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- Ekholm, Tobias, 1970-, et al.
(author)
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A duality exact sequence for legendrian contact homology
- 2009
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In: Duke mathematical journal. - : Duke University Press. - 0012-7094 .- 1547-7398. ; 150:1, s. 1-75
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Journal article (peer-reviewed)abstract
- We establish a long exact sequence for Legendrian submanifolds L⊂P×R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L to P off of itself. In this sequence, the singular homology H* maps to linearized contact cohomology CH*, which maps to linearized contact homology CH*, which maps to singular homology. In particular, the sequence implies a duality between Ker(CH*→H*) and CH*/Im(H*). Furthermore, this duality is compatible with Poincaré duality in L in the following sense: the Poincaré dual of a singular class which is the image of a∈CH* maps to a class α∈CH* such that α(a)=1. The exact sequence generalizes the duality for Legendrian knots in R3 (see [26]) and leads to a refinement of the Arnold conjecture for double points of an exact Lagrangian admitting a Legendrian lift with linearizable contact homology, first proved in [7]
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| 2. |
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| 3. |
- Ekholm, Tobias, 1970-, et al.
(author)
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Isotopies of Legendrian 1-knots and Legendrian 2-tori
- 2008
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In: The Journal of Symplectic Geometry. - 1527-5256 .- 1540-2347. ; 6:4, s. 407-460
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Journal article (peer-reviewed)abstract
- We construct a Legendrian 2-torus in the 1-jet space of S1 × R (orof R2) from a loop of Legendrian knots in the 1-jet space of R. The differential graded algebra (DGA) for the Legendrian contact homologyof the torus is explicitly computed in terms of the DGA of the knot and the monodromy operator of the loop. The contact homology of the torus is shown to depend only on the chain homotopy type of the monodromy operator. The construction leads to many new examples of Legendrian knotted tori. In particular, it allows us to construct a Legendrian torus with DGA which does not admit any augmentation(linearization) but which still has non-trivial homology, as well as two Legendrian tori with isomorphic linearized contact homologies but with distinct contact homologies.
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