aa r X i v : . [ m a t h . S G ] J u l THE GEOMETRY OF b k MANIFOLDS
GEOFFREY SCOTT
Abstract.
Let Z be a hypersurface of a manifold M . The b -tangent bundle of ( M, Z ), whosesections are vector fields tangent to Z , is used to study pseudodifferential operators and stablepoisson structures on M . In this paper we introduce the b k -tangent bundle, whose sections arevector fields with “order k tangency” to Z . We describe the geometry of this bundle and itsdual, generalize the celebrated Mazzeo-Melrose theorem of the de Rham theory of b -manifolds,and apply these tools to classify certain Poisson structurse on compact oriented surfaces. Introduction
Melrose developed the b -calculus to study pseudodifferential operators on noncompact manifolds([Me], [G]). Considering the manifold in question as the interior of a manifold M with boundary, heconstructed the b -tangent bundle b T M whose sections are vector fields on M tangent to ∂M , andthe b -cotangent bundle b T ∗ M , whose sections are differential forms with a specific kind of order-one singularity at ∂M . The authors of [GMP2] have recently applied these ideas to study globalPoisson geometry; in this context, b T M and b T ∗ M are defined on a manifold M with a distinguishedhypersurface Z rather than on a manifold with boundary , and sections of b T M (and b T ∗ M ) arevector fields (and differential forms) tangent to Z (or singular at Z ). In this paper, we generalizethis construction so that vector fields and differential forms with higher order tangency and higherorder singularity may also be realized as sections of bundles.The construction of these bundles in Section 2 is subtle: although we wish to define a b k -vectorfield as a vector field with an “order k tangency to Z ,” there is no straightforward way to rigorouslydefine this notion. To do so, we must include in the definition of a b k -manifold the data of a ( k − Z (and insist that the morphisms in the b k -category preserve this jet). We then define a b k -vector field as a vector field v such that L v ( f ) vanishes to order k for functions f that representthe jet data. Then we define the b k -tangent bundle b k T M as the vector bundle whose sections are b k -vector fields, and the b k -cotangent bundle b k T ∗ M as its dual. When k = 1, these are the familiarconstructions from [Me] and [GMP2].In Section 3 we study the geometry of the fibers of b k T M and b k T ∗ M . Recall from [GMP2] thatthe fibers of b T M and b T ∗ M satisfy b T p M ∼ = (cid:26) T p M for p / ∈ ZT p Z + h y ∂∂y i for p ∈ Z b T ∗ p M ∼ = (cid:26) T ∗ p M for p / ∈ ZT ∗ p Z + h dyy i for p ∈ Z where y is a defining function for Z . Similarly, we show that the fibers of b k T M and b k T ∗ M satisfy b k T p M ∼ = (cid:26) T p M for p / ∈ ZT p Z + h y k ∂∂y i for p ∈ Z b k T ∗ p M ∼ = (cid:26) T ∗ p M for p / ∈ ZT ∗ p Z + h dyy k i for p ∈ Z The author was partially supported by NSF RTG grant DMS-1045119.The author was partially supported by NSF RTG grant DMS-0943832. These competing perspectives can be reconciled by viewing a manifold with boundary M as one half of its double.In doing so, the boundary of M corresponds to a hypersurface of the double. In this paper, we follow the precedentof [GMP2] and define our bundles over manifolds with distinguished hypersurfaces. where y is a local defining function for Z that represents the jet data of the b k -manifold.In Section 4 we define a differential on the complex of b k -forms (sections of the exterior algebraof b k T ∗ M ) and prove a Mazzeo-Melrose type theorem for the cohomology b k H ∗ ( M ) of this complex.(1) b k H p ( M ) ∼ = H p ( M ) ⊕ (cid:0) H p − ( Z ) (cid:1) k However, this isomorphism (like that of the classic Mazzeo-Melrose theorem) is non-canonical. Bydefining the
Laurent Series of a b k -form, which expresses a b k -form as a sum of simpler b ℓ -forms(for ℓ ≤ k ), we show that there is a way to construct the isomorphism in Equation 1 so that the (cid:0) H p − ( Z ) (cid:1) k summand of a b k -cohomology class is canonically defined.In Section 5, we study the geometry of b k -forms of top degree. In [R], the author defined the Liouville volume of a b -form of top degree as a certain principal value of the form. This invariantwas featured in her classification theorem of stable Poisson structures on compact surfaces. Wegeneralize this by defining the volume polynomial of a b k -form of top degree. This polynomialencodes the asymptotic behavior of the integral of the form near Z . We define the Liouville volume as the constant term of this polynomial – it agrees with the classic definition of Liouville volumewhen k = 1. We can also take the Liouville volume of a degree p b k -form along any p dimensionalsubmanifold of M . Citing Poincar´e duality, we define the smooth part of a b k cohomology class [ ω ]to be the de Rham cohomology class whose integrals along p -cycles equal the Liouville volumes of ω along these cycles. At the end of Section 5, we use these tools to realize the abstract isomorphism inEquation 1 with an explicit canonical map. The image of a b k form under this map is its Liouville-Laurent decomposition .In Section 6, we define a symplectic b k -form as a closed b k k = 1,these are frequently called log symplectic forms), and prove the classic Moser theorems in the b k category. We also revisit the classification theorems of stable Poisson structures on compact orientedsurfaces from [R] and [GMP2]. Radko classifies stable Poisson structures using geometric data, whilethe authors of [GMP2] use cohomological data; in this paper, we show how the Liouville-Laurentdecomposition relates the geometric data to the cohomological data.This paper ends in Section 7 with an example of how the theory of b k -manifolds can answerquestions from outside b k -geometry. Let Π be a Poisson structure on a manifold M whose rankdiffers from dim( M ) precisely on a hypersurface Z . We say that Π is of b k -type if it is dual to asymplectic b k -form for some choice of jet data (respectively, we say that the symplectic form on M \ Z is of b k -type ). When M is a surface, this means that the Poisson bivector Π is given by f Π where Π is dual to a symplectic form, and f is locally the k th power of a defining function of Z .We give a condition for two such Poisson structures on a compact surface to be isomorphic in termsof the summands in their respective Liouville-Laurent decompositions. Acknowledgements:
I thank both Daniel Burns and Victor Guillemin for helpful discussions,and the latter for introducing me to Melrose’s b -calculus. Also, to the staff of 1369 Coffee Housewho tolerated my near-residence of their establishment while preparing this paper – thank you.2. Preliminaries
In this section, we establish notation pertaining to jet bundles, review definitions from the theoryof b -manifolds, and generalize these definitions. All manifolds, maps, and vector fields are assumedto be smooth.2.1. Notation.
Let i : Z → M be the inclusion of a hypersurface into a manifold, let C ∞ be thesheaf of smooth functions on M , and let I Z ⊆ C ∞ be the ideal sheaf of Z . HE GEOMETRY OF b k MANIFOLDS 3
Definition 2.1.
The sheaf of germs at Z is i − ( C ∞ ); a germ at Z is a global section of thissheaf. The sheaf of k -jets at Z is J kZ := i − ( C ∞ / I k +1 Z ); a k -jet at Z is a global section of thissheaf.We will write J kZ (or simply J k ) to denote the k -jets at Z , and I Z (or simply I ) to denote theglobal sections of i − ( I Z ). We write [ f ] kZ (or simply [ f ] k ) to denote the k -jet represented by asmooth function f defined in a neighborhood of Z . Also, if j is a k -jet, we write f ∈ j if f represents j and f ∈ I k if f represents an element of I k (equivalently, if [ f ] k − = 0).2.2. Definitions.
In [GMP2], the authors use b -manifolds to study symplectic forms having order-one singularities along a hypersurface. In this paper, we generalize these techniques to study sym-plectic forms having higher-order singularities. We begin by recalling the basic definitions from thetheory of b -manifolds. See [GMP2] for more exposition. Definition 2.2. A b -manifold is a pair ( M, Z ) of a smooth oriented manifold M and an orientedhypersurface Z ⊆ M such that Z = { f = 0 } for some global defining function f : M → R . Definition 2.3. A b -map from ( M, Z ) to ( M ′ , Z ′ ) is a map ϕ : M → M ′ such that ϕ − ( Z ′ ) = Z and ϕ is transverse to Z ′ . Definition 2.4. A b -vector field on ( M, Z ) is a vector field v on M such that v p ∈ T p Z for all p ∈ Z . Definition 2.5.
The b -tangent bundle b T M on (
M, Z ) is the vector bundle whose sections arethe b -vector fields on ( M, Z ). Definition 2.6.
The b -cotangent bundle b T ∗ M is the dual bundle of b T M .The authors of [GMP2] show that sections of the exterior algebra of the b -cotangent bundle aredifferential forms on M with a certain kind of order-one singularity at Z . Towards the goal ofconstructing similar bundles to study differential forms with higher-order singularities, we wish todefine a b k -vector field as a vector field “tangent to order k on Z .” However, the next example showsthat the na¨ıve definition of being “tangent to order k on Z ” (as a vector field v such that L v ( f ) ∈ I k for a defining function f of Z ) is ill-defined. Example 2.7.
On the b -manifold ( M, Z ) = ( { ( x, y ) ∈ R } , { y = 0 } ), two different defining functionsfor Z are given by y and e x y . The vector field v = ∂∂x satisfies L v ( y ) = 0 ∈ I and L v ( e x y ) = e x y / ∈ I so the order of vanishing of the Lie derivative of a defining function depends on the choice of definingfunction.This phenomenon prevents us from emulating the [GMP2] paper mutatis mutandis; we mustendow our b -manifolds with additional data to make possible the definition of a b k -vector field. Definition 2.8.
For k ≥
1, a b k -manifold is a triple ( M, Z, j Z ) where • M is an oriented manifold. • Z ⊆ M is an oriented hypersurface. • j Z is an element of J k − Z that can be represented by a positively oriented local definingfunction y for Z (that is, if Ω Z is a positively oriented volume form of Z , then dy ∧ Ω Z ispositively oriented for M )If k > k − Z , then it itself is a positively oriented local defining function for Z . In this case, any f ∈ j Z is a GEOFFREY SCOTT positively oriented local defining function for Z . When k = 1 the jet data { j Z } is vacuous (becauseany local defining function for Z represents the trivial 0-jet), so the definition of a b -manifold nearly agrees with that of a b -manifold. Definition 2.9. A b k -map from ( M, Z, j Z ) to ( M ′ , Z ′ , j Z ′ ) is a map ϕ : M → M ′ such that ϕ − ( Z ′ ) = Z , ϕ is transverse to Z ′ , and ϕ ∗ ( j Z ′ ) = j Z .The interested reader is invited to check that b k -manifolds and b k -maps form a category. Remark 2.10.
Given a hypersurface Z ⊆ M , a vector field v on M with v p ∈ T p Z for all p ∈ Z ,and f ∈ C ∞ ( M ), the jet [ L v ( f )] k − depends only on [ f ] k − . Proof.
If [ f ] k − = [ f ] k − , then f − f = y k g for a local defining function y and some smooth g .For a vector field v satisfying v p ∈ T p Z , L v ( y ) ∈ I , so[ L v ( f )] k − = [ L v ( f ) + y k L v ( g ) + kgy k − L v ( y )] k − = [ L v ( f )] k − . (cid:3) Remark 2.10 shows that the following definition makes sense.
Definition 2.11. A b k -vector field on ( M, Z, j Z ) is a vector field v with v p ∈ T p ( Z ) for p ∈ Z such that for any f ∈ j Z , L v ( f ) ∈ I k .To check whether a vector field v is a b k -vector field, it suffices (by Remark 2.10) to check that L v ( f ) ∈ I k for just one local defining function f ∈ j Z . The following example shows that Definition2.11 formalizes the notion of a vector field being “tangent to order k ” along a hypersurface. Example 2.12.
On the b k manifold ( R n , Z = { x n = 0 } , [ x n ] k − ), a vector field v = P ni =1 v i ∂∂x i isa b k -vector field iff L v ( x n ) ∈ I k which occurs iff v n ∈ I k . That is, the b k -vector fields are precisely those of the form φ n x kn ∂∂x n + n − X i =1 φ i ∂∂x i for smooth functions φ i .On a b k -manifold ( M, Z, j Z ), each p / ∈ Z is contained in a coordinate neighborhood ( U, { x , . . . , x n } )on which { ∂∂x i } generate the space of b k -vector fields over U as a free C ∞ ( U )-module. For points p ∈ Z , Example 2.12 shows that on a coordinate neighborhood ( U, { x , . . . , x n } ) of p with x n ∈ j Z ,the vector fields (cid:26) ∂∂x , . . . , ∂∂x n − , x kn ∂∂x n (cid:27) generate the space of b k -vector fields over U as a C ∞ ( U )-module. Consequently, b k -vector fieldsform a projective C ∞ module over M , as well as a Lie subalgebra of the algebra of vector fields on M , so we can realize b k -vector fields as the sections of a bundle on M .We call this bundle b k T M the b k -tangent bundle . The dual of this bundle b k T ∗ M is the b k -cotangent bundle . When k = 1 we recover the classic definitions of a b -vector field and the We do not demand that Z = { f = 0 } for some globally -defined f , so the definition of a b -manifold is slightlymore general than the definition of a b -manifold given in [GMP2]. However, any symplectic b -manifold will havethe property that Z is defined by a global function, so the symplectic geometry of b -manifolds coincides with thesymplectic geometry of b -manifolds. HE GEOMETRY OF b k MANIFOLDS 5 b -(co)tangent bundle. We write b k Ω p ( M ) for sections of ∧ p ( b k T ∗ M ). Elements of b k Ω p ( M ) are differential b k -forms . 3. Geometry of the b k -(co)tangent bundle In this section, we describe the fibers of the b k -(co)tangent bundles and study maps between b k -(co)tangent bundles as k varies. These results will prepare us to study the de Rham theory andthe symplectic geometry of b k -manifolds.Let b k Vect( M ) be the space of b k -vector fields and C ∞ p ( M ) be the ideal of functions vanishing at p ∈ M . We can define b k T p M intrinsically as b k T p M ∼ = b k Vect( M ) / ( C ∞ p ( M ) b k Vect( M )) . There is a canonical map that relates the fibers of b k T M to those of
T M .(2) b k T p M ∼ = b k Vect( M ) C ∞ p ( M ) b k Vect( M ) → Vect( M ) C ∞ p ( M )Vect( M ) ∼ = T p M The results of this section will show that for p ∈ Z , there is a canonical element in the kernel of Map2 (and dually a canonical element in the quotient b k T ∗ p M/T ∗ p M ). Instead of proving these resultsusing this intrinsic description of individual fibers, we will take a more global perspective in orderto more closely follow the exposition and results of [GMP2].3.1. Fibers of the b k -(co)tangent Bundle. Similar to the b -manifold case, there are maps be-tween the (co)tangent bundles of Z and the b k -(co)tangent bundles of M restricted to Z . b k T M (cid:12)(cid:12) Z ։ T Z (3) b k T ∗ M (cid:12)(cid:12) Z ← ֓ T ∗ Z (4)Map 3 is induced by the map of sections Γ( M, b k T M ) → Γ( Z, T Z ) given by restricting a b k -vectorfield to Z . Map 4 is dual to Map 3. We study the (co)kernel of these maps, starting with a technicalremark. Remark 3.1.
Let v be a b k -vector field that vanishes on Z when viewed as a section of T M , andlet x n ∈ j Z be a local defining function for Z . Then v also vanishes on Z as a section of b k T M precisely at those points where the k -jet [ L v ( x n )] k vanishes. Proof.
In local coordinates { x , . . . , x n } v = φ n x kn ∂∂x n + X i The kernel of Map 3 has a canonical nowhere vanishing section.Proof. Pick a local defining function y ∈ j Z and a vector field v satisfying dy ( v ) (cid:12)(cid:12) Z = 1. Then[ L y k v ( y )] k = [ y k L v ( y )] k is nonvanishing. By Remark 3.1, y k v is a b k -vector that vanishes on Z as asection of T M but is nowhere vanishing as a section of b k T M . GEOFFREY SCOTT To prove that y k v is canonical, suppose y ∈ j Z and v are different choices of defining functionand vector field. Then y = y (1 + gy k − ) for some smooth g and L v ( y ) = L v ( y ) + gy k − L v ( y ) + y L v ( gy k − )= (1 + kgy k − ) L v ( y ) + y k L v ( g )and [ L y k v − y k v ( y )] k = [ y k L v ( y ) − y k (1 + gy k − ) k L v ( y )] k = (cid:20) y k L v ( y ) − y k (1 + gy k − ) k L v ( y ) − y k L v ( g )1 + kgy k − (cid:21) k = 0By Remark 3.1, y k v − y k v vanishes on Z as a section of b k T M , so y k v and y k v represent the samesection of b k T M (cid:12)(cid:12) Z . (cid:3) Turning our attention to the cotangent bundle, observe that although the differential form y − k dy is not defined on Z as a section of T ∗ M , its pairing with any b k -vector field extends smoothly over Z . Therefore, y − k dy extends smoothly over Z as a section of b k T ∗ M . By pairing y − k dy with arepresentative of a nonvanishing section of ker( b k T M (cid:12)(cid:12) Z → T Z ), we see that y − k dy is nonwherevanishing. This proves the following claim. Claim 3.3. The cokernel of Map 4 has a nowhere vanishing section.The preceding discussion describes of the fibers of the b k -(co)tangent bundle of a b k -manifold( M, Z, [ y ] k − ) as follows. b k T p M ∼ = (cid:26) T p M for p / ∈ ZT p Z + h y k ∂∂y i for p ∈ Z b k T ∗ p M ∼ = (cid:26) T ∗ p M for p / ∈ ZT ∗ p Z + h dyy k i for p ∈ Z Properties of b k -Forms. From the above description of the fibers of b k T p M , we see thatΩ p ( M \ Z ) ∼ = b k Ω p ( M \ Z ). That is, every b k -form restricts to an ordinary differential form on M \ Z .We can therefore interpret a b k -form as a differential form on M \ Z that satisfies certain asymptoticproperties (prescribed by the jet data) around Z . We also see that for any defining function y ∈ j Z ,every b k -form can be written in a neighborhood U of Z in the form(5) ω = dyy k ∧ α + β for differential forms α ∈ Ω p − ( U ) and β ∈ Ω p ( U ). Although the forms α and β appearing inEquation 5 are not uniquely defined by ω , we will show that i ∗ ( α ) is independent of the choice of y, α and β , where i : Z → M is the inclusion. Proposition 3.4. On a b k -manifold, if f , f ∈ j Z are local defining functions for Z , then in aneighborhood U of Z df f k = df f k + β where β ∈ Ω ( U ) .Proof. The proof is technical. See Section 8 for the details. (cid:3) HE GEOMETRY OF b k MANIFOLDS 7 Corollary 3.5. Given a decomposition of ω ∈ b k Ω( M ) as in Equation 5, i ∗ ( α ) is independent of thedecomposition.Proof. Let α and α be the α terms of two such decompositions. Setting the decompositions equaland applying the preceding proposition shows that dyy k ∧ ( α − α )is a smooth form for some local defining function y ∈ j Z , so i ∗ ( α − α ) = 0. (cid:3) This proves the well-definedness of the map ι L : b k Ω p ( M ) → Ω p − ( Z )(6) dyy k ∧ α + β i ∗ ( α )Alternatively, this map can be defined by restricting a form to Z , then contracting with the canonicalsection L described in Proposition 3.2. This motivates the notation ι L for the map.Equation 5 might give us hope that we can define a b k -form without reference to any jet data as“a form ω on M \ Z which admits a decomposition ω = y − k dy ∧ α + β in a neighborhood of Z for some local defining function y ”. However, for a fixed ω the existence of a decomposition ω = y − k dy ∧ α + β depends strongly on [ y ] k − . It turns out that the set of ω ∈ Ω( M \ Z ) which extends over Z withrespect to some [ y ] k − is not even closed under addition. This hopefully motivates (for a secondtime) the necessity of the jet data in the definition of a b k -manifold.3.3. Viewing a b ℓ -Form as a b k -Form. To prepare for the next section, we consider a new familyof maps between the b k -(co)tangent bundles. These maps generalize the fact that any differentialform is naturally a b k -form.For any 0 < ℓ ≤ k , the natural map J k − Z → J ℓ − Z allows us to canonically endow a b k -manifold( M, Z, j Z ) with a b ℓ -manifold structure. Defining b T M := T M and b T ∗ M := T ∗ M for notationalconvenience, a b k -manifold structure on M defines 2 k + 2 different bundles b ℓ T M , b ℓ T ∗ M over M for 0 ≤ ℓ ≤ k . A b k -vector field will also be a b ℓ -vector field for the induced b ℓ -manifold structure.This induces a map(7) b k T M → b ℓ T M and its dual(8) b ℓ T ∗ M → b k T ∗ M ,the latter of which can be described explicitly in terms of the decompositions from Equation 5 as dyy ℓ ∧ α + β dyy k ∧ ( y k − ℓ α ) + β .4. De Rham Theory and Laurent Series of b k -forms We define a differential d : b k Ω p ( M ) → b k Ω p +1 ( M ) by d (cid:18) dyy k ∧ α + β (cid:19) = dyy k ∧ dα + dβ .This definition does not depend on the decomposition. Indeed, d ( ω ) is the unique extension of theimage of the classic de Rham differential d ( ω (cid:12)(cid:12) M \ Z ) ∈ Ω p ( M \ Z ) ∼ = b k Ω p ( M \ Z ) over Z . GEOFFREY SCOTT Definition 4.1. The b k -de Rham complex is ( b k Ω p ( M ) , d ), with b k Ω ( M ) := C ∞ ( M ). The b k -cohomology b k H ∗ ( M ) is the cohomology of this complex. Proposition 4.2. The sequence below, with g given by Map (8) , is exact (9) 0 → b k − Ω p ( M ) g → b k Ω p ( M ) ι L → Ω p − ( Z ) → .Moreover, for any closed α ∈ Ω p − ( Z ) and collar neighborhood ( y, π ) : U → R × Z of Z with y ∈ j Z ,there is a closed form ω ∈ ι − L ( α ) such that ω = dyy k ∧ π ∗ ( α ) in a neighborhood of Z .Proof. The only nontrivial part of the exactness claim is that ker( ι L ) ⊆ im( g ). The kernel of ι L consists precisely of those ω that admit some decomposition ω = dyy k ∧ α + β in a neighborhood of Z for which i ∗ ( α ) = 0. Locally around Z , T ∗ M splits as T ∗ Z + h dy i , so wemay replace α by a form that vanishes on Z without changing ω . Then y − α is a smooth form, and dyy k − ∧ αy + β extends over M to a b k − form in g − ( ω ). Therefore, Sequence 9 is exact.Given a closed α ∈ Ω p − ( Z ) and a collar neighborhood ( y, π ) : U → ( − R, R ) × Z of Z with y ∈ j Z , let e y ∈ C ∞ ( M ) be a function that agrees with y on ( − R/ , R/ × Z and is locally constantoutside U . Then the b k -form ω = e y − k d e y ∧ π ∗ ( α ) extends to a closed b k -form on M that vanishesoutside U and satisfies ι L ( ω ) = α . In ( − R/ , R/ × Z , ω = dyy k ∧ π ∗ ( α )and dπ ∗ ( α ) = π ∗ ( dα ) = 0. (cid:3) One can check that the short exact sequence from Proposition 4.2 is a chain map of complexes,hence induces a long exact sequence · · · → b k − H ∗ ( M ) → b k H ∗ ( M ) → H ∗− ( Z ) → b k − H ∗ +1 ( M ) → . . . By Proposition 4.2, the maps b k H ∗ ( M ) → H ∗− ( Z ) are surjective, so the long exact sequence is acollection of short exact sequences0 → b k − H p ( M ) → b k H p ( M ) → H p − ( Z ) → k , this proves the following proposition. Proposition 4.3. b k H p ( M ) ∼ = H p ( M ) ⊕ (cid:0) H p − ( Z ) (cid:1) k Proof. From the remarks above. (cid:3) So far, this isomorphism is non-canonical: although we can lift every [ α ] ∈ H p − ( Z ) in Equation 10to an element of b k H p ( M ), we do not yet have a preferred choice of lifting, and different choices yieldgenuinely different isomorphisms. Results in Subsection 4.1, where we show that the ( H p − ( Z )) k summand of the image of any [ ω ] ∈ b k H p ( M ) can be canonically defined, will give us partial relieffrom this uncomfortable state of affairs. Finally, in Section 5 we will give an explicit canonical map HE GEOMETRY OF b k MANIFOLDS 9 for the isomorphism in Proposition 4.3, and in doing so we will see a geometric interpretation forthe terms on the right side of the isomorphism.4.1. The Laurent Series of a Closed b k -Form.Definition 4.4. A Laurent Series of a closed b k -form ω is an expression for ω in a neighborhoodof Z of the form ω = k X i =1 dyy i ∧ α − i + β where y ∈ j Z is a positively oriented local defining function and each α − i is closed. Remark 4.5. Every closed b k -form has a Laurent series. In fact, Proposition 4.2 shows that givena collar neighborhood ( y, π ) : U → ( − R, R ) × Z of Z with y ∈ j Z , every closed b k -form ω can bewritten (in a neighborhood of Z ) as the sum of a closed b k − form and dyy k ∧ π ∗ ( ι L ω ).By applying induction on the b k − form, we arrive at a Laurent series of the form ω = k X i =1 dyy i ∧ π ∗ ( γ − i ) + β for closed forms γ − i on Z . Example 4.6. Consider the b k -manifold ( S × S , Z ∪ Z , [ y ] k − ) pictured in Figure 1. U U Z Z y R Figure 1. A b k -manifold with disconnected Z where a collar neighborhood U = U ∪ U of Z is shaded. Let { ( θ i , y ) } be coordinates on U i . Then dθ (respectively dθ ) extends trivially over U (respectively U ) to a form on all of U . Let ω be a b k M . On U , it admits a decomposition ω = dyy k ∧ ( f dθ + gdθ ) + β for smooth functions f, g and a smooth form β . Let π : U → Z be the vertical projection, and for − k ≤ i ≤ − 1, let f i := 1( k + i )! ∂ k + i f∂y k + i (cid:12)(cid:12)(cid:12)(cid:12) Z g i := 1( k + i )! ∂ k + i g∂y k + i (cid:12)(cid:12)(cid:12)(cid:12) Z . Then f = π ∗ ( f − k ) + π ∗ ( f − k +1 ) y + · · · + π ∗ ( f − ) y k − + e fg = π ∗ ( g − k ) + π ∗ ( g − k +1 ) y + · · · + π ∗ ( g − ) y k − + e g for e f , e g ∈ I k . Then ω has a Laurent series ω = k X i =1 dyy i ∧ ( π ∗ ( f i ) dθ + π ∗ ( g i ) dθ ) + β ′ where β ′ is smooth form. Proposition 4.7. The cohomology classes [ i ∗ ( α − i )] ∈ H p − ( Z ) appearing in a Laurent series of ω ∈ b k Ω p ( M ) depend only on [ ω ] .Proof. By Proposition 3.4, we may assume that all our Laurent series are written with respect tothe same local defining function y ∈ j Z . When k = 1, then for ω ∈ b Ω p ( M ), the class [ i ∗ ( α − )] isthe image of [ ω ] in the map appearing in Equation 10, and therefore depends only on [ ω ].For k > 1, assume the proposition is true for k − 1, and let ω ∈ b k Ω p ( M ). Consider Laurent seriesof two representatives of [ ω ], ω = k X i =1 dyy i ∧ α − i + β and ω = k X i =1 dyy i ∧ α ′− i + β ′ Both [ i ∗ ( α − k )] and [ i ∗ ( α ′− k )] are the image of [ ω ] in Equation 10, so are equal. If we can show that k − X i =1 dyy i ∧ α − i + β and k − X i =1 dyy i ∧ α ′− i + β ′ are cohomologous b k − -forms, then we will be done by induction. That is, we must show that(11) ω − dyy k ∧ α ′− k − (cid:18) ω − dyy k ∧ α − k (cid:19) is an exact b k − -form. Because [ ω ] = [ ω ], there is a b k -form η with dη = ω − ω . Moreover,because α − k − α ′− k is a closed form with i ∗ ( α − k − α ′− k ) exact, the relative Poincar´e lemma impliesthat it has a primitive µ . Then η + dyy k ∧ µ is a primitive for the form (11). However, this primitive is a b k -form; to prove that (11) is exact asa b k − -form (and in doing so complete the induction), simply observe that the map b k − H p ( M ) → b k H p ( M )from Sequence 10 is injective, so any b k − -form exact as a b k -form is also exact as a b k − -form. (cid:3) Corollary 4.8. Let ω = k X i =1 dyy i ∧ α − i + β be a Laurent series of the closed b k -form ω . The map b k H p ( M ) → ( H p − ( Z )) k (12) [ ω ] ([ i ∗ ( α − )] , [ i ∗ ( α − )] , . . . , [ i ∗ ( α − k )]) is independent of the choice of Laurent series. HE GEOMETRY OF b k MANIFOLDS 11 Definition 4.9. Given a b k -form ω , the image of [ ω ] under Map 12 is the Laurent Decomposition of [ ω ].The result below strengthens Theorem 4.3. Theorem 4.10. The sequence below, with g, f given by the Map 8 and Map 12 respectively, is exact. (13) 0 → H p ( M ) g → b k H p ( M ) f → ( H p − ( Z )) k → Proof. The map g is the composition of the inclusions b ℓ − H n ( M ) → b ℓ H n ( M )appearing in the short exact sequence (10) for ℓ ≤ k . Therefore, it itself is an inclusion. The proofthat f is surjective follows from the same trick used to create a preimage of a closed α ∈ Ω p − ( Z )in the proof of Proposition 4.2. Exactness at the middle is straightforward. (cid:3) Volume Forms on a b k -manifold Let ( M, Z, j Z ) be a compact b k -manifold, and let ω ∈ b k Ω dim( M ) ( M ). Because ω “blows up” along Z , we cannot expect its integral to be finite. If we remove from M a neighborhood of Z , then theintegral of ω over the remainder is finite, but obviously depends on the choice of neighborhood. Inthis section, we extract a useful invariant of ω by studying the behavior of this integral as the sizeof the removed neighborhood shrinks. We will use this invariant to split the short exact sequence(13), and in doing so make the isomorphism (4.3) canonical.The results from this section apply even to non-compact manifolds; so that we may state theseresults in full generality, we begin by introducing notation for compactly supported de Rham theory. Definition 5.1. The subset b k Ω pc ( M ) ⊆ b k Ω p ( M ) consists of b k -forms with compact support. Theyform a subcomplex of the b k -de Rham complex, the homology of which is called the compact b k -cohomology b k H ∗ c ( M )5.1. Liouville Volume of a b k -form.Definition 5.2. Let ( M, Z, j Z ) be an n -dimensional b k -manifold. Given ω ∈ b k Ω nc ( M ), ǫ > y ∈ j Z , define U y,ǫ = y − (( − ǫ, ǫ )) andvol y,ǫ ( ω ) = Z M \ U y,ǫ ω In [R], Radko proved that when M is a surface and k = 1, lim ǫ → vol y,ǫ ( ω ) converges and isindependent of y . This limit, the Liouville volume of ω , was a key ingredient in her classification ofstable Poisson structures on compact surfaces. When k > 1, this limit will not necessarily convergeto a number, but rather to a polynomial in ǫ − . After proving the existence and well-definedness ofthis polynomial, we will define the Liouville volume of a b k -cohomology class of top degree as theconstant term of this polynomial. Theorem 5.3. For a fixed [ ω ] ∈ b k H nc ( M ) on a b k -manifold ( M, Z, j Z ) with Z compact, there is apolynomial P [ ω ] for which (14) lim ǫ → (cid:0) P [ ω ] (cid:0) ǫ − (cid:1) − vol y,ǫ ( ω ) (cid:1) = 0 for any y ∈ j Z and any ω representing [ ω ] . Although Radko studied b -forms only on surfaces, her proof of the fact stated here works for all n . Proof. We first prove that there is a polynomial P [ ω ] that satisfies Equation 14 for a specific y and ω , then we prove that the polynomial is independent of y , then that the polynomial vanishes forexact ω (so depends only on the b k -cohomology class of ω ).Fix a local defining function y ∈ j Z and a closed collar neighborhood ( y, π ) : U → [ − R, R ] × Z of Z . Because ω is compactly supported, R M \ U ω < ∞ , so to prove the existence of P [ ω ] it suffices toconstruct a polynomial for the case M = U . By Remark 4.5, there exists a Laurent series of ω ofthe form ω = k X i =1 dyy i ∧ π ∗ ( α − i ) + β where α − i ∈ Ω n − ( Z ). Thenvol y,ǫ ( ω ) = Z U \ U y,ǫ k X i =1 dyy i ∧ π ∗ ( α − i ) + Z U \ U y,ǫ β Applying Fubini’s theorem (and cancelling log terms), the first term simplifies to k X i =2 − i − (cid:18) R (cid:19) i − + (cid:18) − ǫ (cid:19) i − − (cid:18) − R (cid:19) i − − (cid:18) ǫ (cid:19) i − ! Z Z α − i = k X i =2 i even (cid:18) − R − i i − (cid:19) Z Z α − i + k X i =2 i even (cid:18) i − Z Z α − i (cid:19) ( ǫ − ) i − and the last term simplifies to Z U β − Z [ − ǫ,ǫ ] × Z β so the polynomial P ( t ) = Z U β + k X i =2 i even (cid:18) − R − i i − (cid:19) Z Z α − i + k X i =2 i even (cid:18) i − Z Z α − i (cid:19) t i − satisfies the conditions of a volume polynomial for this specific choice of y and ω .The proof that this polynomial does not depend on y is techincal; the details can be found inSection 8. To show that the polynomial associated to any exact form is trivial, suppose ω is exactand let η = k X i =1 dyy i ∧ π ∗ η − i + β η be a Laurent series of a primitive of ω . Then Z M \ U y,ǫ ω = Z ∂ ( M \ U y,ǫ ) η = Z ∂ ( M \ U y,ǫ ) β η which approaches 0 as ǫ → (cid:3) Definition 5.4. The polynomial P [ ω ] described in Theorem 5.3 is the volume polynomial of [ ω ].Its constant term P [ ω ] (0) is the Liouville volume of [ ω ].The Liouville volume of [ ω ] can be thought of as the volume that remains of [ ω ] after its singularparts have been carefully discarded. For arbitrary (non- b k ) singularities of a form of top degree,no similar concept exists. In the b k case, the definition is made possible by how well-behaved b k singularities are, as well as by how we use the jet data (when k > 1) to prescribe the asymptoticmanner in which U y,ǫ approaches Z as ǫ → HE GEOMETRY OF b k MANIFOLDS 13 We may also define the Liouville volume of a p < dim( M ) dimensional b k -form ω along a compact p -dimensional submanifold Y ⊆ M transverse to Z : the pullback of ω will be a b k -form of top degreefor the induced b k -structure on Y and therefore has a Louville volume. By Poincar´e duality, thisremark inspires the definition of the smooth part of a b k -form. Definition 5.5. Let [ ω ] ∈ b k H p ( M ). The image of [ ω ] under the map b k H p ( M ) → ( H n − pc ( M )) ∗ ∼ = H p ( M )(15) [ ω ] (cid:0) [ η ] P [ ω ∧ η ] (0) (cid:1) is its smooth part [ ω sm ] ∈ H p ( M ).If [ ω ] is smooth (that is, [ ω ] ∈ H n ( M ) ⊆ b k H n ( M )), then so too is [ ω ∧ η ] smooth for all [ η ] ∈ ( H n − pc ( M )) ∗ . In this case, it follows that P [ ω ∧ η ] (0) equals R M ω ∧ η and that [ ω ] = [ ω sm ]. This remarkshows that Equation 15 splits the short exact sequence from Equation 13, which yields a canonicalisomorphism, the Liouville-Laurent isomorphism, that realizes the (abstract) isomorphism fromProposition 4.3. ϕ : b k H n ( M ) ∼ = H n ( M ) ⊕ (cid:0) H n − ( Z ) (cid:1) k (16) [ ω ] ([ ω sm ] , [ α − ] , . . . , [ α − k ]) Definition 5.6. Let ω be a b k -form of top degree. The Liouville-Laurent decomposition of [ ω ] isits image under Equation 16, ([ ω sm ] , [ α − ] , . . . , [ α − k ]).The following proposition shows that taking the Liouville-Laurent decomposition of a b k -formcommutes with taking its pullback under a b k -map. Proposition 5.7. Let ϕ : ( M, Z, j Z ) → ( M ′ , Z ′ , j Z ′ ) be a b k -map, and [ ω ′ ] ∈ b k H p ( M ′ ) haveLiouville-Laurent decomposition ([ ω ′ sm ] , [ α ′− ] , . . . , [ α ′− k ]) . Then [ ϕ ∗ ( ω ′ )] has Liouville-Laurent de-composition ([ ϕ ∗ ( ω ′ sm )] , [ ϕ (cid:12)(cid:12) Z ∗ ( α ′− )] , . . . , [ ϕ (cid:12)(cid:12) Z ∗ ( α ′− k )]) . Proof. Let y ′ ∈ j Z ′ , and i Z : Z → M , i Z ′ : Z ′ → M ′ be the inclusions. By the definition of a b k -map, y := ϕ ∗ ( y ′ ) represents j Z . Then for a Laurent series of ω ′ , ω ′ = k X i =1 dy ′ y ′ i ∧ π ∗ α ′− i + β ′ , the pullback of ω ′ has Laurent series ϕ ∗ ( ω ′ ) = k X i =1 dyy i ∧ ϕ ∗ ( π ∗ α ′− i ) + ϕ ∗ ( β ′ ) . and we see that [ ϕ ∗ ( ω ′ )] has Laurent decomposition([ i ∗ Z ( ϕ ∗ ( π ∗ α ′− ))] , . . . , [ i ∗ Z ( ϕ ∗ ( π ∗ α ′− k ))]) = ([ ϕ (cid:12)(cid:12) Z ∗ ( i ∗ Z ′ ( π ∗ α ′− ))] , . . . , [ ϕ (cid:12)(cid:12) Z ∗ ( i ∗ Z ′ ( π ∗ α ′− k ))])which proves that the Laurent decomposition commutes with pullback.Let [ η ] ∈ H n − pc ( M ). To prove that [ ϕ ∗ ( ω ′ ) sm ] = [ ϕ ∗ ( ω ′ sm )], it suffices to show that(17) P [ ϕ ∗ ( ω ′ ) ∧ η ] (0) = Z M ϕ ∗ ( ω ′ sm ) ∧ η. Our strategy for proving Equation 17 will be to introduce an auxiliary family of smooth closeddifferential forms ω ′ ǫ ∈ Ω p ( M ′ ) with the property that the Liouville volume of ϕ ∗ ( ω ′ ) ∧ η can becalculated in terms of the asymptotic behavior of R M ϕ ∗ ( ω ′ ǫ ) ∧ η instead of R M \ U y,ǫ ϕ ∗ ( ω ′ ) ∧ η . For ǫ > f ǫ : R → [0 , 1] be a smooth function such that f ǫ (cid:12)(cid:12) R \ ( − ǫ,ǫ ) = 1 and f ǫ (cid:12)(cid:12) ( − ǫ +exp( − ǫ − ) ,ǫ − exp( − ǫ − )) = 0and assume that f ǫ varies smoothly with ǫ . Define ω ′ ǫ = k X i =1 f ǫ ( y ′ ) dy ′ y ′ ∧ π ∗ α ′− i + β ′ and observe that ω ′ ǫ is closed and that R M ϕ ∗ ( ω ′ ǫ ) ∧ η approaches vol y,ǫ ( ϕ ∗ ( ω ′ ) ∧ η ) as ǫ → ϕ ∗ η for a representativeof the pushforward of [ η ] ∈ H n − pc ( M ). Using this notation,0 = lim ǫ → (cid:0) P [ ϕ ∗ ( ω ′ ) ∧ η ] ( ǫ − ) − vol y,ǫ ( ϕ ∗ ( ω ′ ) ∧ η ) (cid:1) = lim ǫ → (cid:18) P [ ϕ ∗ ( ω ′ ) ∧ η ] ( ǫ − ) − Z M ϕ ∗ ( ω ′ ǫ ) ∧ η (cid:19) = lim ǫ → (cid:18) P [ ϕ ∗ ( ω ′ ) ∧ η ] ( ǫ − ) − Z M ′ ω ′ ǫ ∧ ϕ ∗ η (cid:19) = lim ǫ → (cid:0) P [ ϕ ∗ ( ω ′ ) ∧ η ] ( ǫ − ) − P [ ω ′ ∧ ϕ ∗ η ] ( ǫ − ) (cid:1) so P [ ϕ ∗ ( ω ′ ) ∧ η ] (0) = Z M ′ ω ′ sm ∧ ϕ ∗ η = Z M ϕ ∗ ( ω ′ sm ) ∧ η which proves Equation 17. (cid:3) b k Orientation. The notion of orientability of a smooth manifold generalizes in an obviousway to the b k -world. Definition 5.8. A volume b k -form on a b k manifold is a nowhere vanishing b k -form of top degree.A b k -manifold is b k -orientable if it admits a volume b k -form. A b k -orientation on a connectedorientable b k -manifold is a choice of one of the two connected components of the space of volume b k -forms.Although the underlying smooth manifold of every b k -manifold is orientable (an orientation for M is included in the data of a b k -manifold), not all b k -manifolds are b k -orientable. For example, if Z ⊆ M is a meridian of the torus S × S (so M \ Z is connected), the corresponding b -manifoldadmits no volume b -form even though M is orientable. The opposite is also true: if you remove fromthe definition of a b k -manifold the condition that M is oriented, then it remains possible to definethe b k -(co)tangent bundles, and according to these new definitions there would exist b k -manifoldsthat admit a b k -orientation even though the underlying manifold is unorientable. For example,if Z ⊆ M is a meridian of the Klein bottle, there exists a volume b -form on the corresponding b -manifold even though M is not orientable. Although it is possible to study the b k -geometry ofnon-orientable manifolds by modifying the definition of a b k -manifold in this way, omitting the dataof an orientation makes it impossible to define the Liouville volume of a b k -form of top degree. It isfor this reason that we have restricted our attention to b k -structures on oriented manifolds in thispaper.Notice that the image under ι L of a volume b k -form ω will be a smooth volume form on Z . Inthis way, a b k -orientation on ( M, Z, j Z ) induces an orientation on Z which may or may not agreewith the orientation of Z given in the data of a b k -manifold. HE GEOMETRY OF b k MANIFOLDS 15 Definition 5.9. Let ω be a volume b k -form on ( M, Z, j Z ). If the smooth form ι L ( ω ) is positivelyoriented, we say that ω is a positively oriented volume b k -form.Notice that if ω is a volume b k -form which is not positively oriented, one can replace the b k structure on ( M, Z, j Z ) with a different b k structure for which ω is a positively oriented volume b k -form. To do so, reverse the orientations of those components Z ′ of Z for which ι L ( ω ) (cid:12)(cid:12) Z ′ is negativelyoriented, and replace the jet data for those Z ′ with their negatives.6. Symplectic and Poisson Geometry of b k -Forms We begin this section by introducing the notion of a symplectic b k -form and proving Moser’stheorems in the b k -category. We then classify symplectic b k -surfaces, and show how the Liouville-Laurent decomposition of a b -symplectic form on a surface reconciles a classification theorem from[GMP2] with one from [R]. Definition 6.1. A symplectic b k -form on a b k -manifold is a closed b k p ∈ M . Definition 6.2. A symplectic b k -manifold ( M, Z, j Z , ω ) is a b k -manifold ( M, Z, j Z ) with a sym-plectic b k -form ω . Definition 6.3. A b k -symplectomorphism ϕ : ( M, Z, j Z , ω ) → ( M ′ , Z ′ , j Z ′ , ω ′ ) is a b k -map sat-isfying ϕ ∗ ( ω ′ ) = ω . Theorem 6.4. (relative Moser’s theorem) If ω , ω are symplectic b k -forms on ( M, Z, j Z ) with Z compact, ω (cid:12)(cid:12) Z = ω (cid:12)(cid:12) Z , and [ ω ] = [ ω ] , then there are neighborhoods U , U of Z and a b k -symplectomorphism ϕ : ( U , Z, j Z , ω ) → ( U , Z, j Z , ω ) such that ϕ (cid:12)(cid:12) Z = id .Proof. Pick a local defining function y ∈ j Z and Laurent series of ω , ω ω = k X i =1 dyy i ∧ α − i + β ω = k X i =1 dyy i ∧ α ′− i + β ′ . Then i ∗ ( α ′− i − α − i ) ∈ Ω ( Z ) is exact for all i , and i ∗ ( α ′− k − α − k ) = i ∗ ( β ′ − β ) = 0. By the relativePoincar´e lemma there are primitives µ i of ( α ′− i − α − i ) and µ β of ( β ′ − β ) with µ − k (cid:12)(cid:12) Z = µ β (cid:12)(cid:12) Z = 0.Then ω − ω = dµ , where µ = k X i =1 dyy i ∧ µ − i + µ β .Let ω t = tω + (1 − t ) ω , and observe that dω t /dt = dµ . By shrinking our neighborhood around Z , we can assume that ω t has full rank for all t , giving a pairing between b k -vector fields and b k µ is a b k Z (since µ − k (cid:12)(cid:12) Z = 0 and µ β (cid:12)(cid:12) Z = 0), the vector field v t defined by Moser’s equation ι v t ω t = − µ is a b k -vector field that vanishes on Z , the time-one flow of which is the desired b k -symplectomorphism. (cid:3) Theorem 6.5. (global Moser’s theorem) Let ( M, Z, j Z ) be a compact b k -manifold, and ω t := tω +(1 − t ) ω a symplectic b k -form for t ∈ [0 , , with [ ω ] = [ ω ] . Then there is an isotopy ρ t of b k -mapswith ρ ∗ t ( ω t ) = ω for t ∈ [0 , . Proof. Because dω t dt = ω − ω is exact, there is a smooth b k -form µ such that dµ = ω − ω . Because ω t is a b k -form, it defines an pairing between b k b k -vector fields. Therefore, the vectorfield v t defined by Moser’s equation ι v t ω t = − µ is a b k -vector field, so its flow defines an isotopy ρ t of b k -maps with ρ ∗ t ( ω t ) = ω . (cid:3) Classification of Symplectic b k -Surfaces. In [R], the author classifies the space of stablePoisson structures on a connected, compact surface in terms of geometric data. In [GMP2], theauthors demonstrate a correspondence between stable Poisson structures and b -symplectic formson a manifold, and classify b -symplectic forms on a connected, compact surface in terms of their b -cohomology class. Pictorially, we have two sides of the triangle (cid:26) Symplectic b -forms on ( M, Z ) (cid:27)(cid:30) b -symp. (cid:26) L. Vol ∈ R { pd( γ i ) } ri =1 ∈ R r> (cid:27) b H ( M ) [ R ] [ G M P ] where M is a connected, compact surface, { γ i } are the r oriented circles that constitute Z , L. Vol ∈ R is the Liouville volume of ( M, Z, ω ), and pd( γ i ) is the period of the modular vector field on γ i .Theorem 6.6 completes the triangle. That is, it exhibits a direct connection between the coho-mological classification data in [GMP2] and the geometric classification data in [R]. Theorem 6.6. Let [ ω ] = ([ ω sm ] , [ α − ]) be the Liouville-Laurent decomposition of a positively ori-ented b -symplectic form on a connected compact surface. Let { γ r } be the oriented circles that con-stitute Z . Then the Liouville volume of ω is R M ω sm , and the period of the modular vector field on γ r is (cid:18)Z γ r α − (cid:19) − Proof. The fact that the Liouville volume of ω equals R M ω sm follows from the definition of the smoothpart of a b k -form. Let γ i be a connected component of Z . We can find a collar neighborhood U = { ( y, θ ) , | y | < R, θ ∈ [0 , / ∼} R > U ω = c dyy ∧ dθ c > dθ is a positively-oriented volume form on Z . From [R], we know that the period of themodular vector field is c − , and we calculate that Z γ i α − = Z γ i cdθ = c . (cid:3) Theorem 6.7. Let ω , ω be symplectic b k -forms on a compact connected b k -surface ( M, Z, j Z ) . Thefollowing are equivalent (1) There is a b k -symplectomorphism ϕ : ( M, Z, j Z , ω ) → ( M, Z, j Z , ω ) . (2) [ ω ] = [ ω ] HE GEOMETRY OF b k MANIFOLDS 17 (3) The Liouville volumes of ω and ω agree, as do the numbers Z γ r α − i for all connected components γ r ⊆ Z and all ≤ i ≤ k , where α − i are the terms appearingin the Laurent decomposition of the two forms.Proof. (1) ⇐⇒ (2) : This follows from the global Moser’s Theorem (Theorem 6.5) in dimension 2.(2) ⇐⇒ (3) : The isomorphism (16) shows that the cohomology class of a volume b k formis determined by its Liouville-Laurent decomposition, which in turn is determined by itsLiouville volume and the integrals R γ r α − i . (cid:3) Integrable systems and a b k Poincar´e Lemma. We began Section 4 by studying thefollowing complex of sheaves.(18) 0 → C ∞ → b k Ω → b k Ω → b k Ω → . . . Although this complex acts as a generalization of the de Rham complex of smooth forms, there isone remarkable dissimilarity between them: unlike in the smooth case, there exist b k -forms thatare closed but not locally exact. This is true even when k = 1. For example, if y is a localdefining function for Z , then y − dy is a closed b -form, but in no neighborhood of any p ∈ Z is itexact. Informally, it wants to be the differential of log( y ), but log( y ) is not a section of C ∞ in anyneighborhood of p . This “failure of the Poincar´e lemma” plagues only b k b k forms ofhigher degree. Indeed, if ω = k X i =1 dyy i ∧ π ∗ ( α i ) + β is a Laurent decomposition of a b k form ω of degree ≥ Z , then for asufficiently small neighborhood U of any p ∈ Z , there exist primitives η i of α i (cid:12)(cid:12) U ∩ Z and η β of β (cid:12)(cid:12) U ,and k X i =1 dyy i ∧ π ∗ ( η i ) + η β is a primitive of ω . In light of this fact, one way to make peace with this failure of Poincar´e’s lemmais to enlarge the sheaf C ∞ into a new sheaf b k C ∞ whose sections, in any contractible neighborhood,include primitives of all closed b k Definition 6.8. Let ( M, Z, j Z ) be a b k -manifold. The sheaf b k C ∞ on M is defined by b k C ∞ ( U ) := { f ∈ C ∞ ( U \ Z ) | df extends to an element of b k Ω ( U ) } and the differential b k C ∞ → b k Ω is the map that sends f ∈ b k C ∞ ( U ) to the unique element of b k Ω ( U ) that extends df .After replacing C ∞ with b k C ∞ in the complex 18, every closed b k C ∞ ( M ) or b k C ∞ ( M ) is the correct notion of “function” on a b k -manifold ismore than a superficial convention – it raises the question of how to define an integrable system ona b k -manifold. If ω is a symplectic b k -form and f ∈ C ∞ ( M ), then the symplectic gradient X f of f will be a vector field whose restriction to Z is tangent to the leaves of the symplectic foliation of Z defined by ker( ι L ω ). On the other hand, if f ∈ b k C ∞ ( M ), then X f will still be tangent to Z , but not necessarily tangent to the leaves of the symplectic foliation. This difference can also be seen inthe fact that for any p , the map b k C ∞ ( M ) → b k T p ( M )(19) f ( X f ) p is surjective (generalizing the surjection C ∞ ( M ) → T p ( M ) from the classic theory of integrablesystems), but this surjectivity would fail if b k C ∞ ( M ) were replaced with C ∞ ( M ). So far, the onlydefinitions of an “integrable system on a b -manifold” that the author of this paper is aware of havetaken C ∞ functions to be the integrals of motion. It would be exciting to also study integrablesystems whose integrals of motion are b k C ∞ functions.7. Symplectic and Poisson structures of b k -type When the authors of [GMP2] studied the Poisson structures dual to symplectic b -forms, theyfound that b -symplectomorphisms are precisely Poisson isomorphisms of the dual Poisson manifolds.Unfortunately, this observation does not generalize to the b k case: although every symplectic b k -formis dual to a Poisson bivector, not every Poisson isomorphism (with respect to this bivector) is realizedby a b k -map. Similarly, if ( M, Z, j Z , ω ) and ( M, Z, j ′ Z , ω ′ ) are two symplectic b k -manifolds, theremay be a diffeomorphism of ( M, Z ) that restricts to a symplectomorphism ( M \ Z, ω ) → ( M \ Z, ω ′ )even if there is no b k -symplectomorphism ( M, Z, j Z , ω ) → ( M, Z, j ′ Z , ω ′ ). In this section, we showhow to use b k -manifolds to prove statements about objects outside of the b k -category. We begin bydefining the notion of a Poisson (and symplectic) structure of b k -type – these are the Poisson (andsymplectic) structures that are dual to (or equal to) a symplectic b k -form for some choice of jet data.Then we apply the theory of symplectic b k -forms to classify these structures on compact connectedsurfaces. Definition 7.1. Let Z be an oriented hypersurface of an oriented manifold M . Let Π be a Poissonstructure on M having full rank on M \ Z , and let ω ∈ Ω ( M \ Z ) be the symplectic form dual toΠ (cid:12)(cid:12) M \ Z . We say that Π and ω are of b k type if there is some j Z ∈ J k − for which ( M, Z, j Z ) is a b k -manifold on which ω extends to a symplectic b k -form. Remark 7.2. Notice that if Π is a Poisson structure of b k -type on ( M n , Z ) with dual form ω , thenthere will be several distinct jets with respect to which ω is a symplectic b k -form. For example, if j Z = [ y ] is one such jet and f : R → R satisfies f (0) = 0 and f ′ (0) > 0, then the jet j ′ Z := [ f ◦ y ]defines exactly the same b k -(co)tangent bundles as j Z . As such, ω is a symplectic form with respectto both j ′ Z and j Z . However, one can check that the condition of ω n being positively oriented (as avolume b k -form in the sense of Definition 5.9) does not depend upon the chosen jet. Therefore, wesay that Π (or ω ) is a positively oriented Poisson structure (or symplectic form) of b k type if ω extends to a positively oriented volume b k -form for any choice of jet j Z for which ω extends to a b k form.To study Poisson and symplectic structures of b k -type using the tools of b k -geometry, we mustunderstand how a b k -form behaves under diffeomorphisms of ( M, Z ) that are not necessarily b k -maps. Of particular interest to us will be diffeomorphisms of M that restrict to ( z, y ) ( z, P ( y )) ina collar neighborhood Z × R of Z , where P is a polynomial. The following proposition describes howthe Liouville-Laurent decomposition behaves under pullback of such a map (compare this propositionto Proposition 5.7, where we showed that the Liouville-Laurent decomposition commutes with thepullback of a b k -map). Proposition 7.3. Let P be a polynomial with P (0) = 0 and P ′ (0) > . Let ( M, Z, j Z ) be a b k -manifold with positively oriented local defining function y ∈ j Z , and let ϕ : M → M be adiffeomorphism given by id × P ( y ) in a collar neighborhood ( π, y ) : U → Z × R of Z . Then HE GEOMETRY OF b k MANIFOLDS 19 • If ω is a b k -form, then ϕ ∗ ( ω ) is also a b k -form on ( M, Z, j Z ) . • If [ ω ] has Liouville-Laurent decomposition ([ ω sm ] , [ α − ] , . . . , [ α − k ]) and [ ϕ ∗ ( ω )] has Laurentdecomposition ([ ω ′ sm ] , [ α ′− ] , . . . , [ α ′− k ]) , then [ ϕ ∗ ( ω sm )] = [ ω ′ sm ] and [ α − ] = [ α ′− ] .Proof. In a collar neighborhood, let ω = k X i =1 dyy i ∧ π ∗ ( α − i ) + β be a Laurent decomposition of ω . Then(20) ϕ ∗ ( ω ) = k X i =1 P ′ ( y ) dyP ( y ) i ∧ π ∗ ( α − i ) + ϕ ∗ β. Notice that each term P ′ ( y ) P ( y ) i must have a Laurent series with no exponents less than − i : indeed, y i P ′ ( y ) P ( y ) i = (cid:18) yP ( y ) (cid:19) i P ′ ( y )is smooth. By replacing each P ′ ( y ) P ( y ) i in equation 20 with its Laurent series, this proves the first claim.To prove the second claim, first observe that for i = 1, P ′ ( y ) dyP ( y ) i = d (cid:18) − i + 1 P ( y ) − i +1 (cid:19) so the meromorphic function P ′ ( y ) P ( y ) − i has no residue. For i = 1 the function P ′ ( y ) P ( y ) − has aLaurent series with principal part 1 /y . Therefore, by replacing the P ′ ( y ) P ( y ) − i terms in Equation20 with their Laurent series in the variable y , we arrive at a Laurent series of ϕ ∗ ( ω ) that has y − dy ∧ π ∗ ( α − ) as its residue term, proving that [ α − ] = [ α ′− ]. To prove that [ ϕ ∗ ( ω sm )] = [ ω ′ sm ],let [ η ] ∈ b k H n − pc ( M ), where p is the degree of ω and n = dim( M ). It suffices to show that(21) P [ ϕ ∗ ( ω ) ∧ η ] (0) = Z M ϕ ∗ ( ω sm ) ∧ η .Towards this goal, observe that for ǫ > ϕ ( U y,ǫ ) = U y,P ( ǫ ) , so vol y,ǫ ( ϕ ∗ ( ω ∧ ( ϕ − ) ∗ η )) =vol y,P ( ǫ ) ( ω ∧ ( ϕ − ) ∗ η )). Then letting ω ∧ ( ϕ − ) ∗ η = k X i =1 dyy ∧ π ∗ ( e α − i ) + e β be a Laurent series of ω ∧ ( ϕ − ) ∗ η ,vol y,ǫ ( ϕ ∗ ( ω ) ∧ η ) − vol y,ǫ ( ω ∧ ( ϕ − ) ∗ η ) = Z M \ U y,P ( ǫ ) − Z M \ U y,ǫ ! ω ∧ ( ϕ − ) ∗ η = Z Z Z ǫP ( ǫ ) − Z − ǫP ( − ǫ ) ! k X i =1 dyy i π ∗ ( e α − i )+ Z M \ U y,P ( ǫ ) − Z M \ U y,ǫ ! e β As ǫ → 0, this limit approaches an odd function of ǫ , proving that P [ ϕ ∗ ( ω ) ∧ η ] (0) = P [ ω ∧ ( ϕ − ) ∗ η ] (0),from which Equation 21 follows. (cid:3) Lemma 7.4. Let ( a − , . . . , a − k ) ∈ R k with a − k > . There is a polynomial P = P p i y i with p = 0 and p > satisfying k X i =1 a − i P ′ ( y ) P ( y ) i = 1 y k + a − y + Q ( y ) where Q ( y ) is a polynomial.Proof. The proof is technical. See Section 8 for the details. (cid:3) The two results above are the ingredients we need to prove the main theorem of this section. Theorem 7.5. Let Z be an oriented hypersurface of a compact oriented surface M . Let Π , Π ′ betwo positively oriented Poisson structures of b k -type on ( M, Z ) , and ω, ω ′ be the dual b k -symplecticforms (with respect to possibly different b k -structures) with Liouville-Laurent decompositions [ ω ] = ([ ω sm ] , [ α − ] , . . . , [ α − k ])[ ω ′ ] = ([ ω ′ sm ] , [ α ′− ] , . . . , [ α ′− k ]) . If [ ω ′ sm ] = [ ω sm ] ∈ H ( M ) and [ α ′− ] = [ α − ] ∈ H ( Z ) , then there is a Poisson isomorphism ϕ : ( M, Π) → ( M, Π ′ ) .Proof. Let j Z and j ′ Z be the jets of Z with respect to which ω and ω ′ respectively are b k -forms withthe described Liouville-Laurent decompositions, and let y ∈ j Z , y ′ ∈ j ′ Z be positively oriented localdefining functions for Z . Let { γ ℓ } be the oriented circles that constitute the connected componentsof Z . If ϕ : U ℓ → R × S = { ( y, θ ) } ϕ ′ : U ℓ → R × S = { ( y ′ , θ ) } are local coordinate charts for a collar neighborhood U ℓ of γ ℓ , then the map ( ϕ ′ ) − ◦ ϕ is anorientation-preserving map in a neighborhood of γ i , restricts to the identity on γ i , and pulls j ′ Z back to j Z . As such, the collection of these maps (one for each γ ℓ ⊆ Z ) defines a smooth map ina neighborhood of Z that extends to a b k -diffeomorphism ( M, Z, j Z ) → ( M, Z, j ′ Z ). By replacing ω ′ with its pullback under this b k -diffeomorphism and citing Proposition 5.7, we may assume that ω, ω ′ are b k -symplectic forms on the same b k -manifold ( M, Z, j Z ), and that the Liouville-Laurentdecomposititons of ω, ω ′ with respect to this b k structure are as described in the theorem statement.Let π : U ℓ = { ( y, θ ℓ ) } → S be projetion onto the second coordinate. We may assume (by theglobal Moser’s theorem) that ω (cid:12)(cid:12) U ℓ = k X i =1 dyy i ∧ a i π ∗ ( dθ ℓ ) + β where a i ∈ R and a − k > , Π ′ are positively oriented). Then we may apply Lemma 7.4to choose a polynomial P ℓ = P p i y i with p = 0 , p > k X i =1 a − i P ′ ( y ) P ( y ) i = 1 y k + a − y + Q ℓ ( y )for some polynomial Q ℓ ( y ). By replacing ω with its pullback under a diffeomorphism of ( M, Z ) thatis of the form ( y, θ ℓ ) ( P ℓ ( y ) , θ ℓ ) in each U ℓ , we may assume [ ω ] has Liouville-Laurent decomposition([ ω sm ] , [ α − ] , , . . . , , [ dθ ])where dθ is the form on Z that restricts to dθ i on each γ i . Similarly, we may replace ω ′ with a formalso having this Liouville-Laurent decomposition. Finally, we apply the global Moser’s theorem(Theorem 6.5) and the fact that M is a surface to complete the proof. (cid:3) HE GEOMETRY OF b k MANIFOLDS 21 Proof of Technical Results Proof of Proposition 3.4. Proof. The case k = 1 was covered in [GMP2], so we may assume k ≥ 2. Because [ f ] k − = [ f ] k − ,we have f = f (1 + gf k − ) for a smooth function g . Note that (1 + gf k − ) − = (1 + g ′ f k − ) for g ′ = − g (1 + gf k − ) − . Then df f k = df f k (1 + gf k − ) k − + d (1 + gf k − ) f k − (1 + gf k − ) k = (1 + g ′ f k − ) k − df f k + ( k − gdf (1 + gf k − ) k f + β ′ = (1 + ( k − g ′ f k − ) df f k + ( k − gdf (1 + gf k − ) k f + β ′′ = df f k + ( k − g ( − (1 + gf k − ) − + (1 + g ′ f k − ) k ) df f + β ′′ = df f k + ( k − g ( − (1 + g ′ f k − ) + (1 + g ′ f k − ) k ) df f + β ′′ = df f k + β where β ′ = (1 + g ′ f k − ) − k dgβ ′′ = β ′ + k − X i =2 (cid:18) k − i (cid:19) ( g ′ f k − ) i df f k β = β ′′ + ( k − g ( − g ′ f k − + k X i =1 (cid:18) ki (cid:19) ( g ′ f k − ) i ) df f (cid:3) Proof of subclaim of Theorem 5.3. We begin the proof with two technical lemmas. Lemma 8.1. If f ( x ) : R → R satisfies [ f ] k − = [ x ] k − , then its inverse h : ( − ǫ, ǫ ) → R satisfies [ h ] k − = [ x ] k − .Proof. Because [ f ] k − = [ x ] k − , f = ( x + g ( x ) x k ) for some smooth g . Then x = f ( h ( x )) = h ( x ) + g ( h ( x )) h ( x ) k .Because h ( x ) vanishes at 0 (since f does), x − h ( x ) vanishes to order at least k , so [ h ] k − = [ x ] k − . (cid:3) Lemma 8.2. Let f : R → R satisfy [ f ] k − = [ x ] k − . Then for all i ≤ k − x i − − x ) i + 1 f ( − x ) i − f ( x ) i is a smooth function that vanishes at . Proof. Because [ f ] k − = [ x ] k − , f ( x ) = x (1 + gx k − ) for some smooth g . Then h ( x ) := 1 x i − f ( x ) i = (1 + gx k − ) i − x i (1 + gx k − ) i = ( P ij =1 (cid:0) ij (cid:1) g j x j ( k − − i )(1 + gx k − ) i is a smooth function. Equation 22 equals h ( x ) − h ( − x ), so it is a smooth odd function, hencevanishes at zero. (cid:3) Proof. (of subclaim of Theorem 5.3) Let U be a tubular neighborhood ( y, π ) : U → [ − R, R ] × Z ,with y ∈ j Z . Let h be another element of j Z . It suffices to show thatlim ǫ → (vol h,ǫ ( ω ) − vol y,ǫ ( ω )) = 0for the case M = U . To do so, let y h,z : R → R be the function, defined near zero, inverse to h (cid:12)(cid:12) [ − R,R ] ×{ z } . That is, for sufficiently small ǫ , h ( y h,z ( ǫ ) , z ) = ǫ and U h,ǫ = { ( y, z ) ∈ [ − R, R ] × Z | y h,z ( − ǫ ) ≤ y ≤ y h,z ( ǫ ) } Then vol h,ǫ ( ω ) − vol y,ǫ ( ω ) = Z U \ U h,ǫ − Z U \ U y,ǫ ! ω = Z Z Z Ry h,z ( ǫ ) + Z y h,z ( − ǫ ) − R − Z Rǫ − Z − ǫ − R ! k X i =1 dyy i π ∗ ( α − i )+ Z U \ U h,ǫ − Z U \ U y,ǫ ! β = Z Z (cid:18) log (cid:12)(cid:12)(cid:12)(cid:12) y h,z ( − ǫ ) y h,z ( ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) + log (cid:12)(cid:12)(cid:12)(cid:12) ǫ − ǫ (cid:12)(cid:12)(cid:12)(cid:12)(cid:19) π ∗ ( α − )++ k X i =2 − i Z Z (cid:0) − ( y h,z ( ǫ )) − i + ( y h,z ( − ǫ )) − i + ( ǫ ) − i − ( − ǫ ) − i (cid:1) π ∗ ( α − i )+ Z U y,ǫ β − Z U u,ǫ β by the previous lemmas, the limit as ǫ → (cid:3) Proof of Lemma 7.4. Proof. Recall from the proof of Proposition 7.3 that for any polynomial P and i = 1, the expression P ′ ( y ) P ( y ) i has a Laurent series in y with trivial residue term and no exponents less than − i . When i = 1, thesame expression has principal part y − . Therefore, for any polynomial P ,(23) k X i =1 a − i P ′ ( y ) P ( y ) i = k X i =2 b − i y i + a − y + Q ( y )for some b − i ∈ R and some polynomial Q ( y ). In particular, if P ( y ) = ( a − k ) / (1 − k ) y , then astraightforward calculation shows that b − k = 1 in the expression above. However, we wish to find apolynomial P such that not only does b − k = 1 in the expression above, but ( b − k , b − k +1 , . . . , b ) =(1 , , . . . , P = P p i y i so that P (0) = 0, P ′ (0) > 0, and ( b − k , b − k +1 , . . . , b − k + j − ) = (1 , , . . . , 0) in Equation 23 – we aim HE GEOMETRY OF b k MANIFOLDS 23 to find a new P so that P (0) = 0, P ′ (0) > 0, ( b − k , b − k +1 , . . . , b − k + j ) = (1 , , . . . , t ∈ R let e P = P + tP j +1 , we have for some smooth function g , k X i =1 a − i e P ′ ( y ) e P i = k X i =1 a − i P ′ P i (1 + ( j + 1) tP j )(1 + tP j ) i = k X i =1 a − i P ′ P i (cid:16) j + 1 − i ) tp j y j + gy j +1 (cid:17) = 1 y k + k − j X i =2 b − i y i + a − y + Q ( y )+ k X i =1 a − i P ′ P i (cid:16) ( j + 1 − i ) tp j y j + gy j +1 (cid:17) Notice that the y − k + j term of the above expression has coefficient b − k + j + a − k p − k ( j + 1 − k ) tp j = 0if we set t = − b − k + j p k − j − / ( a − k ( j +1 − k )), the y − k + j term vanishes, completing the induction. 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