Lagrangian cobordism. I. (English) Zbl 1272.53071
In symplectic topology, Lagrangian cobordism is a notion initially introduced by Arnold. It was studied by Eliashberg and Audin and later Chekanov. Their work showed that, in full generality, Lagrangian cobordism is a very soft notion although it has some rigidity in the case of monotone cobordism. The main purpose of this paper is using Floer theoretic tools to study a monotone version of Lagrangian cobordism. One of their results is that most Floer-type invariants are preserved by appropriate but quite general Lagrangian cobordisms. This is a remarkable result which turns out that Lagrangian cobordism has much more rigidity than we have expected. Apart from this result, the authors also discuss the quantum homology obstructions to the existences of Lagrangian cobordism. For example, a Lagrangian \(L\) with \(QH(L)\) a division ring cannot be split into two non-narrow parts by a Lagrangian cobordism.
The main idea of the proofs of these rigidity results is to define a Floer-type homology theory for pairs of Lagrangian submanifolds with cylindrical ends – a natural extension of cobordisms. On the other hand, based on the Lagrangian surgery construction, the authors give some examples of non-isotopic but cobordant Lagrangians. These examples contrast with the above mentioned rigidity results. Moreover, the authors construct the cobordism category for a symplectic manifold and describe a functor relating this category to the derived Fukaya category and then interpret their rigidity results from the perspective of category.
The main idea of the proofs of these rigidity results is to define a Floer-type homology theory for pairs of Lagrangian submanifolds with cylindrical ends – a natural extension of cobordisms. On the other hand, based on the Lagrangian surgery construction, the authors give some examples of non-isotopic but cobordant Lagrangians. These examples contrast with the above mentioned rigidity results. Moreover, the authors construct the cobordism category for a symplectic manifold and describe a functor relating this category to the derived Fukaya category and then interpret their rigidity results from the perspective of category.
Reviewer: Hao Ding (Chengdu)
MSC:
| 53D12 | Lagrangian submanifolds; Maslov index |
| 53D40 | Symplectic aspects of Floer homology and cohomology |
| 55N22 | Bordism and cobordism theories and formal group laws in algebraic topology |
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