AARBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I
EMMY MURPHY
Abstract.
Arboreal singularities are an important class of Lagrangian singu-larities. They are conical, meaning that they can be understood by studyingtheir links, which are singular Legendrian spaces in S n − . Loose Legendriansare a class of Legendrian spaces which satisfy an h –principle, meaning thattheir geometric classification is in bijective correspondence with their topo-logical types. For the particular case of the linear arboreal singularities, weshow that constructable sheaves suffice to detect whether any closed set of anarboreal link is loose. Introduction
Arboreal singularities, defined by Nadler [9], are a class of Lagrangian singu-larities L T ⊆ ( R n , ω std ), corresponding to any rooted tree T with no more than n + 1 vertices. We review their definition in Section 2. L T is a conical singularity,meaning that if we define Λ T = L T ∩ S n − , then Λ T is a Legendrian complex in thecontact sphere ( S n − , ξ std ), and L T is the cone of Λ T , along the radial directionin R n std .Loose Legendrians, first defined in [8], are notable because they satisfy an h –principle. This means that two loose Legendrians are isotopic among Legendrianswhenever they are smoothly isotopic in a manner preserving the natural framings.Said differently, they contain certain local models which can be used to “untangle”any interesting geometry, and thus only their topology remains. They only exist indimension 2 n − ≥
5: we will always assume this dimensional restriction throughoutthe paper. We review the definition and properties of loose Legendrians in Section2. Given contemporary tools, it is fairly easy to detect when a given Legendrianis non-loose: as soon as any Fukaya categorical invariant is non-vanishing – suchas the category of constructable sheaves or the Legendrian contact homology – theLegendrian in question cannot be loose. Conversely, detecting when a Legendrianis loose seems to be an extremely difficult problem: all known conditions implyinglooseness are essentially reformulations of the definition. Throughout this paperwe will work with constructable sheaf theory, as defined in [11] building on work[7]. For any Legendrian Λ ⊆ R n − , We denote by Sh(Λ) the derived categoryof constructable sheaves on R n which are compactly supported and have singularsupport on Λ (over any coefficient ring). Here R n is identified as the front projectionof R n − .Links of arboreal singularities Λ T are never loose: Sh(Λ T ) is equivalent to thecategory of derived modules of T (thought of as a quiver), as shown by Nadler [9].In particular there are many non-constant sheaves and it follows that Λ T is notloose. a r X i v : . [ m a t h . S G ] F e b EMMY MURPHY
For a smooth Legendrian manifold, asking questions about the geometry of anynon-trivial subset is not an interesting question. The reason is that any nontrivialclosed subset of a manifold can be isotoped into a neighborhood of a space withpositive codimension, and therefore the h -principle for subcritical isotropic embed-dings gives a complete classification in correspondence with the smooth topology.However, since an arboreal link Λ T is a Legendrian complex it has many interestingsubsets which have nontrivial homology in the top dimension. This paper con-cerns the class of Legendrians obtained by taking any closed set of Λ A n +1 , where T = A n +1 is the linear tree. Theorem 1.1.
Let Λ ⊆ Λ A n +1 be any closed set. Then Λ is loose if and only if Sh(Λ) ∼ = 0 . Here, a Legendrian complex is loose if every top dimensional cell is loose inthe complement of all other cells. This is the natural definition of looseness forLegendrian complexes, as it ensures that the h -principle classification results apply.More generally, we will say that a given cell in a Legendrian complex is loose if thatcell has a loose chart which is disjoint from all other cells.The theorem is phrased in terms of closed sets in Λ A n +1 , but similar to the caseof smooth Legendrian manifolds the topology of these sets are mostly irrelevant.The only data they carry in terms of contact geometry is which cells of Λ A n +1 intersect Λ in a proper set. Thus we will need an effective way to label these cells.Let Q = A n +2 be the linear tree, thought of an appending a new zero object 0 ∈ Q to the tree A n +1 . We think of Q as a quiver, meaning the category with n + 2elements and Mor( x, y ) consisting of a single element if x ≤ y and Mor( x, y ) = ∅ otherwise.We claim then that there is a natural correspondence between the top-dimensionalcells of Λ A n +1 and non-identity elements of Mor( Q ), this is proven in Lemma 3.1.Thus, if W ⊆ Mor( Q ), we can define a Legendrian Λ Q [ W − ] by deleting an openball from any top-dimensional cell of Λ A n +1 which corresponds to (non-identity)elements of W . The following proposition follows quickly from the h -principle forsubcritical isotropics. Proposition 1.2.
Let Λ ⊆ Λ A n +1 be a closed set. We define W ⊆ Mor( Q ) as fol-lows: for each f ∈ Mor( Q ) , f ∈ W if and only if the top-dimensional cell of Λ A n +1 corresponding to f intersects Λ in a proper subset. Then Sh(Λ) = Sh(Λ Q [ W − ] ) .Any given cell of Λ is loose (rel Λ , see Definition 2.8) if and only if the correspond-ing cell of Λ Q [ W − ] is loose (rel Λ Q [ W − ] ). The proof of Theorem 1.1 then follows from the following three results, whichare mostly independent. First we will prove a result generalizing the result from [9].Denote by M od ( Q [ W − ]) the derived category of modules ρ : Q → Ch ∗ sending0 ∈ Q to 0 ∈ Ch ∗ and sending all morphisms in W to quasi-isomorphisms. (i.e. M od ( Q [ W − ]) is the category of derived modules of the localized category Q [ W − ],preserving the initial object 0.) Proposition 1.3.
The category
Sh(Λ Q [ W − ] ) is equivalent to M od ( Q [ W − ]) . The next two results will both concern the notion of 2-out-of-6 closure, whichwill be an important concept in the paper. Given any subset of morphisms W ⊆ Q ,we say that W satisfies the if it contains all identities, andwhenever we have a composition a f → b g → c h → d so that gf ∈ W and hg ∈ W , RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 3 then f , g , h , and hgf are all in W . We note that the 2-out-of-6 property impliesthe weaker 2-out-of-3 property: given a composition a f → b g → c , then whenever anytwo of the morphisms f , g , and gf are in W , the remaining one is as well (we seethis by inserting an identity). In particular if W satisfies the 2-out-of-6 propertythen W is closed under composition.For any subset of morphisms W ⊆ Mor( Q ), we denote by W ⊆ Mor( Q ) the of W , i.e. the smallest subset of morphisms containing W whichsatisfies the 2-out-of-6 property. The second main result of the paper is purelyalgebraic. Proposition 1.4.
Let Q = A n +2 be the linear quiver with initial object andlet W ⊆ Mor( Q ) . For any f ∈ Mor( Q ) , f ∈ W if and only if for every module ρ : Q [ W − ] → Ch ∗ , ρ ( f ) is a quasi-isomorphism. The proof of Proposition 1.4 follows by constructing an explicit model of Q [ W − ](and Yoneda’s Lemma). Finally, the remaining ingredient is to relate the aboveresults to loose Legendrians. Proposition 1.5.
Let D ⊆ Λ Q [ W − ] be a top-dimensional cell, and let f D ∈ Mor( Q ) be the corresponding morphism. Then if f D ∈ W , it follows that D isloose. The converse of the proposition is also true, as follows immediately from [6]and [8]. Together these four propositions prove Theorem 1.1. In fact they prove astronger result, by which we can work with each cell individually.
Theorem 1.6.
Let Λ ⊆ Λ A n +1 be any closed set, and let D ⊆ Λ A n +1 be any top-dimensional cell. If the inclusion functor Sh(Λ \ D ) → Sh(Λ) is an equivalence, itfollows that D is loose rel Λ . Structure of the paper.
In Section 2 we review the necessary background forthe paper, particularly the definitions and basic properties of arboreal singularities,loose Legendrians, and constructible sheaves. The following three sections provethe four main propositions above.In Section 3 we define Λ Q [ W − ] , and prove its basic properties described in Propo-sitions 1.2 and 1.3. Section 4 contains the proof of Proposition 1.4. Finally, Section5 contains the proof of Proposition 1.5. Acknowledgments.
The author is grateful to the American Institute of Math-ematics, for hosting a workshop on arboreal singularities in March 2018, and toD. Alvarez-Gavela, Y. Eliashberg, D. Nadler, and L. Starkston for stimulating dis-cussions. 2.
Background
Throughout the paper we will always work with the contact manifold R n − ,whose contact structure is defined by the kernel of the 1-form dz − (cid:80) n − i =1 y i dx i . Thispaper is concerned with Legendrian spaces in R n − which are more general thansmooth manifolds. While they have a natural cellular structure, the geometry ofthe codimension ≥ EMMY MURPHY
Definition 2.1. A Legendrian complex in a contact manifold (
Y, ξ ) is a subsetΛ ⊆ Y which can be written as Λ = (cid:83) i Λ i , where { Λ i } is a collection of smoothlyembedded, connected Legendrian submanifolds with boundary and corners Λ i ⊆ ( Y, ξ ). { Λ i } are required to be mutually disjoint on their interiors. The set (cid:83) i ( ∂ Λ i )is called the singular set of Λ.There are three basic ingredients necessary for the background of the paper:arboreal singularities, constructable sheaves, and loose Legendrians. For the lattertwo topics our treatment here is essentially standard (taken from [11] and [8] re-spectively). Our treatment of arboreal singularities is somewhat novel, in that wedefine Λ T via its generic front projection, instead of the standard method of usingconormals to hyperplanes in R n [9]. This makes them appear perhaps less natural,but the advantage is that their contact geometry is more explicit.2.1. Arboreal singularities.
Arboreal singularities are a class of Lagrangian sin-gularities which have recently gained interest as a important class to understand,particularly within the context of skeleta of Weinstein manifolds. See [9, 10, 12]for some important applications. We give a definition here which serves as amodel for the links of these singularities, which are Legendrian complexes inside R n − = ∂B n std \ { point } .For a fixed n , let ∆ ⊆ R n − be the standard embedding of the ( n − n vertices of ∆ are equidistant from each other, and each vertexis distance 1 away from the origin. The Venn diagram is a configuration of n roundcopies of B n − ⊆ R n − , whose centers are the vertices of ∆, all with equal radius1 + ε for a small ε >
0. Thus a given k –dimensional face of ∆ contains a vertex v in its closure if and only if the centroid of that face is contained in the ballcorresponding to v . We define r v : R n − → [0 , ∞ ) as the radial distance away fromthe point v ∈ R n − . We also choose a bump function χ : [0 , ∞ ) → [0 ,
1] which isnon-increasing everywhere, equal to r (cid:55)→ (1 + ε − r ) for r ∈ [1 , ε ], equal to 0for r ∈ [1 + ε, ∞ ), and constant near r = 0.Let π : R n − → R n = { ( x , . . . , x n − , z ) } be the front projection and let p : R n → R n − be the projection onto the x i coordinates. Let T be a rooted tree with n + 1 vertices. We choose a bijection between the vertices of ∆ with the non-rootvertices of T . The arboreal link corresponding to T is a Legendrian Λ T ⊆ R n − ,which is homeomorphic to a union of S n − and n copies of D n − , indexed byvertices v ∈ T . We define Λ T by defining its front projection.The root v of T corresponds to S n − ⊆ Λ T , where π ( S n − ) is the standard“flying saucer” front for the Legendrian unknot, see Figure 1. We choose π ( S n − )so that the lower branch of π ( S n − ) coincides with a large ball in the plane { z = 0 } ,enough so that pπ ( S n − ) contains the entire Venn diagram. We also choose theupper branch to have large z -value, in particular over the Venn diagram the upperbranch should satisfy z > n .For all other v ∈ T which are not the root, the disk D n − v ⊆ Λ T is defined sothat p ◦ π : D n − v → R n − takes (the interior of) D n − v diffeomorphically onto theball in the Venn diagram centered at v . For all v , the interior of π ( D n − v ) willbe contained in the open bounded region of R n \ π ( S n − ), and the boundary of π ( D n − v ) only intersects the lower branch of π ( S n − ).We complete the definition inductively with respect to the partial ordering givenby T . We define π ( D n − v ) = { z = (cid:80) w ≤ v χ ( r w ) , r v ∈ [0 , ε ] } . Informally, each RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 5
Figure 1.
The front of the standard Legendrian unknot π ( S n − ). π ( D n − v ) is a “dome” which is stacked on top of all previous domes sitting below itin T . Given two vertices v and v which are incomparable in T , π ( D n − v ) intersects π ( D n − v ) in a ( n −
2) disk, but they are transverse on their interiors and thereforethe interiors of D n − v and D n − v are disjoint in R n − . See Figures 2 and 3. It wouldbe instructive to the reader to make sure they can clearly picture all π (Λ T ) ⊆ R ,corresponding to the 4 distinct rooted trees with n + 1 = 4 vertices. Figure 2.
The front of the standard Λ A , the linear tree withthree vertices. Remark 2.2.
We make a number of technical remarks that may be useful toexperts in the theory, but do not have any bearing on the main result.
EMMY MURPHY
Figure 3.
The front of the standard Λ T where T is the uniquenon-linear rooted tree with three vertices (i.e. the linear tree exceptwith the root being the center vertex.)It is perhaps not immediately obvious that this definition is equivalent to Nadler’soriginal definition from [9]: we claim that, if L T ⊆ B n std is Nadler’s arboreal sin-gularity corresponding to the rooted tree T , then L T ∩ ∂B n std = Λ T , under theidentification R n − ∼ = S n − \ { point } . We will not prove this fact here, but wegive an intuitive sketch. The boundary of the standard R n ⊆ R n is isotopic tothe standard Legendrian unknot, and the coordinate hyperplanes in R n intersectthe sphere at infinity along the Venn diagram in S n − . Similar to the originaldefinition, the front projection is by definition a conormal construction, and so itsuffices to work entirely in R n , but this correspondence involves a stereographicprojection. To make the details precise, one would have to keep track on signswell (to see that “positive conormal” in the original definition corresponds to thepositive conormal implicit in the front projection), and pay detailed attention tothe singularities along the regions where Λ T is glued from disks.Since their original appearance, arboreal singularities have been generalized tosigned versions, see [12]. Our definition here corresponds to arboreal singularitieswhich are purely positive. One can give Legendrian definitions similar to the abovesection for arboreal links with arbitrary signs on the edge set, though we will notdo this here. χ has a discontinuous second derivative at the point 1 + ε , thus the Legendriancurve with front { z = χ ( x ) } is continuous, but not C smooth at x = 1 + ε .Therefore, as an instance of our definition above, if D v is defined by the front { z = χ ( r v ) + χ ( r w ) } , the closed disk D v is not smooth: it has a corner at itsboundary point r v = r w = 1 + ε . More generally, D n − v with have order k cornerswhenever there are k non-root vertices w satisfying w ≤ v . Regardless of n and k ,the front projection π ( D n − v ) is a disk which is C smooth but not C everywhereon its boundary, and p : π ( D n − v ) → R n − is a C diffeomorphism onto a closedround ball. Both π and p ◦ π are C ∞ diffeomorphisms on the interior of D n − v .As a Legendrian complex, Λ T has many singularities itself. The singularities ofΛ T correspond to the intersection points of the spheres in the Venn diagram. In RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 7 particular Λ T has 2 n isolated singularities: the isolated intersections in the Venndiagram are ( n − n − R n − → R n − , each isolated singularity of Λ T givesa isolated Lagrangian singularity: we claim that this is the arboreal singularitycorresponding to the subtree T \ { v } where v is the vertex in T correspondingto the unique disk D n − v whose boundary does not intersect the singularity. Inparticular, even within the self-contained definitions here we could ask whetherthe Legendrian link of the Lagrangian projection of a singularity of Λ T is a lowerdimensional arboreal link. We will also neglect to prove this.If T is a rooted tree with k < n + 1 vertices, we can still define a LegendrianΛ T ⊆ R n − , simply by attaching k copies of D n − to S n − along a k –componentsubdiagram of the Venn diagram. As above, we note without proof that Λ T is thelink of the arboreal singularity L T × R n − k ⊆ R k std × R n − k std .2.2. Constructable sheaves in contact geometry.
In this section we review thebasic material concerning constructable sheaves in contact geometry. Our accounthere principally follows [11].We fix a ring R . Throughout this paper a sheaf F on a manifold M will alwaysrefer to a chain complex of sheaves of R -modules. Such a sheaf is called constructable if it is locally constant with respect to some stratification S of M . For such asheaf F , we will define the singular support of F in terms of stratified Morsetheory. If x ∈ M is contained in a neighborhood U and f : U → R is a smoothfunction, we define the Morse group of ( x, f ) to be the cone of the restrictionmap F ( U ∩ f − ( −∞ , δ )) → F ( U ∩ f − ( −∞ , − δ )), where δ > U andsufficiently small δ (allowed to depend on U ), this chain complex does not dependon U or δ .If F is locally constant with respect to the stratification S and f : U → R is afunction which is stratified Morse with respect to S , then whenever the Morse groupof ( x, f ) is not acyclic we say that the covector df x ∈ T ∗ M is characteristic . Thus acovector p ∈ T ∗ M x is characteristic if there exists a function f with df x = p which isstratified Morse with respect to S and whose Morse group is cohomologically non-trivial. This definition does depend on the choice of S , but the singular support —defined to be the point-set closure of all characteristic covectors — only depends on F . We denote this set by SS ( F ) ⊆ T ∗ M . Then SS ( F ) is Lagrangian everywhereit is smooth [7], furthermore it is conical with respect to fiberwise radial dilation,i.e. the standard Liouville vector field. Thus if S ∗ M is the sphere bundle of T ∗ M (at infinity or with a chosen metric), SS ( F ) ⊆ S ∗ M is a Legendrian space. If π : S ∗ M → M is the front projection and S is any stratification of M for which F is locally constant, then π ( SS ( F )) is contained in the codimension ≥ S . Definition 2.3.
Let Λ ⊆ S ∗ M be a Legendrian complex, then we define thecategory Sh(Λ) as follows. Objects F ∈
Sh(Λ) are chain complexes of sheaves of R -modules, which are constructable and satisfy SS ( F ) ⊆ Λ. We also require that F is cohomologically bounded at each stalk (i.e. perfect), and that it has compactsupport in the case where M is non-compact. Here “subtree” is in the sense of full subcategories of the quiver, rather than subgraphs ofa directed graph: in a rooted tree, any collection of vertices containing the root gives rise to asubtree, which is itself a rooted tree.
EMMY MURPHY
Morphisms in Sh( F ) consist of derived morphisms, that is usual morphisms ofsheaves, where additionally all quasi-isomorphisms are localized.In the particular case where Λ ⊆ R n − (which is the only case we consider),we use the canonical embedding R n − ∼ = S ∗− R n ⊆ S ∗ R n to define Sh(Λ). Here S ∗− R n consists of those ( R + projectivized) covectors which evaluate negatively onthe vector field ∂ z on R n . We note that the front projection π : R n − → R n isequal to the base projection π : S ∗− R n → R n in this correspondence.An important fact, proved in [6], is that Sh(Λ) is a Legendrian invariant: it onlydepends on Λ up to contact isotopy. Theorem 2.4 ([6]) . Any contact isotopy ϕ t : S ∗ M → S ∗ M induces an equivalenceof categories Sh(Λ) → Sh( ϕ (Λ))This theorem is not directly relevant to the results in this paper, but it is signifi-cant in order to quickly prove the converses to the main theorems here: the converseof Theorem 1.6 and the “only if” portion of Theorem 1.1 both follow immediatelyfrom Theorems 2.4 and 2.10. Remark 2.5.
For smooth Legendrians Λ, any family of Legendrian embeddingsis induced by an ambient contact isotopy. For Legendrian complexes this is false.This will not be very relevant for us, since we are always working with Legendrianswhich are subsets of a fixed Legendrian complex Λ A n +1 . In more generality, askingwhether the singular sets of Legendrian complexes are contact isotopic essentiallyreduces to the question of whether they have contactomorphic neighborhoods: sincea neighborhood of the singular set is contained in a small neighborhood of a sub-critical isotropic space, h -principles can promote local contactomorphisms to globalcontact isotopies of the singular sets. Extending these isotopies to the interior ofthe top-dimensional cells then proceeds as in the smooth case.If S is a stratification of R n which refines π (Λ), then any sheaf in Sh(Λ) islocally constant with respect to S (up to quasi-isomorphism). Typically it is easyto arrange that S consists of finitely many cells, and that each cell is contractible.Let Q S be the finite thin category defined by the combinatorics of S : the objectsof Q S are the cells of S , and each Mor( C , C ) contains either one or zero elements,according to whether C is in the closure of C . Let M od ( Q S ) denote the categoryof derived perfect R -modules of Q S . Since any sheaf in Sh(Λ) is constant on eachcell of S , we have a fully faithful embedding Sh(Λ) → M od ( Q S ).We will refrain from stating any general result here (and we discard the notationfrom the previous paragraph), but the intuitive principle is useful to keep in mind:Sh(Λ) is equivalent to a full subcategory of M od ( Q ) , for some finite thin category Q . For Legendrian knots, this is done in detail in Sections 3.3, 3.4, and 3.5 of [11].In more generality figuring out which Q and which full subcategory of M od ( Q )correctly calculates Sh(Λ) involves analyzing the singularities of π (Λ). We do thisfor our relevant cases in Section 3.2.3. Loose Legendrians.
Among Legendrian submanifolds Λ ⊆ ( Y, ξ ) of dimen-sion dim Y = 2 n − ≥ (cid:96) ⊆ R n − , equal to the standard plane { z = y = 0 } outside RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 9 of a compact set, so that Λ is loose if and only if there exists an open set U ⊆ Y , so that the pair ( U, U ∩ Λ) is contactomorphic to ( R n − , Λ (cid:96) ). Thus,while looseness is a global property (Legendrians have no local invariants),it is semi-local in the sense that it can be certified by exhibiting it is looseon a single open set.– Loose Legendrians are classified up to Legendrian isotopy by data which ispurely diffeo-topological. They are typically thought of as being geomet-rically trivial: Fukaya categories and constructable sheaf categories are alltrivial for loose Legendrians. They are useful particularly for constructions:if you want to show that a loose Legendrian Λ satisfies Property X it suf-fices to build any Legendrian (cid:101) Λ which is loose and satisfies Property X . Aslong as the mild topological constraints are satisfied then Λ will be isotopicto (cid:101) Λ, and therefore Λ also satisfies Property X as long as the property isinvariant up to contact isotopy.In order to explain these informal descriptions we will need the notion of a formalLegendrian isotopy . Recall that when ( Y, ξ ) is any contact manifold, the vectorbundle ξ (forgetting its embedding ξ ⊆ T Y ) is equipped with a linear symplecticform, which is well-defined up to conformal scaling, when ξ = ker α this symplecticform is dα | ξ . Since α | Λ = 0 implies dα | Λ = 0, it follows that any Legendrian, simplyby virture of being tangent to ξ everywhere, must in fact be a Lagrangian subspace T Λ x ⊆ ξ x for all x ∈ Λ. Definition 2.6. A formal Legendrian embedding is a pair ( f, F s ), where f : Λ → Y is a smooth embedding, and F s : T Λ → T Y is a homotopy of bundle monomor-phisms covering f for all s ∈ [0 , F s is required to connect F = df and F – amap whose image F ( T Λ) lies inside ξ as a Lagrangian subspace.A Legendrian embedding is precisely a formal Legendrian embedding which isconstant in s . In particular a formal Legendrian isotopy between Legendrian em-beddings f , f : Λ → ( Y, ξ ) is a path in the space of formal Legendrian embeddingsconnecting f to f .We remark that the forgetful map from formal Legendrian embeddings to smoothembeddings is a Serre fibration. Furthermore the homotopy fiber can be identifiedwith something explicit, such as Map(ΩΛ , O n +1 /U n ) for the stably parallelizablecase (the typical case is the gauge group of the Lagrangian Grassmannian of thesymplectic bundle T ∗ Λ ⊗ C → Λ).As alluded to above, a Legendrian Λ ⊆ ( Y, ξ ) is called loose if there is an open U ⊆ Y so that the pair ( U, U ∩ Λ) is contactomorphic to ( R n − , Λ (cid:96) ), where theLegendrian Λ (cid:96) ⊆ R n − is a standard model called a loose chart . The specific geom-etry of the model Λ (cid:96) will be relevant to us soon, we define it below at Proposition2.12. One basic property of Λ (cid:96) is that dim(Λ (cid:96) ) ≥
2, therefore by definition a looseLegendrian must sit inside a contact manifold ( Y, ξ ) of dimension n − ≥ . Thefollowing theorem is the main result from [8].
Theorem 2.7.
Let f , f : Λ → ( Y, ξ ) be two Legendrian embeddings of a connectedsmooth manifold Λ , and assume they are formally Legendrian isotopic. If they areboth loose, then they are Legendrian isotopic. Through the rest of the section we explain how to generalize Theorem 2.7 toLegendrians which are not connected, smooth manifolds.
Definition 2.8.
Let Λ be a smooth connected Legendrian and let A ⊆ Y is someclosed set, possibly intersecting Λ. We say that Λ is loose rel A if there is a loosechart U ⊆ Y for Λ so that U ∩ A ⊆ U ∩ Λ.If Λ = (cid:83) i Λ i is a Legendrian complex, we say that Λ is loose if each Λ i is looserel Λ.For a simple example, if Λ i are disjointly embedded smooth submanifolds, thenΛ is just a Legendrian link with components { Λ i } . Then to say that the link Λis loose means that each component is loose, with a loose chart disjoint from theother components. We note that it is fairly easy to construct a non-loose link,whose components are individually loose.We say that two Legendrian complexes Λ , (cid:101) Λ ⊆ ( Y, ξ ) are formally Legendrianisotopic if, firstly, there is an ambient contact isotopy ϕ t : Y → Y sending an openneighborhood of the singular set of Λ to a neighborhood of the singular set of (cid:101) Λ,and secondly that there is a formal isotopy supported compactly on the interior ofall ϕ (Λ i ) which take ϕ (Λ i ) to (cid:101) Λ i . Theorem 2.9.
Let Λ , (cid:101) Λ ⊆ ( Y, ξ ) be two Legendrian complexes which are formallyLegendrian isotopic. If they are both loose, then they are ambient contact isotopic. The proof of Theorem 2.9 just follows by applying the theorem to each Λ i indi-vidually. We state a more technical version of the loose Legendrian classificationto make this clear. Theorem 2.10 ([8]) . Let f , f : Λ → ( Y, ξ ) be two Legendrian embeddings of aconnected manifold Λ , which are equal on a closed set A ⊆ Y . Suppose further that f = f on an open set U ⊆ Y , U ∩ A = ∅ , and that ( U, U ∩ f (Λ)) = ( U, U ∩ f (Λ)) is a loose chart. Suppose that there is a formal Legendrian isotopy ( g t , g s,t ) between f and f , which is supported on Y \ ( A ∪ U ) . We assume that every connectedcomponent of Λ \ A intersects U .Then there is a Legendrian isotopy f t : Λ → Y connecting f to f , with thefollowing properties. f t is supported on Y \ A , and outside of U f t is C close to g t . From this it is clear that we can work cell by cell to prove Theorem 2.9. Thoughwe did not assume that the loose charts of Λ i were necessarily equal for Λ and (cid:101) Λ,this can be arranged since all Darboux balls with smooth boundary are isotopic,via an isotopy supported away from any Legendrian disjoint from it.One important aspect of loose Legendrian complexes is that the loose charts themselves do not have to be disjoint. If Λ = Λ ∪ Λ is a Legendrian complex andthere are loose charts U , U ⊆ Y with U ∩ Λ = ∅ = U ∩ Λ , and ( U , U ∩ Λ ) ∼ =( R n − , Λ (cid:96) ) ∼ = ( U , U ∩ Λ ), then this implies that Λ is loose in the strongest sense.Even if it is not assumed that U ∩ U = ∅ , we can guarantee that there are otherloose charts U (cid:48) i ⊆ Y for Λ i satisfying the same properties as U i above (for i = 1 , U (cid:48) ∩ U (cid:48) = ∅ .Here is the proof: let (cid:101) U ⊆ Y be any set which is disjoint from Λ , U , and U , and which intersects Λ in the set { z = y i = 0 } ⊆ R n − , i.e. the standardzero section in J R n − . Any small neighborhood of a point is contactomorphicto such a set so this is immediate. Let (cid:101) Λ ⊆ Y be the Legendrian which agreeswith Λ outside of (cid:101) U , and inside (cid:101) U we replace the zero section { z = y i = 0 } withthe compactly supported loose Legendrian (formally representing the zero section). RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 11
Notice that Λ and (cid:101) Λ are both loose Legendrians and they are formally isotopic,therefore they are isotopic.So there is a contact flow ϕ t : Y → Y with ϕ (Λ ) = (cid:101) Λ . In fact since U is aloose chart for both Λ and (cid:101) Λ , we can assume that ϕ t is supported outside of theset V = U ∪ O p ( γ ) ∪ (cid:101) U , where here O p ( γ ) is the neighborhood of an arc in theinterior of Λ , chosen so that V is connected. This follows since the formal isotopybetween Λ and (cid:101) Λ happens in (cid:101) U . Notice also that ϕ t is supported outside of aneighborhood of Λ . Then let U (cid:48) = ϕ − ( (cid:101) U ), and let U (cid:48) = ϕ − ( U ). Proposition 2.11.
Let
Λ = (cid:83) i Λ i be a Legendrian complex, and let Λ k ⊆ Λ be acell with free boundary: i.e. there is an open set U around a point x ∈ ∂ Λ k so that U ∩ Λ i = ∅ for all i (cid:54) = k . Then Λ k is loose rel Λ .Proof. Let U be a small Darboux coordinate ball intersecting Λ k is the zero sectionand disjoint from all other Λ i , and let (cid:101) Λ k be the Legendrian obtained from Λ k byreplacing the portion of Λ k inside U with a Legendrian which is loose and formallyisotopic to the zero section. Certainly (cid:101) Λ is loose rel A , thus it suffices to show thatΛ k is Legendrian isotopic (rel A ) to (cid:101) Λ k .Λ k and (cid:101) Λ k are formally Legendrian isotopy, this implies in particular that theyare formally regular homotopic: i.e. there is a family of maps f t : Λ k interpolatingbetween the inclusion, and the embedding of (cid:101) Λ, and furthermore these maps arecovered by injective bundle maps F t : T Λ k → T Y with Lagrangian image. Thus,the h -principle for Legendrian immersions [5] implies that there is a family of Leg-endrian immersions g t : Λ k → Y , fixed on A , interpolating between the inclusionand the embedding of (cid:101) Λ k .If g t is generically perturbed then all double points of the maps g t : Λ k → Y occur at isolated times t ∈ [0 , k . Let x , . . . x m ∈ Λ k be alist of these points (ignoring the times the double points occur). There is a smoothisotopy through inclusions ϕ t : Λ k → Λ k which is fixed on A ∩ Λ k , and so that x , . . . x m / ∈ ϕ (Λ k ). Here we use the fact that ∂ Λ k \ A (cid:54) = ∅ .Any smooth isotopy of inclusions ϕ t : Λ k → Λ k is realized by a ambient contactisotopy, acting on any contact manifold where Λ k lies as a Legendrian. Thus we canfind a contact isotopy ϕ t : Y → Y , so that ϕ (Λ) ⊆ Λ contains none of the points x , . . . , x m . Define a Legendrian isotopy as follows: first let the contact isotopy ϕ t act from t ∈ [0 , g t ◦ ϕ : Λ k → Y whichconsist of Legendrian embeddings, then finally concatenate with the reverse flow of ϕ t . (cid:3) Notice that this proposition shows that any Legendrian with free boundary sat-isfies an h -principle, even rel boundary . For instance, if we take the unit disk D n − ⊆ R n − = { z = p = 0 } ⊆ R n − , Proposition 2.11 implies that there is a loosechart U ⊆ R n − for D n − with U ∩ ∂D n − = ∅ . Then the h -principle for loose Leg-endrians, applied rel ∂D n − , classifies all Legendrians equal to D n − near a neigh-borhood of the boundary. This cannot be done for example for D n − ⊆ J D n − ,because this Legendrian is not loose.Finally, we will need a concrete definition of a loose chart, in order to understandthe relationship to arboreal singularities in later sections. There are many possibledefinitions, we will use one that is useful for our purpose. Figure 4.
The Legendrian zig-zag (cid:101) Λ ⊆ R . Proposition 2.12 ([8, 1]) . Let (cid:101) Λ ⊆ R be the standard Legendrian zig-zag asgiven in Figure 4. Consider the Legendrian Λ (cid:96) = (cid:101) Λ × D n − ⊆ R × T ∗ D n − .Then ( U, Λ (cid:96) ) is loose. Remark 2.13.
Though Proposition 2.12 is not the original definition of a loosechart, it can be taken as the definition: the pair ( U, Λ (cid:96) ) contains a loose chart(according to the original definition of loose chart), and also any loose chart containsa contactomorphic copy of ( U, Λ (cid:96) ). Since looseness of any Legendrian is defined bycontaining a contactomorphic copy of a loose chart, it follows that a Legendrian isloose if and only if it contains a contactomorphic embedding of the pair ( U, Λ (cid:96) ). Inthis sense it is reasonable to define a loose chart to be the pair ( U, Λ (cid:96) ). This is theperspective we will take throughout the paper.However, while Proposition 2.12 will be important for us as our one touchstoneof a specific Legendrian which is loose, it will not be important to us that a loosechart contains the model ( U, Λ (cid:96) ). Ultimately it is less important what the modelfor a loose chart is , compared to what it does (i.e. Theorem 2.10).3. Pruning
We now focus on arboreal links Λ T in the particular case where T = A n +1 islinear. First we analyze the combinatorial structure of Λ A n +1 . Throughout we workwith open cells C = Int( C ) ⊆ Λ A n +1 .As defined in Section 2.1, the front of the arboreal link Λ A n +1 is defined as aunion of disks D n − v = { z = (cid:80) w ≤ v χ ( r w ) } . In particular, whenever v > v inthe tree T , we see that in the front projection the interior of D n − v is disjoint from D n − v , since the z coordinate is strictly larger everywhere. Since T = A n +1 is linear,this implies that all disks D n − v have disjoint interiors.Thus we see that R n \ π (Λ A n +1 ) has exactly n +1 bounded components: the initialflying saucer π ( S n − ) ⊆ π (Λ A n +1 ) has one bounded component in its complement,and each disk D n − v divides a single component in two. More concretely, since each π ( D n − v ) is attached on top of π ( S n − ) and the previous π ( D n − w ), there is a uniquecomponent U ⊆ R n \ π (Λ A n +1 ) lying below π ( D n − v ) and containing it in its closure.We denote this component by U v − , where v − A n +1 preceding v .In particular, the root v ∈ A n +1 is associated to the bounded component U v lyingbelow every π ( D n − w ). If v ∈ A n +1 is the maximal element, we define U v to be the RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 13 component of R n \ π (Λ A n +1 ) lying above each π ( D n − w ), i.e. the unique componentwhose closure intersects the upper hemisphere of π ( S n − ). See Figure 5. Figure 5.
The open sets U j ⊆ R n . This example corresponds to A = { v → v → v } ; recall that v , the root of A is distinctfrom 0, the initial object of Q .The unbounded component of R n \ π (Λ A n +1 ) will also be important for ourpurposes. Thus we define Q to be the quiver obtained by appending an initialobject 0 to A n +1 . Thus Q = A n +2 , though we avoid this notation to avoid confusionwith the Legendrian in one larger dimension. We define U ⊆ R n \ π (Λ A n +1 ) tobe the unbounded component, and thus the components of R n \ π (Λ A n +1 ) are incorrespondence with the vertices of Q . We claim that the top dimensional cells ofΛ A n +1 are naturally in bijective correspondence with all morphisms of Q . Moregenerally: Lemma 3.1.
Let ≤ k ≤ n + 1 , and let { v , . . . , v k } be any collection of verticesin Q . Then U v ∩ . . . ∩ U v k is the image of the closure of a single cell of Λ A n +1 ,whose codimension in Λ A n +1 is k − . (Here U v denotes the point-set closure of U v .) Every cell of Λ A n +1 arises uniquely in this way, and thus the m -cells of Λ A n +1 are in correspondence with ( n − m + 1) -element subsets of vertices of Q .Proof. If C ⊆ Λ A n +1 is any m -cell, we can define V C = { v ∈ Q ; π ( C ) ⊆ U v } . Thenit is clear that π ( C ) ⊆ (cid:84) v ∈ V C U v . Thus to prove the entire lemma it suffices toshow the reverse inclusion.We have either C ⊆ Int( D n − w ) for some w , or else C ⊆ S n − ⊆ Λ A n +1 . Inthe former case π ( D n − w ) ⊆ U w − , and w − ∈ V C is the minimal element in V C ,by the definition of U w − . In the latter case π ( S n − ) ⊆ U , and again 0 ∈ V C is the minimal element of V C . Since U w − ∩ U v ⊆ π ( D n − w ) for any v > w , itfollows that (cid:84) v ∈ V C U v ⊆ π ( D n − w ). Again addressing the latter case we have that U ∩ π (Λ A n +1 ) = π ( S n − ), and so (cid:84) v ∈ V C U v ⊆ π ( S n − ). Let D = D n − w or D = S n − , according to the cases above so that C ⊆ D . Let x ∈ D ⊆ Λ A n +1 be any point not in C . Then there is a u ∈ A n +1 (not the root)so that ∂D n − u ∩ D separates C from x . The two components of π ( D \ ∂D n − u ) are U u − ∩ D and D \ U u − . Thus π ( C ) ⊆ U u − if and only if π ( x ) / ∈ U u − , and so itfollows that π ( x ) / ∈ (cid:84) v ∈ V C U v . (cid:3) The next lemma shows that the above identification is natural with respect tothe isomorphism Sh(Λ A n +1 ) ∼ = M od ( Q ) from [9]. Recall that M od ( Q ) is defined tobe the derived category of modules sending 0 ∈ Q to 0 ∈ Ch ∗ . Lemma 3.2.
There is an equivalence of categories N : Sh(Λ A n +1 ) → M od ( Q ) satisfying the following property. Let V ⊆ Q be any full subcategory and let U V =Int( (cid:83) v ∈ V U v ) . Then for any two F , F ∈ Sh(Λ A n +1 ) , F | U V ∼ = F | U V if and onlyif N ( F ) | V ∼ = N ( F ) | V .Proof. We start by defining the functor N , so let F ∈
Sh(Λ). For each object v ∈ Q , we define N ( F )( v ) to be the chain complex F ( U v ). For each morphism ϕ : v → v , choose a point x ϕ in the interior of the cell U v ∩ U v . Let U ϕ be asmall neighborhood of the poiont x ϕ , and consider the restriction maps F ( U ϕ ) →F ( U ϕ ∩ U v ) and F ( U ϕ ) → F ( U ϕ ∩ U v ). The former map is necessarily a quasi-isomorphism: its cone is the Morse group of the pair ( x ϕ , f ) for some function f : U ϕ → R which is positive on U ϕ ∩ U v and negative on U ϕ ∩ U v . Since U v lies below U v this means that df x ϕ evaluates positively on ∂ z , and thus thepoint ( x ϕ , P ( df x ϕ )) ∈ S ∗ R n is not contained in R n − = S ∗− R n . In particularthis point is not contained in Λ A n +1 , so it is not in the singular support of F , so F ( U ϕ ) → F ( U ϕ ∩ U v ) is a quasi-isomorphism.Furthermore since π (Λ A n +1 ) is disjoint from U v and U v , the restriction maps F ( U v ) → F ( U ϕ ∩ U v ) and F ( U v ) → F ( U ϕ ∩ U v ) are quasi-isomorphisms. Thuswe have a map (in the derived category) F ( U v ) → F ( U ϕ ∩ U v ) → F ( U ϕ ) → F ( U ϕ ∩ U v ) → F ( U v ) , where the second and fourth maps are inverted quasi-isomorphisms. This defines N ( F )( ϕ ) ∈ Mor( M od ( Q )). To see that N ( F ) respects composition we choosemorphisms ϕ : v → v and ψ : v → v in Q , and choose a point x in the interiorof the cell U v ∩ U v ∩ U v . A neighborhood U x of x in R n is contactomorphic tothe standard trivalent Legendrian front π (Λ ) ⊆ R , extended trivially by R n − .See Figure 6.The analysis of F near the point x ϕ,ψ is similar to the analysis of the cuspsingularity done in [11, Section 3.3]. The combinatorics of the restriction maps isexpressed in the commutative diagram: U v U ψ U v U ψ ◦ ϕ U x U ϕ U v RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 15
Figure 6.
A trivalent vertex, trivially extended.The three downward arrows F ( U ψ ) → F ( U v ), F ( U ϕ ) → F ( U v ) and F ( U ψ ◦ ϕ ) →F ( U v ) must be quasi-isomorphisms, this is the same as the argument above. Since N ( F )( ψ ◦ ϕ ) is the morphism F ( U v ) → F ( U v ) obtained by following the lefthand side of the diagram, and N ( F )( ψ ) ◦ N ( F )( ϕ ) is obtained by following theright hand side, we see that this maps are equal and so N ( F ) is a module. N is a functor in the obvious way: a map between sheaves F → F consistsin particular of maps of chain complexes F ( U v ) → F ( U v ) for all v , since thesemaps commute with all restrictions this defines a map between modules of Q . N is a faithful functor since every stalk of F is naturally quasi-isomorphic to F ( U v )for some v (this holds for any Legendrian in S ∗− R n ).Let M ∈ M od ( Q ). We show that there is a sheaf F M ∈ Sh(Λ A n +1 ) with N ( F M ) = M . Besides showing that N is essentially surjective, the constructionwill be natural and thus show that N is a full functor. It suffices to construct F M in the case where M = Y v = R Mor( v, · ) is the Yoneda module of v , a chosen vertex. Let U ≥ v = (cid:83) w ≥ v U w , and let (cid:101) U ≥ v ⊆ U ≥ v denote the complement of the “bottomboundary”: a point x ∈ ∂U ≥ v is also in (cid:101) U ≥ v unless there is a function f definednear x which is positive on Int( U ≥ v ), negative outside U ≥ v , and satisfying ∂f∂z > R is the constant sheaf on (cid:101) U ≥ v and F = ι ! ( R ), where ι : (cid:101) U ≥ v → R n is the Here we are using the fact that Sh(Λ) consists of perfect sheaves, and M od ( Q ) consists ofperfect modules. In fact in this case the situation is so explicit that this is not necessary: arbitrary C –valued sheaves with singular support in Λ A n +1 are equivalent to Func(Q , C ). But since perfectderived modules are the most relevant structure in symplectic geometry we take advantage of thestructure, since it involves less analysis of the singularities of Λ A n +1 . inclusion map, then the singular support of F M is contained in Λ A n +1 , and clearly N ( F ) = Y v .The statement that N is an equivalence when intertwined with restrictions V ⊆ Q and U V ⊆ R n is obvious from the definition. (cid:3) Let W ⊆ Mor( Q ) be an arbitrary set, which corresponds to some collection oftop-dimensional cells as in Lemma 3.1. Recall that Λ Q [ W − ] is defined to be theLegendrian obtained from Λ A n +1 by deleting a small open ball from the interior ofeach cell corresponding to a morphism of W . Proof of Proposition 1.3:
Since Λ Q [ W − ] ⊆ Λ A n +1 , we have the inclusion functorSh(Λ Q [ W − ] ) → Sh(Λ A n +1 ) which is fully faithful. For a morphism f : v → v define the set U f = Int( U v ∪ U v ), so that pair ( U f , Λ A n +1 ) is diffeomorphicto ( R n , R n − × { } ). If f ∈ W , then U f \ Λ Q [ W − ] is connected, and so any F ∈
Sh(Λ Q [ W − ] ) must be locally constant on U f . Thus N ( F )( f ) is a quasi-isomorphism, and so N : Sh(Λ Q [ W − ] ) → M od ( Q ) factors through M od ( Q [ W − ]).It remains to show that N : Sh(Λ Q [ W − ] ) → M od ( Q [ W − ]) is essentially sur-jective. If F ∈
Sh(Λ A n +1 is any sheaf with N ( F ) ∈ M od ( Q [ W − ]), then for all f ∈ W f : v → v , we have that N ( F ) | { v ,v } is quasi-isomorphic to a con-stant module, and therefore F U f is quasi-isomorphic to a constant sheaf. Thus thesingular support of F is disjoint from U f , and therefore F ∈
Sh(Λ Q [ W − ] ) sinceΛ Q [ W − ] = Λ A n +1 outside of such U f . (cid:3) Proof of Proposition 1.2:
Let Λ ⊆ Λ A n +1 be any closed set, and let W ⊆ Mor( Q )consist of those morphisms f ∈ Mor( Q ) corresponding to those top-dimensionalcells D f so that D f ∩ (Λ A n +1 \ Λ) (cid:54) = ∅ .First, assume that Λ contains the entire singular set of Λ A n +1 it its interior, i.e. allcells of codimension at least 1 are contained in Int(Λ). Then the same proof as aboveshows that the fully faithful image of Sh(Λ) under N is exactly the subcategory M od ( Q [ W − ]): the inclusion Λ ⊆ Λ Q [ W − ] shows that N (Sh(Λ)) ⊆ M od ( Q [ W − ])and the reverse inclusion follows since Λ = Λ A n +1 outside of (cid:83) f ∈ W U f . The in-clusion Λ ⊆ Λ Q [ W − ] shows that every cell which is loose rel Λ Q [ W − ] is also looserel Λ. Since the interior of every top-dimensional cell is diffeomorphic to D n − ,we can choose an ambient contact isotopy ϕ t so that ϕ t (Λ Q [ W − ] ) ⊆ Λ Q [ W − ] and ϕ (Λ Q [ W − ] ) ⊆ Λ: ϕ t is the map which expands the punctures of Λ Q [ W − ] radiallyuntil the boundary of the punctures lie in a small neighborhood of the singular set.Then any cell which is loose rel Λ is also loose rel ϕ (Λ Q [ W − ] ) and therefore alsoloose rel Λ Q [ W − ] . This completes the proof under the stated assumption.Finally, let V ⊆ Λ A n +1 be the closure of a small neighborhood of the singularset, and for arbitrary Λ ⊆ Λ A n +1 let Λ (cid:48) = Λ ∪ V . Then V \ Λ retracts, via contactisotopy fixed on Λ, to an arbitrarily small neighborhood of a subcritical isotropiccomplex. In particular any loose chart which is disjoint from Λ can be made disjointfrom Λ (cid:48) , via contact isotopy fixed on Λ. Also, any sheaf whose singular support iscontained in Λ (cid:48) in fact has singular support contained in Λ. Thus the assumptionin the previous paragraph loses no generality. (cid:3)
RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 17 Localization of quivers
In this section we give a proof of Proposition 1.4. The following is clearly anecessary condition.
Lemma 4.1.
Let W ⊆ Mor( Q ) . Then for any f ∈ W , the image of f under thefunctor Q → Q [ W − ] is an isomorphism only if f ∈ W . In fact the lemma immediately implies Proposition 1.4: if ρ ( f ) is a quasi-isomorphism for all ρ : Q [ W − ] → Ch ∗ the Yoneda lemma implies that f ∈ Q [ W − ]is an isomorphism, by which we conclude f ∈ W from the lemma. The con-verse follows immediately from the definition of 2-out-of-6 closure: if in the com-position a f → b g → c h → d we know that gf and hg are isomorphisms, then g − = f ( gf ) − = ( hg ) − h is a two-sided inverse for g , by which it immediatelyfollows that f , h , and hgf are isomorphisms as well. Proof of Lemma 4.1: If W admitted a calculus of fractions this would be a knownresult [7, 7.1.20]. Since it does not, in order to establish the result we will have toconstruct a model for Q [ W − ]. It is clear that Q [ W − ] = Q [ W − ], so without lossof generality we assume that W = W .This model is as follows: for any a, b ∈ Q , we define Mor( a, b ) to consist ofequivalence classes of diagrams a f → m w ← m g → b , where w ∈ W and f, g arearbitrary morphisms. We denote this morphism by the formal expression gw − f .The equivalence relation is defined by the two horizontal morphisms in the diagrambeing equivalent:(4.1) a m m ba m (cid:48) m (cid:48) b f a x w gx b f (cid:48) w (cid:48) g (cid:48) where w, w (cid:48) ∈ W and all other morphisms are arbitrary. That is, whenever xw = w (cid:48) x , we have ( g (cid:48) x ) w − f = g ( w (cid:48) ) − ( xf ), and in particular g − m f = g (cid:48) − m (cid:48) f (cid:48) whenever g (cid:48) f (cid:48) = gf .Given morphisms a f → m w ← m g → b and b h → n u ← n k → c we need to definethe composition ku − h ◦ gw − f ∈ Mor( a, c ). If y ∈ Mor( m , n ), then hg = uy in Q (since Mor( m , n ) has at most one element), and therefore we define ku − h ◦ gw − f = ( ky ) w − f . Similarly if y ∈ Mor( m, n ) we define ku − h ◦ gw − f = ku − ( yf ). If both y and y exist, the definition is unambiguous because uy = yw ∈ Mor( m , n ). If neither exist, then since Q is linear we must have bothMor( n , m ) and Mor( n, m ) are non-empty, and thus we have a diagram of theform n z → m hg → n z → m . Since hgz = u ∈ W and zhg = w ∈ W , andsince W satisfies the 2-out-of-6 property, it follows that zhgz ∈ W and we define ku − h ◦ gw − f = k ( zhgz ) − f .One easily sees that this composition is well defined on equivalence classes bydoing a case analysis.This defines a category Q [ W − ] with a natural functor Q → Q [ W − ] sending f ∈ Mor( a, b ) to f − a a = 1 b − b f , and whenever f ∈ W we have an inverse in Q [ W − ] given by 1 b f − a . It remains to show that the only f ∈ Mor( Q ) which are sent to isomorphisms are already in W (this also establishes the universal propertyof the localization Q [ W − ], though this is essentially obvious).First we make an auxiliary claim, that the morphism a f → m w ← m g → a isequivalent to the identity only if f, g ∈ W . We show that this property is preservedunder the equivalence given in Diagram 4.1 when a = b . First we suppose z ∈ Mor( m (cid:48) , m ), then m x → m (cid:48) z → m x → m (cid:48) is a 2-out-of-6 diagram with zx = w ∈ W and xz = w (cid:48) ∈ W , so x, x ∈ W . Since f (cid:48) = xf and g = g (cid:48) x , bythe 2-out-of-3 property we see that if either f, g ∈ W or f (cid:48) , g (cid:48) ∈ W , then bothhold. Instead, supposing Mor( m (cid:48) , m ) = ∅ implies that Mor( m, m (cid:48) ) (cid:54) = ∅ and solet z (cid:48) ∈ Mor( m, m (cid:48) ). But then we have a diagram a f → m z (cid:48) → m (cid:48) g (cid:48) → a , so bylinearity of Q we have m = m (cid:48) = a and f = g (cid:48) = 1 a . Thus g = w and f (cid:48) = w (cid:48) , so f, g, f (cid:48) , g (cid:48) ∈ W .Having established the claim, suppose that f ∈ Mor( a, b ) and it has an inverse ku − h in Mor( Q [ W − ]), so ( f k ) u − h = 1 b and ku − ( hf ) = 1 a . The claim impliesthat f k, h, k, hf ∈ W , and thus the 2-out-of-3 property ensures f ∈ W . (cid:3) Criteria for looseness
In this section we present the proof of Proposition 1.5. First we prove a basiclemma which is a geometric model for the 2-out-of-6 property. We let Λ ⊆ R be the Legendrian 1-complex given in Figure 7, this is essentially Λ A except in“long knot” format. Notice in particular that the category (cid:102) Sh(Λ ) consisting ofconstructable sheaves with singular support on Λ and no conditions on the support is equivalent to modules of the category a f → b g → c h → d . First, we show thatany algebraic subcategory can be represented by a geometric slice. Recall that π : R n − → R n is the front projection. Lemma 5.1.
Let
C ⊆ Q be any full subcategory which is equivalent to the cat-egory A = a f → b g → c h → d . Then there is a set U ⊆ R n , so that the pair ( π − ( U ) , π − ( U ) ∩ Λ A n +1 ) is contactomorphic to ( R × T ∗ D n − , Λ × D n − ) , andso that the restriction of constructable sheaves from R n to U realizes the restrictionof modules from Q to C .Proof. This is essentially a corollary of Lemma 3.2, but to be explicit we will proveit more geometrically. Let B n − v ⊆ R n − be the ball in the standard Venn diagramcorresponding to v , where v is any vertex in A n +1 excepting the root. Thus for v ∈ Q , the ball B n − v +1 is defined in the obvious way unless v = 0, or v ∈ Q is themaximal element. In particular B n − b +1 and B n − c +1 definitely exist.We choose a compact arc γ ⊆ R n − as follows. Firstly we require γ is completelydisjoint from B n − v +1 , whenever v is not equal to a, b, c, or d . γ will be completelycontained in B n − a +1 ∩ B n − d +1 as long as both of these balls are defined, if not we require γ to be contained in B n − a +1 or B n − d +1 if only one of these is defined (or else imposenothing if a = 0 and d is the maximum). The first endpoint of γ is required to beoutside B n − b +1 and B n − c +1 . As we follow γ , it is required to enter B n − b +1 , then enter B n − c +1 , then exit B n − b +1 , then exit B n − c +1 . See Figure 8.In R n , the 2-plane γ × R intersects π (Λ A n +1 transversely. Following from thedefinition of Λ A n +1 , we see that the intersection is diffeomorphic as a pair to π (Λ ) ⊆ R , with some additional disjoint arcs lying completely above or below RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 19
Figure 7.
The front π (Λ ). Figure 8.
Possible examples for the arc γ , pictured here in thecase n = 3. Here γ is the example corresponding to the subcate-gory v → v → v → v , and γ corresponds to the subcategory0 → v → v → v . Notice that every such γ is a open set of asmall linking circle of a codimension 2 singularity, as Lemma 3.2would predict.the diffeomorphic copy of π (Λ ). Namely there are additional intersections: a slice of the upper hemisphere of π ( S n − ), a slice of π ( D n − d +1 ) if d is not the maximum,and a slice of the lower hemisphere of π ( S n − ), which is a trivial arc whenever a (cid:54) = 0 (if a = 0 the lower hemisphere of π ( S n − ) forms a portion of our copy ofΛ ). Since all these additional arcs lie above or below the diagram, we can chosea compact piece of 2-plane P ⊆ γ × R so that the only intersection is the copy of π (Λ )Since the intersection is transverse we have a tubular neighborhood (cid:101) P ⊆ R n of P so that (cid:101) P ∩ π (Λ A n +1 ) ∼ = π (Λ × B n − ). Then U = π − ( (cid:101) P ) is the desiredneighborhood. (cid:3) Let Λ ⊆ Λ be the Legendrian 1-complex obtained by puncturing the two edgescorresponding to the morphisms gf and hg . Lemma 5.2. Λ × D n − ⊆ R × T ∗ D n − is loose.Proof. Throughout this proof all isotopies will be with compact support. If insteadof Λ we started with the standard zig-zag (cid:101) Λ ⊆ R (the front of a smooth Legen-drian curve), this is Proposition 2.12. Let Λ = Λ × D n − and (cid:101) Λ = (cid:101) Λ × D n − ⊆ R × T ∗ D n − , which are equal outside of a neighborhood of the codimensional 1stratum of Λ. Let Λ (cid:96) ⊆ R n − be a (smooth) loose Legendrian which is formallyisotopic to the standard zero section, and equal to it outside a compact set. Let (cid:101) Λ be the Legendrian built from (cid:101) Λ by implanting four small copies of Λ (cid:96) , near pointscorresponding to the top-dimensional cells of Λ. See Figure 9. Since both (cid:101)
Λ and (cid:101) Λ are loose Legendrians, there is a Legendrian isotopy taking (cid:101) Λ to (cid:101) Λ . Since the h -principle for loose Legendrians is for parametrized Legendrians, we can assume thatthe isotopy at time 1 fixes (cid:101) Λ pointwise everywhere it fixes it setwise, i.e. outside ofthe implanted copies of Λ (cid:96) .Let ϕ t : R × T ∗ D n − be an extension of this isotopy to an ambient contactisotopy. Λ = (cid:101) Λ outside of an open set U ⊆ R × T ∗ D n − , so that ϕ is the identityon U ∩ (cid:101) Λ. By further ambient isotopy if needed, we can assume that every point ofΛ within the support of ϕ t is ε -close to a point of (cid:101) Λ. Then, since ϕ is continuous,we see that for every point x ∈ Λ ∩ U , ϕ ( x ) is arbitrarily close to x . In particular ϕ (Λ) = ϕ ( (cid:101) Λ) = (cid:101) Λ outside of a small open neighborhood of the codimension 1stratum. In particular the implanted copies of Λ (cid:96) are embedded in ϕ (Λ). Thuseach of the four top-dimensional cells in ϕ (Λ) has a loose chart, disjoint from theother cells. Since ϕ t is a contact isotopy these loose charts also exist in Λ. (cid:3) Finally we prove a lemma that allows us to set up an inductive proof.
Lemma 5.3.
Let Λ be a Legendrian complex (in any contact manifold), and let C (Λ) be the collection of top-dimensional cells of Λ . Let C , C ⊆ C (Λ) be twodisjoint collections. Then, if every cell in C is loose rel Λ , and every cell in C isloose rel Λ \ C , then every cell in C ∪ C is loose rel Λ .Proof. For each D i ∈ C , let U i be a loose chart for D i , which only intersects cellsin C (besides D i itself). Since U i is a ball with smooth boundary, there is noobstruction to finding a smooth isotopy of the singular set of Λ, so that after theisotopy the singular set of Λ is is disjoint from all U i . The singular set is itselfa subcritical isotropic complex, and the space of formal isotropic embeddings isa Serre fibration over the space of smooth embeddings, thus the h -principle for RBOREAL SINGULARITIES AND LOOSE LEGENDRIANS I 21
Figure 9.
The Legendrian ϕ (Λ), which is explicitly loose. Theisotopy ϕ t taking this Legendrian to Λ is built using the fact thatthe large scale structure of Λ , is C close to a zig-zag itself, asexplained in the proof.subcritical isotropics [5] implies that there is a isotopy through isotropics disjoiningthe singular set of Λ from U i . We can then use the isotopy extension theorem tofind a contact isotopy Y → Y which is fixed on Λ \ C , so that after the isotopyeach U i is disjoint from the singular set of Λ. Thus we can assume in the proof thateach U i is disjoint from the singular set of Λ.Let D ∈ C be a cell with loose chart U D , which is disjoint from Λ \ D . Eachloose chart U i contains two disjointly embedded loose charts [8], thus we can assumethat there is a Darboux ball (cid:101) U D which is disjoint from Λ \ D , U D and all U i , andintersects D in the standard plane. We define (cid:101) D to be the Legendrian which isequal to D outside (cid:101) U D , and inside (cid:101) U D (cid:101) D is equal to a loose Legendrian plane whichis formally isotopic to the standard plane and equal to it outside a compact set.Then (cid:101) D is formally isotopic to D via a formal isotopy supported in (cid:101) U D , and both (cid:101) D and D have the same loose chart, U D . Thus D is isotopic to (cid:101) D , via an isotopywhich fixed on Λ \ D . Furthermore, (cid:101) D admits a loose chart which is disjoint fromΛ \ D and also disjoint from all U i , namely the loose chart contained in (cid:101) U D .We now discard the notation involving the tildes and assume that D admits aloose chart U D which is disjoint from Λ \ D and also from each U i , the argumentabove shows that this is no loss of generality.Because U i ⊆ Y is a ball with smooth boundary, there is no obstruction to findinga smooth isotopy, compactly supported on the interior of D \ U D , which disjoins D from all U i . Using the Serre fibration property for formal Legendrian embeddingsover smooth embeddings we see that there is a formal Legendrian isotopy of D with the same property. Since D is loose, Theorem 2.10 implies that there is a Legendrian isotopy ϕ t : D → Y which is compactly supported on the interior of D ,and which is C close to the formal Legendrian isotopy outside of U D . In particular, ϕ ( D ) is disjoint from all U i . Then U (cid:48) i = ϕ − ( U i ) is a loose chart for D i which isdisjoint from D , and also disjoint from any cells of Λ which U i was already disjointfrom.We then apply this argument iteratively to each cell in C , resulting in loosecharts for each D i which are disjoint from all cells in C . (cid:3) Proof of Proposition 1.5:
Let W ⊆ Mor( Q ) be the union of W and the identities.We define W i +1 ⊆ Mor( Q ) as follows: for any composition a f → b g → c h → d in Q sothat gf, hg ∈ W i , then f, g, h, gf, hg, hgf ∈ W i +1 . Then W i +1 ⊇ W i , and whenever W i +1 = W i we see that W i = W . Since Mor( Q ) is finite this happens for somefinite i . Let C i ⊆ C (Λ A n +1 ) consist of all top-dimensional cells so that D ∈ C i exactly when the corresponding morphism f D ∈ Mor( Q ) is contained in W i . Weprove by induction that all cells in C i are loose rel Λ Q [ W − ] .For i = 0 this is just Proposition 2.11. Suppose D ⊆ C i +1 , then Lemma 5.1implies there is a set U ⊆ R n and a contactomorphism ( π − ( U ) , π − ( U ) ∩ Λ A n +1 ) ∼ =( R × T ∗ D n − , Λ × D n − ), so that D is the cell corresponding to one of the themorphisms f, g, h , or hgf in the composition a f → b g → c h → d , and the cellscorresponding to gf and hg are contained in C i . Lemma 5.2 implies that D has aloose chart in π − ( U ), which possibly intersects the cells corresponding to gf and hg but no others. The induction hypothesis is that these cells are themselves loose,and so Lemma 5.3 implies that D is loose rel Λ Q [ W − ] . (cid:3) References [1] R. Casals and E. Murphy,
Legendrian fronts for affine varieties , arXiv:1610.06977[2] K. Cieliebak and Y. Eliashberg,
From Stein to Weinstein and Back: Symplectic Geometry ofAffine Complex Manifolds , Colloquium Publications, 59. AMS, 2012.[3] T. Ekholm, J. Etnyre, and M. Sullivan,
Non-isotopic Legendrian submanifolds in R n +1 , J.Differential Geometry, (2005), 85–128.[4] Y. Eliashberg, Topological characterization of Stein manifolds of dimension >
2, Internat. J.Math., (1990), 29–46.[5] Y. Eliashberg and N. Mishachev, Introduction to the h-Principle , Graduate Studies in Math-ematcs, 48. AMS, 2002.[6] S. Guillermou, M. Kashiwara, P. Schapira,
Sheaf quantization of Hamiltonian isotopies andapplications to non displaceability problems , Duke Math. J. 161 (2012), no. 2, 201—245.[7] M. Kashiwara, P. Schapira,
Categories and sheaves , Grundlehren der MathematischenWissenschaften, 332. Springer-Verlag, Berlin, 2006.[8] E. Murphy,
Loose Legendrian embeddings in high dimensional contact manifolds , preprint,arXiv:1201.2245[9] D. Nadler,
Arboreal singularities , Geom. Topol. 21 (2017), no. 2, 1231—1274.[10] D. Nadler,
Non-characteristic expansions of Legendrian singularities , preprint,arXiv:1507.01513[11] V. Shende, D. Treumann, E. Zaslow,
Legendrian knots and constructible sheaves , Invent.Math. 207 (2017), no. 3, 1031-–1133.[12] L. Starkston,
Arboreal singularities in Weinstein skeleta
Selecta Math. (N.S.) 24 (2018), no.5, 4105-–4140.
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