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The surgery equalities immediately lead to the existence of corresponding Lagrangian cobordisms. Note that in case 1 , a similar cobordism construction was established in  using Lefschetz fibrations independently. Remark 1. We also include a detailed discussion on gradings involved in Lagrangian surgeries. This benefits us in two aspects: we use a grading assumption to exclude bubblings in immersed Floer theory Section 7.
But we emphasize the grading is a vital part of the foundation of Lagrangian surgeries for an intrinsic reason. Consider the simple case when all involved Lagrangians are Z-graded and embedded, according to the cone relation proved in , the algebra instructs a surgery happen only at degree zero cocycles. This principle was noticed first by Paul Seidel . Such a principle interprets several known phenomena in a uniform way. When the resolved intersections have mixed degrees, in many cases this leads to obstructions in Floer theory, as exemplified in [12, Chapter 10].
In better situations when resolved intersections have zero degree mod N, the surgery at least results in collapse of gradings, which can also be checked directly on the Maslov classes. As for our applications, we extend this principle to clean surgeries. This matches well with predictions from homological algebra dictated by Lagrangian Floer theory with clean intersections .
It also extends the surgery exact sequence to clean intersection case. In another direction, since our construction holds for arbitrary symplectic manifolds, when combined with the general framework due to Fukaya-Oh-Ohta-Ono , it yields a proof of Seidel s exact sequence in arbitrary symplectic manifold.
This is part of an ongoing work . As a consequence of the cone relations in functor categories, we also consider the auto-equivalences of AutpFukpW qq, for W a Milnor fiber of ADE-type singularities The generalization of the result from A-type singularities to DE-type singularities was suggested to us by Ailsa Keating. In  it was proved that FukpW q is split generated by the vanishing cycles.
Moreover, in  it is shown that there is a braid group embedded into Symp c pw q when W is an A n -Milnor fiber induced by Dehn twists along the standard vanishing cycles. It is natural to ask whether this braid group indeed is the whole mapping class group, i. On the other hand,  5. As a consequence, we turn to a categorical reduction of the problem.
In other words, we consider Question 1.
Pseudo-Isotopies of Compact Manifolds
We are able to prove a weaker version of Question 1. As Seidel discovered the long exact sequence associated to a Dehn twist along spheres, the spherical twists, as the mirror autoequivalences, also received much attention . Also, such a cone relation on the A-side has become a foundational tool in the study of homological mirror symmetry, especially in the Picard-Lefschetz theory .
It has long been curious since that, what the auto-equivalence corresponding to the Dehn twists along a rank-one symmetric space is.
Pseudo-Isotopies of Compact Manifolds
On the B-side, Huybrechts-Thomas  defined P n -objects on derived categories of smooth algebraic varieties, as well as a corresponding new auto-equivalence called the P n -twist. They then conjectured the P n - twist is exactly the mirror auto-equivalence of the one induced by a Dehn twist along Lagrangian CP n on the Fukaya categories.
Richard Harris studied the problem in A 8 contexts and formulated the corresponding algebraic twist on A-side . The only missing link to the actual geometry of Lagrangian submanifolds, remains unproved for years. As an application of the surgery equalities in Theorem 1.
Catalog Record: Pseudo-isotopies of compact manifolds | HathiTrust Digital Library
Then Huybrechts-Thomas conjecture is true up to determination of connecting maps. The proof of Theorem 1. These are 6. A family version of projective twist is also given, see Theorem 9.
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While it is not difficult to find examples of Lagrangian RP n in problems in symplectic topology , the search of interesting examples of Lagrangian CP n is more intriguing. In  the authors suggested several sources of P n -objects in derived categories. While the role of P n objects on either side of mirror symmetry remains widely open so far, it is interesting to know whether such objects split generate either side of mirror symmetry.
In a different direction, the P n -cone relation should be interested in understanding some basic problems in symplectic topology, such as mapping class groups of a symplectic manifold and the search of exotic Lagrangian submanifolds. For instance, while a Lagrangian CP n -twist is always smoothly isotopic to identity, it is usually not Hamiltonian isotopic to identity. A simplest model result along this line is to generalize Seidel s twisted Lagrangian sphere construction : in the plumbing of three T CP n, the iterated Dehn twists along CP n in the middle should generate an infinite subgroup in the symplectic mapping class group.
With Theorem 1. The only difference between the formulas is the grading shift of the first term, as specified in Theorem 9.
Therefore, the iterated cone relation can be packaged into a long exact sequence without invoking the immersed Floer theory. The idea follows largely that of  and . For the case at hand, we have restricted ourselves to the exact setting for simplification, and mostly followed Alston-Bao s exposition . The upshot is as expected, that the existence of an immersed Lagrangian cobordism incurs a quasi-isomorphism between certain mapping cones coming from Floer theory.
However, the actual proof is far from straightforward. The key issue is the cleanness of self-intersections of the immersed cobordisms, which is required for the well-behaviors of moduli spaces of pseudo-holomorphic curves. It is not hard to establish a cobordism theory naively following Biran-Cornea s definition in embedded categories and assume the required geometric transversality, but this will not even cover the simplest application at hand. L is given by the cotangent fiber at a point, and we assume it passes through the unique immersed point of S.
The cobordism can be constructed so that it naively satisfies Biran-Cornea s definition, i. However, it is evident that the self-intersection cannot be clean since they form a ray. In general any surgery process involving resolution of an immersed point will suffer from the same caveat. Therefore, we need a modification for the Floer theory to be well-defined. Although we only deal with Donaldson-Fukaya categories in our setting, considerable amounts of new issues need to be addressed since most of our curves cannot actually be projected, which is also the main catch to prove statements on the A 8 level.
This will appear independently in the future. Using Theorem 1. A bonus point of this alternative approach is we could compute the connecting maps which is difficult for general Lagrangian cobordisms. The immersed formalism along with a simple algebraic trick extract enough information to determine almost all relevant mapping cones up to quasi-isomorphisms we covered in this paper, see Section 8.
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In particular, when 8. A note on coefficients. Using characteristic zero coefficients is possible up to checking orientations for the general framework on Lagrangian cobordisms for [6, 7]. Acknowledgement This project was initiated from a discussion with Luis Haug, and have benefited significantly from conversations with Octav Cornea, who suggested the key idea of bottleneck among many other invaluable inputs. We thank both of them for their help and interests over the whole period when this paper was being prepared.
All Lagrangians are assumed to be proper, and non-compact exact Lagrangian embeddings are assumed to have cylindrical end, unless specified otherwise. We identify T S with T S by g and switch freely between the two. The following lemma is well-known. Lemma 2. To define Dehn twist, we need to introduce an auxiliary function. We first consider the case when S is not diffeomorphic to a sphere. Example 2. As usual, one may globalize the model Dehn twist.
Definition 2. A Dehn twist along S in M is a compactly supported symplectomorphism defined by the model Dehn twist as above in a Weinstein neighborhood of S and extended by identity outside. Let apsq, bpsq P R. The main property of an admissible curve can be rephrased as follows. We will frequently use the two equivalent descriptions of admissibility interchangeably.
As a consequence, we have Corollary 2. Now, we present an new approach of performing Lagrangian surgery which also motivates the definition of Lagrangian surgery along clean intersections. The conclusion follows. Under the assumption in Lemma 2. Cerf showed that a one-parameter family of functions between two Morse functions can be approximated by one that is Morse at all but finitely many degenerate times.
For the purposes of intuition, here is an analogy. Notice that in stratified spaces, the co-dimension 0 open stratum is open and dense. For notational purposes, reverse the conventions for indexing the stratifications in a stratified space, and index the open strata not by their dimension, but by their co-dimension. In the case of. Cerf's one-parameter theorem asserts the essential property of the co-dimension one stratum. Alexander in Igusa's theorem, Topology 39 MR g Gajer, Riemannian metrics of positive scalar curvature on compact manifolds with boundary, Ann.
MR 89m Gromov and H. Lawson, Jr. MR 81h MR 85g Hatcher and J. Igusa, Higher singularities of smooth functions are unnecessary, Annals of Mathematics, 2nd Ser. MR 85k Igusa, On the homotopy type of the space of generalized Morse functions, Topology 23 , MR 86m Igusa, The space of framed functions, Trans.
Joyce, Compact 8-manifolds with holonomy Spin 7 , Invent. MR 97d Lichnerowicz, Spineurs harmoniques , C.