing the complexes an chain maps Z1 k = C X; Λ k, Z3 k = C SU; Λ k Z2 k = { α, γ C SW ; Λ k C L; Λ k ; i ±α = i ±γ }, φ 2 =, πsu νl, φ 1 = π SW π L ι we efine the total chain space Z by ν SW 1.37 Z k = Z k 1 Z k 1 2 Z k 3 3, D Z = where i ± are the inclusion or attaching maps for the two bounary manifols each canonically iffeomorphic to X L ± of L, either to SW or L, an π SW : SW X, π SU : SU X, an π L : L X enote the various bunle projections. As in Lemma 1.8, neither the push-forwar for α nor γ commute with, however the consistency conition i ±α = i ±γ yiels φ 2 = φ 2 ; together with φ 1 = φ 1 this ensures that D 2 Z =. With the same notation set C ± SW ; Λ = {α C SW ; Λ : i L +α = i L α, i L +α = i L α} φ 1 φ 2 T1 k = C X; Λ k, T2 k = C± SW ; Λ k C X; Λ k 1, T3 k = C SU; Λ k Φ 1 = π SW i X an efine the total chain space T by, Φ 2 = ν SW, πsu 1.38 T k = T k 1 T k 1 2 T k 3 2, D T = Φ 1 Φ 2. Note that D 2 T = because i L + α = i L α guarantees that ν L = ν L. The point of these rather involve constructions of the cohomology with compact supports is that the Chern character erive from the symbolic ata symbol an normal operator of a fully elliptic zero operator has a natural representative in the chain space Z, while the Chern character constructe from the symbols interior an bounary of a fully elliptic operator in the transmission calculus has a natural representative in the chain space T. Lemma 1.9. The cohomology of the complexes 1.37 an 1.38 are isomorphic to the compactly supporte cohomology of W restricte to the interior of X, 1.39 H k T ; D T = H k Z ; D Z = H k c W X\ X. Proof. The map L; Λ k C SU; Λ k 2 γ, β,, γ π L i L +γ, β C k Z C ±

16 16 PIERRE ALBIN AND RICHARD MELROSE fits into the short exact sequence of complexes C SU, L Z C SW, π. This in turn inuces the long exact sequence in cohomology in the top row of... H k SW, π Φ... H k c W i J H k SU, L H k+1 Z ; D Z H k+1 SW, π... H k c Φ L W X δ where the connecting map J is inuce by eφ Hc k+1 W, W X Φ H k+1 c W... C X; Λ k C SW ; Λ k 1 C L; Λ k C SU; Λ k 2 a, α π L i a, νl i α Φ an Φ L are the isomorphisms efine in 1.26 an 1.29 respectively, an Φ is the restriction of Φ to the subcomplex of forms that vanish at W X. The left an right squares above clearly commute, so we only check the commutativity of the mile square. Let u be a k-form on W X with compact support, the map δ is inuce by taking any extension of u into W, say e u, an taking its exterior erivative. It is convenient to fin γ u such that Φe u, γ u, is in Z. To this en choose a trivialization of L, enote by t the fibre variable along L, an note that t γut = 1 k 1 i L u is as require an satisfies γu = i u L π L i u since u is close, an ν L γu = since i t γu =. Thus Φ δ u = Φe u,, = D SW Φe u,, = D Φe u, γ u, +,, i L u, 1k 1 ν L R W u = D Φe u, γ u, +,, Φ L u, which shows that the inuce maps in cohomology commute. It then follows from the Five Lemma that the map Φ is an isomorphism, i.e., H k Z ; D Z = Hc k W, W X. To see that the complex C k T ; D T represents Hc k W, W X consier first the complex C X; Λ C SW ; Λ C X; Λ C SW X ; Λ with ifferential π i i π which in view of 1.2 an 1.3 represents Hc k W, W X an then note that the complex C k T ; D T is obtaine from this complex by applying the push-forwar along L which was shown in Lemma 1.6 to be an isomorphism.

17 RCCBIF Pseuoifferential operators an relative K-theory In the stanar case of Atiyah an Singer, the inex of a vertical family of Freholm operators, A, acting on a superbunle E = E + E on the fibers of a fibration of close manifols M φ B is naturally thought of as an element of the topological K-theory group of B, e.g., [ker A] [cokera] KB when ker A an hence cokera is a bunle over B. On the other han A itself, via its symbol σ A C S M/B; hom E an the clutching construction, efines an element of K c T M/B an the inex factors through this map In Ψ M/B; E [a] K c T M/B K B. In a One way to see this factorization is to start with a family A as above an eliminate properties that the inex oes not see. That is, let K M/B be the set of equivalence classes of vertical pseuoifferential operators acting on superbunles E over M where two operators are consiere equivalent if they can be connecte by a finite sequence of relations: i A Ψ M/B; E B Ψ M/B; F if there is a grae bunle isomorphism Φ : E F over M such that B = Φ 1 AΦ, ii A Ψ M/B; E Ã Ψ M/B; E if A an à are homotopic within elliptic operators, iii A Ψ M/B; E A I Ψ M/B; E C n n where C n n is the trivial superbunle whose Z/2 graing components are both C n. The resulting equivalence classes form a group, K M/B which can be thought of as smooth K-theory an in this case is well-known to coincie with the topological K-theory group K c T M/B. Inee, the equivalence class of an operator only epens on its principal symbol A Ψ k M/B; E = σ A C S M/B; N k hom E where we inclue a trivial line bunle N to hanle the homogeneity in terms of which i-iii give a stanar relative efinition of K c T M/B. Without loss of generality it suffices to consier operators of orer zero. Alternately, pseuoifferential operators of orer acting on C as boune operators acting on L 2 M/B efine, e.g., using a Riemannian metric. These form a -algebra an the closure is a C -algebra, A, containing the compact operators, K. The Freholm operators are the invertibles in A/K, an the smooth K-theory group escribe above is closely relate to the o C K-theory group of this quotient, KC 1 A/K. Inee, the principal symbol extens to a continuous map on A, A A = σ A C S M/B; C,

18 18 PIERRE ALBIN AND RICHARD MELROSE which escens to the quotient A/K i.e., vanishes on K an allows us to ientify the o K-theory group with the stable homotopy classes of invertible maps S M/B C. The most obvious ifference between the smooth K-theory group an the C K-theory group that the former is built up from smooth functions while the latter from continuous functions isappears in the quotient. A more significant ifference comes from the way bunle coefficients are hanle. Stabilization allows us to replace the elliptic elements in Ψ M/B with lim GL N Ψ M/B, the irect limit of the groups of invertible square matrices of arbitrary size an entries in Ψ M/B. Note that an operator A Ψ M/B; E acting on a superbunle E efines an element of the stabilization of Ψ M/B by choosing a vector bunle F such that C j = E F for some j since then à = A I F GL j Ψ M/B ; a ifferent choice of F efines the same element in KC 1 M/B. However if Φ is as in i above, it is possible that Φ 1 AΦ an A will efine istinct elements of KC 1 M/B, an so the ifference between the smooth K-theory groups an the C K-theory groups is essentially that in the former we quotient out by i above. This is further pursue in [1, 2.3]. In the present paper we allow the fibers of the fibration M φ B to have bounary an consier three ifferent calculi of operators on a manifol with bounary, namely the scattering calculus, the zero calculus, an the transmission calculus. In each case we work with the smooth K-theory group as in the previous paragraph enote by K sc M/B, K M/B, an K tm M/B respectively an, for the purposes of inex theory, these groups contain all of the relevant information. These groups have been shown to be isomorphic to the topological K-theory group, K c T M /B, in [17] [21] scattering calculus, [1] -calculus, an [5] transmission calculus. We briefly review what this entails. Scattering calculus. A scattering operator A Ψ sc M/B; E is etermine up to a compact operator by its image uner two homomorphisms: the principal symbol σ A C S M/B; π hom E which is an homomorphism between the lifts of E + an E to S M/B, an the bounary symbol b C T M/B M ; π home which is an homomorphism between the lifts of E + an E to the raial compactification of T M/B over the bounary. These symbols are equal on the common bounary of S M/B an T M/B M, an so can be thought of as jointly representing a section of home lifte to the whole bounary of the compact manifol with corners T M/B. An operator is Freholm on L 2 precisely when both of these symbols are invertible, we call such an operator fully elliptic. A fully elliptic operator A can be eforme by homotopy within such operators operators until b, an a in a neighborhoo of the bounary, are equal to a fixe bunle isomorphism. This isomorphism can then be use to change the bunles so that b is the ientity an a is the ientity near the bounary. This leas to an arbitrary invertible map into home that is the ientity in an neighborhoo of the bounary an this is precisely the information that efines a relative K-theory class, hence 2.1 K sc M/B = K c T M /B. The fully elliptic scattering operators of orer zero with interior symbol equal to the ientity can be reuce by homotopy to perturbations of the ientity by

19 RCCBIF 19 scattering operators of orer. The smooth K-theory of these perturbations of the ientity is enote K sc, M/B an is reaily seen to be equal to the topological K-theory of the bounary, 2.2 K,sc M/B = K c T M/B. Zero calculus. The analogues of 2.1 an 2.2 also hol for the zero calculus, as shown in [1]. However where the scattering calculus is asymptotically commutative as evince in the bounary symbol b C T M/B M ; π home, the zero calculus is asymptotically commutative only in the irections tangent to the bounary an is non-commutative in the irection normal to the bounary. Thus, instea of a bounary symbol, the bounary behavior of a zero operator is capture by a family of operators on a one-imensional space, I, essentially the compactifie normal bunle to the bounary. This family, the reuce normal operator N A C S M/B; Ψ b,c I; E, takes values in the b, c calculus b at one en of the interval, c at the other, an together with the interior symbol, etermines the smooth K-theory class of an operator A Ψ M/B; E. A escription of the reuce normal operator is inclue in Appenix A. As an element of the b, c calculus, N A has three moel operators: its principal symbol, an inicial family at the b-en, an an inicial family at the cusp en. One can think of the smooth K-theory as equivalence classes of zero pseuoifferential operators or alternately as equivalence classes of invertible pairs σ, N C S M/B; hom E C S M/B; Ψ b,c I; E 2.3 The isomorphism s.t. I b N y, η = I b N y, η, I c N y, η ξ = σ, y, ξ ξ, η ξ σ N y, η ω = σ, y, ω,. K, M/B = K c T M/B, between the group of stable equivalence classes of invertible reuce normal families of perturbations of the ientity an a stanar presentation of K c T M/B comes from the contractibility of the unerlying semigroup of invertible operators on the interval [1]. Namely, this allows the reuce normal family to be connecte after stabilization to the ientity through a curve of maps At from S M/B into the invertible b-operators of the form I +A, A of orer an non-trivial only at the one en of the interval. The b-inicial family of At etermines an invertible map from T M/B into hom E equal to the ientity at infinity an hence an element of K c T M/B. The contractibility of the group of invertible group of smooth perturbations of the ientity within the cusp calculus [2] is use to show the isomorphism K M/B = K c T M /B between the group of stable equivalence classes of invertible pairs 2.3 an the stanar representation of K c T M /B. Inee, one can ientify E + an E near the bounary, an, after a homotopy an a smooth perturbation, quantize by a zero operator whose full b-inicial family is the ientity, an then use the,

20 2 PIERRE ALBIN AND RICHARD MELROSE contractibility to take the reuce normal operator to the ientity. The principal symbol of the resulting operator is equal to the ientity near the bounary an classically efines an element of K c T M /B. Transmission calculus. A smooth K-theory class in the Boutet e Monvel calculus is an equivalence class of operators of the form γ + A + B K C X; E + C X; E 2.4 A = : T Q C X; F + C X; F acting on the superbunles E over X an F over X. The principal symbol of A is require to satisfy the transmission conition at the bounary, an in particular this forces the orer of A to be an integer. We can assume without loss of generality that the orer of A an its type are both zero see [5], [7]. The bounary behavior of A is moele by a family of Wiener-Hopf operators parametrize by the cosphere bunle over the bounary. For A as above we enote its bounary symbol by 2.5 h + p + b k C X; H + E y + C X; H + Ey N A y, η = : t q C X; F y + C X; Fy where h + is the projection from the space of functions C R; C that have a regular pole at infinity to the subspace H + of those that vanish at infinity an can be continue analytically to the lower half-plane. Boutet e Monvel showe that any operator A whose principal symbol an bounary symbol are both invertible is homotopic through such operators to one of the form γ + à Q where à is equal to the ientity near the bounary. Thus K tm M/B clearly surjects onto K c T M /B which is all we will nee. However we point out that, using the escription of the C -algebra K-theory from, e.g., [12], [13], an the comparison with smooth K-theory in [1, 2.3], it is possible to show that K tm M/B is actually equal to K c T M /B. 3. Chern character an the families inex theorem The Chern character is a homomorphism from Ch : KX H even X which gives an isomorphism after tensoring with Q. Chern-Weil theory gives a irect representation of the Chern character in erham cohomology for a superbunle E. If is a grae connection on E, i.e. a pair of connections ± on E ± with curvatures ± 2 = 2πiω ± then 3.1 Ch E = e ω+ e ω. Different choices of connection are homotopic an give cohomologous close forms. In the case of a compact manifol with bounary X, the K-theory with compact support in the interior, enote here K c X \ X, is represente by superbunles E where E ± = C N near the bounary. Then the same formula, 3.1, gives the relative

21 RCCBIF 21 Chern character Ch : K c X \ X Hc even X \ X provie the connections are chosen to reuce to the trivial connection,, near the bounary. There are natural isomorphism K c X \ X KX, X an Hc even X \ X H even X, X with the corresponing relative objects. Chains for K c X \ X are given by pairs E, a of a superbunle over X an an isomorphism a : E + E over X. In [6] Feosov gives an explicit formula for the Chern character, in cohomology with compact supports of the cotangent bunle, of the symbol of an elliptic operator acting between vector bunles. This can be moifie to give the relative Chern character in this setting with values in the chain space iscusse in Lemma ChE, a = ChE, Cha Cha = 1 2πi tr a 1 ae wt t, where wt = 1 tω + + ta 1 ω a + 1 2πi t1 ta 1 a 2. Here the bounary term, Cha, is a regularize or improper form of the o Chern character. In the context of the inex formula this also correspons rather naturally to the relative cohomology as iscusse above. Essentially by reinterpretation we fin Proposition 3.1 Feosov [6]. If π : U X is a real vector bunle, E X is a superbunle an a C SU; π home is elliptic i.e. invertible then for any grae connection on E, with curvatures Ω ± = 2πiω ± an inuce connection on home, the class 3.3 ChU, E, a = ChE, Cha C X; Λ even C SU; Λ o, given by the formulæ3.1 an 3.2, represents the relative Chern character 3.4 Kc U H o SU, π Hc even U. Proof. That the pair ChE, Cha is D-close an gives a well-efine class in the cohomology theory follows in essence as in stanar Chern-Weil theory. We inclue such an argument for completeness an for subsequent generalization. Set θ = a 1 a an note that the connection = + [tθ, ] has curvature It follows that e ωt = an hence Thus 3.5 Ω t = 2πiω + + t θ + t 2 θ 2 = 2πiωt e ωt = e ωt t Cha = 1 2πi = 1 2πi = 1 2πi = 1 2πi [ θ, e ωt] [ ] = t e ωt, θ. tr θe ωt t tr θ e ωt θ e ωt t [ ] tr θ e ωt tθ e ωt, θ t θ tr + 2tθ 2 [ ] e ωt t θe ωt, θ t

22 22 PIERRE ALBIN AND RICHARD MELROSE an since the trace vanishes on grae commutators, 3.6 Cha ωt = tr e ωt t = π tre ω π tre ω+ = π ChE. t As is well-known an explaine in 1.1.2, this same formula shows inepenence of the connection an homotopy invariance. That this class actually represents the appropriately normalize Chern character follows from Feosov s erivation in [6]. For an elliptic family of pseuoifferential operators A Ψ M/B; E where ψ : M B is a fibration with typical fibre Z an E is a superbunle over M, the symbol σa C S M/B; π home is invertible by assumption an the iscussion above applies with U = T M/B, the fibre cotangent bunle. The inex formula of Atiyah an Singer is then given as a composite 3.7 K c T M/B Ch H even B in H o S M/B, π TZ H o S M/B, π So, for such a family of operators, 3.8 ChinA = ChT M/B, E, σa TZ S Z where TZ is the To class of Z. That this is well-efine follows from Scattering families inex theorem. Consier next a fibration of manifols with bounary M ψ B. As escribe in 2 the compactly supporte K-theory of W = T M /B can be represente by scattering operators. The K-theory class of a scattering operator is etermine by its two symbol maps, its principal symbol an its bounary symbol. We now explain how to represent the Chern character of the corresponing K-theory class in terms of this ata. In fact given any manifol with bounary M an a bunle W M any class in the compactly supporte K-theory of W M\ M can be represente by a superbunle E M an two invertible maps R S Z a C S W ; π hom E, b C W M ; π hom E. We recall, from 1.5, that one may compute the cohomology H c W M\ M via the complex C M; Λ k { α, β; α C SW ; Λ k 1, β C W M ; Λ k 1 an ι α = ι β }, π D =, where φ = φ π i. Then, choosing a grae connection on E, the forms, again using 3.1 an 3.2, 3.9 ChW, E, a, b = Ch E, Cha, Chb C X; Λ even C SW ; Λ o C W M ; Λ o, represent the Chern character of the K-theory class associate to W, E, a, b.

23 RCCBIF 23 Proposition 3.2. For W, E, a, an b as above, ChW, E, a, b is D-close an its relative cohomology class coincies with the Chern character of the K-theory class efine by W, E, a, b. Proof. From the efinition of the ifferential in 1.5, D-close means that X Ch E =, SW Ch a = π Ch E an W X Ch b = π i Ch E. Thus that the putative Chern character is D-close an homotopy invariant follow just as in Lemma 3.1. It is also invariant uner changes of E by stabilization an bunle isomorphism since this is true of the forms Ch E, Cha, Chb themselves, an hence it only epens on the K-theory class. As explaine in [17] an reviewe in 2, there is a representative of the K-theory class with b = I an a = I near the bounary, an, since in this case 3.9 coincies with Feosov s formula 3.3, we conclue that 3.9 is the usual Chern character map. The inex formula follows similarly. Given a vertical family of fully elliptic scattering pseuoifferential operators acting on a superbunle E M, it is possible to make a homotopy within fully elliptic scaterring operators until the symbols are bunle isomorphisms at an near the bounary. Thus a formula which is homotopy invariant an which coincies with the usual Atiyah-Singer inex formula when the operators are trivial at the bounary must give the inex. Proposition 3.3. The inex in cohomology for a family of fully elliptic scattering pseuoifferential operators on the fibres of a fibration is given by the Atiyah-Singer formula essentially as in 3.7, 3.8: ina = ChE, σa, βa TZ 3.1 = Ch σa TZ + Ch βa ι TZ sc T Z Z sc S Z where TZ C M; Λ even is a erham form representing the To class of the fibres of ψ : M Y. Proof. Homotopy invariance follows from Proposition 3.2 an when the family of operators are trivial near the bounary this clearly coincies with the Atiyah-Singer families inex formula Zero families inex theorem. The non-commutativity of the bounary symbol, in the normal irection, makes the erivation of a formula more challenging, so we first consier the simple case where only the bounary symbol appears. Perturbations of the ientity. Consier a fibration as in 1 where the typical fiber, X, is a manifol with bounary, enote Y. The restriction of the fibrewise cotangent bunle to the bounary T M M/B has a trivial line sub-bunle, the conormal bunle, with the quotient being T M/B. Thus as explaine in 1.3 the compactly supporte cohomology of U = T M/B can be realize as the cohomology H k SU, π of the complex 3.11 C M; Λ C SU; Λ 1, D = π.