Exact Lagrangian submanifolds in simply-connected cotangent bundles

By K. Fukaya, P. Seidel, I. Smith. 2007

116 citations recorded by the source corpus.

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel–Whitney class of the Lagrangian submanifold) we prove such submanifolds are Floer-cohomologically indistinguishable from the zero-section. This implies strong restrictions on their topology. An essentially equivalent result was recently proved independently by Nadler [16], using a different approach.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/4dd43b4117c90bbe38edfe52c20393938543f2d7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

Citations in the corpus

Floer homology on the universal cover, a proof of Audin's conjecture and other constraints on Lagrangian submanifolds (2010, background): “Smith, who proved in [24], [25] that, when L is relatively spin with vanishing Maslov class, its cohomology is isomorphic to the cohomology of K.”
An open string analogue of Viterbo functoriality (2007, methodology): “One could certainly think of applications to the chord conjecture, in line with what has been done for symplectic cohomology, and also to Lagrangian embeddings, in particular fleshing out the last part of Fukaya–Seidel–Smith [14].”
Symplectic cohomology and Viterbo's theorem (2013, background): “Together with Ivan Smith, Fukaya observed that applying such a theory to the easier case of cotangent fibres would yield partial results towards Arnold’s nearby Lagrangian conjecture, but the theory of Lefschetz fibrations [42] gave a way of bypassing the technical problems of defining family Floer ”
Altering symplectic manifolds by homologous recombination (2010, methodology): “We will only give a brief summary, referring to [51, Sections 16 and 19] and [26] for details.”
Parametrized Ring-Spectra and the Nearby Lagrangian Conjecture (2011, methodology): “In the case of vanishing Maslov class and N simply connected Fukaya, Seidel and Smith proved in [9] that p induces a homology isomorphism.”
Quantization of conic Lagrangian submanifolds of cotangent bundles (2012, result): “By the projection formula we deduce Ψε ◦ Ψε(F ) ≃ K ⋆ F , where K = Rqε!((kγ′)[2] ⊕ kγ [1]).”
Microlocal branes are constructible sheaves (2006, methodology): “During the preparation of this paper, we learned that a similar characterization of compact exact branes has recently been obtained by Fukaya-Seidel-Smith [9, 35] by a variety of different methods.”
Microlocal theory of sheaves and Tamarkin's non displaceability theorem (2011, methodology): “Applying the morphism of functors τ0,c : id −→ Tc to β we find τ0,c(Tc∗H) ◦ β = Tc∗β ◦ τ0,c(F [1]).”
T-Duality and Homological Mirror Symmetry of Toric Varieties (2008, methodology): “s [Au]. 2An anti-ample line bundle is the dual of an ample line bundle. 3We employ the “unwrapped” Fukaya category involving normalized geodesic ﬂow, as opposed to the “wrapped” version considered in [FSS]. T-DUALITY AND EQUIVARIANT HOMOLOGICAL MIRROR SYMMETRY 3 these sets, depending on whether the ”