LLAGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY
JEFF HICKS
Abstract.
Lagrangian surgery and Lagrangian cobordism give geometric interpretationsto exact triangles in Floer cohomology. Lagrangian k -surgery modifies an immersed La-grangian submanifold by topological k -surgery while removing a self-intersection pointof the immersion. Associated to a k -surgery is a Lagrangian surgery trace cobordism.We prove that every Lagrangian cobordism is exactly homotopic to a concatenation ofsuspension cobordisms and Lagrangian surgery traces. Furthermore, we show that eachLagrangian surgery trace bounds a holomorphic teardrop pairing the Morse cochain asso-ciated to the handle attachment with the Floer cochain generated by the self-intersection.We give a sample computation for how these decompositions can be used to algorithmicallyconstruct bounding cochains for Lagrangian submanifolds, recover the Lagrangian surgeryexact sequence, and provide conditions for when non-monotone Lagrangian cobordismsyield continuation maps in the Fukaya category. Contents
1. Introduction 12. Background 73. Lagrangian cobordisms are Lagrangian surgeries 114. Teardrops on Lagrangian cobordisms 315. Speculation on Lagrangian cobordisms with immersed ends 456. Computations and applications 53References 601.
Introduction
Since the introduction of Lagrangian intersection Floer cohomology [Flo+88], the tech-niques used to study the geometry of Lagrangian submanifolds L in a symplectic manifold X have grown increasingly sophisticated. Tools which construct new Lagrangian subman-ifolds and give geometric interpretations of Floer theoretic invariants, such as Lagrangianconnect sum, surgery, and cobordism, are particularly useful for applications of Lagrangiansubmanifolds and computation [Pol91; Fuk+07; BC13; Hau20; NT20]. The goal of this pa-per is to show that all Lagrangian submanifolds K ⊂ X × C can be built from Lagrangiansubmanifolds in X via Lagrangian surgery operations. We provide an extension of Fukaya,Oh, Ohta, and Ono’s connect sum exact sequence to Lagrangian k -surgery. As an appli-cation, we show that lowest order contributions to the differential and product structuresof the Floer cohomology of a Lagrangian submanifold K ⊂ X × C can be algorithmicallyrecovered from this decomposition. a r X i v : . [ m a t h . S G ] F e b JEFF HICKS
Context and Previous Work.
Floer Cohomology.
The Floer complex of a Lagrangian submanifold CF • ( L ) is a defor-mation of the cochain group C • ( L ) by incorporating the counts of holomorphic disks withboundary on L into the differential and product structures of C • ( L ). The underlying cochaingroup C • ( L ) can be singular cochains, Morse cochains, or differential forms [Fuk+10; Oh95;ST20]; in this paper, we use Morse cochains CM • ( L ). Various incarnations of this Floer co-homology theory CF • ( L ) have been developed for different applications: a non-exhaustivelist of related constructions include pearly Floer cohomology, cluster complex, or openGromov-Witten algebra of a Lagrangian L [CL06; BC08; LW14; CW19]. The deforma-tion enhances CF • ( L ) with a filtered A ∞ structure counting configurations of holomorphicpolygons with boundary on L . The algebra structure of CF • ( L ) contains an abundance ofdata about the Lagrangian L ; most importantly, if CF • ( L ) is an uncurved- A ∞ algebra, thehomology groups HF • ( L ) are invariant under Hamiltonian isotopy. The non-vanishing of HF • ( L ) is famously related to the non-displaceability of a Lagrangian submanifold L . Inpractice, both the construction and computation of CF • ( L ) are difficult. As evidence, in allknown examples where we have computed the Floer cohomology for connected embeddedmonotone Lagrangian submanifold, this deformation either: • does not modify the Morse differential substantially enough to change the cohomol-ogy so that the Floer cohomology HF • ( L ) (cid:39) H • ( L ) is “wide” or, • modifies the differential so much as to make the Floer complex acyclic, HF • ( L ) = 0,in which case we say the Floer cohomology is “narrow”.It would be especially striking if all monotone Lagrangians are wide or narrow, but con-structing a counterexample requires both tools for building interesting Lagrangian subman-ifolds, and subsequently computing their Floer cohomology.The Floer cohomology of all Lagrangian submanifolds L ⊂ X can be packaged into asingle structure, the Fukaya category of X . The Fukaya category is a filtered A ∞ category,whose objects are Lagrangian submanifolds L . There are a variety of flavors of this categorybut a common feature is that the morphisms between distinct Lagrangians L and L areLagrangian intersection Floer cochains CF • ( L , L ), and that the product structures inthe category come from counting configurations of holomorphic polygons with boundaryon the Lagrangian submanifolds. The relation to the Lagrangian Floer complex of a singleLagrangian comes from taking the automorphisms of an object, CF • ( L, L ), which is eitherdefined as or is shown to be equivalent to the Floer complex CF • ( L ). Lagrangian Cobordisms.
Even when CF • ( L ) is a curved A ∞ algebra, we expect the filtered A ∞ homotopy type of CF • ( L ) to be invariant under Hamiltonian isotopy. In [BC13], Biranand Cornea showed that a weaker form of equivalence, Lagrangian cobordance, also pre-serves the Lagrangian Floer cohomology with appropriate conditions on the Lagrangiancobordism. A Lagrangian cobordism (introduced in [Arn80]) K : L + (cid:32) L − is a La-grangian submanifold K ⊂ X × C with “ends” given by Lagrangians L + , L − ⊂ X (seedefinition 2.1.1) . Given K : L + (cid:32) L − an embedded monotone Lagrangian cobordism, The term Lagrangian cobordism is used to describe two different but related types of Lagrangian sub-manifold in the symplectic geometry literature. In contact geometry, the terminology refers to Lagrangiansubmanifolds inside the symplectization Y × R of a contact manifold Y with ends limiting to Legendrian AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 3 ( q − → q + ) e e ( q − → q + ) T A C CF ( K , ) CF ( K , ) Figure 1.
An immersed null cobordism K , ⊂ C × C of the 0-dimensionalWhitney sphere S ⊂ C . The patterned region represents a holomorphicteardrop pairing Morse cochain with self-intersection.Biran and Cornea show that L + , L − are equivalent objects in the Fukaya category. Inparticular, CF • ( L + ) and CF • ( L − ) are homotopic chain complexes.There is not an algorithm to compute the homotopy equivalence from the data of K ,however, the existence of this homotopy equivalence is enough for most applications. Forinstance, from their construction, it follows that any 2-ended monotone Lagrangian cobor-dism with H • ( L + ) (cid:54) = H • ( L − ) would produce a Lagrangian submanifold which is neitherwide-nor-narrow. This brings us to the first goal of this article, which is to better under-stand the geometry of Lagrangian cobordisms. Our current set of examples is small: theonly monotone embedded 2-ended Lagrangian cobordisms we know are those Hamiltonianisotopic to L − × R ⊂ X × C . Lagrangian Surgeries.
The largest collection of Lagrangian cobordisms come from [Hau20],which introduced a k -surgery operation on Lagrangian submanifolds and an associated k -trace cobordism. Crucially, Haug provides a geometric meaning to the surgery modeldescribed by [ALP94]. These surgeries generalize the Lagrangian connect sum operation of[Pol91]. The simplest example of such a surgery trace, the 0-surgery null-cobordism for apair of points S = { q + , q − } ⊂ C , is drawn in fig. 1. As manifolds, the ends L − , L + ofa Lagrangian surgery trace cobordism K k,n − k +1 : L + (cid:32) L − differ by topological surgery.The “height function” of the Lagrangian trace cobordism, given by projection to the realcoordinate π R : K k,n − k +1 → R , has a single critical point of index k + 1. Additionally, L − has one fewer self-intersection than L + . If K k,n − k +1 is graded, this self intersection livesin degree k + 2. In this sense, Lagrangian surgery “trades” a self-intersection in L + intoadditional topology for L − . Relations between Floer cohomology, surgery, and cobordisms.
It is understood that the geo-metric deformation of a Lagrangian submanifold via Lagrangian surgery should correspondto an algebraic deformation of the Floer cohomology of L . A particularly nice instance ofthis deformation can be understood in the Fukaya category, where it can be recast as “thePolterovich connect sum L q L of transverse Lagrangians L ∩ L = { q } is the mapping submanifolds Λ i ⊂ Y . However, when we refer to Lagrangian cobordism, we will always mean “Lagrangiancobordisms with Lagrangian ends”. JEFF HICKS cone of L and L in the Fukaya category”. This was proven in [Fuk+07] by comparing theholomorphic triangles with boundary on L , L , L to holomorphic strips with boundary onthe surgery configuration L q L , L . In [BC13], this statement is recovered by consideringthe Lagrangian surgery trace cobordism K : ( L ∪ L ) (cid:32) L q L . We wish to understandthis statement for general surgeries.In fig. 1, there exists a holomorphic teardrop (indicated by the patterned region of thediagram) which pairs the minimum e of the height function π R : K , → R with the self-intersection point. At lowest order, this holomorphic teardrop cancels out the contributionsof these generators of the Floer complex. Geometrically, this means that the LagrangianFloer cohomology of K only sees the geometry arising from the ends of the Lagrangiancobordism, and not the interior. This provides a direct computation of the results of [BC13].The full story is complicated by the possible obstructedness of K as an object of the Fukayacategory.These holomorphic teardrops are discussed in [BC20, Section 5.3], where they are implic-itly related to the [Fuk+07] surgery exact triangle. Related work of [NT20; Tan18] showsthat a surgery exact triangle in the Fukaya category arises from the Lagrangian surgerytrace without having to count holomorphic teardrops. There are also many similarities be-tween the Floer theoretic results in this paper for Lagrangian cobordisms with Lagrangianends, and results in [Cha+20] for Lagrangian cobordisms with Legendrian ends. In [PW21],Palmer and Woodward showed that Lagrangian surgery leaves Floer cohomology invariantupon incorporating a bounding cochain. These provide several pieces of evidence that thereshould be a way to explicitly equate the geometric deformation from Lagrangian surgeryto the algebraic deformation of CF • ( L ) from bounding cochain using the machinery of La-grangian cobordisms. This dictionary between Lagrangian surgery, Lagrangian cobordisms,and deformations in the Fukaya category can be observed in geometry arising from wall-crossing formula for the open Gromov-Witten superpotential [Hic19]. Furthermore, sucha dictionary would provide a route to proving that the group of unobstructed Lagrangiancobordisms in [BC13] is exactly the Grothendieck group of the Fukaya category.1.2. Outline and results.
Section 2 reviews known constructions of Lagrangian cobor-disms and Floer theory of Lagrangian cobordisms. We focus on Biran and Corena’s theoremthat “monotone Lagrangian cobordisms provide equivalences in the Fukaya category,” andexplore the limitations that monotonicity places on this theorem. We give a simple exampleof an oriented obstructed Lagrangian submanifold whose ends are non-isomorphic in theFukaya category.The remainder of the paper is organized as follows. • Section 3 gives a characterization of Lagrangian cobordisms. This section only usesstandard techniques in symplectic geometry and does not contain any Floer-theoreticcomputations. • Section 4 demonstrates the existence of holomorphic teardrops with boundary onour Lagrangian cobordism, which naively pairs the surgery handles of the cobor-dism with the self-intersections. We include a computation demonstrating howobstructability of Lagrangian cobordisms is recovered from teardrop counts. • The final two sections, sections 5 and 6, look at conjectural applications to the Floercohomology of immersed Lagrangian cobordisms. In a future project, we plan toexpand on section 5, which outlines some expected properties of Floer cohomology
AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 5 for Lagrangian cobordisms with immersed ends. This conjectural framework isrequired to perform the computation in section 6.1, which exhibits an algorithmconstructing continuation maps in the Fukaya category from Lagrangian cobordisms.Finally, section 6.2 discusses future applications of these techniques to LagrangianFloer cohomology.In section 3.1, we provide some standard tools for decomposing Lagrangian cobordisms.In short: when decomposing a Lagrangian cobordism K ⊂ X × C , we can consider decom-positions that have boundaries fibering over the X or C coordinate. In proposition 3.1.4 weshow that Lagrangian cobordisms are decomposable along the C -coordinate, and in proposi-tion 3.1.7 we give a method for decomposing Lagrangian cobordisms along the X -coordinate.These decompositions are used in section 3.2 to construct the standard surgery handle. Wereview the parameterization of the Whitney sphere and the Lagrangian null-cobordism forthe Whitney sphere in section 3.2.1. The parameterization is compared to the Lagrangiansurgery handle from [ALP94]. In addition to the parameterization of the surgery handle,figs. 7 to 11 provide plots of these handles as projections to the C coordinate and as sets ofcovectors in T ∗ R n ; we hope that these examples provide the reader with intuition on theconstruction and geometry of surgery handles. We use this particular parameterization ofthe surgery handle in section 3.3, where we prove the main result of this section. Theorem (Restatement of theorem 3.3.1) . Let K : L + (cid:32) L − be a Lagrangian cobordism. K is exactly homotopic to the concatenation of surgery trace cobordisms and suspensions ofexact homotopies. This requires showing that every Lagrangian cobordism can be placed into a good positionby an exact homotopy. We additionally comment on the relation between Lagrangiansurgery and anti-surgery.Section 4 investigates the relation between self-intersections of Lagrangian submanifoldsand topology of the Lagrangian surgery trace. In section 4.1, we show that each surgeryhandle of a Lagrangian cobordism can be paired with a self-intersection of the slice andthat this pairing respects the index of these objects as Floer cochains whenever K is agraded Lagrangian submanifold. It follows that whenever K : L − → L + is embedded andgraded that χ ( L − ) = χ ( L + ). Sections 4.2 and 4.3 review the theory of Floer cochainsfor immersed Lagrangian submanifolds and Lagrangian cobordisms. In section 4.4 we givea precise statement showing that the self-intersection to handle pairing extends to Floercohomology. Theorem (Restatement of theorem 4.4.3) . The standard Lagrangian surgery trace bounds aholomorphic teardrop pairing the critical point of the surgery trace with the self-intersectionin Floer cohomology.
Section 4.5 uses this pairing to justify why the example Lagrangian cobordism given infig. 5 does not construct a continuation map in the Fukaya category.The remainder of the paper looks towards the future. Section 5 addresses expectationsfor how immersed Lagrangian Floer theory interacts with Lagrangian cobordisms with bot-tlenecks. The technical details of adapting the immersed Lagrangian intersection Floertheory of [PW21] to our setting will be outlined in a future work. Section 6.1 applies thisconjectural framework to a computation yielding a continuation map associated with a La-grangian surgery trace cobordism in specific examples (figs. 2 and 25). In these examples,
JEFF HICKS S E S − E A (cid:48) B (cid:48) L E b Figure 2.
A Lagrangian cobordism K A,B : L E b (cid:32) ( S E (cid:116) S − E ) correspond-ing to a Lagrangian surgery which does not yield a continuation map in theFukaya category. The Lagrangian cobordism is obstructed.the holomorphic teardrop contributes to a curvature term m : Λ → CF • ( K A,B ) in Floercohomology. The existence of bounding cochain or obstruction of CF • ( K A,B ) either yieldsor precludes the construction of a continuation map on Floer cohomology between L E b and S E ∪ S − E . In the case of fig. 2, the Lagrangians L E b and S E ∪ S − E are disjoint, so the resultis obvious; however the comparison to the setting of fig. 25 where a continuation map canbe constructed is illustrative.The last section, section 6.2, explores future applications of theorems 3.3.1 and 4.4.3.These include: proving and generalizing the [Fuk+07] surgery exact triangle without neck-stretching methods; algorithms for determining unobstructedness of Lagrangian cobordisms;extending [BC13] to the unobstructed setting; comparing shadows of Lagrangian cobordismsto valuations of differentials in Floer cohomology; and computing Lagrangian Floer coho-mology of Lagrangians in Lefschetz fibration via Morse decomposition.1.3. Acknowledgments.
I would particularly like to thank Luis Haug with whom I dis-cussed many of the ideas of this paper; much of my inspiration comes from his work in[Hau20]. Additionally, the catalyst of this project was a discussion with Nick Sheridan atthe 2019 MATRIX workshop on “Tropical geometry and mirror symmetry,” and the majorideas of this paper were worked out during an invitation to ETH Z¨urich from Ana Can-nas da Silva for a mini-course on “Lagrangian submanifolds in toric fibrations”. Finally,I’ve benefitted from many conversations with Paul Biran, Octav Cornea, Mark Gross, An-drew Hanlon, Ailsa Keating, Cheuk Yu Mak, and Ivan Smith on the topic of Lagrangiancobordisms. Graphics in this paper were created using the matplotlib library [Hun07].This work is supported by EPSRC Grant EP/N03189X/1 (Classification, Computation,and Construction: New Methods in Geometry).
AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 7 Background
Notation. X will always denote a symplectic manifold of dimension 2 n . There will bemany Lagrangian submanifolds of varying dimensions in this article. The dimension of asubmanifold will be determined by reverse alphabetical order, sodim( J ) − K ) − L ) = n = dim( M ) + 1 . We will work with Lagrangian submanifolds which may bound holomorphic disks. Whenwe say that a Lagrangian submanifold is tautologically unobstructed , we will mean that itbounds no holomorphic disks for geometric reasons. A unobstructed
Lagrangian submanifoldwill be a Lagrangian submanifold whose space of Maurer-Cartan (definition 4.2.10) solutionsis non-empty.In this paper, we will frequently take local coordinates for a Lagrangian submanifold U ⊂ L ⊂ X , and identify the Weinstein neighborhood with a neighborhood of U ⊂ R n ⊂ C n .We will denote the coordinates near U by ( q i + p i ).2.1. Lagrangian Homotopy and Cobordism.
A homotopy of Lagrangian submanifoldsis a smooth map i t : L × R → X with the property that at each t ∈ R , i t : L → X is an immersed Lagrangian submanifold. For each t ∈ R , a homotopy of Lagrangiansubmanifolds yields a closed cohomology class Flux t ( i t ) ∈ H ( L, R ), called the flux classof i t at t . The value of the flux class on chains c ∈ C ( L, R ) is defined byFlux t ( i t )( c ) := (cid:90) i t : c × [0 ,t ] → X ω. If Flux t ( i t ) is exact for all t ∈ R , we say that this homotopy is an exact homotopy. In thecase that i t is an exact isotopy, there exists a time dependent Hamiltonian H t : X × R → R with the following properties: • H t | L is a primitive for the flux class in the sense that dH t | L = Flux t ( i t ). • The isotopy is generated by the Hamiltonian flow φ t : X × R → X in the sense that i t ( L ) = φ t ( i ( L )) . Lagrangian cobordisms are an extension of the equivalence relation of exact homotopy.
Definition 2.1.1 ([Arn80]) . Let L + , L − be (possibly immersed) Lagrangian submanifolds of X . A with ends L + , L − is a (possibly immersed) Lagrangiansubmanifold K ⊂ ( X × C , ω X + ω C ) for which there exists a compact subset D ⊂ C so that : K \ ( π − C ( D )) = ( L + × R >t + ) ∪ ( L − × R The shadow projection of a Lagrangian cobordism K ⊂ T ∗ S × C to the C factor. This Lagrangian cobordism is the suspension of a Hamil-tonian isotopy. We chose the color brown for our Lagrangian cobordismsbased on the commonly held belief that “a Lagrangian cobordism looks likea hairy potato”.a diagram of a Lagrangian cobordism. We call the R -component of X × C the cobordismparameter , and the projection to this coordinate will be denoted by π R : X × C → R . Givena Lagrangian cobordism K , we will abuse notation and use π R : K → R to denote thecobordism parameter restricted to K . Claim 2.1.2. Let t ∈ R be a regular value of the projection π R : K → R . The slice of K at t , K | t := π X ( π − R ( t ) ∩ K ) is a (possibly immersed) Lagrangian submanifold of X . The slices of a Lagrangian cobordism show that the equivalence relation of Lagrangiancobordance extends the equivalence relation of exactly homotopic. We call a Lagrangiancobordism cylindrical if π R : K → R has no critical points. Proposition 2.1.3 ([ALP94]) . Let L + , L − ⊂ X be two Lagrangian submanifolds. L + and L − are exactly homotopic if and only if there exists a cylindrical Lagrangian cobordism K : L + (cid:32) L − between these two Lagrangians. Given an exact isotopy i t : L × R → X whose primitive H t : L → R has compact support,there is a cylindrical Lagrangian cobordism called the suspension of i t parameterized by: L × R → X × C ( q, t ) (cid:55)→ ( i t ( q ) , t + H t ( q )) ∈ X × C . For the purpose of providing some geometric grounding to our discussions, we give anexample of a Lagrangian cobordism which is not an exact homotopy. We first note thatevery compact Lagrangian submanifold K ⊂ X × C gives an example of a Lagrangiancobordism K : ∅ (cid:32) ∅ . While these Lagrangian cobordisms are not very interesting from aFloer-theoretic perspective (as they can be displaced from themselves), they are useful forunderstanding the kinds of geometry which can appear in a Lagrangian cobordism. AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 9 K | K | K | K | K | Figure 4. Scatter plots of randomly selected points lying on the shearedproduct torus K = φ ( L T ). Top: The shadow projection of the Lagrangiansubmanifold K . Bottom: Several slices of K = φ ( L T ) at different real val-ues. The critical values of the cobordism parameter π R : K → R are at ± . ± . 5. We use this opportunity to highlight a common misconception:although all slices K | t are immersed, the Lagrangian K is embedded. Example 2.1.4 (Sheared Product Torus) . Consider C × C with coordinates ( q , p , q , p ) .The product Lagrangian torus L T is the submanifold parameterized by ( θ , θ ) (cid:55)→ (cos( θ ) + sin( θ ) , cos( θ ) + sin( θ )) ⊂ C × C . We apply a linear symplectic transformation φ : C → C ( q , p , q , p ) (cid:55)→ (cid:18) q + 12 q + p − p , q + 12 q + p − p (cid:19) so that K := φ ( L T ) is in general position. After taking this shear, π R : K → R is aMorse function with four critical values corresponding to the standard maximum, saddles,and minimum on the torus. Several slices and the shadow of the Lagrangian cobordism aredrawn in fig. 4. Previous Work: Anti-surgery. Lagrangian anti-surgery, introduced by [Hau20],gives a method for embedding the Lagrangian cobordism handle from [ALP94]. Given aLagrangian L ⊂ X , an isotropic anti-surgery disk for L is an embedded isotropic disk i : D k +1 → X with the following properties: • Clean Intersection: The boundary of D k +1 is contained in L . Additionally, theinterior of D k +1 is disjoint from L , and the outward pointing vector field to D istransverse to L . • Trivial Normal Bundle: Over D k +1 , we can write a splitting ( T D k +1 ) ω = T D k +1 ⊕ E . Furthermore, we ask that there is a symplectic trivialization D k +1 × C n − k − → E so that over the boundary, E | ∂D k +1 is contained in T L | ∂D k +1 .Given an isotropic anti-surgery disk D k +1 with boundary on L , [Hau20] produces a La-grangian α D k +1 ( L ), the anti-surgery of L along D k +1 , along with a Lagrangian anti-surgerytrace cobordism K α Dk +1 : L (cid:32) α D k +1 ( L ) . As a manifold, α D k +1 ( L ) differs from L by k -surgery along ∂D k +1 , and the cobordism pa-rameter π R : K α Dk +1 → R provides a Morse function with a single critical point of index k + 1. When compared to L , the anti-surgery α D k +1 ( L ) posses a single additional self-intersection q D n . The construction is inspired by an analogous construction for Legendriansubmanifolds in [Riz16]. The terminology “anti-surgery” is based on the following obser-vation: given a Lagrangian anti-surgery disk D n for L , the Polterovich surgery [Pol91] of α D n ( L ) at the newly created self-intersection point q D n is Lagrangian isotopic to L . In thissense, anti-surgery and surgery are inverse operations on Lagrangian submanifolds. Accord-ingly, if L arises from L (cid:48) by anti-surgery along a disk D k +1 , Haug states that L arises from L (cid:48) by Lagrangian n − k + 1 surgery.These surgeries and anti-surgeries appear in fig. 4, which decomposes the product torusinto slices related by the creation/deletion of Whitney spheres, surgeries, and anti-surgeries.Some higher-dimensional examples of anti-surgery are in figs. 10 and 11, which draws La-grangians related by anti-surgery in the cotangent bundle of R . In these figures, we’ve high-lighted the isotropic anti-surgery disk corresponding modifications in red; the Lagrangianson the right-hand side all exhibit a single self-intersection at the origin.2.3. Floer Theoretic Properties of Lagrangian Cobordisms. Our motivation forstudying Lagrangian cobordisms comes from their Floer theoretic properties. A funda-mental result states that cobordant Lagrangians have homotopic Floer theory. Theorem 2.3.1 ([BC13]) . Suppose that K : L + (cid:32) L − is a monotone embedded Lagrangiansubmanifold. Let L (cid:48) ⊂ X be a monotone test Lagrangian submanifold. Then the chaincomplexes CF • ( L + , L (cid:48) ) and CF • ( L − , L (cid:48) ) are chain homotopic. More generally, Biran and Cornea prove that a Lagrangian cobordism with k -inputs { L + i } ki =1 and output L − yields a factorization of L − into an iterated mapping cone of the L + i .In the setting of two-ended monotone Lagrangian cobordisms, applications of theorem 2.3.1are limited by lack of examples. In fact, [Su´a17] shows that under the stronger conditionthat K is exact, every embedded exact Lagrangian cobordism K : L (cid:32) L of dim( K ) ≥ topology of L × R . It is still currently unknown if all such Lagrangian cobordismsare Hamiltonian isotopic to suspensions of Hamiltonian isotopies.It is expected that theorem 2.3.1 should extend to more general settings than monotoneLagrangian submanifolds. One of the broader extensions is to the class of unobstructed im-mersed Lagrangian cobordisms . Roughly, unobstructed Lagrangians are those whose countsof holomorphic disks can be made to cancel in cohomology (see definition 4.2.10). The Floer AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 11 theoretic property of unobstructedness is absolutely necessary to obtain a continuation mapin the Fukaya category. We given an example of a Lagrangian cobordism which cannot givea continuation map in the Fukaya category. Example 2.3.2. Let S E ⊂ T ∗ S be the section E π dθ which bounds an annulus of area E with the zero section. Consider the Lagrangian submanifold which is the disjoint union oftwo circles S E ∪ S − E ⊂ T ∗ S , as drawn in fig. 5. By applying anti-surgery along theinterval { e } × [ − E / π, E/ π ] , we obtain a Lagrangian cobordism to L E , where L E is aLagrangian double section of T ∗ S intersecting the zero section at (0 , . We subsequentlyapply Lagrangian surgery at this self intersection to obtain S E (cid:116) S − E . Let K ⊂ T ∗ S × C be the Lagrangian cobordism built from concatenating the anti-surgery and surgery tracecobordism. We explain why K does not necessarily satisfy the conditions of theorem 2.3.1.The Lagrangian cobordism K bounds two holomorphic teardrops of area A and A , givenby the portion of { (0 , } × C } ⊂ T ∗ S × C which is bounded by K . When these teardropshave differing area, they collectively contribute a non-trivial m curvature term to the Floercohomology CF • ( K ) , which cannot be cancelled by a bounding cochain. In this setting, E (cid:54) = E , and so the Lagrangians S E ∪ S − E and S E ∪ S − E are disjoint. Since both S E ∪ S − E and S E ∪ S − E are non-trivial objects of the Fukaya category, the Lagrangiancobordism K cannot hope to yield a continuation map. This example demonstrates that understanding when Lagrangians are unobstructed isessential for building meaningful continuation maps from Lagrangian cobordisms. In pre-vious work, the author [Hic19] showed that bounding cochains for Lagrangian cobordismscould be used to compute wall-crossing transformations for Lagrangian mutations. Despitethe substantial analytic difficulties in defining CF • ( K ) for the class of Lagrangian cobor-disms with immersed ends, the goal of section 6.1 is to compute bounding cochains andcontinuation maps for specific examples of K with immersed ends.3. Lagrangian cobordisms are Lagrangian surgeries In this section we prove that every Lagrangian cobordism can be decomposed into acomposition of Lagrangian surgery traces and exact homotopy suspensions. Section 3.1gives some constructions for decomposing Lagrangian cobordisms. In section 3.2 we describethe standard Lagrangian surgery handle. Finally, in section 3.3 we show that a Lagrangiancobordism can be exactly homotoped to good position, and subsequently decomposed intosurgery traces.3.1. Decompositions of Lagrangian cobordisms. We consider two types of decompo-sitions for Lagrangian cobordisms K ⊂ X × C : across the cobordism parameter C andacross the X -coordinate.In the cobordism parameter: given Lagrangian cobordisms K +0 : L + (cid:32) L K − : L (cid:32) L − there exists a concatenation cobordism K − ◦ K +0 : L + (cid:32) L − . The exact homotopy classof the concatenation does not depend on the neck length. The concatenation operationshows that Lagrangian cobordance is an equivalence relation on the set of Lagrangian sub-manifolds. In the setting of differentiable manifolds, concatenation can be used to providea decomposition of any cobordism into a sequence of standard surgery handles. ↑ ↑ ↑ ↑↑ ↑ ↑ ↑ Surgery Exact Homotopy Anti-Surgery C − E /π − E /πE /π E /πT ∗ S A A (a) Slices(b) Shadow Figure 5. An oriented immersed Lagrangian cobordism. If A (cid:54) = A , then K is an obstructed Lagrangian submanifold. The slashed regions correspondto the images of holomorphic teardrops with boundary on K under theprojection π C . The boundary of these teardrops obstruct the solution to theMaurer-Cartan equation for K .When describing surgery and trace cobordisms, it is also important to consider cobor-disms with boundary. We call this decomposition along the X -coordinate. Definition 3.1.1. Let L + ⊂ X be a Lagrangian submanifold with boundary M . Let M × [0 , (cid:15) ) ⊂ X be a collared neighborhood of the boundary. A cobordism of Lagrangians withfixed boundary K : L + (cid:32) L − is a Lagrangian submanifold K ⊂ X × C whose boundary hasa collared neighborhood of the form M × [0 , (cid:15) ) × R ⊂ X × C . We will use cobordisms of Lagrangians with fixed boundary to describe local modificationsto Lagrangian submanifolds. Let L + = L + ↓ ∪ M L + ↑ be a decomposition of a Lagrangiansubmanifolds along a surface M = ∂L + ↑ . Given K ↑ : L + ↑ → L −↑ a cobordism of Lagrangianswith fixed boundary M , we can obtain a Lagrangian cobordism K ↑ ∪ M × R ( L + ↓ × R ) : L + (cid:32) ( L −↑ ∪ M L + ↓ ) . In this case, we say that the Lagrangian L − := L −↑ ∪ M L + ↓ arises from modification of L + atthe set L + ↑ . We say that K : L + (cid:32) L − decomposes across the X -coordinate along M ⊂ L +AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 13 if there exist Lagrangian cobordisms with fixed boundary MK ↑ : L + ↑ (cid:32) L −↑ K ↓ : L + ↓ (cid:32) L −↓ so that L − := L −↑ ∪ M L −↓ ⊂ X is a Lagrangian submanifold and K := K ↑ ∪ M × R K ↓ . In this setting, we additionally have a exchange relation making the Lagrangian submani-folds ( K ↑ ∪ M × R ( L −↓ × R )) ◦ (( L + ↑ × R ) ∪ M × R K ↓ )(( L −↑ × R ) ∪ M × R K ↓ ) ◦ ( K ↑ ∪ M × R ( L + ↓ × R )) . exactly homotopic.As we will frequently work with immersed Lagrangian cobordisms, we need the followingreplacement of Weinstein neighborhoods. Definition 3.1.2. A local symplectic embedding is a map φ : Y → X so that around every y ∈ Y there exists a neighborhood U on which φ | U : U → X is a symplectic embedding. Let i : L → X be an immersed Lagrangian submanifold. A local Weinstein neighborhood is amap from a neighborhood of the zero section in the cotangent bundle of L to Xφ : B ∗ (cid:15) L → X which is a local symplectic embedding, and whose restriction to the zero section so that thediagram L XB ∗ (cid:15) L i φ commutes. For both decomposition along the X coordinate and cobordism parameter, we will usethe following generalization of proposition 2.1.3. Lemma 3.1.3 (Generalized Suspension) . Let I be a finite indexing set. Suppose that we aregiven for each α ∈ I an exact Lagrangian homotopy i αt : L α × R → X α , whose flux primitiveis H αt : L α × R → R . Let Y be another manifold, and pick functions ρ α : Y → R . For fixed q α ∈ L α , let ( y, dρ α ( y )) : Y → T ∗ Y be the parameterization of the exact Lagrangian sectionwhose primitive is ρ α ( y )( q α ) : Y → R . Then j ρ I : (cid:32)(cid:89) α ∈ I L α (cid:33) × Y → (cid:32)(cid:89) α ∈ I X α (cid:33) × T ∗ Y ( q α , y ) (cid:55)→ (cid:32) i αρ α ( y ) ( q α ) , y, (cid:88) α ∈ I H αρ α ( y ) ( q α ) dρ α ( y ) (cid:33) parameterizes a Lagrangian submanifold. We note that the usual suspension construction is recovered by taking I = { } , ρ ( y ) = t and Y = R . Proof. For convenience, write j for j ρ I . Pick local coordinates y , . . . y n for Y , so that wemay locally identify T ∗ Y with C n and write sections of the cotangent bundle as Y → T ∗ Yy i (cid:55)→ (cid:32) y i + (cid:88) α ∈ I H αρ α ( y ) ( q α ) ∂ρ α ∂y i (cid:33) . Let ddt i αt ∈ T X α be the vector field along the image of i αt associated to the isotopy i αt . Let( q αj ) denote local coordinate on the L α . Since i αt is a exact Lagrangian homotopy with fluxprimitive given by H αt , we have that: ι ddt i αt ω X α | L α = (cid:88) i ∂ q j H αt dq αj . Let ∂ v α ∈ T L α , and ∂ y i ∈ T Y be vectors. We compute j ∗ ( ∂ y j ) and j ∗ ( (cid:80) α ∈ I c α ∂ v α ) :( j ) ∗ ( ∂ y j ) = (cid:32) ∂ρ α ∂y j ddt i αρ α ( y ) , δ ij + (cid:88) α ∈ I (cid:18) ∂ ρ α ∂y i ∂y j H ρ α ( y ) + ∂ρ α ∂y i ∂ρ α ∂y j ddt H αρ α ( y ) (cid:19)(cid:33) ( j ) ∗ (cid:32)(cid:88) α ∈ I c α ∂ v α (cid:33) = (cid:32)(cid:88) α ∈ I c α ( i ρ ( s ) ) ∗ ∂ v α , (cid:88) α ∈ I c α ∂ρ α ∂y i ∂ αv H αρ α ( y ) (cid:33) We compute the vanishing of the symplectic form on (cid:80) α ∈ I b α ∂ v α , (cid:80) β ∈ I c β ∂ w β ∈ (cid:81) α ∈ I T L α .This term vanishes as T L α is a Lagrangian subspace of ω αX . ω ( j ) ∗ (cid:32)(cid:88) α ∈ I b α ∂ v α (cid:33) , ( j ) ∗ (cid:88) β ∈ I c β ∂ w β = (cid:88) α = β ∈ I b α c β ω X α (cid:0) ( i ρ ( s ) ) ∗ ∂ v α , ( i ρ ( s ) ) ∗ ∂ w β (cid:1) The vanishing of the symplectic form on ( j ) ∗ ( ∂ y i ) , ( j ) ∗ (cid:0)(cid:80) α ∈ I c α ∂ v α (cid:1) comes from the as-sumption that the Lagrangian homotopies L αt have flux primitive given by H αt . ω (cid:32) ( j ) ∗ ( ∂ y i ) , ( j ) ∗ (cid:32)(cid:88) α ∈ I c α ∂ v α (cid:33)(cid:33) = (cid:88) α ∈ I c α (cid:18) ω X α (cid:18) ∂ρ α ∂y j ddt i αρ α ( y ) , ( i ρ ( s ) ) ∗ ∂ v α (cid:19)(cid:19) + (cid:88) α ∈ I c α ω T ∗ Y (cid:18) δ ij , ∂ρ α ∂y j ∂ αv H αρ α ( y ) (cid:19) = (cid:88) α ∈ I c α (cid:18) ∂ρ α ∂y i (cid:18) ι ddt i αρα ( y ) ω X α + dH αρ α ( y ) (cid:19) ∂ v α (cid:19) = 0 AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 15 − (cid:15) − (cid:15) (cid:15) (cid:15) − (cid:15) (cid:15) | | | | | | | −− ts Figure 6. Contour plot of the truncation profile ρ ( t, s ).The vanishing of the symplectic form on j ∗ ( ∂ y i ) , j ∗ ( ∂ y j ) corresponds to the fact closedsections of the cotangent bundle are Lagrangian sections. ω ( j ∗ ( ∂ y i ) , j ∗ ( ∂ y j )) = (cid:88) α ∈ I ω X α (cid:18) ∂ρ α ∂y i ddt i αρ α ( y ) , ∂ρ α ∂y j ddt i αρ α ( y ) (cid:19) + ω Y (cid:16) δ ik + (cid:80) α ∈ I (cid:18) ∂ ρ∂yi∂yk H ρ ( y ) + ∂ρα∂yi ∂ρα∂yk ddt H αρα ( y ) (cid:19) , δ jk + (cid:80) α ∈ I (cid:18) ∂ ρ∂yj∂yk H ρ ( y ) + ∂ρα∂yj ∂ρα∂yk ddt H αρα ( y ) (cid:19) (cid:17) =0 (cid:3) Decomposition across the cobordism parameter. Cobordisms can be decomposed intosmaller cobordisms along any regular level set of Morse function. We show an analogousdecomposition for Lagrangian cobordisms. Proposition 3.1.4. Let K : L + (cid:32) L − be a Lagrangian cobordism with fixed boundary M .Suppose that π R : K → R has isolated critical values, and that ∈ R is a regular value ofthe projection π R : K → C , so that L := K | ⊂ X is a Lagrangian submanifold. Thenthere exists Lagrangian cobordisms with fixed boundary MK (cid:107) ( −∞ , : L (cid:32) L − K (cid:107) [0 , ∞ ) : L + (cid:32) L , so that K and K (cid:107) ( −∞ , ◦ K (cid:107) [0 , ∞ ) are exactly homotopic Lagrangian cobordisms.Proof. We construct a cylindrical Lagrangian cobordism J ⊂ ( X × C ) × C with ends J : K (cid:107) ( −∞ , ◦ K (cid:107) [0 , ∞ ) (cid:32) K which will be the suspension of our exact homotopy. Consider the decomposition as sets K = K | t ≤− (cid:15) ∪ t = − (cid:15) K | [ − (cid:15),(cid:15) ] ∪ t = (cid:15) K | [ (cid:15), ∞ ) , where K | t ≤− (cid:15) = π − R (( −∞ , − (cid:15) ]) K | [ − (cid:15),(cid:15) ] = π − R ([ − (cid:15), (cid:15) ]) K | [ (cid:15), ∞ ) = π − R ([ (cid:15), ∞ ))The piece K | [ − (cid:15),(cid:15) ] is cylindrical so by proposition 2.1.3 there exists a primitive H t : L × [ − (cid:15), (cid:15) ] → R so that we can parameterize K | [ − (cid:15),(cid:15) ] as: L × [ − (cid:15), (cid:15) ] → X × C ( q, t ) (cid:55)→ ( i t ( q ) , t + H t ( q )) We choose a truncation profile ρ ( t, s ) : [ − (cid:15), (cid:15) ] × R → [ − (cid:15), (cid:15) ] which satisfies the followingconditions (as indicated in fig. 6) : ρ ( t, s ) | s< = t ρ ( t, s ) | | t | > (cid:15)/ = tρ ( t, | | t | <(cid:15)/ = 0 ∂ρ∂s (cid:12)(cid:12)(cid:12)(cid:12) s> = 0Consider the Lagrangian submanifold J | [ − (cid:15),(cid:15) ] × R given by the generalized suspension fromlemma 3.1.3, j ρ : L × [ − (cid:15), (cid:15) ] × R → X × T ∗ [ − (cid:15), (cid:15) ] × T ∗ R This is a Lagrangian cobordism over the s parameter with collared boundaries in both the t and s directions: • In the s direction, K | | t | = (cid:15) is a boundary for K | [ − (cid:15),(cid:15) ] := J | [ − (cid:15),(cid:15) ] ×{ s =0 } . This extendsto a collared boundary K | | t | > (cid:15)/ , and K | | t | > (cid:15)/ × R s is a collared boundary for J | [ − (cid:15),(cid:15) ] × R s . Therefore, J | [ − (cid:15),(cid:15) ] × R s , as a cobordism in the s direction, is a Lagrangiancobordism with fixed boundary. • At each value of s , the collar M × [0 , (cid:15) (cid:48) ) × [ − (cid:15), (cid:15) ] t ⊂ K | [ − (cid:15),(cid:15) ] is a collared boundaryof the slice J | [ − (cid:15),(cid:15) ] ×{ s } .We’re interested in the positive end of this Lagrangian cobordism, which we call K + | [ − (cid:15),(cid:15) ] ,so that J | [ − (cid:15),(cid:15) ] × R : K + | [ − (cid:15),(cid:15) ] (cid:32) K | [ − (cid:15),(cid:15) ] is a cylindrical Lagrangian cobordism with collared boundary. The Lagrangian K + | [ − (cid:15),(cid:15) ] iscylindrical, and parameterized by( q, t ) (cid:55)→ (cid:18) i ρ ( t, ( q ) , t + dρ ( t, dt H ρ ( t, (cid:19) We can therefore form a Lagrangian cobordism J : K + (cid:32) K , where K + = K | ( −∞ ,(cid:15) ] ∪ t = − (cid:15) K + | [ − (cid:15),(cid:15) ] ∪ t = (cid:15) K | [ (cid:15), ∞ ) . By construction, π i R ( K + | [ − (cid:15)/ ,(cid:15)/ ) = 0, and so we may assemble Lagrangian cobordisms: K (cid:107) ( −∞ , := K + | ( ∞ , ∪ L ( L × [0 , ∞ )) : L (cid:32) L − K (cid:107) [0 , ∞ ) := K + | [0 , ∞ ) ∪ L ( L × ( −∞ , L + (cid:32) L . Clearly, K + = K (cid:107) ( −∞ , ◦ K (cid:107) [0 , ∞ ) . Additionally, since K + fixes the boundary M , both K (cid:107) ( −∞ , and K (cid:107) [0 , ∞ ) are Lagrangian submanifolds which fix the boundary M . (cid:3) Since this construction occurs away from the critical locus, we additionally have a match-ing of Morse critical pointsCrit( π R : K → R ) = Crit( π R : K (cid:107) ( −∞ , ◦ K (cid:107) [0 , ∞ ) → R )= Crit( π R : K (cid:107) [0 , ∞ ) → R ) ∪ Crit( π R : K (cid:107) ( −∞ , → R )Recall that the shadow of a Lagrangian cobordism Area( K ) is the infimum of areas ofsimply connected regions containing the image of π C : K → C . If K is cylindrical arising AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 17 from exact homotopy H t , then the shadow can be explicitly computed via the Hofer norm:Area( K ) = (cid:90) R (cid:32) sup q ∈ L + H t ( q ) − inf q ∈ L + H t ( q ) (cid:33) dt The shadow contribution of K | [ − (cid:15),(cid:15) ] and K + | [ − (cid:15),(cid:15) ] are equal after a change of coordinates:Area( K + | [ − (cid:15),(cid:15) ] ) = (cid:90) [ − (cid:15),(cid:15) ] (cid:32) sup q ∈ L + (cid:18) dρ ( t, dt H ρ ( t, ( q ) (cid:19) − inf q ∈ L + (cid:18) dρ ( t, dt H ρ ( t, ( q ) (cid:19)(cid:33) dt = (cid:90) [ − (cid:15),(cid:15) ] (cid:32) sup q ∈ L + H t ( q ) − inf q ∈ L + H t ( q ) (cid:33) dt = Area( K | [ − (cid:15),(cid:15) ] ) . From this it follows that the shadow of a Lagrangian cobordism is preserved under thedecomposition of proposition 3.1.4. The same method allows us to construct Lagrangiancobordisms from Lagrangian submanifolds K ⊂ X × C . Definition 3.1.5. Let K ⊂ X × C be a Lagrangian submanifold (not necessarily a La-grangian cobordism) with fixed boundary. Suppose that t − , t + are regular values of theprojection π R : K → R , and that the critical values of π R are isolated. The truncation of K to [ t − , t + ] is the Lagrangian cobordism with fixed boundary K (cid:107) [ t − ,t + ] : K | t + (cid:32) K | t − . Decomposition across the X -coordinate. We now look at how to “isolate” a portion ofa Lagrangian cobordism across the X -coordinate so that we can present it as a Lagrangianmodification. Definition 3.1.6. Let K : L + (cid:32) L − be an embedded Lagrangian cobordism with π R : K → R having isolated critical points. A dividing hypersurface for K is an embedded hypersurface M ⊂ L with the following properties: • M divides L in the sense that L = L − ∪ M L . • M ⊂ L ⊂ K contains no critical points of π R : K → R . A dividing hypersurface allows for the following decomposition of our Lagrangian cobor-dism. Proposition 3.1.7. Let M ⊂ L be a dividing hypersurface for K : L + (cid:32) L − . Then thereexists a decomposition of Lagrangian cobordisms up to exact homotopy: K ∼ ˜ K − ◦ K M ◦ ˜ K + so that K M is a Lagrangian cobordism which decomposes along M , and K M | = L .Proof. Let i : K → X × C be the parameterization of our Lagrangian cobordism. Consider asmall collared neighborhood M × I s ⊂ L . We take B ∗ (cid:15) ( M × I s ) a small neighborhood of thezero section inside the cotangent bundle of ( M × I s ). There exists a map φ : B ∗ (cid:15) ( M × I s ) → X ,which is locally a symplectic embedding, and sends the zero section to i ( M × I s ). For H : M × I s → X with | dH | < (cid:15) and the support of H contained on an interior subset of M × I , denote by dH ⊂ X the (possibly immersed) submanifold parameterized by M × I df −→ B ∗ (cid:15) ( M × I s ) φ −→ X. There exists an open neighborhood U ⊂ K containing M ⊂ L ⊂ K with the property that( U | t ) ∩ B ∗ (cid:15) ( M × I s ) is a section of the cotangent ball for each t . Let H t ( q, s ) : M × I s → R be the primitive of this section for each t so we can parameterize i : U → X × C by M × I s × I t → X × C ( q, s, t ) (cid:55)→ (cid:18) dH t , t + ddt H t (cid:19) We now consider a function ρ ( s, t ) : I s × I t → R which is constantly 1 in a neighborhood of ∂ ( I s × I t ), and constantly 0 on an interior set ( − (cid:15) (cid:48) , (cid:15) (cid:48) ) s × ( − (cid:15) (cid:48)(cid:48) , (cid:15) (cid:48)(cid:48) ) t ⊂ I s × I t . Consider theLagrangian suspension cobordism˜ i : M × I s × I t → X × C ( q, s, t ) (cid:55)→ (cid:18) d ( ρ ( s, t ) H t ) , t + ddt ρ ( s, t ) H t (cid:19) and by abuse of notation, let ˜ i : K → X × C be the Lagrangian cobordism where wehave replaced K | U with the chart parameterized above. For t ∈ ( − (cid:15) (cid:48)(cid:48) , (cid:15) (cid:48)(cid:48) ) and ( q, s ) ∈ M × ( − (cid:15) (cid:48) , (cid:15) (cid:48) ), we have that ddt ˜ i ( q, s, t ) = 0. Therefore, ˜ K (cid:107) ( − (cid:15) (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ) admits a decompositionin the X factor along M . We define K M := ˜ K (cid:107) ( − (cid:15) (cid:48)(cid:48) ,(cid:15) (cid:48)(cid:48) ) ˜ K − := ˜ K (cid:107) < − (cid:15) (cid:48)(cid:48) ˜ K + := ˜ K (cid:107) <(cid:15) (cid:48)(cid:48) . (cid:3) We write this decomposition as K M = K M − ∪ M K M + . We note that no part of this construction modifies the height function, soCrit( π R : K → R ) = Crit( π R : ˜ K − ◦ K M ◦ ˜ K + → R ) . As in the setting of decomposition along the C coordinate, we can show that in good casesthat this decomposition does not modify the Lagrangian shadow. Suppose that M ⊂ L is a dividing hypersurface for K : L + (cid:32) L − . Furthermore, suppose that over the chart M × I s × I t ⊂ K considered in the proof of proposition 3.1.7, we havesup q ∈ K | π R ( q )= t π R i ( q ) > sup q ∈ M × I s ×{ t } π R i ( q )inf q ∈ K | π R ( q )= t π R i ( q ) < inf q ∈ M × I s ×{ t } π R i ( q ) . Then Area( K ) = Area( ˜ K − ◦ K M ◦ ˜ K + ) . Standard Lagrangian surgery handle. In this section we give a description of astandard Lagrangian surgery handle. We include many figures in the hope of making thegeometry of Lagrangian surgery apparent and start with the simple example of Lagrangiannull-cobordism for Whitney spheres. AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 19 L 2, 04/3 immersed in T * 2 L 2, 04/3 to 1.00.50.00.51.0 Figure 7. The Whitney Sphere L , , +4 / ⊂ C . On the left: a projection ofrandomly sampled points on L , , +4 / to the first complex coordinate. The areaof the projection is 2 · / 3. On the right: L , , +4 / drawn as a subset of T ∗ R .3.2.1. Null-Cobordism and the Whitney Sphere. We first give a definition of the Whitneysphere in higher dimensions, and show that this is null-cobordant. Definition 3.2.1. The Whitney sphere of area A is the Lagrangian submanifold L n, , + A ⊂ C n which is parameterized by i n, , + A : S nr → C n ( x , . . . , x n ) (cid:55)→ ( x + x x , x + x x , . . . , x n + x x n ) . where S n = { ( x , . . . x n ) | (cid:80) ni =0 x i = r } , and r = (cid:113) A . This Lagrangian has a single transverse self-intersection at the pair of points ( ± r, , . . . , ∈ S nr . We call these points q ± ∈ L n, , + A . The quantity A describes the area of the projectionof L n, , + A to the first complex coordinate,Area( π C ( L n, , + A ))2 = 2 · (cid:18)(cid:90) r t (cid:112) r − t dt (cid:19) = A. The Lagrangian submanifold L , , + A is the figure eight curve. An example of the Whitney1-sphere is drawn in the fig. 8 as the slice K , | = L , / . In fig. 7 we give a plot of theWhitney 2-sphere, L , , +4 / ⊂ C , presented as a set of covectors in the cotangent bundle T ∗ R . K 1, 1 loc | K 1, 1 loc | K 1, 1 loc | K 1, 1 loc | Figure 8. A plot of points on the null-cobordism K , ⊂ C of the Whitneysphere L , , +4 / ⊂ C . Points are consistently colored between figures. Top:The shadow projection of this null-cobordism. Bottom: Slices of the null-cobordism at different values of the cobordism parameter.The Whitney sphere can be extended in one dimension higher to a Lagrangian subman-ifold parameterized by the disk. Let r ( x , . . . , x n ) = (cid:80) ni =0 x i . The parameterization j n, : R n +1 → C n × C ( x , . . . , x n ) (cid:55)→ ( i n, r ( x , . . . , x n ) , r − x ) . gives an embedded Lagrangian disk K n, ⊂ C n × C which has the following properties: • When r < 0, the slice K n, | r is empty; • The slice K n, | is not regular; • When r > K n, | r is a Whitney sphere of area A = r .We will prove that this is a Lagrangian submanifold in section 3.2.2. In fig. 8 we draw thisLagrangian null-cobordism and its slices, which are Whitney spheres of decreasing radius.While K n +1 , is not a Lagrangian cobordism (as it does not fiber over the real line outsideof a compact set,) it should be thought of as a model for the null-cobordism of the Whitneysphere. If we desire a Lagrangian cobordism, we may apply definition 3.1.5 to truncate thisLagrangian submanifold and obtain a Lagrangian cobordism.3.2.2. Standard Surgery and Anti-surgery Handle. The standard Lagrangian surgery handle[ALP94] is a Lagrangian R n × R inside T ∗ R n × T ∗ R , where T ∗ R is identified with C by( q, p ) (cid:55)→ p + q AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 21 Let σ i,k be +1 if i ≤ k and − G k,n − k +1 : R n +1 → R ( x , x , . . . , x n ) (cid:55)→ x (cid:32) n (cid:88) i =1 σ i,k x i (cid:33) + 13 x The graph of dG k,n − k +1 parameterizes a Lagrangian submanifold inside the cotangent bun-dle, R n +1 → T ∗ R n × C ( x , x , . . . , x n ) (cid:55)→ (cid:32) x + σ ,k x x , . . . , x n + σ n,k x n x , x + (cid:32) x + n (cid:88) i =1 σ i,k x i (cid:33)(cid:33) , whose projection to the π R coordinate is a Morse function with a single critical point ofindex k + 1. By multiplying the last coordinate by − , we interchange the real andimaginary parts of the shadow projection. Definition 3.2.2. For k ≥ , the local Lagrangian ( k, n − k + 1) surgery trace is theLagrangian submanifold K k,n − k +1 loc ⊂ ( C ) n × C parameterized by j k,n − k +1 : R n +1 → T ∗ R n × C ( x , x , . . . , x n ) (cid:55)→ (cid:32) x + σ ,k x x , . . . , x n + σ n,k x n x , x + n (cid:88) i =1 σ i,k x i − x (cid:33) . The positive and negative slices of this Lagrangian submanifold will be denoted L k,n − k, + loc := K k,n − k +1 loc | L k +1 ,n − k, − loc := K k,n − k +1 | − . For k = − , we define K − ,n +2 loc := ( K n, loc ) − . This will be the local model for Lagrangian cobordism surgery trace, which we constructin section 3.2.3. A particularly relevant example is K ,n +1 loc ⊂ C n +1 , which gives a localmodel for the Polterovich surgery trace (see fig. 9 for the example K , loc ).The slice L k,n − k, + loc is an immersed Lagrangian submanifold with a single double point, π X ◦ j k,n − k +1 ( ± , , . . . , 0) = (0 , . . . , , . We denote these points q ± ∈ L k,n − k, + loc . A useful observation is that when the positive endof the surgery trace is restricted to the first k -coordinates, L k,n − k, + loc | C k = L k, , + loc ⊂ C k we see an isotropic Whitney sphere. The other end of the Lagrangian surgery trace, L k +1 ,n − k − , − loc is embedded. Furthermore, L k,n − k, − loc | C k = ∅ . Our convention for the Morse index of a critical point is the dimension of the upward flow space of thepoint. The following mnemonics may be useful to the reader: the positive end of the surgery cobordism isimmersed and locally looks like the character “+”. K 0, 2 loc | K 0, 2 loc | K 0, 2 loc | K 0, 2 loc | K 0, 2 loc | Figure 9. A Lagrangian Surgery cobordism K , loc ⊂ C . Top: The shadowprojection of the surgery 1-handle K , loc . Bottom: Slices of the surgery. Theleft-most and right-most pictures are models of the surgery and anti-surgeryneck.This allows us to interpret K k,n − k +1 loc as a null-cobordism of a Whitney isotropic in thefirst k -coordinates. According to our convention (which is that Lagrangian cobordisms gofrom the positive end to the negative end), the Lagrangian cobordism K k,n − k +1 loc resolves aself-intersection of the input end. For this reason, we say that K k,n − k +1 loc provides a localmodel of Lagrangian surgery. We call the inverse Lagrangian submanifold, ( K k,n − k +1 loc ) − the local model for Lagrangian anti-surgery. Example 3.2.3 (Lagrangian Surgery Handle K , loc ) . In fig. 10 we draw slices of the La-grangian cobordism K , loc . In the surgery interpretation, the Lagrangian self-intersectionpoint is an isotropic Whitney sphere L , ⊂ C (highlighted in blue). We resolve the self-intersection by replacing this S with a S -family of K , null-cobordisms for the Whitneysphere.In the anti-surgery interpretation, the isotropic Lagrangian disk highlighted in red is con-tracted, collapsing the S boundary to a transverse self-intersection. Example 3.2.4 (Lagrangian Surgery Handle K , loc ) . In fig. 11 we draw slices of the La-grangian cobordism K , loc . In the surgery interpretation, we resolve the isotropic Whitney S This is a slightly deceptive characterization, as not all Whitney isotropic are null-cobordant. See re-mark 4.2.5. AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 23 K 0, 3 loc | K 0, 3 loc | Figure 10. Slices of the surgery 2-handle K , loc ⊂ C from before and afterthe critical point. These correspond to models of the surgery and anti-surgery neck. K 1, 2 loc | K 1, 2 loc | Figure 11. Slices of the surgery 2-handle K , loc ⊂ C from before and afterthe critical point. These correspond to models of the surgery and anti-surgery neck. sphere highlighted in blue by replacing it with two copies (an S family) of the null-cobordism D , .In the anti-surgery interpretation, the isotropic Lagrangian disk highlighted in red iscontracted, collapsing the immersed S boundary and yielding a Lagrangian with a self-intersection. A key take away is that while the cobordisms K k,n − k +1 loc and K n − k,k +1 loc are topologicallyinverses, they are not inverses of each other as Lagrangian cobordisms . This is easily seenby comparing figs. 10 and 11. Lagrangian surgery trace. We now apply section 3.1.2 to build from K k,n − k +1 loc aLagrangian cobordism with fixed boundary. Pick a radius A ∈ R > . Let L k,n − k +1 , = K k,n − k +1 loc | ; see for instance fig. 8. As a set of points in R n × R , L k,n − k +1 , = { ( x , x , . . . , x n ) | x + n (cid:88) i =1 σ i,k x i = 0 } . We then take hypersurface M k,n − k +1 ⊂ L k,n − k +1 , cut out by x + x + · · · x n = 1. As inthe proof of proposition 3.1.7, take an extension M k,n − k +1 × I s × I t ⊂ K which is disjointfrom the subset V = x + x + · · · x n = 1 / 4. By using proposition 3.1.7 to perform adecomposition across the X -coordinate along M k,n − k +1 , we obtain a Lagrangian submani-fold K k,n − k +1 loc M k,n − k +1 − where we take the component of the decomposition containing theorigin in D n × D . Since K k,n − k +1 loc M k,n − k +1 − | V = K k,n − k +1 loc | V , we have that for (cid:18) K k,n − k +1 loc M k,n − k +1 − ∩ V (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) π − R (1 / = (cid:16) K k,n − k +1 loc (cid:17)(cid:12)(cid:12)(cid:12) π − R (1 / | V . We define the standard Lagrangian trace of area to be K k,n − k +1 := K k,n − k +1 loc M k,n − k +1 − (cid:13)(cid:13)(cid:13)(cid:13) [ − / , / and the standard Lagrangian surgery trace of area A to be the rescaling (under the z (cid:55)→ c · z on C n ) of our the previously constructed Lagrangian submanifold, K k,n − k +1 A := 6 A · K k,n − k +1 . The ends of the standard Lagrangian surgery trace of area A will be denoted: K k,n − k +1 A : L k,n − k +1 , + A (cid:32) L k,n − k, − A . Remark 3.2.5. Note that in the case of k = − , n , this simply corresponds to truncation K k,n − k +1 A = K k,n − k +1 loc (cid:107) Theorem 3.2.6 (Properties of the standard Lagrangian Surgery Trace) . The Lagrangiansurgery trace K k,n − k +1 A : L k,n − k, + A (cid:32) L k,n − k, − A has the following properties: • L k,n − k, + A is a Lagrangian S k × D n − k with a single self-intersection. Its intersectionwith the first k -coordinates is a Whitney isotropic L k, , + A • L k,n − k, − A an embedded Lagrangian D k +1 × S n − k − . • π R : K k,n − k +1 A → R is Morse, with a single critical point of index k + 1 . AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 25 Remark 3.2.7. When k = − , we have a Lagrangian cobordism K − ,n +2 A = ( K n, ) − .This case differs slightly from the standard Lagrangian Surgery trace in that the positiveend L − ,n +1 , + A is empty, and the negative end L ,n, − A is a Whitney sphere. The construction of a standard Lagrangian surgery handle allows us to define the standardLagrangian surgery trace. Definition 3.2.8. We say that K : L + (cid:32) L − is a standard Lagrangian surgery trace if itadmits a decomposition across the X coordinate as K = K − ∪ S n − k × S n K k,n − k +1 A , where K − is a cylindrical Lagrangian cobordism with collared boundary. While the standard Lagrangian surgery trace is a useful cobordism to have, a geometricsetup for performing Lagrangian surgery on a given Lagrangian L + is desirable. Such acriterion is given in [Hau20] by the anti-surgery disk. In that paper, it was noted thatthe presence of a Whitney isotropic k sphere was a necessary but not sufficient conditionfor implanting a Lagrangian surgery handle. We give a sufficient characterization in re-mark 4.2.5.3.3. Cobordisms are iterated surgeries. Having described the Lagrangian surgery op-eration and trace cobordism, we show that all Lagrangian cobordisms decompose into aconcatenation of surgery traces and exact homotopies. This characterization is analogousto the handle body decomposition of cobordisms from the data of a Morse function. Theorem 3.3.1. Let K : L + (cid:32) L − be a Lagrangian cobordism. Then there is a sequenceof Lagrangian cobordisms K H it : L − i +1 (cid:32) L + i for i ∈ { , . . . , j } K k i ,n − k i +1 i : L + i (cid:32) L − i for i ∈ { , . . . j } which satisfy the following properties: • L − j +1 = L + and L +0 = L − • Each K k i ,n − k i +1 is a Lagrangian surgery trace; • Each K H it is the suspension of an exact homotopy and; • There is an exact homotopy between K ∼ K H jt ◦ K k j ,n − k j +1 j ◦ K H j − t ◦ · · · ◦ K H t ◦ K k ,n − k +11 ◦ K H t . The decomposition comes from using the function π R : K → R to provide a handle bodydecomposition of K .3.3.1. Morse Lagrangian Cobordisms. We first must show that K can be placed into generalposition by exact homotopy so that π R is a Morse function (as in example 2.1.4). Claim 3.3.2 (Morse Lemma for Lagrangian Cobordisms) . Let K ⊂ X be a Lagrangiancobordism. There exists K (cid:48) , a Lagrangian cobordism exactly homotopic to K , with π R : K (cid:48) → R a Morse function.Proof. Let i : K → X × C be our embedding, and let φ : B ∗ (cid:15) K → X be a local Weinsteinneighborhood. Let q := π R ◦ φ p := π R ◦ φ be the pullback of the real and imaginary coordinates to the local Weinstein neighborhood.Since φ is locally a diffeomorphism, the splitting of coordinates on X × C locally lifts tosplitting of coordinates on B ∗ (cid:15) K . Let C ∞ (cid:15) ( B ∗ (cid:15) K ) be the smooth functions B ∗ (cid:15) K → R withcompact support disjoint from the boundary. Let C ∞ cob ( K ; R ) be the functions which agreewith π R : K → R outside of a compact set. Given H ∈ C ∞ (cid:15) ( B ∗ (cid:15) K ), let ψ tH be the time t Hamiltonian flow of H , and let i tH = φ ◦ ψ tH be the corresponding exactly homotopicimmersion of K . We obtain a map P : C ∞ (cid:15) ( B ∗ (cid:15) K ) → C ∞ cob ( K ; R ) H (cid:55)→ π R ◦ i H . so that P ( H ) is the real coordinate of the immersion i tH . We will show that this mapis a submersion, and in particular open. Let f : K → R be a function with compactsupport, representing a tangent direction of C ∞ cob ( K ; R ). As K ⊂ B ∗ (cid:15) K is embedded, f can be extended to a compactly supported function F : B ∗ (cid:15) K → R so that F | K = f , and F ∈ C ∞ (cid:15) ( B ∗ (cid:15) K ).Take coordinates ˜ q , ˜ p on B ∗ (cid:15) K which are the lift of the q and p coordinates on X inthe sense that φ ∗ ∂ ˜ q = ∂ q φ ∗ ∂ ˜ p = ∂ p . In these coordinates, the flow of H in the ˜ q coordinate is d ˜ q dt = dHd ˜ p . We define our Hamiltonian H f : B ∗ (cid:15) K → R by the integral H f ( x ) := (cid:90) π X ( x ) × ( −∞ , ˜ p ( x )) F d ˜ p . With this choice of Hamiltonian, the Hamiltonian flow at time zero of the real coordinateat a point x ∈ K ⊂ B ∗ (cid:15) K is given by ddt ( π R ◦ i tH ) | t =0 ( x ) = d ˆ˜ q dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =0 ( x ) = f ( x ) . This shows that P is a submersion at 0. Since every open set of C ∞ cob ( K ; R ) contains aMorse function, and the image of P is open, there is a choice of Hamiltonian H near 0 sothat P ( H ) = π R ◦ i H is a Morse function on K . (cid:3) A similar argument shows that every K is exactly homotopic to K (cid:48) with the propertythat Crit( π R : K (cid:48) → R ) is disjoint from I si ( K (cid:48) ), the set of self-intersections of K (cid:48) . If π R : K → R is a Morse function whose critical points are disjoint from its self-intersections,we say that the Lagrangian cobordism is a Morse-Lagrangian cobordism .3.3.2. Placing Cobordisms in good position. The Lagrangian condition forces a certainamount of independence between the π R ◦ i and π R ◦ i projections of the Lagrangiancobordism. Claim 3.3.3. Let i : K → X × C be a Morse-Lagrangian cobordism. Then x ∈ K cannotbe a critical point of both π R ◦ i and π R ◦ i . AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 27 K 1, 3 loc | K 1, 3 loc | K 1, 3 loc | K 1, 3 loc | K 1, 3 loc | Figure 12. The “zipper” cobordism, a degenerate Morse-Lagrangian cobor-dism which can be perturbed to either surgery of anti-surgery trace. Proof. If so, then i ∗ ( T p K ) ⊂ T X ⊂ T X × T C . Since i ∗ ( T p K ) is a Lagrangian subspace, itcannot be contained in any proper symplectic subspace of T ( X × C ). (cid:3) Even when K is a Morse-Lagrangian cobordism, it need not be the case that at a criticalpoint q ∈ Crit( π R : K → R ) that K | q − (cid:15) is obtained from K | q + (cid:15) by surgery. In the simplestcounterexample, K | q − (cid:15) could be obtained from K | q + (cid:15) by anti-surgery. However, there alsodegenerate cases where K is neither a surgery or anti-surgery trace, but locally look like“zippers”; these Lagrangian cobordisms can be built by rotating the cobordism parameterto obtain a cobordism halfway between surgery and anti-surgery (see fig. 12). In orderto rule out these zipper cobordisms, we need to apply another exact isotopy based on aninterpolation between Morse functions. Claim 3.3.4 (Interpolation of Morse Functions) . Let f, g : R n → R be Morse functions, witha single critical point of index k at the origin and f (0) = g (0) = 0 . Take V any neighborhoodwhich contains the origin. There exists a smooth family of functions h c : R n × [0 , → R which satisfies the following properties. • In the complement of V , h c | R n \ V = f | R n \ V ; • There exists a small neighborhood U of the origin so that h | U = g | U . • h is Morse with a single critical point; • At time 0, h ( x ) = f ( x ) and; Proof. Pick coordinates x , . . . , x n , and y , . . . , y n so that in a neighborhood of the origin, f = n (cid:88) i =1 σ i,k x i g = n (cid:88) i =1 σ i,k y i . Let φ : R n → R n be a linear map so that φ ∗ ( ∂ y i ) = ∂ x i . Pick φ c : R n × [0 , / → R n anisotopy of linear maps smoothly interpolating between φ = id and φ = φ . Take U ⊂ V a small ball around the origin with the property that for all c , φ c ( U ) ⊂ V . Now consideran path of diffeomorphisms ˜ φ c : R n → R n satisfying the constraints˜ φ c | R n \ V = id ˜ φ = id ˜ φ | U = φ | U For c ∈ [0 , / 2] we define h c := f ◦ ˜ φ c .We now define h c for c ∈ [1 / , U ⊂ U a neighborhood of the origin with theproperty that for every q ∈ U and ∂ v ∈ T q R n : | ( d ( f ◦ φ ) − dg )( ∂ v ) | ≤ | df ( ∂v ) | . Take an interior subset U ⊂ U which is a neighborhood of the origin. Let ρ be a bumpfunction, which is constantly 1 on U , and 0 outside U . Let τ : [1 / , → [0 , 1] be anincreasing function smoothly interpolating between τ (1 / 2) = 0 and τ (1) = 1. For c ∈ [1 / , h c := (1 − τ ( c ) ρ ) · h / + τ ( c ) ρg. It remains to show that h is Morse, with a unique critical point at the origin. For any q ∈ R n \ U , we have that dh = ( ˜ φ ) ∗ df , which is nonvanishing. For any q ∈ U , we have dh = dg , which vanishes if and only if q = 0. For q ∈ U \ U , take ∂ v ∈ T q R n with theproperty that dρ ( ∂ v ) = 0. Then | dh ( ∂ v ) | = | (1 − ρ ) df ( ∂ v ) + ρdf ( ∂ v ) | > | df ( ∂v ) | > . This proves that q is not a critical point of h . (cid:3) Proposition 3.3.5. Let K be a Morse Lagrangian cobordism. Let q ∈ K be a critical pointof the projection π R : K → R of index k + 1 . There exists • A neighborhood of the origin U ⊂ T ∗ D n × C , and a symplectic embedding φ : U → X × C which respects the splitting so that φ (0) = q and • K (cid:48) a Morse Lagrangian cobordism exactly homotopic to K so that K (cid:48) | U = K k,n − k +1 loc | U . Furthermore, the critical points of K (cid:48) are in bijection with thecritical points of K .Proof. At q take the Lagrangian tangent space T q K ⊂ T π X ( q ) X ⊕ T π C ( x ) C . Since q isa critical point of π R , we have that T q K ⊂ T π X ( q ) X ⊕ R . By dimension counting, anyset of vectors ∂v , ∂ v , . . . , ∂ v n ∈ T q K with the property that ( π X ) ∗ ( ∂ v ) (cid:54) = 0 for all v i cannot form a basis of a Lagrangian subspace for π X ( T q K ); therefore, there exists a vector ∂ s ∈ T q K so that π X ( ∂ s ) = 0, and we may split T q K = ( π X ) ∗ ( T q K ) ⊕ R . Choose aDarboux chart U ⊂ T ∗ R n × C , φ : U → X × C which respects the product decomposition,and has φ ∗ ( R n × R ) = T q K . Write K for R n × R | U . Because K and K have the sametangent space, we can further restrict to a Weinstein neighborhood of K so that K is an AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 29 s − s VD nr D nr D nr K ------- WU Figure 13. Nested neighborhoods for interpolatingprimitives. At the end of interpolation: our La-grangian handle matches the standard surgery han-dle over U ; the region V is used to interpolate be-tween g = ∂ s G and f = ∂ s F ; the region W is sub-sequently used to correct h c so that its integral overthe s -coordinate matches F at the boundary of K .exact section of the tangent bundle B ∗ (cid:15) K . By taking a possibly smaller neighborhood, wewill identify K = D nr × [ − s , s ], where the s -coordinate denotes the R direction. Let F be the primitive of this section, so that K | B ∗ K = dF .If we let s be the coordinate on K which travels in the R direction, then we can compute π R : K → R at q ∈ K by f ( q ) := π R ( q ) = ∂ s F ( q ) , which is a Morse function on K with a single critical point.We now implement the handle in this neighborhood. Let G k,n − k +1 : K → R be theprimitive for the handle from definition 3.2.2, so that g := ∂ s G k,n − k is a Morse function on K . We will use claim 3.3.4 to obtain a function h c interpolating between f and g , and definea preliminary primitive H prec := (cid:82) h c ds . The section associated to H prec will satisfy all theMorse properties we desire; however it does not agree with F outside near the boundaryof K . This is because while f and h c agree at the boundary of K , there is no reason for (cid:82) ss h c ( x, s ) ds match (cid:82) ss f ( x, s ) ds = F ( x, s ) near the boundary of K . Therefore, we need toadd a correcting term to h c in order to make these integrals agree near the boundary of K .We set up this correction using a neighborhood as drawn in fig. 13. Take a set W = D nr × [ s / , s / ⊂ K . Let ρ : W → R ≥ r < r so that for all x, y ∈ D nr , ρ ( x, s ) = ρ ( y, s ).Furthermore, assume that (cid:82) { x }× [ s / , s / ρ ( x, s ) ds = 1. Let α = sup ( x,s ) ∈ K | dρ | . Let β =inf ( x,s ) ∈ W | df | . Since f has no critical points in W , this is greater than 0. To each choice of V ⊂ D nr × [ − s / , s / 2] and associated interpolation h c : V → R , we can define a function A c ( x ) := (cid:90) { x }× [ − s / ,s / ( h c ( x, s ) − f ( x, s )) ds. We may choose V small enough so that our interpolation satisfiessup x,c | A c ( x ) | < β α sup x | dA ( x ) | < β H c ( x, s ) = (cid:90) { x }× [ − s ,s ] h c ( x, s ) − A c ( x ) ρ ( x, s ) ds. Then ∂ s H ( x, s ) = h c ( x, s ) − A c ( x ) ρ ( x, s ). We have that d∂ s H = dh c − d ( A c ( x ) ρ ). Byconstruction | d ( A c ( x ) ρ ) | < | dh c | inside the region W , and A c ( x ) ρ vanishes outside of W .It follows that ∂ s H ( x, s ) has no critical points outside of V . The derivative ∂ s H ( x, s ) isMorse, agrees with g in a neighborhood of the origin, where it has a single critical point.Furthermore, near the boundary of K , we have H c ( x, s ) = F ( x, s ) for all ( x, s ) ∈ K \ (cid:0) D nr × [ − s / , s / (cid:1) . Consider the Lagrangian section of T ∗ K given by dH . This Lagrangian section is exactlyisotopic to K | T ∗ K , with exact primitive vanishing at the boundary. We therefore have anexactly homotopic family of Lagrangian cobordisms K c := K \ ( K | T ∗ K ) ∪ dH c , where Crit( π R : K → R ) = Crit( π R : K → R ) and K | T ∗ K = K k,n − k +1 loc | T ∗ K . (cid:3) Cobordisms are concatenations of surgeries. We now prove that every Lagrangiancobordism is exactly homotopic to the concatenation of standard surgery handles. Proof of theorem 3.3.1. Let K : L + (cid:32) L − be a Lagrangian cobordism. After applicationof claim 3.3.2, we obtain K (cid:48) , a Lagrangian cobordism exactly homotopic to K with theproperty that π R : K (cid:48) → R is Morse with distinct critical values. Enumerate the criticalpoints { q i } li =1 = Crit( π R : K (cid:48) → R ). By proposition 3.3.5, we may furthermore assumethat K (cid:48) is constructed so that there exists symplectic neighborhoods U i = T ∗ D n × T ∗ D so that K (cid:48) | U i = K k i ,n − k i +1 loc .For each q i take (cid:15) i small enough so that the ( q i − (cid:15) i , q i + (cid:15) i ) are disjoint. Take r i smallenough so that the ball of radius r i centered at the critical point q i is contained within thecharts U i . K (cid:48) is exactly homotopic to the composition K (cid:107) ( −∞ ,q l − (cid:15) ] ◦ K (cid:107) [ q l − (cid:15),q l + (cid:15) ] ◦ K (cid:107) [ q l + (cid:15),q l − + (cid:15) ] ◦ · · · ◦ K (cid:107) [ q + (cid:15),q − (cid:15) ] ◦ K (cid:107) [ q − (cid:15),q + (cid:15) ] ◦ K (cid:107) [ q + (cid:15), ∞ ) . By applying proposition 3.1.7 on K (cid:48) (cid:107) ( q i − (cid:15) i ,q i + (cid:15) i ) at the dividing hypersurface M i := ( x , . . . , x n ) ∈ U i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) n (cid:88) j =0 x j ≤ r i we obtain an exact homotopy K (cid:48) (cid:107) ( q i − (cid:15) i ,q i + (cid:15) i ) ∼ K − i ◦ K iM i ◦ K + i where K ± i are cylindrical, and K k i ,n − k +1 i := K iM i are Lagrangian surgery traces.For each i , let K H it = K (cid:107) [ q i +1 + (cid:15),q i − (cid:15) ] ◦ K + i +1 ◦ K − i be the suspension of an exact homotopy.Then K ∼ K H jt ◦ K k j ,n − k j +1 j ◦ K H j − t ◦ · · · ◦ K H t ◦ K k ,n − k +11 ◦ K H t . (cid:3) Finally, we make a remark about anti-surgery versus surgery. We’ve shown that every La-grangian cobordism can be decomposed as a sequence of exact homotopies and Lagrangiansurgery traces; in particular, the Lagrangian anti-surgery trace K − k,n − k +1 can be rewrittenas a Lagrangian surgery and exact isotopy. An anti-surgery takes an embedded Lagrangian AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 31 Exact HomotopySurgeryExact Homotopy Figure 14. Rewriting an anti-surgery as exact homotopies and surgeries. L and adds a self-intersection; one can equivalently think of this as starting with an em-bedded Lagrangian, applying an exact homotopy to obtain a pair of self-intersection points,and then surgering away one of the self-intersections. This is drawn in fig. 144. Teardrops on Lagrangian cobordisms One of the main observations about this decomposition is that every critical point of π R : K → R is paired with a self intersection of K . On the chain level, this reflects thatwhenever K : L + (cid:32) L − is an oriented embedded Lagrangian cobordism, χ ( L + ) = χ ( K ) = χ ( L − ).4.1. Grading of Self-intersections, and an observed pairing on cochains. We re-cover this equality of Euler characteristic on the chain level for Lagrangian cobordisms withself-intersections. Suppose that X has a nowhere vanishing section Ω of Λ n C ( T ∗ M, J M ). Wesay that i : L → X is graded if there exists a function θ : L → R so that the determinantmap det : L → S q (cid:55)→ Ω( T q L ) ⊗ / | Ω( T q L ) | can be expressed as det( q ) = e πθ . In this setting, we can define a generalized Eulercharacteristic for immersed Lagrangian submanifolds. Let I si ( L ) = { ( p → q ) | p, q ∈ L, i ( p ) = i ( q ) } be the set of ordered self-intersections . Note that each self-intersection of L gives rise totwo elements of I si . The index of a self intersection ( p → q ) is defined asind( p → q ) := n + θ ( q ) − θ ( p ) − (cid:93) ( i ∗ T p L, i ∗ T q L ) , where (cid:93) ( V, W ) is the K¨ahler angle between two Lagrangian subspaces. We particularlysuggest reading the exposition in [AB20] on computation of this index. Claim 4.1.1 (Index of Handle Self Intersections) . Consider the local Lagrangian handle j k,n − k : L k,n − k, + → C n . The notation reflects our interpretation of each element ( p → q ) as being a short Hamiltonian chordstarting at p and ending at q . Equip C n with the standard holomorphic volume form. The index of the self intersectionsare ind( q − → q + ) = n − k − . ind( q + → q − ) = k + 1 . Proof. To reduce clutter, we write j for j k,n − k +1 . The tangent subspace to the surface x + (cid:80) ni =1 σ i,k x i = 1 in D n × D at a point γ ( θ ) = (sin( θ ) , , . . . , , cos( θ )) is spanned bythe basis { cos( θ ) ∂ − sin( θ ) ∂ , ∂ , ∂ , . . . , ∂ n } . Let e , . . . , e n be the standard basis of T C C n .Then i ∗ ( ∂ ) = (cid:88) i σ i,k x i e i j ∗ ( ∂ i ) =(1 + 2 σ i,k x ) e i for i = 1 , . . . , n so that the tangent space at T j ( γ ( θ )) L k,n − k, + loc is spanned by vectors (cid:126)v i ( θ ), where (cid:126)v ( θ ) = (cid:32) cos( θ ) + 2 (cid:32) n (cid:88) i =1 σ i,k (cos ( θ ) − sin ( θ ) (cid:33)(cid:33) e (cid:126)v i ( θ ) =(1 + 2 σ i,k cos( θ )) e i For i = 2 , . . . , n .Let z i ( θ ) be coefficients so that (cid:126)v i ( θ ) = z i ( θ ) e i . Then arg( z i ( θ )) is decreasing for i ≤ k andincreasing for i > k . The endpoints of the z i are given by z (0) = 1 + 2 z ( π ) = − z i (0) = 1 + 2 σ i,k z i ( π ) = 1 − σ i,k . To compute the index of ( q − → q + ), we complete the path z i ( θ ) / | z i ( θ ) | to a loop bytaking the short path, and sum the total argument swept out by each of the z i . For ease ofcomputation, let α = arctan(2). • When i = 1, the loop z ( θ ) / | z ( θ ) | sweeps out − π − α radians; the short pathcompletion yields a contribution of − π to the total index of this loop. • When 1 < i ≤ k , the loop z i ( θ ) / | z i ( θ ) | sweeps out − α radians; the short pathcompletion yields a total contribution of 0 from this loop. • When k < i ≤ n , the loop z i ( θ ) / | z i ( θ ) | sweeps out 4 α radians; the short pathcompletion yields a total contribution of 2 π from this loop.The total argument swept out is ( n − k − · π . The index of the self-intersection isind( q − → q + ) = n − k − . By similar computation (or using duality) we see thatind( q + → q − ) = k + 1 . (cid:3) Let L be a graded Lagrangian submanifold, and f : L → R be a Morse function. The setof Floer generators is I ( L ) = Crit( f ) ∪I si ( L ). For each x ∈ Crit( f ), let ind( x ) be the Morseindex. Define the self-intersection Euler characteristic to be χ si ( L ) := (cid:80) x ∈I ( L ) ( − ind( x ) . Proposition 4.1.2. Let K : L + → L − be a Lagrangian submanifold. Then χ si ( L − ) = χ si ( L + ) . AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 33 Index I si ( L + ) Crit( f + ) Crit( f − ) k x + k + 1 ( q + → q − ) n − k − q − → q + ) x − (a) Floer cochains which belong solely to I ( L + ) and I ( L − ) for a Lagrangian surgery trace K k,n − k +1 : L + (cid:32) L + , when k < n Index I si ( L + ) Crit( f + ) − q − → q + )0 en xn + 1 ( q + → q − ) (b) In the degenerate case K n, , I ( L − ) isempty. Index I si ( L − ) Crit( f − ) − q − → q + )0 en xn + 1 ( q + → q − ) (c) In the degenerate case K − ,n +2 , I ( L + )is empty. Table 1. Chain level differences in Floer generators before and aftersurgery.As was pointed out to me by Ivan Smith, this also follows in the case that K is embeddedby the much simpler argument that the Euler characteristic is the signed self-intersection,and noting that we can choose a Hamiltonian push-off so the intersections of K ∩ φ ( K ) arein index preserving bijections with intersections L ± ∩ φ (cid:48) ( L ± ). Nevertheless, we give proofusing decomposition as this will motivate section 4.4. Proof. As each exact homotopy preserves χ si ( L − ), we need only check the case that K k,n − k +1 : L + (cid:32) L − is a surgery trace. Choose f ± : L ± → R to be the standard Morsefunctions for a surgery configuration, with critical points x + ∈ Crit( L + ) and x − ∈ Crit( L − )representing the cohomology classes of the surgery handle. We break into three cases:when k = − , ≤ k < n and k = n . The differences between I ( L + ) , I ( L − ) are listed intable 1. From the values listed in table 1 it follows that χ si ( L − ) = χ si ( L + ). (cid:3) This computation leads to the following question: in what sense do the generators ( q + → q − ) and x + cancel on the level of cohomology, and can we extend proposition 4.1.2 to anequivalence of Floer theory. This requires a standard form for Lagrangian cobordism witha self intersection.4.2. Overview of immersed Floer cohomology. Problematically, the decompositiongiven by theorem 3.3.1 is not very useful for understanding Floer cohomology for Lagrangiancobordisms with immersed ends, as such Lagrangian cobordisms will not have transverseself-intersections. This is because the standard definition of Lagrangian cobordisms (def-inition 2.1.1) does not allow us to easily work with immersed Lagrangian ends. Rather,[MW18] gives a definition for a bottlenecked Lagrangian cobordism which gives a method forconcatenating Lagrangian cobordisms with immersed ends in a way that preserves transver-sality of self-intersections.4.2.1. Bottlenecked Lagrangian Cobordisms. Definition 4.2.1. Let i : L → X be an immersed Lagrangian with transverse self in-tersections. A bottleneck data for L is an extension of the immersion to the suspension ofan exact homotopy i t : L × I ⊂ X × T ∗ I with primitive H t : L × I → R satisfying thefollowing conditions: • Bottleneck: H = 0 and there exists a bound C ∈ R so that | dHdt | ≤ C everywhere. • Immersed away from 0: If i t ( q ) = i t ( q ) , then either q = q or t = 0 .For simplicity of notation , we will denote the data of a bottleneck by ( L, H t ) . We will frequently say that a Lagrangian submanifold K ⊂ X × C is bottlenecked at time t if K | [ t − (cid:15),t + (cid:15) ] is a bottleneck. Example 4.2.2 (Whitney Sphere) . The first interesting example of a bottleneck comesfrom the Whitney n -sphere L n, A ⊂ C n . We treat L n, A as a Lagrangian cobordism inside of C n − × C . The shadow projection π C : L n, A → C is drawn in fig. 7. The bottleneck occursat t = 0 , and the bottleneck Lagrangian is the Whitney ( n − -sphere L n − , A ⊂ C n − . Thisbottleneck corresponds to the exact homotopy of i n − , A ( t ) : S n − r ( t ) → C n − where A ( t ) = (1 − t ) , and r ( t ) = √ − t . The primitive for this exact homotopy is H t = 2 x t . Each L n, A ( t ) contain a pair of distinguished points, q ± = ( ± r ( t ) , , . . . , , whichcorrespond to the self intersection of the Whitney ( n − sphere. Note that q + is themaximum of H t on each slice, and q − is the minimum of H t on each slice. Given an immersed Lagrangian i : L → X with transverse self-intersections, there existsa standard way to produce a bottleneck. Pick a local Weinstein neighborhood φ : B ∗ r L → X . Let h : L → X be a function such that | dh | < r , and that h ( p ) (cid:54) = h ( q ) whenever( p → q ) ∈ I si ( L ). Let H t = t · h . Then the Lagrangian submanifold parameterized by L × ( − (cid:15), (cid:15) ) → X × T ∗ ( − (cid:15), (cid:15) )( q, t ) (cid:55)→ ( φ ( d ( t / · h ) q , t + H t ( q )))is an example of a bottleneck. At each self-intersection ( q → q ) ∈ I si ( L ), it will either bethe case that h ( q ) > h ( q ) or h ( q < q ).More generally, we say that ( q → q ) ∈ I si ( L ) has a maximum grading from the bot-tleneck ( K, H t ) if dH t dt ( q , > dH t dt ( q , 0) ; otherwise, we say that this generator receives anminimum grading in the base from the bottleneck. Remark 4.2.3. Our convention for maximal/minimal grading from the base is likely relatedto the convention of positive/negative perturbations chosen in [BC20, Remark 3.2.1]. Example 4.2.4 (Whitney Sphere, revisited) . For the bottleneck i n − , A ( t ) : S n − r ( t ) → C n − constructed in example 4.2.2 , dH t dt ( q ± ) (cid:12)(cid:12)(cid:12)(cid:12) t =0 = ± so that ( q − → q + ) has a maximum grading in the base. The primitive of an exact homotopy doesn’t determine the exact homotopy i t ; however, many propertiesof the bottleneck are determined by H t . AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 35 We can construct another bottleneck data so that ( q − → q + ) has a minimum grading inthe base. As H ( S n − ) = 0 , all Lagrangian homotopies are exact homotopies. Consider thehomotopy i n − , s ( t ) : L n − , B ( t ) → C n − , where B ( t ) = 1+ t , and t ∈ [ − , . This bottleneck is theexact homotopy of which first decreases the radius of the Whitney sphere, then increases theradius of the Whitney sphere. For this bottleneck, ( q − → q + ) inherits a minimum gradingfrom the base.While both i n − , A ( t ) , i n − , B ( t ) provide bottlenecks for the Whitney sphere, they are really quitedifferent as Lagrangians in C n − × C . The first bottleneck can be completed to a null-cobordism by simply adding in two caps (yielding the Whitney n -sphere in C n ); the secondbottleneck cannot be closed off without adding in either a handle or another self-intersection. Remark 4.2.5. Although not relevant to the discussion of bottlenecks, the above examplegives us an opportunity to address the discussion at the end of [Hau20, Section 3.4] re-lated to Whitney degenerations and Lagrangian surgery. The question Haug asks is: Doescontaining a Whitney isotropic sphere suffice for implanting a surgery model? Haug showsthat this is not sufficient condition. The specific example considered is the Whitney sphere L , , + ⊂ C , which contains a -Whitney isotropic L , , + ⊂ L , , + ⊂ C . If there was aLagrangian surgery trace K , : L , , + (cid:32) L − which collapsed the 1-Whitney isotropic, then L − would have the topology of an embedded pair of spheres. Since no such Lagrangian sub-manifold exists in C , we conclude that possessing a -Whitney isotropic does not sufficefor implanting a surgery handle.Upon a closer examination, we see that the Whitney 1-isotropic has a small normalneighborhood L , , + × I ⊂ L , , + ⊂ C × C which gives it the structure of a Lagrangianbottleneck. This is the bottleneck i n − , A ( t ) described above.Consider instead a Lagrangian submanifold L + which contains a Whitney k -isotropic L k, , + × I ⊂ L with a neighborhood giving it the structure of the i n − , B ( t ) bottleneck. Thenthere exists a Lagrangian surgery trace K k,n − k : L + (cid:32) L − . This can be immediatelyobserved for instance in fig. 11 — the right hand side is exactly isotopic to i , B ( t ) (with thecobordism parameter in the vertical direction). As the example shows, the data constructing a bottleneck involves taking non-trivialchoices of gradings on the self-intersection points of the bottleneck. We therefore use doublebottlenecks instead. Definition 4.2.6. Let i : L → X be an immersion with transverse self-intersections, and h : L → R , φ : B ∗ r L → X as in the construction of a standard bottleneck. Furthermore,assume that dh = 0 at each self-intersection point. Let ρ ( t ) = t ( t + (cid:15) )( t − (cid:15) ) , and ˜ H t ( q ) := ρ (cid:48) ( t ) h ( q ) . A standard double bottleneck is the suspension of an exact homotopy: i t : L × ( − (cid:15), (cid:15) ) → X × T ∗ ( − (cid:15), (cid:15) )( q, t ) (cid:55)→ ( φ ( d ( ρ ( t ) · h ) q , t + ˜ H t )) . with the property that π X i t ( q ) = π X i t ( q ) if and only if i ( q ) = i ( q ) and t = ± (cid:15)/ √ . Each self-intersection ( q → q ) ∈ I si ( L ) corresponds to two self-intersections(( q , ± (cid:15)/ √ → ( q , ± (cid:15)/ √ I si ( L, H t ); if (( q , (cid:15)/ √ → ( q , (cid:15)/ √ q , − (cid:15)/ √ → ( q , − (cid:15)/ √ the base (and vice versa). To each immersed point ( q → q ) ∈ I si ( L ), we can associate avalue: A ( q → q ) := (cid:90) (cid:15)/ √ (cid:15)/ √ ˜ H t ( q ) − ˜ H t ( q ) dt, Finally, we observe that for each ( q → q ), the curves t + ˜ H t ( q ) and t + ˜ H t ( q ) bounda strip in u : R × I → C whose area is A ( q → q ) . Since q , q are critical points of h , theyare fixed by the homotopy and we obtain a holomorphic strip { i ( q ) } × u : R × I → X × C with boundary on the double bottleneck.4.2.2. Immersed Lagrangian Floer Cohomology. We begin this section with a disclaimerthat the following discussion is based on expectations for the Floer cochains of an immersedLagrangian submanifold. There are several models based on the work of [AJ08] whichproduce a filtered A ∞ algebra associated to an immersed Lagrangian i : L → X . We follownotation from [PW21] adapted to the Morse cochain setting, although much of our intuitionfor this filtered A ∞ algebra come from [FO97; BC08; LW14; CW19]. The construction of afiltered A ∞ algebra tailored to the particular set-up that we required will be the topic of afuture work which defines the theory of Lagrangian cobordisms with immersed ends.Let f : L → R be a Morse function, i : L → X be a graded Lagrangian immersion withtransverse self-intersections, and I si ( i ) = { ( p → q ) | p, q ∈ L, i ( p ) = i ( q ) } be the set ofordered preimages of transverse self-intersections. Assume that Crit( f ) ∩ I ( L ) = ∅ . Weassume that X is aspherical. Definition 4.2.7 ([Fuk+10]) . Let R be a commutative ring with unit. The universalNovikov ring over R is the set of formal sums Λ ≥ := (cid:40) ∞ (cid:88) i =0 a i T λ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ R, λ i ∈ R ≥ , lim i →∞ λ i = ∞ . (cid:41) Let k be a field. The Novikov Field is the set of formal sums Λ := (cid:40) ∞ (cid:88) i =0 a i T λ i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a i ∈ k, λ i ∈ R , lim i →∞ λ i = ∞ (cid:41) . An energy filtration on a graded Λ -module A • is a filtration F λ i A k so that • Each A k is complete with respect to the filtration and has a basis with valuation zeroover Λ . • Multiplication by T λ increases the filtration by λ . Associated to this data is a filtered graded Λ module generated on critical points of f and the ( p → q ). CF k ( L ) = (cid:77) x ∈ Crit( f )ind( x )= k Λ x ⊕ (cid:77) ( p → q ) ∈I si ( K )ind( p → q )= k Λ ( p → q ) . We will write I ( L ) = Crit( f ) ∪ I si ( L ) for the set of generators of CF • ( L ). The Floercohomology of L is a filtered A ∞ algebra whose cochains are CF • ( L ) and whose productoperations deform the Morse algebra structure by counts of holomorphic polygons withboundary on L . AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 37 More concretely, an abstract perturbation P for a Lagrangian submanifold L with Morsefunction f is an assignment of a manifold with boundary and corners M P ( L, β, x i )to each choice of homology class β ∈ H ( X, L ) and x = x , x , . . . , x k ∈ Crit( f ). Thisperturbation is graded ifdim( M P ( L, β, x i )) = 2 − k + ind( x ) − k (cid:88) i =1 ind( x k ) . We say this perturbation is coherent if the boundary stratification of the moduli spaces is ∂ ¯ M P ( L, β, x ) = (cid:91) β + β = βy ◦ j z = x M P ( L, β , y ) × M P ( L, β , z )where y ◦ j z = x means that z = y j and { z ; z , . . . , z j − , y , . . . , y m , z j +1 , . . . , z n } = x We associate to a coherent abstract perturbation P for a Lagrangian submanifold L an A ∞ algebra CF •P ( L ) by taking count of points in the zero-dimensional moduli spaces, m k ( x , . . . , x k ) = (cid:88) β ∈ H ( X,L ) T ω ( β ) M P ( L, β, x i ) · x . When the choice of abstract perturbation is unimportant, we will simple write CF • ( L ). Aperturbation scheme is a collection of coherent perturbations P = {P} which satisfy someadditional properties. A main criteria is that these perturbation should extend geometricdata from L whenever we are able to find regularizing data explicitly computing the modulispace of treed disks. Following [Hic19], we make the following definition: Definition 4.2.8 (Geometric Pearly Flowlines) . Let ( X, ω, J ) be a symplectic manifoldequipped with a choice of almost complex structure and L ⊂ X a Lagrangian submanifold.Let x , x ∈ I ( L ) . We say that L has regular pearly flow lines at energy c if whenever β ∈ H ( X, L ) is a homology class with ω ( β ) ≤ c , the moduli space of pearly flow lines M J ( L, β, x , x ) is regularly cut out by the ¯ ∂ J equation without perturbations. We say thata perturbation scheme is geometric if whenever L has regular pearly flow lines at energy c ,there exists a perturbation P ∈ P satisfying the property that whenever ω ( β ) < c , M P ( L, β, x , x ) = M J ( L, β, x , x ) . For example, the space of Morse flow lines is always transversely cut out (provided that f is Morse-Smale), so L always has regular pearly flow lines at energy 0. Assumption 4.2.9 (Existence of a geometric extension) . There exists a geometric pertur-bation scheme for Lagrangian submanifolds L equipped with Morse functions f . Figure 15. m structure on CF • ( S n ) with n ≥ 2. There are noMorse flow lines contributing the differential on the Floer cohomol-ogy. However, the presence of holomorphic teardrops contributesto differential (drawn in red) cancelling all homology classes. 0 ex ( q + → q − )( q − → q + ) nn + 1 − CF • ( L ) is a deformation of CM • ( L ) ⊕ (cid:76) ( p → q ) ∈I si Λ ( p → q ) .We will later add additional hypothesis on our perturbation scheme to make homotopy typeof CF • ( L ) independent of perturbation chosen.Lagrangian surgery can be interpreted as a geometric deformation of a Lagrangian sub-manifold. To understand the conjectured equivalent deformation on Floer cohomology, weneed to discuss deformations of filtered A ∞ algebras. Definition 4.2.10. Let A be a filtered A ∞ algebra. Let d ∈ A be an element with val( d ) > .The d -deformed algebra ( A, d ) is the filtered A ∞ algebra whose chains match A , and whoseproduct is given by m k d ( a , . . . a k ) = ∞ (cid:88) i =0 (cid:88) i + ... + i k = i m k ( d ⊗ i ⊗ a ⊗ d ⊗ i ⊗ · · · ⊗ d ⊗ i k − ⊗ a k ⊗ d ⊗ i k ) . We say that b is a bounding cochain or is a solution to the Maurer-Cartan equation if m b = 0 . When discussing the Lagrangian Floer cohomology, we will write CF • ( L, d ) for the La-grangian Floer cohomology deformed by the element d .4.2.3. Example Computation: Whitney Sphere. We review a computation from [AB20] com-puting the Floer complex for the Whitney sphere L n, A ⊂ C n . We assume that n ≥ L n, A is a graded Lagrangian submanifold. We take a Morse function for the L n, A givenby the x coordinate. Then Crit( f ) = { e, x } , where e is the maximum of f in degree 0,and x is a generator in degree n . We take the points q ± = ( ± r ( A ) , , . . . , ∈ S nr ( A ) tobe the preimages of the self intersections of this Whitney sphere. The computation fromsection 4.1 shows that ( q + → q − ) has degree n + 1, and ( q − → q + ) has degree − p = ( x , . . . , x n , ∈ S n ⊂ R n × R be a point on the equatorialsphere. Then we can construct a holomorphic teardrop with boundary on the Whitneysphere L n, A ⊂ C n which is parameterized by u p : D α → C n z (cid:55)→ ( x z, x z, . . . , x n z )where D α = { a + b | a ∈ [0 , , | b | ≤ a √ − a } . Let β ∈ H ( X, L ) be the homology classof this teardrop. The parameter A of the Whitney sphere is constructed so that ω ( β ) = A. AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 39 ( π → e (0 → π ) xT E − T E Figure 16. The chain complex CF • ( L E ). The black lines representthe Morse Flow lines, while the red edges represent contributions tothe differential coming from holomorphic strips. Claim 4.2.11. The moduli space of holomorphic teardrops with boundary on S n has onecomponent, M J ( L n, A , β, ( q − → q + )) = S n − where β is the class of the teardrop [ u p ] . The evaluation map ev : M J ( L n, A , β, ( q − → q + )) × ( D \ { } ) → L n, A is a homeomorphism onto L n, A \ { q + , q − } . We now place this computation in context with the observation of table 1. In that table,a matching was given on the chain level between critical points of the surgery handles andthe self-intersections of the handles. The above computation shows that in the settingof L n, A , holomorphic teardrops geometrically realize the pairing between self-intersectionsand critical points of the surgery handle. Figure 15 contains a summary of generators anddifferentials of CF • ( L n, A ) which shows homology of the Whitney sphere vanishes. Thegoal of section 4.4 is to formalize this intuition by characterizing holomorphic teardropson the standard Lagrangian surgery trace. These holomorphic teardrops can be used inexamples of immersed Lagrangian cobordism with transverse self-intersection to constructcontinuation maps; a toy computation is given in section 4.5.4.2.4. Running example: Multisection of T ∗ S . We return to the Lagrangian submanifold L E ⊂ T ∗ S first defined in example 2.3.2 , parameterized by S → T ∗ S θ (cid:55)→ (cid:18) θ, E θ ) (cid:19) This is a Lagrangian submanifold with 1-self intersection where θ = 0 and θ = π . Withrespect to the standard holomorphic form on T ∗ S = C , the generator ( π → 0) has degree0 and (0 → π ) has degree 1. There are two holomorphic strips of area E from the ( π → → π ) generator. In this case there are no holomorphic disks or teardropswith boundary on CF • ( L E ) so we may compute HF • ( L E ), which is isomorphic as a vectorspace to H • ( S ) ⊕ H • ( S ). We note that we do not expect CF • ( L E ) to be homotopic as adifferential graded algebra or filtered A ∞ algebra to C • ( S ∪ S ).A computation which will be useful later is understanding Lagrangian intersection Floercohomology between L E and a section S E (cid:48) of the cotangent bundle. Here, S E (cid:48) is theLagrangian section parameterized by θ (cid:55)→ (cid:18) θ, E (cid:48) π (cid:19) , E (cid:48) π (0 → π ) pq A (0 → π ) pq C (0 → π ) pq B Figure 17. Areas of various holomorphic strips and triangles appearing inthe computation of CF • (( L E , b , S E (cid:48) ))as drawn in fig. 17. Let b := (cid:80) i a i T D i (0 → π ) ∈ CF • ( L E ) be a deforming cochain for L E , whose lowest order term is a T D . We compute CF • (( L E , b ) , S E (cid:48) ). The intersectionbetween these two Lagrangian submanifolds consists of two points, L E ∩ S E (cid:48) = { p, q } . There are two holomorphic strips with boundary on L E and S E (cid:48) which have area A and B .Additionally, there exists a holomorphic triangle with ends limiting to q, p and (0 → π ) ofarea C . The constants A, B, C, E, E (cid:48) satisfy the relation: B + E (cid:48) = E C The differential on CF • (( L E , b , S E (cid:48) )) is given by m ( p ) = T A − T B + a T D · T C + O (min( A, D + C )) . For this differential to vanish we obtain the constraint at lowest order that: D = E − E (cid:48) . Furthermore, if this condition is met, there exists some extension of T D (0 → π ) to abounding cochain b so that HF • (( L E , b , S E (cid:48) )) = H • ( S ). We will recover this boundingcochain from the Lagrangian surgery trace cobordism in section 6.1. This will require us tounderstand the Lagrangian Floer cohomology of cobordisms.4.3. Floer theory of Lagrangian cobordisms. Let K : L + (cid:32) L − be a Lagrangiancobordism, and let t + , t − be real values so that K | t ± = L ± are the ends of the cobordism.An admissible Morse function is a function f : K → R with the following properties: • Outward Pointing: For any s ∈ ( t + , ∞ ), we have grad( f ) · π ∗ R ( ∂ t ) | s > 0. Similarly,for any s ∈ ( −∞ , t − ), we have that grad( f ) · π ∗ R ( ∂ t ) | s < • Bottlenecks: grad( f ) is parallel to the submanifold K | t ± , equivalently ( π R ) ∗ grad( f ) =0.As a result, f | L ± are Morse functions for the ends, and we have a decomposition of Morsecochains as a vector space CM • ( K ) = CM • ( L + ) ⊕ CM • ( L − ) ⊕ (cid:77) x ∈ Crit( f ) ∩ π − R ( t − ,t + ) Λ x . AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 41 We now state a principle which we will later generalize to Floer theory. An admissible Morsefunction f can be considered as the perturbation of a Morse-Bott function with Morse-Bottmaximum along L ± . Observation 4.3.1 (Morse Bottleneck principle) . Let f : K → R be an admissibleMorse function for a cobordism K : L + (cid:32) L − . Consider a tuple x = y, x , . . . , x k , with y ∈ Crit( f | L + ) . Whenever at least one of the x i (cid:54)∈ Crit( f | L + ) , the space of Morse trees M J ( K, , x ) is empty. This allows us to conclude that CM • ( L ± , f | L ± ) ⊕ (cid:76) x ∈ Crit( f ) ∩ π − R ( t − ,t + ) Λ x are A ∞ idealsof CM • ( K, f ). Furthermore, the perturbations used to construct the Morse-tree modulispaces can be chosen so that whenever x ⊂ Crit( f | L ± ), we have an agreement of modulispace: M J ( L ± , , x ) = M J ( K, , x ) . As a result, the projection on chains π ± : CM • ( K, f ) → CM • ( L ± , f | L ± )is a morphism of A ∞ algebras.We expect the bottleneck principle to extend to the setting of Lagrangian cobordisms K : L + (cid:32) L − with transverse self-intersections. Since K is a Lagrangian cobordism withtransverse self-intersections , L ± must be embedded. We have a decomposition of the setof chains as: CF • ( K ) = CM • ( L + ) ⊕ CM • ( L − ) ⊕ (cid:77) x ∈I ( K ) ∩ π − R ( t − ,t + ) Λ x . The projection of π C : K | | t | >t ± → C fibers over the real axis, so by the open mappingprinciple holomorphic polygons with boundary on K must be contained within the fibers.This constraint gives us the Floer-bottleneck principle. Observation 4.3.2 (Floer-Bottleneck Principle) . Let K be an embedded Lagrangian cobor-dism. Suppose that f : K → R is an admissible Morse function. Consider a tuple x = y, x , . . . , x k , with y ∈ Crit( f | L + ) . If at least one of the x i (cid:54)∈ Crit( f | L + ) , then forthe standard choice of split almost complex structure the space of treed holomorphic disks M J ( K, β, x ) is empty. The second observation — that f is a perturbation of a Morse-Bott function whichthereby gives a matching of moduli spaces of Morse flow-trees — is expected to hold for thesetting of Lagrangian Floer cohomology as well. We state this as an assumption. Definition 4.3.3 (Floer-Bott Principle) . We say that abstract perturbation schemes P ± for L ± and P for K satisfy the Floer-Bott principle if given abstract perturbation P ± for L ± there exists a perturbation P for K so that for every tuple x = y, x , . . . , x k ∈ I ( K ) ,with y ∈ Crit( f | L ± ) , and β ∈ H ( X, L ± ) we have M P ( K, ( i ±∗ ) β, x ) = (cid:26) M P ± ( L ± , β, x ) if for all i , x i ∈ I ( L ± ) ∅ otherwise. Here, we are considering honest Lagrangian cobordisms, not Lagrangian cobordisms with bottlenecks,which will be addressed in section 5. K 1, 2 loc | K 1, 2 loc | Figure 18. Left: The shadow of K , | as projected to the first complexcoordinate, with bottleneck at the origin. Right: The Lagrangian K , | drawn as a section of T ∗ R . We say that abstract perturbation schemes P i for Lagrangians L i are mutually extendible if whenever there exists a Lagrangian cobordism K : L i (cid:32) L j , there exists a perturbationscheme P ij for K satisfying the Floer-Bott principle. Assumption 4.3.4. All of our abstract perturbation schemes are mutually extendible. From assumption 4.3.4 it follows that there exists A ∞ projections π ± : CF • ( K ) → CF • ( L ± ) which are deformations of the Morse homomorphisms, and from the arguments in[Hic19], mutually extendible geometric abstract perturbation schemes produce A ∞ algebrasfiltered A ∞ homotopy type is independent of the choice of perturbation.4.4. Existence of holomorphic teardrops. We will now look at immersed Lagrangiansubmanifolds with transverse self-intersections which can be written as the composition ofbottlenecked Lagrangian submanifolds. The methods used in proposition 3.1.4 can similarlybe used to give a decomposition of a Lagrangian submanifold K ⊂ X into bottleneckedLagrangian submanifolds. Proposition 4.4.1. Let K ⊂ X × C be a Lagrangian submanifold, and t − , t + regular valuesof the projection π R : K → R . Suppose that the exact homotopy of K | [ t ± − (cid:15),t ± + (cid:15) ] is generatedby h t : L ± × [ t ± − (cid:15), t ± + (cid:15) ] → R . Furthermore, suppose that over this region, h t ( x ) (cid:54) = h t ( y ) for each double point ( q → q ) of the slice K | t . Then there exist choices of Lagrangianbottlenecks ( L ± , H ± ) and a Lagrangian cobordism with doubled bottleneck: ( K (cid:107) bot [ t − ,t + ] , t + , t − ) : ( L + , H + t ) (cid:32) ( L − , H − t ) . While the standard surgery handle does not have transverse self-intersections, there isa geometrically pleasing construction of a bottlenecked Lagrangian surgery trace. The La-grangian submanifold K k +1 ,n − k +1 , + ⊂ C n × C can also be considered as a Lagrangiansurgery trace, as π R : K k +1 ,n − k +1 , + | π R < / → R has a single critical point. Furthermore, AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 43 this Lagrangian has a single self-intersection, forming a bottleneck under the C projec-tion: see the shadow projection drawn in fig. 18. By taking applying an exact isotopyto π R : K k +1 ,n − k +1 , + | π R < / , we obtain a double bottlenecked Lagrangian surgery trace K k,n − k +1 A,B ⊂ C n +1 which has 2 self intersections, and a single critical point of the height func-tion. The quantity A measures the symplectic area of the class of the teardrop on K k,n − k +1 A,B ,while B is the area of holomorphic strip associated with a doubled bottleneck. By followingthe method of proof in theorem 3.3.1, we obtain a doubled-bottleneck decomposition ofLagrangian cobordisms. Corollary 4.4.2. Let K : ( L + , ˜ H + t ) (cid:32) ( L − , ˜ H − t ) be a Lagrangian cobordism with doubledbottlenecks. Then there a sequence of Lagrangian cobordisms with doubled bottlenecks K H it :( L − i +1 , ˜ H − i +1 ,t ) (cid:32) ( L + i , ˜ H + i,t ) For i ∈ { , . . . , j } K k i ,n − k i +1 A i ,B i :( L + i , ˜ H + i,t ) (cid:32) ( L − i , ˜ H − i,t ) For i ∈ { , . . . j } which satisfy the following properties: • L − j +1 = L + and L +0 = L − • Each K k i ,n − k i +1 A i ,B i is a Lagrangian surgery trace with doubled bottlenecks; • Each K H it is the suspension of an exact homotopy and; • There is an exact homotopy between K ∼ K H jt ◦ K k j ,n − k j +1 A j ,B j ◦ K H j − t ◦ · · · ◦ K H t ◦ K k ,n − k +1 A ,B ◦ K H t . Teardrops on Surgery Traces. We now examine the Floer cohomology of the bot-tlenecked surgery handle K k,n − k +1 A,B . With the standard choice of Morse function π R : K k,n − k +1 A,B → R , this handle has a single critical point y of index k . Additionally, thishandle has 2 self-intersections, giving 4 generators for the Lagrangian Floer cohomologyrelative ends. Two of the generators correspond to the negative end of the doubled bottle-neck, and two generators sits at the positive end of the doubled bottleneck. We discard thegenerator with a minimum grading from the base above the positive end of the bottleneck,and call the remaining generators ( q − → q + ) − , ( q + → q − ) − and ( q − → q + ) + . Their indexesare given in fig. 19. We enhance proposition 4.1.2 by computing the moduli space of treeddisks. Theorem 4.4.3. Let K k,n − k +1 A,B be the local model of the doubled bottleneck Lagrangiansurgery trace. Take the standard Morse function π R : K k,n − k +1 A,B → R , and standard choiceof almost complex structure on C n +1 . Let β A be the class of teardrop with boundary on K k,n − k +1 A,B which has boundary on the Whitney isotropic contained in K k,n − k +1 A,B | . The modulispace M J ( K k,n − k +1 A,B , β A , ( q − → q + ) − , y ) is comprised as a single regular treed teardrop.Proof. The space of holomorphic teardrops with input on ( q − → q + ) − can be described bythe space of holomorphic teardrops with boundary on the isotropic k -sphere contained in K k,n − k +1 , + A . By claim 4.2.11, the moduli space of such teardrops has an evaluation mapwhich sweeps out the homology class of y . (cid:3) In section 5 we will justify why the generator ( q + → q − ) + is not included. ( q − → q + ) y y ( q + → q − ) − C Index k + 2 k + 1 n − k − q − → q + ) − ( q + → q − ) + T B n − k T A m if k = 0 T B Figure 19. Floer cohomology of the bottlenecked surgery trace K k,n − k +1 A,B .The holomorphic teardrop is marked in orange and has area A , the holomor-phic strip from the bottleneck is marked in blue and has area B .This leads to the appearance of a term in the differential of the Floer complex of K k,n − k +1 A,B ;see fig. 18. The most interesting example of this occurs when we take the trace of thestandard Polterovich surgery, when k = 0. In this setting, the Whitney isotropic is a 1-dimensional sphere, and so the space of holomorphic teardrops is zero-dimensional. Thisgives us an additional contribution which we can count. Claim 4.4.4. For the standard choice of J , M J ( K ,n +1 A,B , β A , ( q + → q − ) − ) consists of asingle holomorphic teardrop. The presence of this contribution will turn on a non-trivial curvature term in the A ∞ algebra which needs to be cancelled. A necessary criteria for this cancellation is B < A .4.5. A return to example 2.3.2. We now return to a variation of example 2.3.2. Con-sider the Lagrangian cobordism K A + ,A − which is obtained concatenating the following La-grangian submanifolds: • A Lagrangian anti-surgery trace ( K , A + ) − : S E + ∪ S − E + → L E , where L E is animmersed Lagrangian submanifold in T ∗ S . • A Lagrangian surgery trace K , A − : L E (cid:32) S E − ∪ S − E − .The slices and shadow of the Lagrangian cobordism K A + ,A − are drawn in fig. 20 (a) and(b). The relations between E + , E − , A + , A − and E are E − E + = A + E − E − = A − . The immersed Floer cohomology of K A + ,A − is a deformation of the Morse cohomology of K A + ,A − . We take the standard Morse function for the ends of the cobordism, which isgenerated on critical points: CM • ( S E ± ∪ S − E ± ) = Λ (cid:104) e ± , e ± , x ± , x ± (cid:105) . AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 45 K A + ,A − topologically is two pairs of pants sharing a common neck; we take the our Morsefunction to be a perturbation of the Morse-Bott function which has maximums along theends, and a minimally graded S along the common neck. The generators and 0-dimensionalflow lines contributing to m : CM • ( K A + ,A − ) → CM • ( K A + ,A − ) are drawn in fig. 20 (c).Pairs of cancelling flow lines are indicated with dotted lines. In addition to these Morseflow trees, the product structure on CF • ( K A + ,A − ) counts configurations of holomorphicteardrops. By theorem 4.4.3 both the surgery and anti-surgery trace contribute holomorphicteardrops with output on ( q + → q − ). The lowest order contributions to m and m arelisted in fig. 20 (d). The most important term is the curvature term, m = ( T A − T A )( q + → q − ) + O (min( A , A )) . We break into two cases.4.5.1. Case 1: A (cid:54) = A . In the event where A (cid:54) = A , the m : Λ → CF • ( K A + ,A − ) term isnon-zero. Furthermore, since min( A , A ) is the smallest area of a holomorphic teardrop,val( (cid:104) m k ( x , . . . , x k ) , ( q + → q − ) (cid:105) ) > min( A , A )for all x i satisfying val( x i ) > 0. It follows that there is no solution to the Maurer-Cartanequation, and CF • ( K A + ,A − ) is obstructed. Furthermore, clearly S E + ∪ S − E + is not isomor-phic to S E − ∪ S − E − as an object of the Fukaya category.4.5.2. Case 2: A = A . The more interesting example is to consider is when A = A . Inthis setting, the curvature term vanishes, and CF • ( K A + ,A − ) is tautologically unobstructed.A computation shows that the chain complex CF • ( K A + ,A − ) is a mapping cylinder; by usinghomotopy transfer theorem one can construct a A ∞ morphism ˆ i − : CF • ( S E − ∪ S − E − ) → CF • ( K A + ,A − ). The compositionΦ = π + ◦ ˆ i − : CF • ( S E − ∪ S − E − ) → CF • ( S E + ∪ S − E + )is a continuation map, reflecting that K A + ,A − should provide an equivalence of objects inthe Fukaya category.5. Speculation on Lagrangian cobordisms with immersed ends A general goal suggested by the previous two sections is that one should be able tocompute the Floer cohomology of a Lagrangian cobordism by decomposing it into a con-catenation of bottlenecked surgery traces and subsequently gluing together the associatedcontinuation maps over each component. We will not lay out this computation in this pa-per, which would require a rigorous understanding the Lagrangian Floer cohomology of aLagrangian cobordism with bottlenecks. However, we lay out our expectations for what CF • ( K ) should be for Lagrangian cobordisms K with bottlenecks here, so that (assumingthese expectations) we may proceed with a computation in section 6.1. The computationin section 6.1 is a proof-of-concept for the methods explored in section 6.2. ↑ ↑ ↑ ↑↑ ↑ ↑ ↑ Surgery Exact Homotopy Anti-Surgery C − E + /π − E − /πE − /π E + /πT ∗ S A − A + e − x +0 E − + x − y − e +0 x +0 E ++ x +1 e x ( q + → q − )( q + → q − ) y + e − x +0 E − + x − y − e +0 x +0 E ++ x +1 e x ( q + → q − )( q + → q − ) y + T A − T A + T A − T A + m m T A − T A + (a) Slices(b) Shadow(c) Morse Differential(d) Floer Differential Figure 20. Anti-surgery followed by surgery. AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 47 Speculation on bottlenecked Lagrangians. Our setup of Lagrangian intersectionFloer cohomology for Lagrangian cobordism with transverse self-intersections will not ex-tend to the Lagrangian surgery trace, as the end of the Lagrangian surgery trace does nothave a transverse self-intersection. The data of a bottleneck allows us to transfer some ofthese methods to the immersed setting. Remark 5.1.1. We stress at this junction that this framework is set-up to allow the com-putation of example of section 6.1. There remains a lot to be understood about immersedLagrangian submanifolds inside of Louiville domains, and therefore the theory of immersedLagrangian cobordisms. We anticipate that the “correct way” to define this Floer cohomol-ogy is to use a partially wrapped Fukaya category, where X × C is equipped with two stopspreventing wrapping in the base C component. However, technology from [GPS20; Syl19]is not currently adapted to handle the natural complications which arise from working withimmersed Lagrangians. Already the simple example of defining the wrapped Floer cohomol-ogy for L , × R ⊂ C × C is problematic due to the possibility of a non-compact family ofholomorphic teardrops upon choosing a perturbation (see example 5.1.2). Therefore, we usethe model of immersed Lagrangian cohomology from [PW21], and impose (admittedly arti-ficial) constraints on our Lagrangian submanifolds in order to perform some computations.It is our intention to replace this with a less ad-hoc computational method when developinga general theory of Lagrangian cobordisms with immersed ends, based on observations from[BC20]. We first give an example highlighting the limitations of naively employing the standarddefinition of CF • ( K ) for Lagrangian submanifolds with bottlenecks. Example 5.1.2. Consider the exact homotopy i n − , r ( t ) : S n − r ( t ) → C n − , where ˜ r ( t ) = (cid:112) − tan − ( t ) /π . The suspension of this exact homotopy is a Lagrangian cobordism with abottleneck at t = 0 . The single self-intersection gives rise to two generators in q + and q − ; thegenerator ( q − → q + ) receives a maximal grading in the base. The suspension K i n − , r ( t ) is thelimit of a rescaling of the Whitney sphere L n, ⊂ C n . The set of holomorphic teardrops withboundary on the Whitney sphere becomes a non-compact family of holomorphic teardropswith boundary on K i n − , r ( t ) and output on ( q + → q − ) . Already this example shows some of the subtleties of immersed Lagrangian submanifolds.For instance, it is a general expectation that the Lagrangian Floer cohomology should obeya Kunneth-type formula; for instance, if L ⊂ X and L ⊂ X are compact embeddedLagrangian submanifolds, there should be a Kunneth formula computing CF • ( L × L ) interms of CF • ( L ) and CF • ( L ). If we drop the conditions of compact and embedded, thenthe above example relates to the Lagrangian submanifold L , × R ⊂ C × C ; under theexpectation of a Kunneth formula, CF • ( L , × R ) should be homotopic to CF • ( L , ).In example 5.1.2, we can fix the non-compactness problem by removing the ( q + → q − )generator. If we do this, we create another problem, as the generators for CF • ( K ) wouldno longer match with the generators of the Floer theory of the bottleneck.We propose a remedy by employing doubled bottlenecks. Let ( L, ˜ H t ) be a doubledbottleneck, and let K ˜ H t ⊂ X × C be the suspension of the exact homotopy. We equip K ˜ H t with a Morse function f : K ˜ H t → R which is a perturbation of the Morse-Bott function ××× ×− u Figure 21. Required boundary conditions for a holomorphic polygon tohave inputs and outputs in I si,max . The intersections of the form ( • → • )receive a maximum grading in the base from the bottleneck. For the standardsplit holomorphic J , there is no u parameterizing a holomorphic polygon withoutput limiting to a maximum. − ( t − (cid:15)/ √ : K ˜ H t → R so we have a chain isomorphism CM • ( K ˜ H t ,f ) → CM • ( L, f | ).Let I si,max ( K ˜ H t ) denote the set of generators that have a maximal grading from the base. Claim 5.1.3. Let K ˜ H t ⊂ X × C be a doubled bottleneck, with bottlenecks at ± (cid:15)/ √ , and J be a split almost complex structure for X × C , so that the projection π C : X × C → C is J -holomorphic. Let u : D \ { ξ k } nk =1 → X × C be a regular holomorphic polygon withboundary on K ˜ H t and strip-like ends limiting to points in I si,max ( K ˜ H t ) . Then this polygonis completely contained within the fiber, that is ( π C ◦ u )( z ) ∈ {± (cid:15)/ √ } .Proof. Suppose for contradiction that π C ◦ u is not constant. By maximum principle andunique continuation, Re( π C ◦ u ) ⊂ R \ {± (cid:15)/ √ } . Without loss of generality, assume thatRe( π C ◦ u ) ⊂ [ (cid:15)/ √ , ∞ ). Let ( x → y ) ∈ I si,max ( K ˜ H t ) be the limit of the strip-like endcorresponding to the output, with π R ( x ) = π R ( y ) = (cid:15)/ √ 3. Let U x , U y ⊂ K ˜ H t be smalldisjoint neighborhoods of x and y . Then for sufficiently small c > 0, we have that u ( e πθ + c ) ∈ U x , and u ( e πθ − c ) ∈ U y . Since ( x → y ) has maximum grading in the base, for sufficientlysmall c > 0, Im( π C ◦ u )( e πθ + c ) > Im( π C ◦ u )( e πθ − c ) . Additionally, as c → 0, we have that Re( e πθ + c ) is decreasing. It follows that the map u isnot orientation preserving, and cannot be holomorphic. See fig. 21. (cid:3) This observation gives the following candidate definition for Lagrangian cobordisms withimmersed ends. Definition 5.1.4. A graded Lagrangian cobordism with doubled bottlenecks is a collectionof data ( K, t + , t − ) : ( L + , ˜ H + t ) (cid:32) ( L + , ˜ H + t ) , where K ⊂ X × R is a graded Lagrangiansubmanifold whose restrictions to neighborhoods of t ± are the doubled bottlenecks K | [ t ± − (cid:15)/ √ ,t ± + (cid:15)/ √ = ( L ± , ˜ H ± t ) . AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 49 Let f : K → R a Morse function bottlenecked at t + , t − . The bottlenecked Floer generators of ( K, t + , t − ) are: I bot ( K ) = Crit( f ) ∪ I si ( K | [ t + ,t − ] ) ∪ I si,max ( K | [ t + , ∞ ) ) ∪ I si,max ( K | ( −∞ ,t − ] ) . The double bottlenecked Floer complex is the graded Λ module CF • bot ( K ) = Λ (cid:104)I bot ( K ) (cid:105) . Remark 5.1.5. We now justify why only include the self-intersections that inherit a max-imal grading from the base as generators. Consider a Lagrangian cobordism K : L − → L + .If we were to compute the self-Lagrangian intersection Floer cohomology K , we would takea small Hamiltonian push off φ ( K ) which displaces the ends at infinity. When we choosea push off φ which is small (so that CF • ( K (cid:48) , φ ( K (cid:48) )) may be compared to the CF • bot ( K (cid:48) ) )but displaces the ends are appropriately wrapped over the C coordinate, the intersections of K ∩ φ ( K (cid:48) ) are in bijection with I bot ( K ) . We now provide some observations on the double bottlenecked Floer Complex. By essen-tially the same argument as claim 5.1.3, we obtain compactness of moduli spaces of treeddisks for split almost-complex structures. Claim 5.1.6. Let x ⊂ I bot ( K ) be a set of Floer cochains. If M J ( K, β, x ) is regular foralmost complex J which splits, then M J ( K, , x ) is compact. Showing that this criterion is enough to ensure compactness of moduli spaces upon choos-ing a regularizing perturbation scheme (which may not be split) requires an understandingof how coherence is achieved within the regularizing perturbation scheme.We now examine how polygons with boundary on K ˜ H t compare to polygons with bound-ary on L . Suppose we have a regular holomorphic strip u : D \ {− , } → X withboundary on L and input limiting to ( x → y ) ∈ I ( L ) and output limiting to ( x → y ) ∈ I ( L ). Furthermore, let us assume that this strip is index 0 and therefore isolated.Then u × { (cid:15)/ √ } : D \ {− , } → X × C parameterizes possibly non-regular holomorphicstrip with boundary on K ˜ H t . The regularity of this polygon is dependent on whether thegenerators (( x i , (cid:15)/ √ → ( y i , (cid:15)/ √ • If (( x i , (cid:15)/ √ → ( y i , (cid:15)/ √ ∈ I si,max ( K ˜ H t ), then the strip u × { (cid:15)/ √ } has index 0and is regular. • If (( x i , (cid:15)/ √ → ( y i , (cid:15)/ √ ∈ I si,min ( K ˜ H t ), then the strip u × { (cid:15)/ √ } has index 0and is regular. • If (( x , (cid:15)/ √ → ( y , (cid:15)/ √ ∈ I si,max ( K ˜ H t ), and (( x , (cid:15)/ √ → ( y , (cid:15)/ √ ∈I si,min ( K ˜ H t ) then the strip u × { (cid:15)/ √ } has index 1 and is regular. • If (( x , (cid:15)/ √ → ( y , (cid:15)/ √ ∈ I si,min ( K ˜ H t ), and (( x , (cid:15)/ √ → ( y , (cid:15)/ √ ∈I si,max ( K ˜ H t ), then the strip u × { (cid:15)/ √ } has index − − u of index 1) is of more interest. Themoduli space of holomorphic strips is a 1-dimensional, and is expected to have boundarygiven by breaking of holomorphic strips. The natural candidates for breaking are either of the points (( x i , − (cid:15)/ √ → ( y i , − (cid:15)/ √ • ( x , − (cid:15)/ √ → ( y , − (cid:15)/ √ 3) has minimal grading, it cannot be the input of a stripending at ( x , (cid:15)/ √ → ( y , ± (cid:15)/ √ • Similarly, as ( x , − (cid:15) √ → ( y , − (cid:15) √ 3) has maximal grading, it cannot be the outputof a strip starting at ( x + (cid:15)/ √ , y + (cid:15)/ √ J , the strip u sits in a one-dimensional family breaking at( x , − (cid:15) √ → ( y , (cid:15) √ Example 5.1.7. Consider the space T ∗ T . For a choice of Morse function f : T → R ,the section K f := df and the zero section K intersect transversely, and the Lagrangian in-tersection Floer cohomology CF • ( K f , K ) can be directly compared to the Morse Homology CM • ( T , f ) . Consider the Morse function f ( θ , θ ) = sin( θ )(2 + cos( θ )) ; this is the “stan-dard” height function for T . Notably, this is a non-example of a Morse-Smale function, asthere is a non-regular flow line between the two index one critical points.We identify T ∗ T with T ∗ S × C ∗ , and consider the projection of K f the second C ∗ coordinate. Let U = { z ∈ C ∗ | arg( z ) ∈ [ − (cid:15), π + (cid:15) ] } . The restrictions K f | U and K | U are examples of Lagrangian suspension cobordisms of sections of T ∗ S (where the cobordismparameter is given by the π θ , as opposed to π R , coordinate). Furthermore, K := ( K f ∪ K ) | U is an example of a double bottleneck, with bottlenecks occurring at θ = 0 and θ = π . Welabel the critical points of f as Crit( f ) = { e = (0 , , x π = (0 , π ) , e π = ( π, π ) , x = ( π, } ⊂ K f , and let { e (cid:48) , x (cid:48) , e (cid:48) π , x (cid:48) π } ⊂ K be the corresponding points in the zero section. The corre-sponding pairs (i.e. ( e → e (cid:48) ) ) label elements of I si ( K ) . We focus on the self-intersectionsof K that start on the K f sheet and end on the K sheet. Above bottleneck, we see selfintersections ( e → e (cid:48) ) and ( x → x (cid:48) ) ,which have maximal and minimal grading from thebase respectively. Above the π bottleneck, we see self intersections ( e π → e (cid:48) π ) and ( x π → x (cid:48) π ) ,which have minimal and maximal grading from the base respectively. There exists a holo-morphic strip from ( e → e (cid:48) ) to ( e π → e (cid:48) π ) and a holomorphic strip from ( x π → x (cid:48) π ) to ( x → x (cid:48) ) . There are two non-regular holomorphic strips starting at ( e π → e (cid:48) π ) andending at ( x π → x (cid:48) π ) . Between ( e → e (cid:48) ) and ( x → x (cid:48) ) there is a 1-dimensional fam-ily of strips. Upon choosing an appropriate perturbation of K (arising from perturbingthe Morse-Smale function f to a Morse function), we obtain a pair of regular holomor-phic strip from ( e → e (cid:48) ) to ( x π → x (cid:48) π ) . We note that the subcomplex on the generators Λ (cid:104) ( e → e (cid:48) )( x π → x (cid:48) π ) (cid:105) is isomorphic to the Lagrangian intersection Floer cohomologybetween the zero section and a push off in T ∗ S . This is summarized in fig. 23. These toy models lead us to the following conjecture: Conjecture 5.1.8 (Morse-Bott perturbations for doubled Bottlenecks) . Let ( L , ˜ H t ) bethe data of a doubled bottleneck. Consider the bijection i : I ( L ) → I bot ( K ˜ H t ) . Let P bea choice of regular abstract perturbation data for L . Then there exists a choice of regular AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 51 K f K ( e → e (cid:48) )( x (cid:48) → x )( e (cid:48) π → e π ) ( x π → x (cid:48) π )Index012 Figure 22. Generators arising from self intersection, starting on the K f branch and ending on the K branch of K .( e → e (cid:48) )( e π → e (cid:48) π )( x → x (cid:48) ) ( x π → x (cid:48) π ) (a) Before picking a regularizing perturba-tion data, we have both regular holomorphicstrips, and irregular holomorphic strips (rep-resented with a dashed line). The thick lineis a 1-dimensional family of strips. ( e → e (cid:48) )( e π → e (cid:48) π )( x → x (cid:48) ) ( x π → x (cid:48) π ) (b) After picking a regularizing perturbation,we recover holomorphic strips between ( e → e (cid:48) ) and ( x π → x (cid:48) π ). There now exists an A ∞ projection to the ideal spanned by ( e → e (cid:48) )and ( x π → x (cid:48) π ). Figure 23. Holomorphic strips with boundary on K ∪ K f before and afterchoosing regularizing perturbations. abstract perturbation data P for K ˜ H t so that for all β and x , M P ( L , β, x ) = M P ( K, β, i ( x )) . Provided that conjecture 5.1.8 holds, we can carry over much of the construction forembedded Lagrangian cobordisms to immersed Lagrangian cobordisms. We shall from here on out assume the conjecture. Let K ˜ H t be a double bottleneck, and CF • bot ( K ˜ H t ) be the A ∞ algebra generated on Crit( f ) ⊕ I max ( K ˜ H t ). For each self intersection ( x → y ), let A ( x → y ) bethe area of the holomorphic strip in the doubled bottleneck (as in the discussion followingdefinition 4.2.6). Then the map π : CF • bot ( K ˜ H t ) → CF • ( L )( x, (cid:15)/ (cid:112) (3)) (cid:55)→ x For x ∈ Crit( f )( x, (cid:15)/ √ → ( y, (cid:15)/ √ (cid:55)→ ( x → y ) For intersections in the right bottleneck( x, − (cid:15)/ √ → ( y, − (cid:15)/ √ (cid:55)→ T A ( x → y ) ( x → y ) For intersections in the left bottleneck . is (tautologically) an A ∞ projection. This leads us to the following definition for LagrangianFloer theory of bottlenecked Lagrangians. Definition 5.1.9. Let K : ( L + , ˜ H + t ) (cid:32) ( L − , ˜ H − t ) be a Lagrangian cobordism with doublebottleneck at t ± . Let f : K → R be an admissible Morse function. The relative Floercochains of K is the Λ -module CF • bot ( K ) generated on CF • bot ( K ) := Λ (cid:10)(cid:0) I si,min ( K ) ∩ π − ([ t − , t + ]) (cid:1) ∪ I si,max ( K ) ∪ Crit( f ) (cid:11) . We will now assume that there exists a version of the pearly model extending to CF • bot ( K ), and that π ± : CF • bot ( K ) → CF • ( L ± ) is a filtered A ∞ homomorphism.5.2. Example Computation: Continuation of from exact homotopy. Consider theLagrangian submanifold L E ⊂ C ∗ from section 4.2.4. We can take an exact homotopy K : L + → L − as drawn in fig. 24; this is a Lagrangian cobordism with doubled bottlenecks. Thesymplectic area of holomorphic strips in the slices are E i , and the areas of the holomorphicstrips transversing slices are C k . The relations between the E i and C k are E − C = E E + 2 C = E E − C = E For this example, we only examine the portion of CF • bot ( K ) which is generated by theself intersections, as the Morse component of CF • bot ( K ) is clearly cylindrical. These selfintersections are labelled in fig. 24. We label the generators of the ends of the cobordism I si ( L ± ) = { (0 → π ) ± , ( π → ± } , This is a mapping cylinder, whose projections to theends π ± : CF • bot ( K ) → CF • ( L ± ) are defined by their action on generators as: π − (( π → ) = ( π → − π + (( π → ) = T C (1 → ) + π − ((0 → π ) ) = T C (0 → π ) − π + ((0 → π ) ) = (0 → π ) + The homotopy inverse to π + is the inclusion i − : CF • ( L + ) → CF • bot ( K ) which is definedon generators by i + (( π → + ) = T − C ( π → + T − C + C − C ( π → i + ((0 → π ) + ) = ( π → + T C − C (0 → π ) with homotopy H : CF • bot ( K ) → CF • bot ( K ) defined up to first order by H ((0 → π ) ) = T − C (0 → π ) + O ( − C ) H (( π → ) = T − C ( π → + O ( − C ) AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 53 One point to be especially cautious about is the negative Novikov valuation in the homotopyinverse and homotopy. To construct the A ∞ map from the data of i + and H we use acurved A ∞ homotopy lemma. The requirement for convergence of the maps constructedis that H ◦ m has positive valuation. Since m = 0, this automatically holds. Let ˆ i + : CF • ( L + ) → CF • bot ( K ) be the A ∞ homomorphism constructed using the homotopy transfertheorem. Note that ∆ := E − E = − C + C − C is greater than zero. At lowest order, π − ◦ ˆ i + : CF • ( L + ) → CF • bot ( K ) sends: π − ◦ ˆ i + (( π → + ) = T ∆ ( π → − + O (∆) π − ◦ ˆ i + ((0 → π ) + ) = T − ∆ (0 → π ) − + O (∆) . Finally, we note that there is a pushforward map on Maurer-Cartan solutions ( π − ◦ ˆ i + ) ∗ : M C ( L + ) → M C ( L − ). In the case that this map has negative valuation, we can onlyensure convergence for elements b ∈ M C ( L + ) such that val(( π − ◦ ˆ i + )( b )) > 0. This occurswhen b = T B (0 → π ) + O ( B ), and B > ∆. At lowest order, ( π − ◦ ˆ i + ) ( T B (0 → π ) ) = T B − ∆ (0 → π ) − + O ( B − ∆), identifying( L E + , T B (0 → π ) + + O ( B )) (cid:39) ( L E − , T B − ∆ (0 → π ) − + O ( B − ∆))at lowest order as objects of the Fukaya category. This computation is verified by sec-tion 4.2.4 by checking the Lagrangian intersection Floer cohomology against a test La-grangian L E . 6. Computations and applications We conclude with a sample computation demonstrating some applications of theo-rems 3.3.1 and 4.4.3. The main idea is to show that each standard surgery trace yields aspecific continuation map in the Fukaya category, and subsequently provide an algorithmfor computing continuation maps from general Lagrangian cobordisms. The explicitconstruction of this continuation map will allow for the comparison of numerical invariants,such as boundary gap and shadow metric. Applications will extend beyond the setting ofLagrangian cobordisms to Lagrangians in Lefschetz fibrations.6.1. Continuations, obstructions, and mapping cones. We now use this to computecontinuation maps for immersed Lagrangian cobordisms from handle body decomposition.Consider the double-section Lagrangian L E b ⊂ T ∗ S discussed in section 4.2.4. Let S ± E bethe pair of sections ± E π dθ of T ∗ S . L E b has a single self-intersection that we may resolvevia surgery. This surgery comes with a bottlenecked surgery trace cobordism K A,B : L E b (cid:32) S E (cid:116) S − E . Here, A is the area of the small teardrop of the surgery, and B is the area ofthe strip associated to the bottleneck; the slice above the second bottleneck is denoted L E a .The relations between A, B, E a , E b and E are E a − E b = 2 B E a − E = 2 A. See fig. 25 (a, b) for slice and shadow of the Lagrangian cobordism. We equip K A,B with an admissible Morse function; this has critical points e , e , x , x near the negativeend, e a , e b , x a , x b corresponding to the Morse critical points of the slices L E a and L E b , andan additional critical point y from the surgery handle. The Lagrangian K A,B has two L E L E L E L E C C C ( π → (0 → π ) ( π → ( π → (0 → π ) (0 → π ) T C T C T C T C ± T E + C ± T E + C Index Figure 24. For a Lagrangian cobordism K : L E + (cid:32) L E − , the bottleneckedLagrangian Floer cohomology gives a mapping cylinder. The highlightedregions correspond to the generators living above each bottleneck. Only theportion of the complex generated by self intersections is drawn here.self intersections; adopting the notation from fig. 24 we call the 3 generators of the Floercohomology ( π → a , (0 → π ) a , and (0 → π ) b .6.1.1. Computation of low order products. As it is difficult to compute the full productstructure for Floer cochains of K A,B , we restrict ourselves to computing smallest-ordercontributions to the m and m terms on CF • bot ( K A,B ). These lowest order computationsare sufficient to prove unobstructedness of Floer groups by standard arguments using thefiltration of CF • bot ( K A,B ). • The differential on the Morse complex of CM • ( K A,B ) can be fully computed. TheMorse flowlines are drawn in fig. 25(c). The differential is represented by black dot-ted and solid arrows in fig. 25(d). The dotted arrows have canceling contributions. • By theorem 4.4.3, there exists a holomorphic teardrop which can either be consideredas having input on (0 → π ) a or output on ( π → a with area A . This contributesto both the m and m terms (as in claim 4.4.4): – The teardrop with output appears in the m term in the A ∞ structure m = T A · (0 → π ) a + ( O A ) AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 55 e x e x S r S r C ∗ C ∗ y A Bex e (cid:48) x (cid:48) e x x e yy e a x a e b x b ×× ×× a a b b m e x e x y e a x a (0 → π ) a ( π → a (0 → π ) b e b x bT B T A T A T A S E (cid:116) S − E L E a L E b ± T B + E b Index012(a) Slices(b) Shadow(c) Morse Function(d) Floer Complex Figure 25. Slices, shadow, Morse function and Floer complex of the surgerytrace cobordism. This is represented with the highlighted red term in fig. 25(d). Additionally,this teardrop contributes to the differential, (cid:104) m ( x ) , (0 → π ) a (cid:105) = T A + O ( A ) . – The teardrop with input on ( π → 0) gives a contribution to the differential (cid:104) m ( π → a , y (cid:105) = T A + O ( A ) . • The holomorphic strips contributing to the differential on CF • bot ( K A,B ) arise fromthe strips which appear on a doubled bottleneck. In particular, the strips from con-jecture 5.1.8 and constructed in this particular setting in example 5.1.7 contributeat lowest order: (cid:104) m ( π → a , (0 → π ) b (cid:105) = T B + E b − T B + E b + O ( B + E b ) . The doubled bottleneck also has a holomorphic strip pairing the generator which is“doubled” with maximal and minimal grading from the base: (cid:104) m (0 → π ) b , (0 → π ) a (cid:105) = T B + O ( B ) . These contributions are represented by red arrows in fig. 25(d). We obtain two cases whichwe consider separately: when A > B , and A < B . We discuss the Floer theoretic andgeometric implications below.6.1.2. Continuation maps and Unobstructedness: A > B . We first note that whenever A > B , the curvature term m can be cancelled out at lowest order by the differential, as m ( T A − B (0 → π ) b ) = T A (0 → π ) a + O ( A ) . We can build a bounding cochain order by order using the filtration on CF • bot ( K A,B ),whose first term is b A,B := T A − B (0 → π ) b + O ( A − B ) ∈ M C ( K A,B ) . This gives us a sufficient condition for this Lagrangian cobordism to be unobstructed: theflux swept out by the exact homotopy (as measured by the area B of the holomorphicstrip) must not be greater than the area of the surgery neck taken (as measured by theholomorphic teardrop A ). Since π + : CF • bot ( K A,B ) → CF • ( L E b ) is an A ∞ homomorphism,we obtain a bounding cochain π + ∗ b A,B for CF • ( L E,b ). We can verify this computation bychecking the Lagrangian intersection Floer cohomology against a test Lagrangian. Recallfrom section 4.2.4, the complex CF • ( L E b , π + ∗ b A,B ) , S E ) is acyclic whenever the lowest orderterm of π + ∗ ( b A,B ) is not T E − E (cid:48) . In this case, the lowest order term of b is T A − B = T Ea − E − Ea − Eb = T Eb − E . So at lowest order, CF • ( L E b , π + ∗ b A,B ) , S E ) has nontrivial homology.With this first-order verification that K A,B provides a continuation in the Fukaya cate-gory, we construct a continuation between the ends of the cobordism. First, we note that the AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 57 projection π + : CF • bot ( K A,B , b ) → CF • ( L E b ) is a homotopy equivalence. The homotopycan be described on generators as follows: let I − = { e , x , e , x }I = { y, e a , x a , (0 → π ) a }I + = { ( π → a , e b , x b , (0 → π ) b } . Let CF • bot ( K A,B ) | I ∗ be the subspace spanned by the appropriate set of generators. This isgenerally not an A ∞ ideal, however CF • bot ( K A,B ) | I ⊕ CF • bot ( K A,B ) | I + , is an A ∞ ideal. We additionally consider the map H : CF • bot ( K A,B ) | I + → CF • bot ( K A,B ) | I which on the given basis sends: H ( e a ) = e − e H ( y ) = e H ((0 → π ) a ) = T − A ( x − x ) H ( x a ) = x Let i + : CF • ( L | E b ) → CF • bot ( K A,B ) be the chain-level inclusion i + ( e b ) = e b + e − e i + ( x b ) = x b + x i + (( π → b ) = T − B (( π → a + T A e ) i + (0 → π ) b = (0 → π ) b + T B − A ( x − x )At lowest order, the map i + is a homotopy inverse to π + with homotopy given by H .By using the homotopy transfer theorem we may extend i + to an A ∞ homomorphismˆ i + : CF • ( L E b , ( π + ) ∗ b A,B ) → CF • bot ( K A,B , b A,B ). We combine the inclusion map with theprojection π − : CF • bot ( K A,B , b A,B ) → CF • ( L − , ( π − ) ∗ b A,B ) to obtain a continuation mapΦ := π − ◦ ˆ i + : CF • ( L E b , ( π + ) ∗ b A,B ) → CF • ( S E (cid:116) S − E , ( π − ) ∗ b A,B ) . One can furthermore use the continuation map to study how modifications of the bound-ing cochain on ( L E b , b ) relate to local systems on the Lagrangian S E , (cid:116) S − E . As Φ is aweakly filtered A ∞ homomorphism of energy loss B − A , there exists a pushforward mapon degree 1 Maurer-Cartan solutions with valuation greater than A − B Φ ∗ : { d ∈ M C ( L E b , b ) , val( (cid:104) d , (0 → π ) b (cid:105) ) > A − B } → M C ( S E (cid:116) S − E ) . Given an element d = c ( T A − B + (cid:15) + O ( A − B + (cid:15) ))((0 → π ) b ) + 2 c ( T (cid:15) + O ( (cid:15) )) x b , we cancompute the pushforward to first order:Φ ∗ ( d ) = c ( T (cid:15) + O ( (cid:15) )) x + c ( T (cid:15) + O ( (cid:15) )) x showing that Φ ∗ is a surjection. If we interpret graded deformations in M C (( L E b , b )) as alocal system, this suggests that all local systems on S E (cid:116) S − E are realized by deformationsof ( L E b , b )). There may be some concern that the valuation of H is negative. This is in general not a problem as longas H ◦ m has positive valuation. Since we have canceled out m by incorporation of a bounding cochain,we can use the homotopy transfer theorem here. Obstructedness: B > A . The other setting of interest is when B ≥ A , as drawn infig. 2. Then the Lagrangian cobordism K A,B is obstructed, in the sense that M C ( K A,B )is empty. In this setting, L E b is never isomorphic to S E (cid:116) S − E for any choice of deformingcochain, as E b < E . This is easily observed as L E b can be displaced from S E (cid:116) S − E by aHamiltonian isotopy.6.2. Future directions and applications. We outline some future directions of researchbased on both the decomposition given in theorem 3.3.1, teardrop existence proven intheorem 4.4.3, and sample computations from section 6.1.6.2.1. Theory of Lagrangian cobordisms with immersed ends. The sketch laid out in section 5describing the theory of Lagrangian cobordisms with immersed ends should developed insubstantially more detail. There are several approaches to handling the theory of Lagrangiancobordisms with immersed ends. One approach is to simply cap off the ends of Lagrangiancobordism with surgery handles, as to make the ends embedded. However, it is desirableto have a self-contained theory which does not require this kind of modification. In theevent where K is tautologically unobstructed, the framework in [BC20] provides a Floercohomology for Lagrangian cobordisms with immersed ends.We can see two possible approaches to defining a Floer theory for Lagrangian cobordismswith immersed ends. One approach is to use a Lagrangian intersection Floer model, insteadof a pearly model, for CF • ( L, L ). This has the potential upshot that the choices use toobtain compact moduli spaces are more-easily justified (as they match the choices usedin to define Lagrangian intersection Floer cohomology in the partially wrapped setting).Additionally, the existence of a projections π ± : CF • ( K, K ) → CF • ( L ± , L ± ) should followmore easily than in other methods. A difficulty remains in that teardrop bubbling in thissetting is still not well understood.A second approach would be to find a geometric justification for the choices made inthe definition of CF • bot ( K ). One possibility is that domain dependent Hamiltonian per-turbations can be chosen so that the doubled bottleneck naturally appears, perhaps byfollowing the techniques laid out in [BC20]. This would require a better understanding ofhow perturbation schemes for construction CF • bot ( L ) are coherently constructed.6.2.2. Surgery Exact Triangles. The clearest application of this machinery is to formalizethe connection between the surgery exact triangles from [Fuk+07] and [BC13], extendingwork of [Tan18]. Suppose Lagrangians L and L intersect transversely at a single point q .Then [Fuk+07] proves the existence of an exact triangle in the Fukaya category L → L → L q L , by showing that holomorphic n -gons with boundary on L , L , . . . , L n − are in bijectionwith n − L L ) , L , . . . , L n − . The proof uses a neck-stretching argument toshow that the corner of the triangle can be rounded off at the surgery neck. While the factthat L q L is a mapping cone of L → L can be recovered using [BC13], the rounding thecorner proof additional identifies the morphism of the mapping cone as q ∈ CF • ( L , L ).Theorem 4.4.3 provides a route to an alternate method of proof that avoids these neck-stretching arguments. The Lagrangian surgery cobordism K A,B ( L ∪ L ) → L q L can beequipped with a bounding cochain b , and theorem 4.4.3 states that the lowest order term of AGRANGIAN COBORDISMS AND LAGRANGIAN SURGERY 59 this bounding cochain must be T A − B q . This makes ( L ∪ L , π −∗ b ) isomorphic to L q L ,and clearly ( L ∪ L , π −∗ b ) is the A ∞ mapping cone of L q −→ L .6.3. Continuation maps from Lagrangian cobordism. Theorem 3.3.1 gives a methodfor recovering the iterated mapping cone decomposition of [BC08] at the lowest order bytaking an exact homotopy to an iterated composition of surgery traces. By first decomposinginto a concatenation of bottlenecked Lagrangian surgery traces, and then computing theholomorphic teardrops on each of those traces, we can hope to compute the iterated mappingcone decomposition associated to a Lagrangian cobordism inductively. The upshot is two-fold.Firstly, the machinery of bounding cochains will be naturally be incorporated into thedecomposition. While there is some progress in understanding unobstructed Lagrangiancobordisms [Hic19], the computation from section 6.1 lays out a general method for de-termining if a Lagrangian cobordism is unobstructed. After applying the decompositionfrom theorem 3.3.1 each surgery trace cobordism K iA,B : L i +1 (cid:32) L i places a constraint onsolutions to the Maurer Cartan solutions on K A,B at lowest order. Similarly, each exacthomotopy induces a continuation map on Maurer-Cartan solutions at the lowest order. Bytaking the intersection over all such constraints, we can determine if the K has a boundingcochain b .Secondly, the decomposition gives us an algorithmic method for computing continuationmaps. We expect that to every unobstructed ( K, b ) this algorithm proves the existence of acontinuation map CF • ( L + , π + b ) → CF • ( L − , π + b ), thereby extending the results of [BC08].This method has already been used in [Hic19] to show that the wall-crossing formula foropen GW invariants naturally arises from considering the Lagrangian mutation cobordism.This algorithm would additionally provide a foothold for attacking the conjectured relationbetween the cobordism group of the Fukaya category and the Grothendieck group of theFukaya category.6.3.1. Lagrangian Shadows and Metrics on Categories. Biran, Cornea, and Shelukhin definea metric on the space of Lagrangian submanifolds by taking the infimum over all areas ofLagrangian shadows of cobordisms K : L + (cid:32) L . We note that the decomposition fromtheorem 3.3.1 can be performed to modify the area of the shadow projection by as smallan amount as desired. On each component of the decomposition, the shadow area can bedirectly related to Floer theoretic energies as follows: • Over each exact homotopy, the shadow area is given by the H¨ofer norm and, • Over each surgery trace, the shadow area is given by the teardrop area.The above algorithm for computing continuation maps provides quantitative data on theFloer cohomology of Lagrangian submanifolds, in that the quantities A i , B i associated to thesurgery traces K A i ,B i explicitly determine lowest level valuations appearing the continuationmaps. We expect that this explicit data will be connected to the pseudo-metrics on theFukaya category defined by [CS+19]. This would potentially relate these pseudo-metricsto the shadow metric. In particular, the valuation of the continuation map associatedto the trace cobordism is exactly related to the teardrop area (and therefore shadow).Furthermore, we believe that the upcoming language of triangulated persistence categories(Biran, Cornea, and Zhang) will be the natural structure to formalize the computationsgiven in section 5. Novel Constructions of Lagrangian submanifolds. A major question in the study ofLagrangian submanifolds is simply what Lagrangian submanifolds are out there. For in-stance, there are no known examples of 2-ended non-cylindrical monotone Lagrangian cobor-disms. Provided that this property can be understood in terms of immersed Lagrangiancobordisms, the decomposition from theorem 3.3.1 should provide a route to the construc-tion of interesting 2-ended Lagrangian cobordisms. Such a cobordism would be of interestto the general symplectic geometry community as an example of a neither-wide-nor-narrowLagrangian submanifold, that is a monotone Lagrangian submanifold with HF • ( L ) (cid:54) = 0 , H • ( L ) . Already we can use this surgery viewpoint on Lagrangian submanifolds to produce inter-esting Lagrangian submanifolds. For example, in the very enjoyable paper [Eva20], Evansconjectures that there is a relation between the width c of( CP × CP , ω ⊕ ( cω )) , and the minimal genus of a (necessarily non-orientable) Lagrangian submanifold whose ho-mology class is (0 , ∈ H ( CP × CP , Z / Z ). A set of examples exhibiting this behavioris given using the machinery of tropical Lagrangian submanifolds. However, we note that CP × ( CP \ { , ∞} ) ⊂ CP × C , and therefore many examples can be considered as La-grangian cobordisms. In a discussion with Paul Biran, the author learned that a cobordismconstruction of the Klein bottle, and its relation to the shadow of a Lagrangian cobordism,had already been considered. Using Lagrangian surgery handles, we can construct a set ofexamples whose genus is c + 2. It would be interesting to learn if this bound is sharp fromconsiderations of Floer cohomology and Lagrangian shadow.6.3.3. Floer theory of Lagrangians in Lefschetz fibrations. Finally, we remark that we ex-pect the majority of the tools developed here to extend over to the setting of Lagrangiansin Lefschetz fibrations. In that environment, we conjecture that every Lagrangian sub-manifold can be decomposed into surgery traces, exact homotopies, and addition/removalof vanishing cycles. This decomposition would provide a geometric realization of the gen-eration statement from [Sei08]. Furthermore, in addition to yielding a decomposition ofevery Lagrangian submanifold into an iterated mapping cone complex of vanishing cycles,the methods outlined in section 6.1 would provide an algorithm for computing the Floercohomology of these Lagrangians at the lowest order. This would provide a computationaltool which addresses a wide set of Lagrangian submanifolds of general interest. References [AB20] Garrett Alston and Erkao Bao. Immersed Lagrangian Floer cohomology viapearly trajectories . 2020. arXiv: .[AJ08] Manabu Akaho and Dominic Joyce. “Immersed Lagrangian Floer theory”. Jour-nal of Differential Geometry Holomorphic curves in symplectic geometry .Springer, 1994, pp. 271–321.[Arn80] Vladimir Igorevich Arnol’d. “Lagrange and Legendre cobordisms. I”. FunctionalAnalysis and Its Applications EFERENCES 61 [BC08] Paul Biran and Octav Cornea. Lagrangian Quantum Homology . 2008. arXiv: .[BC13] Paul Biran and Octav Cornea. “Lagrangian cobordism. I”. 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