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Register a new userTwisted generating functions and the nearby Lagrangian conjecture
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We prove that, for closed exact embedded Lagrangian submanifolds of cotangent bundles, the homomorphism of homotopy groups induced by the stable Lagrangian Gauss map vanishes. In particular, we prove that this map is null-homotopic for all spheres. The key tool that we introduce in order to prove this is the notion of twisted generating function and we show that every closed exact Lagrangian can be described using such an object, by extending a doubling argument developed in the setting of sheaf theory. Floer theory and sheaf theory constrain the type of twisted generating functions that can appear to a class which is closely related to Waldhausens tube space, and our main result follows by a theorem of Bokstedt which computes the rational homotopy type of the tube space.
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We prove that any closed connected exact Lagrangian manifold L in a connected cotangent bundle T*N is up to a finite covering space lift a homology equivalence. We prove this by constructing a fibrant parametrized family of ring spectra FL parametrized by the manifold N. The homology of FL will be (twisted) symplectic cohomology of T*L. The fibrancy property will imply that there is a Serre spectral sequence converging to the homology of FL and the product combined with intersection product on N induces a product on this spectral sequence. This product structure and its relation to the intersection product on L is then used to obtain the result. Combining this result with work of Abouzaid we arrive at the conclusion that L -> N is always a homotopy equivalence.
We study a cylindrical Lagrangian cobordism group for Lagrangian torus fibres in symplectic manifolds which are the total spaces of smooth Lagrangian torus fibrations. We use ideas from family Floer theory and tropical geometry to obtain both obstructions to and constructions of cobordisms; in particular, we give examples of symplectic tori in which the cobordism group has no non-trivial cobordism relations between pairwise distinct fibres, and ones in which the degree zero fibre cobordism group is a divisible group. The results are independent of but motivated by mirror symmetry, and a relation to rational equivalence of 0-cycles on the mirror rigid analytic space.
We study configurations of disjoint Lagrangian submanifolds in certain low-dimensional symplectic manifolds from the perspective of the geometry of Hamiltonian maps. We detect infinite-dimensional flats in the Hamiltonian group of the two-sphere equipped with Hofers metric, prove constraints on Lagrangian packing, find instances of Lagrangian Poincar{e} recurrence, and present a new hierarchy of normal subgroups of area-preserving homeomorphisms of the two-sphere. The technology involves Lagrangian spectral invariants with Hamiltonian term in symmetric product orbifolds.
An exact Lagrangian submanifold $L$ in the symplectization of standard contact $(2n-1)$-space with Legendrian boundary $Sigma$ can be glued to itself along $Sigma$. This gives a Legendrian embedding $Lambda(L,L)$ of the double of $L$ into contact $(2n+1)$-space. We show that the Legendrian isotopy class of $Lambda(L,L)$ is determined by formal data: the manifold $L$ together with a trivialization of its complexified tangent bundle. In particular, if $L$ is a disk then $Lambda(L,L)$ is the Legendrian unknot.
We study Dehn twists along Lagrangian submanifolds that are finite quotients of spheres. We decribe the induced auto-equivalences to the derived Fukaya category and explain its relation to twists along spherical functors.
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